WEBVTT 00:00:00.000 --> 00:00:02.470 align:middle line:90% [SQUEAKING] 00:00:02.470 --> 00:00:04.446 align:middle line:90% [RUSTLING] 00:00:04.446 --> 00:00:05.434 align:middle line:90% [CLICKING] 00:00:05.434 --> 00:00:10.822 align:middle line:90% 00:00:10.822 --> 00:00:12.280 align:middle line:84% GIAN PAOLO BERETTA: Good afternoon. 00:00:12.280 --> 00:00:18.610 align:middle line:84% So we've been studying properties of stable equilibrium 00:00:18.610 --> 00:00:22.270 align:middle line:84% states, relations that can be extracted 00:00:22.270 --> 00:00:27.630 align:middle line:84% from the fundamental relation of any system 00:00:27.630 --> 00:00:29.890 align:middle line:84% and, therefore, if the system is, 00:00:29.890 --> 00:00:33.090 align:middle line:84% for example, a substance of a pure substance 00:00:33.090 --> 00:00:36.820 align:middle line:90% or mixtures of substances. 00:00:36.820 --> 00:00:45.190 align:middle line:84% And yes, it is a lot of math and mathematical manipulations. 00:00:45.190 --> 00:00:48.180 align:middle line:84% And today we will have a little more of that. 00:00:48.180 --> 00:00:52.320 align:middle line:84% And then we'll stop about the mathematics, at least 00:00:52.320 --> 00:00:54.150 align:middle line:90% for this part. 00:00:54.150 --> 00:01:01.970 align:middle line:84% But I wanted to stress the fact that we are extracting 00:01:01.970 --> 00:01:07.070 align:middle line:84% a lot of information out of just the idea 00:01:07.070 --> 00:01:12.470 align:middle line:84% that there exists a fundamental relation for stable equilibrium 00:01:12.470 --> 00:01:17.000 align:middle line:84% states from which we can extract properties. 00:01:17.000 --> 00:01:21.680 align:middle line:84% And it's precisely the mathematical manipulations 00:01:21.680 --> 00:01:26.630 align:middle line:84% that is not just by itself but with 00:01:26.630 --> 00:01:29.990 align:middle line:90% proper physical interpretation. 00:01:29.990 --> 00:01:31.490 align:middle line:90% And that's the important aspect. 00:01:31.490 --> 00:01:32.760 align:middle line:90% It's not just the mathematics. 00:01:32.760 --> 00:01:35.060 align:middle line:84% It's also to interpret the mathematics 00:01:35.060 --> 00:01:38.530 align:middle line:84% and see what the useful part of it is. 00:01:38.530 --> 00:01:43.510 align:middle line:84% It's from these that you can extract a lot of information. 00:01:43.510 --> 00:01:55.220 align:middle line:84% For example, in economics, manipulations of this sort 00:01:55.220 --> 00:01:58.850 align:middle line:84% have earned the Nobel Prize to Paul Samuelson, who 00:01:58.850 --> 00:02:04.010 align:middle line:84% was a professor here at MIT, because he realized that, 00:02:04.010 --> 00:02:05.900 align:middle line:90% essentially also in economics-- 00:02:05.900 --> 00:02:08.340 align:middle line:84% don't ask me what part of economics-- 00:02:08.340 --> 00:02:10.789 align:middle line:84% but in any case, in economics, there 00:02:10.789 --> 00:02:14.360 align:middle line:84% are functions like that that can be manipulated by the tools 00:02:14.360 --> 00:02:16.190 align:middle line:84% that we are developing from which 00:02:16.190 --> 00:02:18.260 align:middle line:84% you can extract a lot of information 00:02:18.260 --> 00:02:22.840 align:middle line:90% about economics in that case. 00:02:22.840 --> 00:02:29.590 align:middle line:84% So mathematics is a powerful tool, 00:02:29.590 --> 00:02:35.380 align:middle line:84% a way that allows you to think straight. 00:02:35.380 --> 00:02:38.650 align:middle line:84% So, right, we've done the Legendre transform. 00:02:38.650 --> 00:02:43.700 align:middle line:84% And we have applied it to derive the Helmholtz free energy, 00:02:43.700 --> 00:02:46.840 align:middle line:84% the Gibbs free energy, the enthalpy, and also 00:02:46.840 --> 00:02:54.050 align:middle line:84% this Euler free energy that we'll return to also 00:02:54.050 --> 00:02:57.810 align:middle line:90% today and the next time. 00:02:57.810 --> 00:03:00.330 align:middle line:84% Also we developed this Maxwell relations 00:03:00.330 --> 00:03:07.080 align:middle line:84% coming from the reciprocity of the second order derivatives 00:03:07.080 --> 00:03:09.890 align:middle line:90% of the various functions. 00:03:09.890 --> 00:03:14.430 align:middle line:84% Today I want to introduce quickly, or review quickly 00:03:14.430 --> 00:03:18.860 align:middle line:84% if you've seen it already, this other tool which 00:03:18.860 --> 00:03:21.090 align:middle line:90% is Jacobian determinants. 00:03:21.090 --> 00:03:23.930 align:middle line:90% 00:03:23.930 --> 00:03:29.020 align:middle line:84% So if you have a system of two functions 00:03:29.020 --> 00:03:31.750 align:middle line:84% and you take the matrix in which you 00:03:31.750 --> 00:03:36.400 align:middle line:84% put all the partial derivatives in that fashion, 00:03:36.400 --> 00:03:40.000 align:middle line:84% the determinant of that matrix is called the Jacobian 00:03:40.000 --> 00:03:40.970 align:middle line:90% determinant. 00:03:40.970 --> 00:03:42.410 align:middle line:90% The matrix is the Jacobian. 00:03:42.410 --> 00:03:44.200 align:middle line:84% And the determinant of the Jacobian 00:03:44.200 --> 00:03:50.250 align:middle line:84% is we call it the Jacobian anyway, just for short. 00:03:50.250 --> 00:03:55.340 align:middle line:84% And we use this symbol, which resembles a partial 00:03:55.340 --> 00:03:59.780 align:middle line:84% because somehow it's a partial of a vector of functions 00:03:59.780 --> 00:04:05.600 align:middle line:84% with respect to a vector of independent variables. 00:04:05.600 --> 00:04:07.160 align:middle line:84% And it has these properties, which 00:04:07.160 --> 00:04:11.980 align:middle line:84% you can infer easily by the properties of determinants. 00:04:11.980 --> 00:04:17.300 align:middle line:84% You exchange row, exchange a column, you change the sign, 00:04:17.300 --> 00:04:22.470 align:middle line:84% and so you can easily prove these relations. 00:04:22.470 --> 00:04:25.260 align:middle line:84% You can also notice that if one of the functions 00:04:25.260 --> 00:04:33.420 align:middle line:84% like, for example, g is the simple function g equals y, 00:04:33.420 --> 00:04:39.840 align:middle line:84% then the second row is just the derivative 00:04:39.840 --> 00:04:42.000 align:middle line:90% of y with respect to x. 00:04:42.000 --> 00:04:46.390 align:middle line:84% Since y and x are independent variables, that's 0. 00:04:46.390 --> 00:04:51.100 align:middle line:84% And the derivative of y with respect to y is 1. 00:04:51.100 --> 00:04:55.690 align:middle line:84% So that's 0 and 1, which leaves for the determinant just 00:04:55.690 --> 00:04:58.120 align:middle line:84% this term here, which is the partial derivative 00:04:58.120 --> 00:05:01.800 align:middle line:90% of f with respect to x. 00:05:01.800 --> 00:05:06.450 align:middle line:84% So this is a useful identity because you 00:05:06.450 --> 00:05:09.180 align:middle line:84% can pass from partial derivatives that 00:05:09.180 --> 00:05:14.920 align:middle line:84% might be want to evaluate to an equivalent Jacobian. 00:05:14.920 --> 00:05:19.095 align:middle line:84% And then we will see that we can manipulate that further. 00:05:19.095 --> 00:05:22.060 align:middle line:90% 00:05:22.060 --> 00:05:24.620 align:middle line:84% Of course, if also f is equal to x, 00:05:24.620 --> 00:05:32.471 align:middle line:84% then this becomes simply the identity, I mean 1. 00:05:32.471 --> 00:05:33.900 align:middle line:90% OK. 00:05:33.900 --> 00:05:35.880 align:middle line:84% The other theorem about Jacobians 00:05:35.880 --> 00:05:40.110 align:middle line:84% is that if you consider a function of a single variable 00:05:40.110 --> 00:05:45.210 align:middle line:84% and you invert the function so you have f as a function of x, 00:05:45.210 --> 00:05:48.820 align:middle line:84% and you invert it as x as a function of f, 00:05:48.820 --> 00:05:53.430 align:middle line:84% you know that the derivative of this inverted function 00:05:53.430 --> 00:05:55.980 align:middle line:84% is equal to the inverse of the derivative 00:05:55.980 --> 00:05:58.960 align:middle line:90% of the original function. 00:05:58.960 --> 00:06:05.110 align:middle line:84% And the equivalent for a system of two functions of two 00:06:05.110 --> 00:06:13.740 align:middle line:84% variables is that the Jacobian of the inverted system, when 00:06:13.740 --> 00:06:16.500 align:middle line:84% you solve the system, this system of equations, 00:06:16.500 --> 00:06:21.120 align:middle line:84% to put in evidence x and y as a function of x and g. 00:06:21.120 --> 00:06:26.920 align:middle line:84% So if you change the independent variables, and so you invert-- 00:06:26.920 --> 00:06:30.120 align:middle line:84% well, the Jacobian of the inverted system 00:06:30.120 --> 00:06:34.620 align:middle line:84% is equal to the inverse of the Jacobian of the original system. 00:06:34.620 --> 00:06:38.490 align:middle line:84% And that also shows that if you prove-- 00:06:38.490 --> 00:06:42.930 align:middle line:84% if you choose the function g of y 00:06:42.930 --> 00:06:51.430 align:middle line:84% as before, g equals y as before, you can prove by this relation 00:06:51.430 --> 00:06:58.200 align:middle line:90% the fact that the inverse-- 00:06:58.200 --> 00:07:02.310 align:middle line:84% I mean the partial derivative of the inverse with respect 00:07:02.310 --> 00:07:07.060 align:middle line:84% to f, at fixed y, is equal to the inverse 00:07:07.060 --> 00:07:11.380 align:middle line:84% of the original partial derivative. 00:07:11.380 --> 00:07:18.680 align:middle line:84% So formally, you see this is x over y, at constant y. 00:07:18.680 --> 00:07:21.020 align:middle line:90% Let's read it loosely. 00:07:21.020 --> 00:07:24.130 align:middle line:90% Well, you can just put it-- 00:07:24.130 --> 00:07:26.470 align:middle line:90% consider the x and the y. 00:07:26.470 --> 00:07:30.170 align:middle line:84% And consider it as a fraction and put it under a denominator 00:07:30.170 --> 00:07:31.630 align:middle line:90% and you get this function. 00:07:31.630 --> 00:07:33.730 align:middle line:90% And it is fair. 00:07:33.730 --> 00:07:35.230 align:middle line:84% In other words, this proves that you 00:07:35.230 --> 00:07:38.150 align:middle line:90% can do that easy manipulation. 00:07:38.150 --> 00:07:40.910 align:middle line:90% 00:07:40.910 --> 00:07:42.500 align:middle line:90% All right. 00:07:42.500 --> 00:07:44.540 align:middle line:84% And the other item is that if you 00:07:44.540 --> 00:07:48.650 align:middle line:84% have a function f of x, so x is your independent variable, 00:07:48.650 --> 00:07:50.120 align:middle line:84% but suppose you want to represent 00:07:50.120 --> 00:07:54.200 align:middle line:84% your independent variable in parametric form. 00:07:54.200 --> 00:07:59.510 align:middle line:84% So x is a function of another variable, say a, x of a. 00:07:59.510 --> 00:08:01.500 align:middle line:84% So you substitute in the function. 00:08:01.500 --> 00:08:03.590 align:middle line:84% And so you have a function of a function. 00:08:03.590 --> 00:08:05.195 align:middle line:90% This a function of a. 00:08:05.195 --> 00:08:08.370 align:middle line:84% Now a has become the independent variable. 00:08:08.370 --> 00:08:11.220 align:middle line:84% So the chain rule for derivatives 00:08:11.220 --> 00:08:14.910 align:middle line:84% says for single variables that the derivative 00:08:14.910 --> 00:08:17.490 align:middle line:84% of f with respect to a is the derivative 00:08:17.490 --> 00:08:20.430 align:middle line:84% of f with respect to x times the derivative of x with respect 00:08:20.430 --> 00:08:21.795 align:middle line:90% to a. 00:08:21.795 --> 00:08:29.260 align:middle line:84% The equivalent for our system of two variables-- 00:08:29.260 --> 00:08:34.400 align:middle line:84% so when you change your system f and g, 00:08:34.400 --> 00:08:37.460 align:middle line:84% instead of as a function of x and y, 00:08:37.460 --> 00:08:41.110 align:middle line:84% but as a function of a and b, so the x and y 00:08:41.110 --> 00:08:46.140 align:middle line:84% are in turn functions of a and b, 00:08:46.140 --> 00:08:49.330 align:middle line:84% then you have a situation similar to that. 00:08:49.330 --> 00:08:54.100 align:middle line:84% And the derivative actually, the Jacobian 00:08:54.100 --> 00:08:58.640 align:middle line:84% of f and g with respect to the new variables, 00:08:58.640 --> 00:09:04.360 align:middle line:84% is equal to the Jacobian with respect 00:09:04.360 --> 00:09:08.110 align:middle line:84% to the old variables times the Jacobian of the old variables 00:09:08.110 --> 00:09:10.720 align:middle line:90% with respect to the new ones. 00:09:10.720 --> 00:09:18.950 align:middle line:84% Informally, you can write this result-- 00:09:18.950 --> 00:09:20.950 align:middle line:90% suppose you delete these two. 00:09:20.950 --> 00:09:27.160 align:middle line:84% So we view this result as simply the Jacobian of f with respect 00:09:27.160 --> 00:09:28.860 align:middle line:90% to a. 00:09:28.860 --> 00:09:30.840 align:middle line:90% So you write it this way. 00:09:30.840 --> 00:09:33.480 align:middle line:84% And then you have empty spaces here 00:09:33.480 --> 00:09:36.180 align:middle line:84% where you can put anything that cancels out. 00:09:36.180 --> 00:09:41.370 align:middle line:84% So for example, you can put the Jacobian of x, y with respect 00:09:41.370 --> 00:09:44.050 align:middle line:84% to x, y, which is equal to 1, the identity. 00:09:44.050 --> 00:09:47.410 align:middle line:90% We just proved it before. 00:09:47.410 --> 00:09:54.540 align:middle line:84% And thus the formal manipulation becomes easy 00:09:54.540 --> 00:09:59.430 align:middle line:84% now because then you interpret this one as the Jacobian 00:09:59.430 --> 00:10:00.900 align:middle line:90% of f with respect to x, y. 00:10:00.900 --> 00:10:06.005 align:middle line:84% And this is the Jacobian of x, y in respect to a, b. 00:10:06.005 --> 00:10:08.880 align:middle line:90% 00:10:08.880 --> 00:10:16.190 align:middle line:84% Let's give an example that brings us to a useful relation 00:10:16.190 --> 00:10:19.160 align:middle line:84% useful identity for two properties that 00:10:19.160 --> 00:10:24.180 align:middle line:84% are among the list that we have seen are measurable. 00:10:24.180 --> 00:10:30.440 align:middle line:84% So take the Hess determinant, the determinant 00:10:30.440 --> 00:10:32.390 align:middle line:84% of the Hess relation-- of the Hess 00:10:32.390 --> 00:10:35.720 align:middle line:84% matrix of the fundamental relation with respect 00:10:35.720 --> 00:10:38.480 align:middle line:90% to entropy and volume. 00:10:38.480 --> 00:10:42.200 align:middle line:84% So it's the matrix of the second order 00:10:42.200 --> 00:10:46.700 align:middle line:84% derivatives of energy with respect to entropy and volume. 00:10:46.700 --> 00:10:52.497 align:middle line:84% Now since this second derivatives are derivatives of, 00:10:52.497 --> 00:10:54.330 align:middle line:84% for example, we remember that the derivative 00:10:54.330 --> 00:10:56.247 align:middle line:84% of the first derivative of E with respect to S 00:10:56.247 --> 00:10:58.350 align:middle line:84% is the temperature, you can also write them 00:10:58.350 --> 00:11:06.630 align:middle line:84% in this way in terms of the conjugate variables, temperature 00:11:06.630 --> 00:11:11.680 align:middle line:84% for entropy minus the pressure for volume. 00:11:11.680 --> 00:11:12.200 align:middle line:90% OK. 00:11:12.200 --> 00:11:16.970 align:middle line:84% So this is just applying those definitions. 00:11:16.970 --> 00:11:20.030 align:middle line:84% And then here you recognize that, apart from the minus sign 00:11:20.030 --> 00:11:28.160 align:middle line:84% here that can be taken outside of the determinant, 00:11:28.160 --> 00:11:31.460 align:middle line:84% this is just the Jacobian of temperature and pressure 00:11:31.460 --> 00:11:37.370 align:middle line:84% versus with respect to S and V. So now 00:11:37.370 --> 00:11:41.520 align:middle line:84% let's do this trick of shifting the denominator, 00:11:41.520 --> 00:11:42.960 align:middle line:90% leaving the empty spaces. 00:11:42.960 --> 00:11:45.475 align:middle line:84% And let's put here whatever we want. 00:11:45.475 --> 00:11:48.140 align:middle line:90% 00:11:48.140 --> 00:11:50.780 align:middle line:90% So I'll do it in two ways. 00:11:50.780 --> 00:11:54.385 align:middle line:90% I put S and p here and there. 00:11:54.385 --> 00:11:57.620 align:middle line:90% 00:11:57.620 --> 00:12:03.590 align:middle line:84% And then I notice that I have chosen my-- 00:12:03.590 --> 00:12:09.530 align:middle line:84% made my choice so that in each of these determinant-- 00:12:09.530 --> 00:12:15.650 align:middle line:84% I mean Jacobian, there is a repeated pressure 00:12:15.650 --> 00:12:18.440 align:middle line:90% above and below. 00:12:18.440 --> 00:12:24.490 align:middle line:84% So that is the derivative of T with respect to S at constant p. 00:12:24.490 --> 00:12:29.350 align:middle line:84% And this other side has S, which is repeated. 00:12:29.350 --> 00:12:37.910 align:middle line:84% So that is the derivative of p with respect to V at constant S. 00:12:37.910 --> 00:12:40.640 align:middle line:90% But then if you look up-- 00:12:40.640 --> 00:12:45.290 align:middle line:90% and let me remind it here. 00:12:45.290 --> 00:12:50.780 align:middle line:84% We define the heat capacity at constant pressure, 00:12:50.780 --> 00:12:52.640 align:middle line:90% a few days ago. 00:12:52.640 --> 00:12:55.760 align:middle line:84% And that was defined as the derivative 00:12:55.760 --> 00:12:59.840 align:middle line:84% of the enthalpy with respect to temperature at p. 00:12:59.840 --> 00:13:05.810 align:middle line:84% But if you look at how the differential for the enthalpy 00:13:05.810 --> 00:13:13.340 align:middle line:84% is, if you keep p and n constant, dH is equal to TdS. 00:13:13.340 --> 00:13:16.070 align:middle line:84% So you can substitute the dH with the TdS 00:13:16.070 --> 00:13:19.910 align:middle line:90% here at constant p and n. 00:13:19.910 --> 00:13:23.900 align:middle line:84% So Cp, or actually, you see here that the derivative 00:13:23.900 --> 00:13:26.360 align:middle line:84% of S with respect to T at constant p 00:13:26.360 --> 00:13:34.210 align:middle line:84% is Cp over T. This derivative is therefore the inverse of Cp 00:13:34.210 --> 00:13:36.850 align:middle line:90% over T. Why the inverse? 00:13:36.850 --> 00:13:41.130 align:middle line:84% Because we just proved that these derivatives 00:13:41.130 --> 00:13:48.710 align:middle line:84% here is equal to the inverse of its reciprocal, so to speak. 00:13:48.710 --> 00:13:51.110 align:middle line:84% Also this other one we can recognize 00:13:51.110 --> 00:13:55.165 align:middle line:84% as one of the properties that we have defined or almost. 00:13:55.165 --> 00:14:01.700 align:middle line:90% 00:14:01.700 --> 00:14:04.040 align:middle line:84% It's the derivative of V with respect 00:14:04.040 --> 00:14:07.890 align:middle line:84% to pressure with the minus in front. 00:14:07.890 --> 00:14:11.450 align:middle line:90% And so it's the compressibility. 00:14:11.450 --> 00:14:12.980 align:middle line:84% Actually, the other day, we defined 00:14:12.980 --> 00:14:14.960 align:middle line:84% the isothermal compressibility, so when 00:14:14.960 --> 00:14:18.200 align:middle line:84% you compress a material at constant temperature, 00:14:18.200 --> 00:14:21.020 align:middle line:84% but you could also compress it at constant entropy. 00:14:21.020 --> 00:14:25.230 align:middle line:84% That defines the isentropic compressibility. 00:14:25.230 --> 00:14:28.230 align:middle line:90% 00:14:28.230 --> 00:14:32.440 align:middle line:84% And so here you recognize the inverse 00:14:32.440 --> 00:14:35.700 align:middle line:84% of the isentropic compressibility. 00:14:35.700 --> 00:14:36.200 align:middle line:90% OK. 00:14:36.200 --> 00:14:44.220 align:middle line:84% So first result is that since the determinant of the Hess 00:14:44.220 --> 00:14:48.650 align:middle line:84% matrix has to be non-negative by the conditions of stability 00:14:48.650 --> 00:14:54.950 align:middle line:84% that we have already proved, we have this inequality. 00:14:54.950 --> 00:14:58.930 align:middle line:84% But that's not the main object here. 00:14:58.930 --> 00:15:03.040 align:middle line:84% I want to do now the same but by substituting 00:15:03.040 --> 00:15:09.890 align:middle line:84% here in the empty spaces, another pair of variables, again 00:15:09.890 --> 00:15:15.350 align:middle line:84% in such a way that numerator and denominator have 00:15:15.350 --> 00:15:19.350 align:middle line:84% one same variable like here the temperature and here the volume. 00:15:19.350 --> 00:15:22.350 align:middle line:90% 00:15:22.350 --> 00:15:25.330 align:middle line:84% Then this becomes the partial of p with respect 00:15:25.330 --> 00:15:26.830 align:middle line:84% to volume with constant temperature. 00:15:26.830 --> 00:15:30.790 align:middle line:84% And this becomes a partial of T with respect to S at constant V. 00:15:30.790 --> 00:15:35.660 align:middle line:84% And by the same definitions that we just reviewed, 00:15:35.660 --> 00:15:40.650 align:middle line:84% you can show that this is equal to T over the heat 00:15:40.650 --> 00:15:44.610 align:middle line:84% capacity at constant volume and this other 00:15:44.610 --> 00:15:50.820 align:middle line:84% is 1 over the volume times the isothermal compressibility. 00:15:50.820 --> 00:15:54.815 align:middle line:84% Also this group must be greater than or equal to 0. 00:15:54.815 --> 00:15:57.820 align:middle line:90% 00:15:57.820 --> 00:16:03.870 align:middle line:84% But since they came from the manipulation of the same stuff, 00:16:03.870 --> 00:16:09.120 align:middle line:84% these groups must also be equal to one another. 00:16:09.120 --> 00:16:15.250 align:middle line:84% And that by eliminating T and V, which cancel if you equate that 00:16:15.250 --> 00:16:19.980 align:middle line:84% to this, you obtain this result, that the ratio of the heat 00:16:19.980 --> 00:16:24.030 align:middle line:84% capacities, constant pressure and volume, 00:16:24.030 --> 00:16:29.580 align:middle line:84% is equal to the ratio of the compressibilities, 00:16:29.580 --> 00:16:36.000 align:middle line:84% at constant temperature versus at constant entropy. 00:16:36.000 --> 00:16:42.160 align:middle line:84% For example, if you remember like for air 00:16:42.160 --> 00:16:53.140 align:middle line:84% or for a diatomic gas, how much is Cp over CV? 00:16:53.140 --> 00:16:54.350 align:middle line:90% Anybody remembers? 00:16:54.350 --> 00:16:58.000 align:middle line:90% 00:16:58.000 --> 00:17:05.210 align:middle line:90% For air, it's 1.4. 00:17:05.210 --> 00:17:11.700 align:middle line:90% It is 7/5, 7 over 5. 00:17:11.700 --> 00:17:19.430 align:middle line:84% So that means that also the ratio of this compressibility 00:17:19.430 --> 00:17:21.475 align:middle line:90% is 1 point 4. 00:17:21.475 --> 00:17:24.339 align:middle line:90% 00:17:24.339 --> 00:17:28.960 align:middle line:84% We will see in a while that this gives us-- 00:17:28.960 --> 00:17:33.410 align:middle line:84% this has some importance in a practical application, 00:17:33.410 --> 00:17:41.710 align:middle line:84% which is the air springs that are used in many transport 00:17:41.710 --> 00:17:42.460 align:middle line:90% devices-- 00:17:42.460 --> 00:17:48.130 align:middle line:84% I mean transport systems like trains, trucks, cars, and so on, 00:17:48.130 --> 00:17:53.830 align:middle line:90% even bicycles, air springs. 00:17:53.830 --> 00:18:00.580 align:middle line:84% But before we get to that, let's keep going with another relation 00:18:00.580 --> 00:18:06.110 align:middle line:84% that we can prove by, again, using 00:18:06.110 --> 00:18:08.930 align:middle line:90% the properties of Jacobians. 00:18:08.930 --> 00:18:14.825 align:middle line:84% And here is a situation in which you have a surface. 00:18:14.825 --> 00:18:17.690 align:middle line:90% 00:18:17.690 --> 00:18:22.520 align:middle line:84% For example, a function of x, y, and z equals 0, 00:18:22.520 --> 00:18:25.640 align:middle line:84% represents a surface that you can represent 00:18:25.640 --> 00:18:27.840 align:middle line:90% in three dimensions x, y, z. 00:18:27.840 --> 00:18:33.920 align:middle line:84% So of course, you can view that surface 00:18:33.920 --> 00:18:36.080 align:middle line:84% from various points of view by choosing 00:18:36.080 --> 00:18:38.540 align:middle line:84% what your base reference frame is 00:18:38.540 --> 00:18:40.320 align:middle line:84% and what the independent variable is. 00:18:40.320 --> 00:18:43.550 align:middle line:84% So you can represent that surface also 00:18:43.550 --> 00:18:49.210 align:middle line:84% in these three alternative ways, x as a function of x, y. 00:18:49.210 --> 00:18:50.080 align:middle line:90% No. 00:18:50.080 --> 00:18:51.260 align:middle line:90% You see what I'm saying? 00:18:51.260 --> 00:19:05.950 align:middle line:84% If I have this surface here which is f of x, y, z equals 0, 00:19:05.950 --> 00:19:07.780 align:middle line:90% I can represent it. 00:19:07.780 --> 00:19:11.740 align:middle line:90% Suppose this is x, y, and z. 00:19:11.740 --> 00:19:12.590 align:middle line:90% OK. 00:19:12.590 --> 00:19:14.950 align:middle line:90% So this is also-- 00:19:14.950 --> 00:19:19.060 align:middle line:84% For every x and y, I have a value of z 00:19:19.060 --> 00:19:25.030 align:middle line:84% so I can view it as a function z of x and y. 00:19:25.030 --> 00:19:28.780 align:middle line:84% But I can also view it also from here. 00:19:28.780 --> 00:19:32.950 align:middle line:84% I fix x, I fix z, and that gives me a y. 00:19:32.950 --> 00:19:39.640 align:middle line:84% So that's y function of z and x and so on. 00:19:39.640 --> 00:19:43.300 align:middle line:84% So these are alternative ways of representing, 00:19:43.300 --> 00:19:48.040 align:middle line:84% in thermodynamics, the typical surface that is treated this way 00:19:48.040 --> 00:19:50.560 align:middle line:90% is the pVT relation. 00:19:50.560 --> 00:19:53.630 align:middle line:90% The equation of state. 00:19:53.630 --> 00:19:59.260 align:middle line:84% That particular equation, that, for a class of states, 00:19:59.260 --> 00:20:03.670 align:middle line:84% like single phase states, is equivalent 00:20:03.670 --> 00:20:07.900 align:middle line:90% to the fundamental relation. 00:20:07.900 --> 00:20:12.220 align:middle line:84% And so it's useful because it's in terms 00:20:12.220 --> 00:20:16.030 align:middle line:84% of easily measurable properties like volume, temperature, 00:20:16.030 --> 00:20:16.610 align:middle line:90% and pressure. 00:20:16.610 --> 00:20:24.700 align:middle line:90% 00:20:24.700 --> 00:20:26.470 align:middle line:90% So here we do-- 00:20:26.470 --> 00:20:28.140 align:middle line:90% what do we do? 00:20:28.140 --> 00:20:35.440 align:middle line:84% We take, essentially, ratios of stuff that are equal to 1 00:20:35.440 --> 00:20:36.900 align:middle line:90% or actually say minus 1. 00:20:36.900 --> 00:20:39.820 align:middle line:90% 00:20:39.820 --> 00:20:45.970 align:middle line:84% So say x, z, and z, x in the denominator. 00:20:45.970 --> 00:20:47.870 align:middle line:84% By the properties of determinants, 00:20:47.870 --> 00:20:49.510 align:middle line:90% that's equal to minus 1. 00:20:49.510 --> 00:20:56.550 align:middle line:84% If I switch one column, it changes sign. 00:20:56.550 --> 00:21:00.460 align:middle line:84% So I write this, which are easy identities, 00:21:00.460 --> 00:21:04.180 align:middle line:84% then I rearrange them in a more interesting way 00:21:04.180 --> 00:21:11.280 align:middle line:84% so that, again, I have the same variable numerator 00:21:11.280 --> 00:21:12.450 align:middle line:90% and denominator. 00:21:12.450 --> 00:21:15.210 align:middle line:84% So for example, this is the partial of x with respect 00:21:15.210 --> 00:21:17.160 align:middle line:90% to y at constant z. 00:21:17.160 --> 00:21:19.660 align:middle line:84% This is the partial of x with respect-- 00:21:19.660 --> 00:21:21.810 align:middle line:84% I'm sorry, y respect to z at constant x. 00:21:21.810 --> 00:21:23.700 align:middle line:84% And this is partial of z with respect 00:21:23.700 --> 00:21:28.710 align:middle line:84% to x at constant y, which is written here. 00:21:28.710 --> 00:21:32.880 align:middle line:84% And if you multiply minus 1 three times, you get minus 1. 00:21:32.880 --> 00:21:36.450 align:middle line:84% So this is called the cyclic relation. 00:21:36.450 --> 00:21:42.330 align:middle line:84% And if you apply it to the pVT relation, 00:21:42.330 --> 00:21:44.645 align:middle line:84% equation of state, here is what you get. 00:21:44.645 --> 00:21:49.340 align:middle line:90% 00:21:49.340 --> 00:21:52.430 align:middle line:84% And for example, you could apply it-- 00:21:52.430 --> 00:21:57.230 align:middle line:84% remember that one of the Maxwell relations that we said 00:21:57.230 --> 00:22:01.580 align:middle line:84% could be useful to measure entropy because it gives entropy 00:22:01.580 --> 00:22:06.150 align:middle line:84% changes that you can measure by measuring volume 00:22:06.150 --> 00:22:08.490 align:middle line:84% changes, pressure changes, temperature 00:22:08.490 --> 00:22:11.140 align:middle line:84% changes at constant temperature and constant volume. 00:22:11.140 --> 00:22:16.260 align:middle line:84% So this is somehow the basis of an instrument 00:22:16.260 --> 00:22:20.160 align:middle line:84% that you could call an entropy meter. 00:22:20.160 --> 00:22:24.460 align:middle line:84% Well, by applying this cyclic relation-- 00:22:24.460 --> 00:22:29.030 align:middle line:84% and I'm not going to the details but I'll let you show that-- 00:22:29.030 --> 00:22:31.840 align:middle line:90% 00:22:31.840 --> 00:22:34.990 align:middle line:84% you prove that this derivative is 00:22:34.990 --> 00:22:38.080 align:middle line:84% equal to the ratio of alpha p over kappa T. 00:22:38.080 --> 00:22:45.880 align:middle line:84% So it's the isobaric coefficient of expansion alpha 00:22:45.880 --> 00:22:48.310 align:middle line:84% over the isothermal compressibility 00:22:48.310 --> 00:22:51.460 align:middle line:90% of your material. 00:22:51.460 --> 00:23:00.040 align:middle line:84% For example, for a gas, I mean, if I can write the ideal gas 00:23:00.040 --> 00:23:07.600 align:middle line:84% relation, you can view that as a V 00:23:07.600 --> 00:23:13.360 align:middle line:84% as a function of T, and p, and n. 00:23:13.360 --> 00:23:18.260 align:middle line:90% 00:23:18.260 --> 00:23:24.050 align:middle line:90% So it's nRT over p. 00:23:24.050 --> 00:23:28.390 align:middle line:84% So it's easy to compute, for example, alpha p. 00:23:28.390 --> 00:23:33.250 align:middle line:84% alpha p is 1 over V, the derivative of V with respect 00:23:33.250 --> 00:23:37.430 align:middle line:90% to T at constant pressure. 00:23:37.430 --> 00:23:44.390 align:middle line:84% So it's how much the expansion for a given 00:23:44.390 --> 00:23:46.610 align:middle line:90% increase in temperature. 00:23:46.610 --> 00:23:47.310 align:middle line:90% All right. 00:23:47.310 --> 00:23:53.600 align:middle line:84% So it's 1 over V. The derivative of V with respect to T is easy. 00:23:53.600 --> 00:23:58.260 align:middle line:90% It's nR over p. 00:23:58.260 --> 00:24:15.080 align:middle line:84% And if you use that again, this is 1 over T. kappa T is minus 1 00:24:15.080 --> 00:24:19.880 align:middle line:84% over V, the derivative of V with respect to p at constant T. 00:24:19.880 --> 00:24:20.970 align:middle line:90% All right. 00:24:20.970 --> 00:24:25.810 align:middle line:90% We can also do that derivative. 00:24:25.810 --> 00:24:28.840 align:middle line:84% The derivative of this with respect to p 00:24:28.840 --> 00:24:34.480 align:middle line:90% is minus nRT over p squared. 00:24:34.480 --> 00:24:39.100 align:middle line:90% 00:24:39.100 --> 00:24:42.440 align:middle line:90% I apply that again. 00:24:42.440 --> 00:24:44.840 align:middle line:84% And the minus goes away with the minus. 00:24:44.840 --> 00:24:46.380 align:middle line:90% One V goes away. 00:24:46.380 --> 00:24:47.820 align:middle line:90% And I'm left with 1 over p. 00:24:47.820 --> 00:24:50.710 align:middle line:90% 00:24:50.710 --> 00:24:58.270 align:middle line:84% So alpha and kappa for ideal gases are relatively simple. 00:24:58.270 --> 00:25:06.220 align:middle line:84% And so this gives that the derivative here 00:25:06.220 --> 00:25:13.680 align:middle line:90% is equal to p over T for a gas. 00:25:13.680 --> 00:25:20.610 align:middle line:90% 00:25:20.610 --> 00:25:24.000 align:middle line:84% There is another interesting relation 00:25:24.000 --> 00:25:32.750 align:middle line:84% that you can prove not so easily but relatively straightforward 00:25:32.750 --> 00:25:38.630 align:middle line:90% by using the Jacobians. 00:25:38.630 --> 00:25:41.510 align:middle line:90% 00:25:41.510 --> 00:25:44.060 align:middle line:84% It's the Mayer relation which relates 00:25:44.060 --> 00:25:49.410 align:middle line:84% Cp, CV, alpha p and kappa T, and, of course, temperature 00:25:49.410 --> 00:25:51.940 align:middle line:90% and volume. 00:25:51.940 --> 00:25:58.740 align:middle line:84% And here you see this is just the definition of Cp. 00:25:58.740 --> 00:26:02.310 align:middle line:84% We write this derivative in terms of a Jacobian 00:26:02.310 --> 00:26:06.030 align:middle line:84% with the property that is kept constant, 00:26:06.030 --> 00:26:08.950 align:middle line:84% repeated numerator and denominator. 00:26:08.950 --> 00:26:12.910 align:middle line:84% Then we shift, leave the empty space, and then we search. 00:26:12.910 --> 00:26:17.080 align:middle line:84% And now the choice of what to put there is sort of a smart-- 00:26:17.080 --> 00:26:21.780 align:middle line:84% you have to use just trial and error or some experience 00:26:21.780 --> 00:26:24.180 align:middle line:90% on how to choose it. 00:26:24.180 --> 00:26:30.660 align:middle line:84% But here if you, for example, we choose to select T and V, 00:26:30.660 --> 00:26:34.530 align:middle line:84% the second one has a repeated variable numerator 00:26:34.530 --> 00:26:37.140 align:middle line:84% and denominator so it is a partial derivative. 00:26:37.140 --> 00:26:41.390 align:middle line:84% And again, we find that it's related to kappa T. 00:26:41.390 --> 00:26:44.820 align:middle line:84% The first determinant has four different variables. 00:26:44.820 --> 00:26:50.520 align:middle line:84% So we write it here in terms of its definition. 00:26:50.520 --> 00:26:54.550 align:middle line:90% 00:26:54.550 --> 00:26:57.190 align:middle line:84% And then these various entries can be 00:26:57.190 --> 00:26:59.840 align:middle line:90% recognized as those properties. 00:26:59.840 --> 00:27:06.540 align:middle line:84% For example, this is CV over T. This we just proved 00:27:06.540 --> 00:27:11.500 align:middle line:90% is alpha over kappa T. 00:27:11.500 --> 00:27:17.500 align:middle line:84% This is the same because of the symmetry of the Hess determinant 00:27:17.500 --> 00:27:21.230 align:middle line:84% or because of the Maxwell relations, which is the same. 00:27:21.230 --> 00:27:25.380 align:middle line:84% And this is one over kappa T. So now we compute the determinant. 00:27:25.380 --> 00:27:26.660 align:middle line:90% You get that. 00:27:26.660 --> 00:27:29.510 align:middle line:90% You place it here. 00:27:29.510 --> 00:27:32.155 align:middle line:84% And out comes the Mayer relation. 00:27:32.155 --> 00:27:35.270 align:middle line:90% 00:27:35.270 --> 00:27:36.560 align:middle line:90% OK. 00:27:36.560 --> 00:27:37.080 align:middle line:90% Yeah. 00:27:37.080 --> 00:27:39.100 align:middle line:84% So what's the Mayer relation for a gas? 00:27:39.100 --> 00:27:43.900 align:middle line:90% 00:27:43.900 --> 00:27:52.215 align:middle line:84% What's the relation between Cp and CV for the ideal gas? 00:27:52.215 --> 00:27:56.045 align:middle line:90% 00:27:56.045 --> 00:27:56.545 align:middle line:90% No? 00:27:56.545 --> 00:28:10.590 align:middle line:90% 00:28:10.590 --> 00:28:15.220 align:middle line:90% How about R, the gas constant? 00:28:15.220 --> 00:28:20.450 align:middle line:84% Usually, you don't do heat capacities. 00:28:20.450 --> 00:28:24.330 align:middle line:90% We do specific heats. 00:28:24.330 --> 00:28:26.060 align:middle line:90% So that's CP. 00:28:26.060 --> 00:28:31.280 align:middle line:90% Small c is capital C over n. 00:28:31.280 --> 00:28:36.410 align:middle line:90% And cV, small c, heat capacity. 00:28:36.410 --> 00:28:41.580 align:middle line:84% I'm sorry, specific heat, is the CV over n. 00:28:41.580 --> 00:28:47.340 align:middle line:84% And then the Meyer relation is that CP equals CV plus the gas 00:28:47.340 --> 00:28:51.630 align:middle line:84% constant, but let's prove it because we 00:28:51.630 --> 00:28:54.840 align:middle line:90% said that we have it there. 00:28:54.840 --> 00:28:58.610 align:middle line:90% So we have to compute V-- 00:28:58.610 --> 00:29:06.760 align:middle line:84% what is it-- the VT alpha squared over kappa T, 00:29:06.760 --> 00:29:11.260 align:middle line:84% all right, which is this term here, 00:29:11.260 --> 00:29:14.260 align:middle line:84% which is the difference between the two capacities. 00:29:14.260 --> 00:29:15.790 align:middle line:90% So alpha. 00:29:15.790 --> 00:29:16.580 align:middle line:90% So that's VT. 00:29:16.580 --> 00:29:19.380 align:middle line:90% 00:29:19.380 --> 00:29:23.240 align:middle line:84% Alpha is 1 over T. So we got T squared here. 00:29:23.240 --> 00:29:25.250 align:middle line:90% And kappa is 1 over p. 00:29:25.250 --> 00:29:28.360 align:middle line:90% So I've got the p there. 00:29:28.360 --> 00:29:31.310 align:middle line:90% So that goes with that. 00:29:31.310 --> 00:29:38.510 align:middle line:84% And hopefully, since pV is equal to nRT, divided by T gives nR. 00:29:38.510 --> 00:29:44.640 align:middle line:90% 00:29:44.640 --> 00:29:51.900 align:middle line:84% I see a lot of rust in your mind about previous thermo. 00:29:51.900 --> 00:29:52.475 align:middle line:90% That's OK. 00:29:52.475 --> 00:29:56.665 align:middle line:90% 00:29:56.665 --> 00:29:57.165 align:middle line:90% All right. 00:29:57.165 --> 00:30:00.420 align:middle line:90% 00:30:00.420 --> 00:30:03.760 align:middle line:90% So done with these properties. 00:30:03.760 --> 00:30:10.480 align:middle line:84% I want to go back to the subject of available energy. 00:30:10.480 --> 00:30:15.380 align:middle line:84% You remember what we've done I forget now 00:30:15.380 --> 00:30:17.525 align:middle line:84% if it was the previous lecture or the one before. 00:30:17.525 --> 00:30:24.410 align:middle line:90% 00:30:24.410 --> 00:30:27.090 align:middle line:90% Available energy. 00:30:27.090 --> 00:30:31.650 align:middle line:84% This is the setup in which I have a system and the reservoir. 00:30:31.650 --> 00:30:35.340 align:middle line:84% And they undergo a weight process. 00:30:35.340 --> 00:30:38.160 align:middle line:84% And we know that if the process, the weight process 00:30:38.160 --> 00:30:49.390 align:middle line:84% is reversible, we can extract the maximum amount of energy 00:30:49.390 --> 00:30:52.630 align:middle line:84% when the system ends in a mutual equilibrium 00:30:52.630 --> 00:30:57.460 align:middle line:84% state with the environment, with the reservoir. 00:30:57.460 --> 00:30:59.620 align:middle line:84% Here instead we are going to adopt 00:30:59.620 --> 00:31:03.640 align:middle line:84% two different states, two arbitrary states, initial 00:31:03.640 --> 00:31:05.330 align:middle line:90% and final. 00:31:05.330 --> 00:31:11.060 align:middle line:84% And you ask yourself, well, if my system has to go from A1 00:31:11.060 --> 00:31:17.330 align:middle line:84% to A2, and it has available the interaction with this thermal 00:31:17.330 --> 00:31:24.520 align:middle line:84% reservoir, what is the maximum weight lift that it can do, 00:31:24.520 --> 00:31:32.160 align:middle line:84% or if it cannot lift the weight, what is the minimum weight 00:31:32.160 --> 00:31:37.460 align:middle line:84% lowering that is required so as to achieve that change? 00:31:37.460 --> 00:31:42.840 align:middle line:90% 00:31:42.840 --> 00:31:45.045 align:middle line:90% And we have seen that-- 00:31:45.045 --> 00:31:47.630 align:middle line:90% 00:31:47.630 --> 00:31:51.440 align:middle line:84% I mean this class of problems-- what's the maximum work, 00:31:51.440 --> 00:31:53.970 align:middle line:90% what's the minimum work-- 00:31:53.970 --> 00:31:57.240 align:middle line:84% Are always solved by writing an energy 00:31:57.240 --> 00:32:02.330 align:middle line:84% balance, an entropy balance, and also 00:32:02.330 --> 00:32:05.360 align:middle line:84% inserting as much as we know about the properties 00:32:05.360 --> 00:32:06.620 align:middle line:90% of the systems involved. 00:32:06.620 --> 00:32:08.120 align:middle line:84% In this case, we have a reservoir 00:32:08.120 --> 00:32:10.460 align:middle line:84% for which we know the fundamental relation is 00:32:10.460 --> 00:32:12.850 align:middle line:84% this linear relationship between energy and entropy. 00:32:12.850 --> 00:32:15.680 align:middle line:90% 00:32:15.680 --> 00:32:20.660 align:middle line:90% And by the way, usually in-- 00:32:20.660 --> 00:32:24.420 align:middle line:90% 00:32:24.420 --> 00:32:30.510 align:middle line:84% often, especially in physics language, 00:32:30.510 --> 00:32:35.180 align:middle line:84% what we call thermal reservoir is called a heat bath. 00:32:35.180 --> 00:32:35.680 align:middle line:90% OK. 00:32:35.680 --> 00:32:42.210 align:middle line:84% Now you realize that we defined the heat interaction 00:32:42.210 --> 00:32:57.980 align:middle line:84% so that from heat from A to B so that if you 00:32:57.980 --> 00:33:05.110 align:middle line:84% want how much entropy is transferred, 00:33:05.110 --> 00:33:09.790 align:middle line:84% you get it because the ratio between energy and entropy 00:33:09.790 --> 00:33:11.590 align:middle line:84% transferred in a heat interaction 00:33:11.590 --> 00:33:16.070 align:middle line:84% is equal to the temperature at which that transfer occurs. 00:33:16.070 --> 00:33:18.480 align:middle line:90% OK. 00:33:18.480 --> 00:33:24.970 align:middle line:84% Now the reservoir-- you can write the fundamental relation 00:33:24.970 --> 00:33:29.140 align:middle line:90% for the reservoir this way. 00:33:29.140 --> 00:33:38.140 align:middle line:90% 00:33:38.140 --> 00:33:42.190 align:middle line:84% So you see why they call it heat bath 00:33:42.190 --> 00:33:48.660 align:middle line:84% because the heat reservoir is ready to provide the heat 00:33:48.660 --> 00:33:49.720 align:middle line:90% interaction. 00:33:49.720 --> 00:33:55.380 align:middle line:84% If you extract energy at its temperature, 00:33:55.380 --> 00:34:00.810 align:middle line:84% then it provides also the entropy and in the right 00:34:00.810 --> 00:34:03.840 align:middle line:84% ratio so that you have a heat interaction at the temperature 00:34:03.840 --> 00:34:06.170 align:middle line:90% of the reservoir. 00:34:06.170 --> 00:34:12.350 align:middle line:84% So it's a heat bath in the sense that it can provide heat 00:34:12.350 --> 00:34:14.146 align:middle line:90% so that at least justifies-- 00:34:14.146 --> 00:34:16.650 align:middle line:90% 00:34:16.650 --> 00:34:18.929 align:middle line:90% that's the reading of-- 00:34:18.929 --> 00:34:22.659 align:middle line:84% my justification of that language. 00:34:22.659 --> 00:34:28.670 align:middle line:84% It is not a reservoir of heat, though in the sense 00:34:28.670 --> 00:34:35.100 align:middle line:84% that heat is not a form of energy. 00:34:35.100 --> 00:34:38.580 align:middle line:90% It is a form of energy transfer. 00:34:38.580 --> 00:34:40.270 align:middle line:90% You cannot store heat. 00:34:40.270 --> 00:34:42.360 align:middle line:90% You store energy. 00:34:42.360 --> 00:34:42.860 align:middle line:90% OK. 00:34:42.860 --> 00:34:45.810 align:middle line:84% You transfer energy in the form of heat 00:34:45.810 --> 00:34:49.159 align:middle line:84% when you have an interaction of that form. 00:34:49.159 --> 00:34:51.590 align:middle line:90% So the language is important. 00:34:51.590 --> 00:34:55.380 align:middle line:84% And loose language is OK if you understand what you're doing. 00:34:55.380 --> 00:34:58.910 align:middle line:84% If it is too loose and if you assign 00:34:58.910 --> 00:35:02.910 align:middle line:84% to this language too popular a meaning, 00:35:02.910 --> 00:35:05.040 align:middle line:90% then you get a lot of confusion. 00:35:05.040 --> 00:35:07.590 align:middle line:90% 00:35:07.590 --> 00:35:13.030 align:middle line:84% This is why one of our objectives 00:35:13.030 --> 00:35:17.565 align:middle line:84% here is to try to be as careful as possible with the language. 00:35:17.565 --> 00:35:21.800 align:middle line:90% 00:35:21.800 --> 00:35:29.060 align:middle line:84% And by the way, my sincere congratulations for how good 00:35:29.060 --> 00:35:31.440 align:middle line:90% you're doing in your homeworks. 00:35:31.440 --> 00:35:36.587 align:middle line:90% 00:35:36.587 --> 00:35:37.087 align:middle line:90% OK. 00:35:37.087 --> 00:35:39.769 align:middle line:90% 00:35:39.769 --> 00:35:46.450 align:middle line:90% So the work that-- 00:35:46.450 --> 00:35:48.710 align:middle line:84% by solving this system of equations, 00:35:48.710 --> 00:35:57.570 align:middle line:84% we already said that we find an expression for the work which 00:35:57.570 --> 00:36:00.150 align:middle line:84% has a term related to irreversibility-- entropy 00:36:00.150 --> 00:36:03.150 align:middle line:84% produced by irreversibility, which is not there 00:36:03.150 --> 00:36:04.870 align:middle line:90% if the process is reversible. 00:36:04.870 --> 00:36:08.540 align:middle line:84% And that's typically the best you can do. 00:36:08.540 --> 00:36:18.500 align:middle line:84% Whether you have to spend energy or produce a mechanical energy, 00:36:18.500 --> 00:36:22.340 align:middle line:84% you get the best always when you do it reversibly. 00:36:22.340 --> 00:36:25.410 align:middle line:84% So if you have to extract energy, 00:36:25.410 --> 00:36:27.930 align:middle line:84% if you can extract energy, you extract the maximum 00:36:27.930 --> 00:36:29.700 align:middle line:84% if you do the reversible process. 00:36:29.700 --> 00:36:31.400 align:middle line:84% If you have to spend energy in order 00:36:31.400 --> 00:36:38.000 align:middle line:84% to obtain a change of state, for example, heating up some water, 00:36:38.000 --> 00:36:43.680 align:middle line:84% that's a task that requires spending energy, 00:36:43.680 --> 00:36:49.130 align:middle line:84% you can ask yourself, what is the minimum work 00:36:49.130 --> 00:36:58.190 align:middle line:84% that you need to spend in order to heat up that water. 00:36:58.190 --> 00:37:04.410 align:middle line:84% And if you do that, maybe when we do the next time 00:37:04.410 --> 00:37:13.300 align:middle line:84% they review on exergies, we'll see 00:37:13.300 --> 00:37:17.160 align:middle line:84% how efficient are certain ways we have to heat up water. 00:37:17.160 --> 00:37:22.790 align:middle line:90% 00:37:22.790 --> 00:37:24.848 align:middle line:90% OK. 00:37:24.848 --> 00:37:26.870 align:middle line:84% Now the maximum, the reversible work, 00:37:26.870 --> 00:37:31.400 align:middle line:84% is the difference between the available energies of the two 00:37:31.400 --> 00:37:32.725 align:middle line:90% states of your system. 00:37:32.725 --> 00:37:35.270 align:middle line:90% 00:37:35.270 --> 00:37:40.680 align:middle line:84% It's also the difference in this availability function gamma 00:37:40.680 --> 00:37:44.760 align:middle line:84% that is energy minus entropy multiplied by the temperature 00:37:44.760 --> 00:37:48.020 align:middle line:90% of the reservoir. 00:37:48.020 --> 00:37:52.850 align:middle line:84% We called it canonical availability function. 00:37:52.850 --> 00:37:58.310 align:middle line:84% And that we have seen that if you try to represent it 00:37:58.310 --> 00:38:04.180 align:middle line:84% graphically, it acquires an absolute minimum 00:38:04.180 --> 00:38:05.880 align:middle line:84% at the state in which the system is 00:38:05.880 --> 00:38:09.270 align:middle line:84% in mutual equilibrium with the reservoir. 00:38:09.270 --> 00:38:17.080 align:middle line:84% So by exploiting this idea that this is a minimum, 00:38:17.080 --> 00:38:20.360 align:middle line:84% we extract information about, for example, 00:38:20.360 --> 00:38:31.710 align:middle line:84% the concavity of the fundamental relation by devising changes. 00:38:31.710 --> 00:38:34.690 align:middle line:84% You start from this mutual equilibrium state. 00:38:34.690 --> 00:38:39.840 align:middle line:84% And you move anywhere else in one 00:38:39.840 --> 00:38:44.850 align:middle line:84% of the possible states of your system A. Wherever you move, 00:38:44.850 --> 00:38:48.080 align:middle line:90% gamma must increase. 00:38:48.080 --> 00:38:53.810 align:middle line:84% And so, for example, we chose a particular changes. 00:38:53.810 --> 00:38:57.830 align:middle line:84% And we proved that with those particular changes, 00:38:57.830 --> 00:39:02.840 align:middle line:84% you can find that the second differential 00:39:02.840 --> 00:39:14.030 align:middle line:84% of the energy at fixed n and V, second differential, so 00:39:14.030 --> 00:39:18.180 align:middle line:90% the concavity here is positive. 00:39:18.180 --> 00:39:20.410 align:middle line:84% You can view it also from the other point of view. 00:39:20.410 --> 00:39:23.490 align:middle line:84% So you find that the second differential of S 00:39:23.490 --> 00:39:27.840 align:middle line:90% at fixed V and n is negative. 00:39:27.840 --> 00:39:30.155 align:middle line:90% OK. 00:39:30.155 --> 00:39:38.840 align:middle line:84% Now, yeah, we also applied that to prove that, 00:39:38.840 --> 00:39:40.790 align:middle line:90% for example, the specific-- 00:39:40.790 --> 00:39:45.036 align:middle line:84% I mean the heat capacities are non-negative. 00:39:45.036 --> 00:39:49.110 align:middle line:90% 00:39:49.110 --> 00:39:53.990 align:middle line:90% And we proved the first part. 00:39:53.990 --> 00:39:57.140 align:middle line:84% This is just the first part of the theorem 00:39:57.140 --> 00:39:58.620 align:middle line:90% by LeChatelier and Braun. 00:39:58.620 --> 00:40:01.960 align:middle line:90% 00:40:01.960 --> 00:40:06.230 align:middle line:84% Today we'll do also a second part, 00:40:06.230 --> 00:40:11.280 align:middle line:90% which is even more interesting. 00:40:11.280 --> 00:40:16.620 align:middle line:84% So the first part was, anyway, that if you perturb 00:40:16.620 --> 00:40:19.710 align:middle line:84% the state of your system, which is initially 00:40:19.710 --> 00:40:21.990 align:middle line:84% in mutual equilibrium with the reservoir, 00:40:21.990 --> 00:40:25.080 align:middle line:84% you put it off the mutual equilibrium 00:40:25.080 --> 00:40:28.650 align:middle line:84% by, for example, giving energy, the system 00:40:28.650 --> 00:40:33.170 align:middle line:84% will react by increasing its temperature. 00:40:33.170 --> 00:40:36.560 align:middle line:84% And the temperature, as we know, is the escaping tendency 00:40:36.560 --> 00:40:37.290 align:middle line:90% for energy. 00:40:37.290 --> 00:40:41.960 align:middle line:84% So it increases its willingness to give away energy. 00:40:41.960 --> 00:40:47.510 align:middle line:84% And since the reservoir is there, 00:40:47.510 --> 00:40:54.050 align:middle line:84% the increasing in temperature prepares for a heat interaction 00:40:54.050 --> 00:40:57.810 align:middle line:84% in which the system gives energy to the reservoir, 00:40:57.810 --> 00:41:03.810 align:middle line:84% thereby contrasting the change with which you took it away 00:41:03.810 --> 00:41:04.600 align:middle line:90% from equilibrium. 00:41:04.600 --> 00:41:07.140 align:middle line:84% It tries to spontaneously return to equilibrium 00:41:07.140 --> 00:41:09.130 align:middle line:90% by increasing its temperature. 00:41:09.130 --> 00:41:15.170 align:middle line:90% 00:41:15.170 --> 00:41:17.510 align:middle line:90% So that's one thing. 00:41:17.510 --> 00:41:18.290 align:middle line:90% All right. 00:41:18.290 --> 00:41:30.410 align:middle line:84% So to see a little more, let's broaden our class or category 00:41:30.410 --> 00:41:33.350 align:middle line:84% of systems that we call thermal reservoirs 00:41:33.350 --> 00:41:37.940 align:middle line:84% or, say, heat baths because, in practice, 00:41:37.940 --> 00:41:42.915 align:middle line:84% we often have also reservoirs that are variable volume. 00:41:42.915 --> 00:41:46.570 align:middle line:90% 00:41:46.570 --> 00:41:52.420 align:middle line:84% Often, the atmosphere where we live is the environment. 00:41:52.420 --> 00:41:55.860 align:middle line:84% And it's something that would accept a change in volume. 00:41:55.860 --> 00:42:01.060 align:middle line:90% 00:42:01.060 --> 00:42:04.810 align:middle line:84% If the change in volume is reasonably small 00:42:04.810 --> 00:42:09.880 align:middle line:84% and also the changes in energy are reasonably small, 00:42:09.880 --> 00:42:13.360 align:middle line:84% you can add or subtract the volume and energy 00:42:13.360 --> 00:42:15.910 align:middle line:84% to the environment without changing its temperature 00:42:15.910 --> 00:42:17.830 align:middle line:90% and pressure. 00:42:17.830 --> 00:42:19.990 align:middle line:84% So if the system is large enough, 00:42:19.990 --> 00:42:25.630 align:middle line:84% like the air in the environment of a town, 00:42:25.630 --> 00:42:30.970 align:middle line:84% or the water in the sea, or in the lake, or in the river, 00:42:30.970 --> 00:42:33.230 align:middle line:90% with respect to-- 00:42:33.230 --> 00:42:34.900 align:middle line:84% if it is large enough with respect 00:42:34.900 --> 00:42:38.590 align:middle line:84% to the amounts of energies, entropies and volume 00:42:38.590 --> 00:42:42.250 align:middle line:84% that you exchange, then you may consider that 00:42:42.250 --> 00:42:46.630 align:middle line:84% as a thermal reservoir, so a system 00:42:46.630 --> 00:42:52.390 align:middle line:84% that maintains fixed values of the temperature 00:42:52.390 --> 00:42:53.290 align:middle line:90% and the pressure. 00:42:53.290 --> 00:42:58.430 align:middle line:84% In this case, we consider a system which has variable volume 00:42:58.430 --> 00:43:02.260 align:middle line:90% but still the amounts are fixed. 00:43:02.260 --> 00:43:07.270 align:middle line:84% Then the fundamental relation for a system 00:43:07.270 --> 00:43:15.850 align:middle line:90% is also linear except that now-- 00:43:15.850 --> 00:43:16.750 align:middle line:90% OK. 00:43:16.750 --> 00:43:25.810 align:middle line:84% Before we had energy, entropy, and this is the temperature, TR. 00:43:25.810 --> 00:43:48.540 align:middle line:84% Now we add volume as a possibility to change then, 00:43:48.540 --> 00:43:59.880 align:middle line:84% we have an equation of the plane of which the slopes are 00:43:59.880 --> 00:44:05.430 align:middle line:84% TR when you take the slope at constant V and minus pR 00:44:05.430 --> 00:44:10.730 align:middle line:84% when you take the slope at constant S. 00:44:10.730 --> 00:44:13.080 align:middle line:84% So this is the equation of a plane. 00:44:13.080 --> 00:44:16.810 align:middle line:84% And that's the fundamental relation 00:44:16.810 --> 00:44:20.040 align:middle line:84% for that thermal reservoir at variable volume. 00:44:20.040 --> 00:44:25.600 align:middle line:90% 00:44:25.600 --> 00:44:27.210 align:middle line:90% So we ask the same question now. 00:44:27.210 --> 00:44:27.730 align:middle line:90% OK. 00:44:27.730 --> 00:44:33.960 align:middle line:84% What if I'm changing state for my system 00:44:33.960 --> 00:44:39.120 align:middle line:84% but it exchanges volume with the reservoir. 00:44:39.120 --> 00:44:42.800 align:middle line:84% So maybe the system changes also the volume. 00:44:42.800 --> 00:44:45.230 align:middle line:84% And where does it take that volume? 00:44:45.230 --> 00:44:48.530 align:middle line:90% From the reservoir. 00:44:48.530 --> 00:44:49.760 align:middle line:90% Or where does it give it? 00:44:49.760 --> 00:44:52.870 align:middle line:90% To the reservoir. 00:44:52.870 --> 00:44:57.790 align:middle line:84% You want to ask the maximum work or the minimum work requirement 00:44:57.790 --> 00:44:59.590 align:middle line:90% for doing that change. 00:44:59.590 --> 00:45:04.180 align:middle line:84% Well, energy balance, entropy balance, fundamental relation 00:45:04.180 --> 00:45:05.230 align:middle line:90% for the reservoir. 00:45:05.230 --> 00:45:07.540 align:middle line:84% And here you also have to do a volume balance 00:45:07.540 --> 00:45:09.460 align:middle line:84% because you have an additional variable that 00:45:09.460 --> 00:45:15.450 align:middle line:84% has to be kept fixed for the composite system. 00:45:15.450 --> 00:45:19.870 align:middle line:84% The overall volume is conserved in order 00:45:19.870 --> 00:45:24.330 align:middle line:84% to have no other external effects like in a weight 00:45:24.330 --> 00:45:26.610 align:middle line:90% process. 00:45:26.610 --> 00:45:29.250 align:middle line:90% So when you do the-- 00:45:29.250 --> 00:45:36.370 align:middle line:84% when you solve this problem by eliminating the variables that 00:45:36.370 --> 00:45:41.366 align:middle line:84% relate to the reservoir, here is the result. 00:45:41.366 --> 00:45:45.180 align:middle line:84% It is almost like before except that there is now 00:45:45.180 --> 00:45:50.160 align:middle line:84% an extra term here, which is PR times the difference in volume 00:45:50.160 --> 00:45:52.220 align:middle line:90% of your system. 00:45:52.220 --> 00:45:56.180 align:middle line:84% And again, it's a reversible term 00:45:56.180 --> 00:46:01.470 align:middle line:84% minus a term related to irreversibility. 00:46:01.470 --> 00:46:06.850 align:middle line:90% 00:46:06.850 --> 00:46:09.340 align:middle line:84% The reversible term can be seen as the difference 00:46:09.340 --> 00:46:15.280 align:middle line:90% in two properties. 00:46:15.280 --> 00:46:19.060 align:middle line:84% You could consider this as a definition of another property 00:46:19.060 --> 00:46:24.020 align:middle line:84% that we still call, for short of names, 00:46:24.020 --> 00:46:27.040 align:middle line:84% available energy with respect to that particular reservoir 00:46:27.040 --> 00:46:29.780 align:middle line:90% at variable volume. 00:46:29.780 --> 00:46:36.350 align:middle line:84% It's like the one before plus this additional PR times delta 00:46:36.350 --> 00:46:40.340 align:middle line:84% V. And also this one, if you notice, 00:46:40.340 --> 00:46:46.160 align:middle line:84% may be written as a difference in an availability function 00:46:46.160 --> 00:46:53.300 align:middle line:84% E minus TRS plus PRV, which we are going to call just 00:46:53.300 --> 00:46:55.380 align:middle line:90% with the name capital Psi. 00:46:55.380 --> 00:47:00.160 align:middle line:90% 00:47:00.160 --> 00:47:05.540 align:middle line:84% And we may notice that this is not-- 00:47:05.540 --> 00:47:09.910 align:middle line:84% that it looks like but it's not the Gibbs free energy 00:47:09.910 --> 00:47:15.460 align:middle line:84% except when the system is in mutual equilibrium-- 00:47:15.460 --> 00:47:16.960 align:middle line:84% in the state of mutual equilibrium 00:47:16.960 --> 00:47:18.530 align:middle line:90% with the environment. 00:47:18.530 --> 00:47:24.770 align:middle line:84% In that case, TR is also the temperature of your system A 00:47:24.770 --> 00:47:27.410 align:middle line:84% and therefore this is the Gibbs free energy, 00:47:27.410 --> 00:47:36.680 align:middle line:84% whereas in any other state, this TR and this PR 00:47:36.680 --> 00:47:43.635 align:middle line:84% may not be the temperature of your system A. 00:47:43.635 --> 00:47:48.510 align:middle line:84% And actually, if the state is not an equilibrium state, 00:47:48.510 --> 00:47:52.790 align:middle line:84% the system A might not even have a temperature defined. 00:47:52.790 --> 00:47:56.570 align:middle line:84% Actually, it does not if it is not stable equilibrium. 00:47:56.570 --> 00:48:01.350 align:middle line:84% Yet this availability function is defined. 00:48:01.350 --> 00:48:05.160 align:middle line:84% And it gives you these answers to questions 00:48:05.160 --> 00:48:13.440 align:middle line:84% of minimum or maximum work to do or to obtain. 00:48:13.440 --> 00:48:20.320 align:middle line:90% 00:48:20.320 --> 00:48:21.870 align:middle line:90% Yes. 00:48:21.870 --> 00:48:25.170 align:middle line:84% Exactly with the same procedure that we 00:48:25.170 --> 00:48:34.010 align:middle line:84% have done to obtain these conditions for the stability. 00:48:34.010 --> 00:48:36.200 align:middle line:90% You can do it also here. 00:48:36.200 --> 00:48:41.710 align:middle line:84% Now, since I have several variables, it's not-- 00:48:41.710 --> 00:48:42.210 align:middle line:90% I'm sorry. 00:48:42.210 --> 00:48:44.060 align:middle line:90% Let me go back one moment here. 00:48:44.060 --> 00:48:50.200 align:middle line:84% So you should think of a similar procedure 00:48:50.200 --> 00:48:54.100 align:middle line:84% only in a three-dimensional setup 00:48:54.100 --> 00:48:56.750 align:middle line:84% in which you start from a mutual equilibrium state 00:48:56.750 --> 00:49:01.800 align:middle line:90% and you move away from it. 00:49:01.800 --> 00:49:04.050 align:middle line:84% And then you apply the condition that gamma 00:49:04.050 --> 00:49:07.820 align:middle line:90% must increase if you move away. 00:49:07.820 --> 00:49:17.520 align:middle line:84% Now whereas here you could move away only on the S plane, 00:49:17.520 --> 00:49:25.240 align:middle line:84% in the new setup, where also the volume can be changed, 00:49:25.240 --> 00:49:31.210 align:middle line:84% yes, you can move at constant V or constant E, 00:49:31.210 --> 00:49:33.740 align:middle line:84% but you can also move in any other direction 00:49:33.740 --> 00:49:36.370 align:middle line:90% so you have more choices. 00:49:36.370 --> 00:49:40.250 align:middle line:84% And whatever choice you make, you still 00:49:40.250 --> 00:49:46.325 align:middle line:84% have to have this property to increase. 00:49:46.325 --> 00:49:48.830 align:middle line:90% 00:49:48.830 --> 00:49:55.100 align:middle line:84% So here, I'll let you, if you're interested, watch the details. 00:49:55.100 --> 00:50:00.440 align:middle line:84% I chose, for example, to move by changing 00:50:00.440 --> 00:50:05.750 align:middle line:84% the entropy and the volume, keeping the n fixed. 00:50:05.750 --> 00:50:08.750 align:middle line:84% And it turns out that when you compute the change 00:50:08.750 --> 00:50:13.940 align:middle line:84% in the variable Psi, it is exactly equal to 1/2 the second 00:50:13.940 --> 00:50:21.120 align:middle line:84% differential of E. And therefore you prove that this is greater 00:50:21.120 --> 00:50:23.670 align:middle line:90% than or equal to 0. 00:50:23.670 --> 00:50:27.180 align:middle line:84% Or if I choose instead to change the energy and the volume 00:50:27.180 --> 00:50:30.060 align:middle line:84% and therefore-- and to move along the stable equilibrium 00:50:30.060 --> 00:50:33.210 align:middle line:84% state curve so that the entropy is determined 00:50:33.210 --> 00:50:36.060 align:middle line:84% by the fundamental relation, then delta Psi 00:50:36.060 --> 00:50:39.420 align:middle line:84% becomes equal to minus the second differential 00:50:39.420 --> 00:50:41.810 align:middle line:90% of the entropy. 00:50:41.810 --> 00:50:44.400 align:middle line:84% And since that must be greater than 0, 00:50:44.400 --> 00:50:53.140 align:middle line:84% it gives again that the d2S at fixed n (is negative). 00:50:53.140 --> 00:51:00.090 align:middle line:84% Now this d2S if you want to represent it, 00:51:00.090 --> 00:51:06.320 align:middle line:84% it's not just the second derivative 00:51:06.320 --> 00:51:14.070 align:middle line:84% because you see when we have d2S at constant V and n, 00:51:14.070 --> 00:51:18.600 align:middle line:84% this was equal to the second derivative of S 00:51:18.600 --> 00:51:23.670 align:middle line:84% with respect to E times dE squared. 00:51:23.670 --> 00:51:26.680 align:middle line:90% 00:51:26.680 --> 00:51:35.380 align:middle line:84% But this other d2S at constant n only, you can vary both E and V. 00:51:35.380 --> 00:51:40.330 align:middle line:84% Then it's given by d2S with respect 00:51:40.330 --> 00:51:50.470 align:middle line:84% to E squared times dE squared plus twice the second derivative 00:51:50.470 --> 00:51:54.970 align:middle line:84% of S with respect to E and with respect to V times 00:51:54.970 --> 00:52:01.090 align:middle line:84% dE dV plus the second with respect 00:52:01.090 --> 00:52:06.020 align:middle line:90% to V twice times dV squared. 00:52:06.020 --> 00:52:09.850 align:middle line:84% So it's a quadratic form which you 00:52:09.850 --> 00:52:22.180 align:middle line:84% can express also in this form using matrices, dE dV. 00:52:22.180 --> 00:52:24.350 align:middle line:84% Here, you put the Hess determinant, 00:52:24.350 --> 00:52:28.610 align:middle line:84% which is the determinant made up of the second derivatives. 00:52:28.610 --> 00:52:37.665 align:middle line:90% 00:52:37.665 --> 00:52:41.260 align:middle line:90% And this is equal. 00:52:41.260 --> 00:52:45.280 align:middle line:84% And then you multiply this by this other vector made up 00:52:45.280 --> 00:52:48.490 align:middle line:90% of two elements dE and dV. 00:52:48.490 --> 00:52:51.550 align:middle line:84% If you do matrix multiplication of this-- 00:52:51.550 --> 00:52:54.760 align:middle line:90% let's be more elegant here-- 00:52:54.760 --> 00:52:59.230 align:middle line:90% you get the above result. OK. 00:52:59.230 --> 00:53:04.215 align:middle line:90% 00:53:04.215 --> 00:53:15.040 align:middle line:84% And the fact that this has to be negative for arbitrary 00:53:15.040 --> 00:53:22.590 align:middle line:84% choice of your vector dE dV, which means no matter 00:53:22.590 --> 00:53:27.930 align:middle line:84% what direction you want to move, this 00:53:27.930 --> 00:53:33.860 align:middle line:84% has to be negative, that requires that the matrix inside 00:53:33.860 --> 00:53:35.790 align:middle line:90% be non-negative definite. 00:53:35.790 --> 00:53:39.470 align:middle line:90% 00:53:39.470 --> 00:53:44.220 align:middle line:84% And the matrix is a non-negative definite. 00:53:44.220 --> 00:53:44.720 align:middle line:90% When? 00:53:44.720 --> 00:53:47.440 align:middle line:90% 00:53:47.440 --> 00:53:48.250 align:middle line:90% Sorry. 00:53:48.250 --> 00:53:50.620 align:middle line:90% Non-positive. 00:53:50.620 --> 00:53:53.880 align:middle line:90% So it's negative semidefinite. 00:53:53.880 --> 00:53:55.660 align:middle line:90% OK. 00:53:55.660 --> 00:53:56.515 align:middle line:90% When is a matrix-- 00:53:56.515 --> 00:54:01.420 align:middle line:90% 00:54:01.420 --> 00:54:02.630 align:middle line:90% let's start with positive. 00:54:02.630 --> 00:54:05.009 align:middle line:84% When is a matrix positive definite? 00:54:05.009 --> 00:54:12.050 align:middle line:90% 00:54:12.050 --> 00:54:18.650 align:middle line:90% I have a matrix a11, a12. 00:54:18.650 --> 00:54:20.790 align:middle line:90% And let's make it symmetric. 00:54:20.790 --> 00:54:29.210 align:middle line:90% 00:54:29.210 --> 00:54:30.030 align:middle line:90% OK. 00:54:30.030 --> 00:54:40.170 align:middle line:84% So this, if it is positive semi-definite, 00:54:40.170 --> 00:54:44.526 align:middle line:84% the diagonal entries must be non-negative. 00:54:44.526 --> 00:54:51.390 align:middle line:90% 00:54:51.390 --> 00:54:55.050 align:middle line:84% The determinant must be non-negative. 00:54:55.050 --> 00:55:01.650 align:middle line:84% So a11 a22 minus a12 squared because it's 00:55:01.650 --> 00:55:05.930 align:middle line:84% a symmetric must be greater than or equal to 0. 00:55:05.930 --> 00:55:11.400 align:middle line:90% 00:55:11.400 --> 00:55:13.980 align:middle line:90% You want to make it-- 00:55:13.980 --> 00:55:18.050 align:middle line:84% if the matrix, instead of being positive 00:55:18.050 --> 00:55:20.800 align:middle line:84% semi-definite is negative semidefinite, 00:55:20.800 --> 00:55:28.580 align:middle line:84% so this is less than or equal to, 00:55:28.580 --> 00:55:38.190 align:middle line:84% so the entries on the diagonal should be non-positive. 00:55:38.190 --> 00:55:42.350 align:middle line:90% 00:55:42.350 --> 00:55:44.703 align:middle line:90% How about the determinant? 00:55:44.703 --> 00:55:45.745 align:middle line:90% Should I change the sign? 00:55:45.745 --> 00:55:50.190 align:middle line:90% 00:55:50.190 --> 00:55:58.980 align:middle line:84% Not for a 2 by 2 matrix because you see also, 00:55:58.980 --> 00:56:03.660 align:middle line:84% the more fundamental property of a non-negative-- 00:56:03.660 --> 00:56:09.970 align:middle line:84% of a non-positive-- of a negative semi-definite matrix 00:56:09.970 --> 00:56:17.410 align:middle line:84% is that its eigenvalues are all negative. 00:56:17.410 --> 00:56:20.320 align:middle line:84% So it means-- let's call them lambda. 00:56:20.320 --> 00:56:23.980 align:middle line:90% So lambda 1 is less than 0. 00:56:23.980 --> 00:56:25.905 align:middle line:90% Lambda 2 is less than 0. 00:56:25.905 --> 00:56:33.110 align:middle line:90% 00:56:33.110 --> 00:56:37.670 align:middle line:84% And for a 2 by 2 matrix, the determinant 00:56:37.670 --> 00:56:40.940 align:middle line:84% is equal to the product of the two eigenvalues. 00:56:40.940 --> 00:56:45.850 align:middle line:84% So this is also equal to lambda 1 times lambda 2. 00:56:45.850 --> 00:56:48.110 align:middle line:84% So if you multiply two negative numbers, 00:56:48.110 --> 00:56:49.370 align:middle line:90% you get a positive number. 00:56:49.370 --> 00:56:54.620 align:middle line:84% And so you get the determinant is greater than zero. 00:56:54.620 --> 00:56:59.200 align:middle line:84% If instead you had a 3 by 3 by 3 matrix, then 00:56:59.200 --> 00:57:01.900 align:middle line:84% you have three negative numbers that 00:57:01.900 --> 00:57:05.440 align:middle line:84% multiply to give the determinant in that case 00:57:05.440 --> 00:57:08.160 align:middle line:90% with an odd number of entries. 00:57:08.160 --> 00:57:11.110 align:middle line:90% 00:57:11.110 --> 00:57:13.870 align:middle line:84% Negative times negative times negative makes negative. 00:57:13.870 --> 00:57:25.920 align:middle line:84% So the determinant of a negative semi-definite matrix is positive 00:57:25.920 --> 00:57:32.370 align:middle line:84% for odd number of rows and columns and negative for-- 00:57:32.370 --> 00:57:33.660 align:middle line:90% what did I say? 00:57:33.660 --> 00:57:37.930 align:middle line:84% Positive for even number and negative for odd number. 00:57:37.930 --> 00:57:38.430 align:middle line:90% OK. 00:57:38.430 --> 00:57:45.110 align:middle line:90% 00:57:45.110 --> 00:57:47.770 align:middle line:84% We'll need that in a short while. 00:57:47.770 --> 00:57:52.380 align:middle line:90% 00:57:52.380 --> 00:57:53.890 align:middle line:90% Here-- oh yeah. 00:57:53.890 --> 00:58:11.960 align:middle line:84% Let me just go back to a simple question about-- 00:58:11.960 --> 00:58:15.910 align:middle line:90% 00:58:15.910 --> 00:58:16.410 align:middle line:90% all right. 00:58:16.410 --> 00:58:22.530 align:middle line:90% Suppose you have a vacuum. 00:58:22.530 --> 00:58:24.090 align:middle line:90% So we have a piston. 00:58:24.090 --> 00:58:31.640 align:middle line:84% You have a container in which you manage to have nothing. 00:58:31.640 --> 00:58:32.150 align:middle line:90% Vacuum. 00:58:32.150 --> 00:58:35.240 align:middle line:84% Vacuum means no particles, nothing. 00:58:35.240 --> 00:58:37.100 align:middle line:90% OK. 00:58:37.100 --> 00:58:45.870 align:middle line:84% So the question is, is that a situation-- 00:58:45.870 --> 00:58:51.270 align:middle line:84% and this vacuum is in an environment which 00:58:51.270 --> 00:59:01.130 align:middle line:84% I can model as a reservoir at pressure pR, also temperature TR 00:59:01.130 --> 00:59:05.970 align:middle line:84% but the temperature doesn't play much of a role here. 00:59:05.970 --> 00:59:10.110 align:middle line:84% So it's like I have this container in the environment 00:59:10.110 --> 00:59:11.250 align:middle line:90% here. 00:59:11.250 --> 00:59:13.650 align:middle line:90% I managed to make vacuum. 00:59:13.650 --> 00:59:17.240 align:middle line:84% What's the minimum work required to make that vacuum? 00:59:17.240 --> 00:59:19.980 align:middle line:90% 00:59:19.980 --> 00:59:23.770 align:middle line:84% Or if I have that vacuum, can I do something useful? 00:59:23.770 --> 00:59:25.410 align:middle line:90% Can I lift a weight? 00:59:25.410 --> 00:59:27.755 align:middle line:90% What's the maximum lift? 00:59:27.755 --> 00:59:33.820 align:middle line:90% 00:59:33.820 --> 00:59:37.800 align:middle line:84% Well, let's apply this formula here. 00:59:37.800 --> 00:59:43.250 align:middle line:90% 00:59:43.250 --> 00:59:46.830 align:middle line:84% Assume that we do things the best thermodynamic way. 00:59:46.830 --> 00:59:52.100 align:middle line:84% So that means without generating entropy by irreversibility. 00:59:52.100 --> 00:59:57.860 align:middle line:84% And suppose I want my initial state, state one. 00:59:57.860 --> 01:00:04.910 align:middle line:84% So let me say that I started with my piston down below here. 01:00:04.910 --> 01:00:13.620 align:middle line:84% And this is my state 1 and this is my state 2 and I pull. 01:00:13.620 --> 01:00:18.980 align:middle line:90% 01:00:18.980 --> 01:00:23.780 align:middle line:84% Here I have the pressure of the environment. 01:00:23.780 --> 01:00:29.620 align:middle line:84% So if I pull, how much work do I have to do? 01:00:29.620 --> 01:00:33.970 align:middle line:84% I have to-- if you look at the piston on one side, 01:00:33.970 --> 01:00:36.400 align:middle line:84% you have zero pressure because nothing here. 01:00:36.400 --> 01:00:37.805 align:middle line:90% On the other side, you have PR. 01:00:37.805 --> 01:00:40.720 align:middle line:90% 01:00:40.720 --> 01:00:53.065 align:middle line:84% So if the volume change is V2 minus V1, you can view that as-- 01:00:53.065 --> 01:00:56.170 align:middle line:90% 01:00:56.170 --> 01:00:57.850 align:middle line:90% let's call x this-- 01:00:57.850 --> 01:01:00.470 align:middle line:90% 01:01:00.470 --> 01:01:00.970 align:middle line:90% OK. 01:01:00.970 --> 01:01:05.350 align:middle line:84% Let's go call z the location of the weight-- 01:01:05.350 --> 01:01:07.270 align:middle line:90% I'm sorry, of the piston. 01:01:07.270 --> 01:01:11.445 align:middle line:84% And let's call little a the area of the piston. 01:01:11.445 --> 01:01:16.230 align:middle line:90% 01:01:16.230 --> 01:01:24.550 align:middle line:84% So the force that you have to apply 01:01:24.550 --> 01:01:28.630 align:middle line:90% is equal to the pressure times-- 01:01:28.630 --> 01:01:33.190 align:middle line:84% which is force per unit area times the area. 01:01:33.190 --> 01:01:36.860 align:middle line:90% So that's the force. 01:01:36.860 --> 01:01:44.008 align:middle line:90% The work is f times x. 01:01:44.008 --> 01:01:44.550 align:middle line:90% I'm sorry. z. 01:01:44.550 --> 01:01:52.310 align:middle line:90% 01:01:52.310 --> 01:02:00.470 align:middle line:90% So that's pR times a z. 01:02:00.470 --> 01:02:06.560 align:middle line:84% But a z is the volume so that's pR times V. But actually, it's 01:02:06.560 --> 01:02:13.270 align:middle line:84% V2 minus V1 because this is V2 and this is 01:02:13.270 --> 01:02:15.830 align:middle line:90% V1 is equal to 0 in our case. 01:02:15.830 --> 01:02:25.300 align:middle line:84% So it seems that the work that you 01:02:25.300 --> 01:02:30.670 align:middle line:84% need to spend in order to go from 1 to 2 in a reversible way 01:02:30.670 --> 01:02:43.870 align:middle line:84% is equal to that, which is written here except that now it 01:02:43.870 --> 01:02:48.780 align:middle line:84% has the wrong sign because this represents 01:02:48.780 --> 01:02:54.330 align:middle line:84% the work with the arrow out instead 01:02:54.330 --> 01:02:56.680 align:middle line:84% if we change the direction of the arrow 01:02:56.680 --> 01:03:02.430 align:middle line:84% so let's say that this is the work that I need to spend. 01:03:02.430 --> 01:03:12.600 align:middle line:84% And of course, this I could also write the same thing 01:03:12.600 --> 01:03:14.040 align:middle line:90% by changing the arrow. 01:03:14.040 --> 01:03:16.150 align:middle line:84% If I change the arrow, I change also the sign. 01:03:16.150 --> 01:03:21.120 align:middle line:90% 01:03:21.120 --> 01:03:26.010 align:middle line:84% So essentially, the minimum work required to produce the vacuum 01:03:26.010 --> 01:03:29.910 align:middle line:84% is equal to the pressure of the environment times the volume 01:03:29.910 --> 01:03:32.990 align:middle line:90% of that vacuum. 01:03:32.990 --> 01:03:36.350 align:middle line:84% That's an easy result. We could have-- 01:03:36.350 --> 01:03:38.750 align:middle line:84% I mean, we actually got to that result 01:03:38.750 --> 01:03:43.170 align:middle line:84% without all the complicated long story that we're telling. 01:03:43.170 --> 01:03:52.070 align:middle line:84% But that's just to show the consistency of this relation 01:03:52.070 --> 01:03:53.795 align:middle line:90% with what we know. 01:03:53.795 --> 01:04:00.740 align:middle line:90% 01:04:00.740 --> 01:04:08.140 align:middle line:84% So now let's think of another kind of reservoir 01:04:08.140 --> 01:04:14.140 align:middle line:84% which, again, can be typically-- is a typical model 01:04:14.140 --> 01:04:21.696 align:middle line:84% of the air in the environment or some natural environment. 01:04:21.696 --> 01:04:24.990 align:middle line:90% 01:04:24.990 --> 01:04:27.030 align:middle line:84% So here is a particular reservoir 01:04:27.030 --> 01:04:31.240 align:middle line:84% that can exchange through a semi-permeable membrane 01:04:31.240 --> 01:04:35.270 align:middle line:84% only one type of substance with your system. 01:04:35.270 --> 01:04:41.000 align:middle line:84% We may call it an osmotic reservoir or a thermal reservoir 01:04:41.000 --> 01:04:47.670 align:middle line:84% or we just don't give any names but just 01:04:47.670 --> 01:04:52.600 align:middle line:90% define it by just these words. 01:04:52.600 --> 01:04:54.510 align:middle line:90% All right. 01:04:54.510 --> 01:04:57.190 align:middle line:84% You run the same machinery, energy balance, entropy, 01:04:57.190 --> 01:04:57.880 align:middle line:90% balance. 01:04:57.880 --> 01:05:00.750 align:middle line:84% Now you also have to make a balance 01:05:00.750 --> 01:05:05.020 align:middle line:84% for the amounts of constituents of the kind of substance 01:05:05.020 --> 01:05:06.946 align:middle line:90% that can be transferred. 01:05:06.946 --> 01:05:10.320 align:middle line:90% 01:05:10.320 --> 01:05:18.640 align:middle line:84% For example, you may be interested in-- 01:05:18.640 --> 01:05:26.830 align:middle line:84% some people are devising methods or plans 01:05:26.830 --> 01:05:30.910 align:middle line:84% that want to extract CO2 from the atmosphere. 01:05:30.910 --> 01:05:31.410 align:middle line:90% All right. 01:05:31.410 --> 01:05:35.450 align:middle line:84% So you want to ask, well, what's the minimum amount of work I 01:05:35.450 --> 01:05:43.500 align:middle line:84% need to spend if I want to extract CO2 from the atmosphere 01:05:43.500 --> 01:05:48.040 align:middle line:84% because that minimum defines the target. 01:05:48.040 --> 01:05:51.340 align:middle line:84% Defines also the way in which you determine 01:05:51.340 --> 01:05:54.880 align:middle line:84% the efficiency, the second law efficiency 01:05:54.880 --> 01:06:01.720 align:middle line:90% of that particular equipment. 01:06:01.720 --> 01:06:04.360 align:middle line:84% And it's a question very similar to this one 01:06:04.360 --> 01:06:14.190 align:middle line:84% because one way to do that, not saying it's the reversible way, 01:06:14.190 --> 01:06:17.335 align:middle line:84% but I'm saying that one way to do it is first you do vacuum. 01:06:17.335 --> 01:06:20.100 align:middle line:90% 01:06:20.100 --> 01:06:27.400 align:middle line:84% And then you open up the permeability of the piston 01:06:27.400 --> 01:06:29.830 align:middle line:90% to just CO2. 01:06:29.830 --> 01:06:34.370 align:middle line:84% And so CO2 will fill this region. 01:06:34.370 --> 01:06:39.490 align:middle line:84% And you'll have just CO2 because the other parts, 01:06:39.490 --> 01:06:43.840 align:middle line:84% oxygen and nitrogen, cannot go through this particular 01:06:43.840 --> 01:06:45.650 align:middle line:90% semi-permeable membrane. 01:06:45.650 --> 01:06:48.290 align:middle line:90% And that way you have separated. 01:06:48.290 --> 01:06:51.630 align:middle line:90% So one way is produce vacuum. 01:06:51.630 --> 01:06:54.893 align:middle line:84% And we know how much work it requires. 01:06:54.893 --> 01:06:57.430 align:middle line:90% 01:06:57.430 --> 01:07:02.970 align:middle line:84% Use a semi-permeable membrane and there you have your CO2. 01:07:02.970 --> 01:07:06.900 align:middle line:84% Your CO2 eventually will reach mutual equilibrium 01:07:06.900 --> 01:07:07.980 align:middle line:90% with the reservoir. 01:07:07.980 --> 01:07:14.890 align:middle line:90% 01:07:14.890 --> 01:07:18.130 align:middle line:84% At that stage, I have blocked my piston. 01:07:18.130 --> 01:07:21.010 align:middle line:90% Cannot go up and down. 01:07:21.010 --> 01:07:22.450 align:middle line:90% And I change the reservoir. 01:07:22.450 --> 01:07:26.080 align:middle line:84% Instead of having a reservoir at variable volume, 01:07:26.080 --> 01:07:31.570 align:middle line:84% I have a reservoir that can exchange-- 01:07:31.570 --> 01:07:37.110 align:middle line:84% at constant chemical potential of that substance, CO2. 01:07:37.110 --> 01:07:42.440 align:middle line:90% 01:07:42.440 --> 01:07:49.640 align:middle line:84% The final state will be that, instead of vacuum, 01:07:49.640 --> 01:07:52.710 align:middle line:90% I will just have CO2 pure. 01:07:52.710 --> 01:07:56.950 align:middle line:84% Let's say that this will have a chemical potential 01:07:56.950 --> 01:08:01.920 align:middle line:90% that we will call mu double i. 01:08:01.920 --> 01:08:05.010 align:middle line:84% And it will have a pressure that corresponds 01:08:05.010 --> 01:08:09.160 align:middle line:84% to what we defined when they are in mutual equilibrium, 01:08:09.160 --> 01:08:14.040 align:middle line:84% equality of chemical potentials defines the partial pressure. 01:08:14.040 --> 01:08:17.310 align:middle line:84% So the CO2 will be at the partial pressure 01:08:17.310 --> 01:08:23.850 align:middle line:84% that it has in air viewed as a mixture. 01:08:23.850 --> 01:08:24.350 align:middle line:90% OK. 01:08:24.350 --> 01:08:27.000 align:middle line:90% 01:08:27.000 --> 01:08:31.830 align:middle line:84% This second part of the process from after I made the vacuum, 01:08:31.830 --> 01:08:36.380 align:middle line:84% and I stopped the piston, and I opened the pore, the second part 01:08:36.380 --> 01:08:43.399 align:middle line:84% of the process is spontaneous because, initially, you 01:08:43.399 --> 01:08:46.550 align:middle line:84% are in a nonequilibrium state because here you 01:08:46.550 --> 01:08:50.240 align:middle line:84% have no CO2, whereas, in order for equilibrium, 01:08:50.240 --> 01:08:52.830 align:middle line:84% you need to have chemical potentials to be equal. 01:08:52.830 --> 01:08:56.700 align:middle line:84% So this is an irreversible process. 01:08:56.700 --> 01:08:58.750 align:middle line:84% There will be entropy produced by reversibility. 01:08:58.750 --> 01:09:01.460 align:middle line:84% So this is not the best way to do it. 01:09:01.460 --> 01:09:05.640 align:middle line:84% You could also evaluate how much is the efficiency, 01:09:05.640 --> 01:09:10.220 align:middle line:84% if you could compare it with what's the minimum work that 01:09:10.220 --> 01:09:11.390 align:middle line:90% is required. 01:09:11.390 --> 01:09:14.930 align:middle line:84% And the minimum work that is required 01:09:14.930 --> 01:09:18.609 align:middle line:84% is given here by this formula here. 01:09:18.609 --> 01:09:19.109 align:middle line:90% I'm sorry. 01:09:19.109 --> 01:09:23.560 align:middle line:90% By this formula here. 01:09:23.560 --> 01:09:30.950 align:middle line:84% So there is, again, another available energy function-- 01:09:30.950 --> 01:09:35.520 align:middle line:84% I'm sorry, available energy property 01:09:35.520 --> 01:09:43.080 align:middle line:84% for this particular kind of osmotic reservoirs, 01:09:43.080 --> 01:09:45.479 align:middle line:84% the difference of which tells you 01:09:45.479 --> 01:09:51.800 align:middle line:84% the minimum or the maximum works for producing or exploiting 01:09:51.800 --> 01:09:53.160 align:middle line:90% a situation. 01:09:53.160 --> 01:09:56.980 align:middle line:90% 01:09:56.980 --> 01:10:02.740 align:middle line:84% So we will return later on to this question of minimum work 01:10:02.740 --> 01:10:08.060 align:middle line:84% for separating a substance from, say, the atmosphere. 01:10:08.060 --> 01:10:12.240 align:middle line:90% 01:10:12.240 --> 01:10:16.350 align:middle line:84% Here this combination that appears 01:10:16.350 --> 01:10:21.330 align:middle line:84% in the expression for this new version of the available energy, 01:10:21.330 --> 01:10:26.550 align:middle line:84% namely E minus TRS minus the chemical potential times 01:10:26.550 --> 01:10:29.520 align:middle line:84% the amount of that substance that 01:10:29.520 --> 01:10:32.910 align:middle line:84% can cross between the system and the reservoir, 01:10:32.910 --> 01:10:39.570 align:middle line:84% we may give it a name or not, simply an availability function. 01:10:39.570 --> 01:10:42.630 align:middle line:84% Maybe we could call it osmotic availability function. 01:10:42.630 --> 01:10:46.620 align:middle line:84% Maybe we could use this letter or another letter. 01:10:46.620 --> 01:10:48.080 align:middle line:90% It's just a matter of choice. 01:10:48.080 --> 01:10:54.210 align:middle line:90% 01:10:54.210 --> 01:10:59.830 align:middle line:84% As for the other availability functions, 01:10:59.830 --> 01:11:03.150 align:middle line:90% there is also this minimum. 01:11:03.150 --> 01:11:10.200 align:middle line:84% So also from this one, you can derive by the same method 01:11:10.200 --> 01:11:15.380 align:middle line:84% as before, same procedure, conditions 01:11:15.380 --> 01:11:19.240 align:middle line:90% for stability of equilibrium. 01:11:19.240 --> 01:11:21.470 align:middle line:84% For the second differential of the energy, 01:11:21.470 --> 01:11:27.560 align:middle line:84% now it will be like this but not with volume 01:11:27.560 --> 01:11:31.430 align:middle line:90% but with n i that substitutes. 01:11:31.430 --> 01:11:38.610 align:middle line:84% And that's the expression for these second order 01:11:38.610 --> 01:11:39.760 align:middle line:90% differentials. 01:11:39.760 --> 01:11:44.600 align:middle line:90% 01:11:44.600 --> 01:11:48.690 align:middle line:84% And finally, we could have a grand situation, so 01:11:48.690 --> 01:11:57.300 align:middle line:84% a grand reservoir with which your system can 01:11:57.300 --> 01:12:00.180 align:middle line:90% exchange anything it wants-- 01:12:00.180 --> 01:12:03.800 align:middle line:84% amounts, volume, energy, and entropy. 01:12:03.800 --> 01:12:06.387 align:middle line:90% 01:12:06.387 --> 01:12:06.887 align:middle line:90% OK. 01:12:06.887 --> 01:12:09.869 align:middle line:90% 01:12:09.869 --> 01:12:12.660 align:middle line:84% So you have to run again the same machinery-- energy 01:12:12.660 --> 01:12:14.030 align:middle line:90% balance, entropy, balance. 01:12:14.030 --> 01:12:17.060 align:middle line:84% Fundamental relation for the reservoir. 01:12:17.060 --> 01:12:19.230 align:middle line:84% Notice that-- of course, in the previous case, 01:12:19.230 --> 01:12:20.910 align:middle line:84% maybe I didn't emphasize it enough, 01:12:20.910 --> 01:12:24.590 align:middle line:84% but what changes from reservoir to reservoir 01:12:24.590 --> 01:12:28.460 align:middle line:84% is the expression for the fundamental relations. 01:12:28.460 --> 01:12:37.370 align:middle line:84% These terms are new, peculiar to this reservoir, that 01:12:37.370 --> 01:12:40.670 align:middle line:84% acts as a reservoir not only for exchanges of energy 01:12:40.670 --> 01:12:43.708 align:middle line:84% and entropy but also for if you change volume, 01:12:43.708 --> 01:12:45.000 align:middle line:90% it doesn't change the pressure. 01:12:45.000 --> 01:12:46.790 align:middle line:84% If you change the amounts, it doesn't 01:12:46.790 --> 01:12:48.630 align:middle line:90% change the chemical potentials. 01:12:48.630 --> 01:12:52.160 align:middle line:90% 01:12:52.160 --> 01:12:55.580 align:middle line:84% And then you have to remember also 01:12:55.580 --> 01:13:02.070 align:middle line:84% to do the volume balance and the balances 01:13:02.070 --> 01:13:06.820 align:middle line:84% for all of the amounts of the various constituents that 01:13:06.820 --> 01:13:08.690 align:middle line:90% are conserved. 01:13:08.690 --> 01:13:10.045 align:middle line:90% No chemical reactions here. 01:13:10.045 --> 01:13:14.600 align:middle line:90% 01:13:14.600 --> 01:13:22.470 align:middle line:84% Again, eliminate the terms that relate to the reservoir. 01:13:22.470 --> 01:13:24.380 align:middle line:84% And you get this expression, which 01:13:24.380 --> 01:13:29.560 align:middle line:84% is another generalization of the available energy with respect 01:13:29.560 --> 01:13:34.540 align:middle line:84% to this completely open reservoir. 01:13:34.540 --> 01:13:42.780 align:middle line:84% And this is the available energy, the expression, 01:13:42.780 --> 01:13:48.890 align:middle line:84% which can also be written as a difference in two values. 01:13:48.890 --> 01:13:51.920 align:middle line:84% So the work, the maximum or minimum, the reversible work 01:13:51.920 --> 01:13:56.630 align:middle line:84% is the difference in the values of this new availability 01:13:56.630 --> 01:13:59.570 align:middle line:84% function, which I like to call Euler availability 01:13:59.570 --> 01:14:01.400 align:middle line:84% function for the reason I already 01:14:01.400 --> 01:14:05.470 align:middle line:90% explained the other time. 01:14:05.470 --> 01:14:11.120 align:middle line:84% And that will be also clear later on. 01:14:11.120 --> 01:14:16.800 align:middle line:84% So this E minus TRS plus PRV minus mu Rn. 01:14:16.800 --> 01:14:19.850 align:middle line:84% This availability function is defined for all states, 01:14:19.850 --> 01:14:22.080 align:middle line:90% equilibrium and nonequilibrium. 01:14:22.080 --> 01:14:30.330 align:middle line:84% When your system is in mutual equilibrium, 01:14:30.330 --> 01:14:35.240 align:middle line:84% so it is in the state at temperature TR, pressure PR, 01:14:35.240 --> 01:14:38.620 align:middle line:84% and with the chemical potential, so complete equilibrium 01:14:38.620 --> 01:14:45.740 align:middle line:84% between system and reservoir, then this availability function 01:14:45.740 --> 01:14:48.920 align:middle line:84% corresponds to that characteristic function 01:14:48.920 --> 01:14:58.370 align:middle line:84% that we call the Euler free energy that I just reviewed 01:14:58.370 --> 01:15:01.190 align:middle line:90% at the beginning of today. 01:15:01.190 --> 01:15:03.580 align:middle line:90% OK. 01:15:03.580 --> 01:15:13.930 align:middle line:84% And from this one, this also is minimum at mutual equilibrium. 01:15:13.930 --> 01:15:17.480 align:middle line:90% 01:15:17.480 --> 01:15:25.830 align:middle line:84% And you can do the same again to extract 01:15:25.830 --> 01:15:29.930 align:middle line:90% conditions for stability. 01:15:29.930 --> 01:15:36.850 align:middle line:84% This time, this expression d2E is with respect 01:15:36.850 --> 01:15:42.350 align:middle line:84% not to just two variables, so this matrix, 01:15:42.350 --> 01:15:44.970 align:middle line:84% the Hess matrix will not be just 2 by 2 01:15:44.970 --> 01:15:50.330 align:middle line:84% but will be 2 plus r because it also can 01:15:50.330 --> 01:15:51.980 align:middle line:90% change all the constituents. 01:15:51.980 --> 01:15:57.830 align:middle line:84% And if you have r constituents is like 2 plus r by 2 plus r. 01:15:57.830 --> 01:16:04.570 align:middle line:84% So it's this matrix that we have already anticipated a while ago. 01:16:04.570 --> 01:16:09.690 align:middle line:90% And this therefore proves-- 01:16:09.690 --> 01:16:16.650 align:middle line:84% I mean the details here prove in more detail also 01:16:16.650 --> 01:16:18.886 align:middle line:84% these expressions that we had anticipated. 01:16:18.886 --> 01:16:22.730 align:middle line:90% 01:16:22.730 --> 01:16:23.920 align:middle line:90% But you can also choose-- 01:16:23.920 --> 01:16:27.270 align:middle line:90% 01:16:27.270 --> 01:16:33.450 align:middle line:84% since now you-- think that you are in a mutual equilibrium 01:16:33.450 --> 01:16:38.470 align:middle line:84% state, and then you can move your state 01:16:38.470 --> 01:16:41.680 align:middle line:90% in various directions. 01:16:41.680 --> 01:16:53.610 align:middle line:84% And one way is that you choose another state that still 01:16:53.610 --> 01:16:56.770 align:middle line:84% has the same temperature and pressure of your reservoir 01:16:56.770 --> 01:16:59.940 align:middle line:90% but you change the amounts. 01:16:59.940 --> 01:17:04.395 align:middle line:84% So you choose to move away from mutual equilibrium 01:17:04.395 --> 01:17:06.270 align:middle line:84% because the conditions for mutual equilibrium 01:17:06.270 --> 01:17:09.780 align:middle line:84% with the reservoir not only now that it can exchange not 01:17:09.780 --> 01:17:12.230 align:middle line:90% only energy but also amounts. 01:17:12.230 --> 01:17:15.210 align:middle line:84% It's not only the temperature equality. 01:17:15.210 --> 01:17:17.610 align:middle line:84% It's also the pressure equality for the volume 01:17:17.610 --> 01:17:20.190 align:middle line:84% change and the chemical potentials, 01:17:20.190 --> 01:17:22.380 align:middle line:90% all chemical potentials. 01:17:22.380 --> 01:17:26.250 align:middle line:84% So I can move away from mutual equilibrium 01:17:26.250 --> 01:17:29.460 align:middle line:84% while keeping some of the mutual equilibrium conditions 01:17:29.460 --> 01:17:32.760 align:middle line:84% satisfied, like same temperature and same pressure, 01:17:32.760 --> 01:17:34.760 align:middle line:90% and moving only the amounts. 01:17:34.760 --> 01:17:37.690 align:middle line:90% 01:17:37.690 --> 01:17:43.240 align:middle line:84% If you do that and move along the stable equilibrium state 01:17:43.240 --> 01:17:50.820 align:middle line:84% surface, constant temperature and pressure, 01:17:50.820 --> 01:17:52.710 align:middle line:90% the condition that delta-- 01:17:52.710 --> 01:18:03.380 align:middle line:84% that this Euler availability function must increase along 01:18:03.380 --> 01:18:11.170 align:middle line:84% that path turns out to be the change in this variable Xi 01:18:11.170 --> 01:18:17.240 align:middle line:84% is equal to the second differential of the Gibbs 01:18:17.240 --> 01:18:17.910 align:middle line:90% free energy. 01:18:17.910 --> 01:18:20.780 align:middle line:90% 01:18:20.780 --> 01:18:27.320 align:middle line:84% You can prove it by remembering that the Gibbs free energy is 01:18:27.320 --> 01:18:29.450 align:middle line:90% E minus S minus TR. 01:18:29.450 --> 01:18:33.450 align:middle line:84% And here TR and PR remain constant. 01:18:33.450 --> 01:18:34.915 align:middle line:90% So it remains defined. 01:18:34.915 --> 01:18:38.420 align:middle line:90% 01:18:38.420 --> 01:18:46.560 align:middle line:84% So this proves that the Gibbs free energy 01:18:46.560 --> 01:18:49.250 align:middle line:90% has a minimum if you move-- 01:18:49.250 --> 01:18:52.520 align:middle line:84% so the second differential is non-negative. 01:18:52.520 --> 01:18:53.020 align:middle line:90% Yes. 01:18:53.020 --> 01:18:56.580 align:middle line:84% It is non-negative if you move at constant-- 01:18:56.580 --> 01:19:06.370 align:middle line:84% if you change the composition at constant T and P. 01:19:06.370 --> 01:19:09.370 align:middle line:84% If you keep only temperature fixed 01:19:09.370 --> 01:19:13.460 align:middle line:84% and change the volume instead of keeping the pressure constant, 01:19:13.460 --> 01:19:16.190 align:middle line:90% you get a similar relation. 01:19:16.190 --> 01:19:18.350 align:middle line:84% This is not the Gibbs free energy. 01:19:18.350 --> 01:19:30.460 align:middle line:84% It's the Helmholtz free energy at constant T. 01:19:30.460 --> 01:19:35.270 align:middle line:84% So remember that we have seen the-- 01:19:35.270 --> 01:19:39.740 align:middle line:84% that from these conditions here, we 01:19:39.740 --> 01:19:43.730 align:middle line:84% obtain a first part of the LeChatelier-Braun principle. 01:19:43.730 --> 01:19:50.540 align:middle line:84% Now, yes, another little piece of math. 01:19:50.540 --> 01:19:51.655 align:middle line:90% Sorry about that. 01:19:51.655 --> 01:19:55.470 align:middle line:90% 01:19:55.470 --> 01:19:57.960 align:middle line:84% But that will get us to the second part of the Le 01:19:57.960 --> 01:20:04.690 align:middle line:84% Chatelier-Braun principle, which is a more general statement 01:20:04.690 --> 01:20:14.695 align:middle line:84% which requires a little brush up or maybe first-time-you-see 01:20:14.695 --> 01:20:15.195 align:middle line:90% math. 01:20:15.195 --> 01:20:22.050 align:middle line:90% 01:20:22.050 --> 01:20:27.400 align:middle line:84% So let's say we have a function of two variables. 01:20:27.400 --> 01:20:31.510 align:middle line:84% I put also the other z here just to say that the z's 01:20:31.510 --> 01:20:33.020 align:middle line:90% are going to be constant. 01:20:33.020 --> 01:20:38.390 align:middle line:84% So I have several variables but I keep constant all except two 01:20:38.390 --> 01:20:40.990 align:middle line:90% of them, which are x and y. 01:20:40.990 --> 01:20:45.510 align:middle line:90% 01:20:45.510 --> 01:20:47.410 align:middle line:84% And then, for example, I have that function. 01:20:47.410 --> 01:20:51.140 align:middle line:90% I can write the differential. 01:20:51.140 --> 01:20:55.740 align:middle line:84% So that's the partial of A with respect to x times dx 01:20:55.740 --> 01:20:58.600 align:middle line:84% plus the partial of A with respect to y times dy. 01:20:58.600 --> 01:21:04.890 align:middle line:84% I'm going to use the notation comma x for partial with respect 01:21:04.890 --> 01:21:09.190 align:middle line:84% to x, which is often used in fluid mechanics 01:21:09.190 --> 01:21:12.830 align:middle line:84% so maybe you've seen it already or elsewhere. 01:21:12.830 --> 01:21:16.760 align:middle line:90% 01:21:16.760 --> 01:21:21.050 align:middle line:84% And for example, comma double x is the second derivative 01:21:21.050 --> 01:21:24.890 align:middle line:90% with respect to x twice. 01:21:24.890 --> 01:21:28.520 align:middle line:84% comma xy is the derivative with respect 01:21:28.520 --> 01:21:30.660 align:middle line:90% to x and with respect to y. 01:21:30.660 --> 01:21:34.660 align:middle line:90% 01:21:34.660 --> 01:21:36.720 align:middle line:90% OK. 01:21:36.720 --> 01:21:41.130 align:middle line:84% So this is the partial derivative with respect to x. 01:21:41.130 --> 01:21:46.170 align:middle line:84% And here is the full glory of the representation 01:21:46.170 --> 01:21:50.680 align:middle line:84% of the derivatives where these subscripts, as you remember, 01:21:50.680 --> 01:21:56.630 align:middle line:84% mean the other variables of the original function 01:21:56.630 --> 01:21:59.980 align:middle line:84% that you're taking the partial derivative of. 01:21:59.980 --> 01:22:00.480 align:middle line:90% OK. 01:22:00.480 --> 01:22:02.440 align:middle line:90% Good. 01:22:02.440 --> 01:22:08.410 align:middle line:84% Then we have, like we've seen there for d2S, 01:22:08.410 --> 01:22:15.980 align:middle line:84% here's d2 of this variable A. Can think of S 01:22:15.980 --> 01:22:19.240 align:middle line:84% but it could be also others like the Gibbs free energy, 01:22:19.240 --> 01:22:22.080 align:middle line:84% or the Helmholtz free energy, or whatever you need. 01:22:22.080 --> 01:22:26.050 align:middle line:90% 01:22:26.050 --> 01:22:29.290 align:middle line:84% And this is the quadratic form that 01:22:29.290 --> 01:22:37.780 align:middle line:84% represents the explicit form of the second differential. 01:22:37.780 --> 01:22:38.390 align:middle line:90% OK. 01:22:38.390 --> 01:22:41.980 align:middle line:84% So now that we have a quadratic form, 01:22:41.980 --> 01:22:48.040 align:middle line:84% the trick is to put it in the so-called canonical form 01:22:48.040 --> 01:22:53.480 align:middle line:84% because, you see, this quadratic form has a square here, 01:22:53.480 --> 01:22:58.060 align:middle line:84% a square there, and then it has a product dx dy. 01:22:58.060 --> 01:23:01.410 align:middle line:84% This doesn't have a definite sign. 01:23:01.410 --> 01:23:04.190 align:middle line:90% 01:23:04.190 --> 01:23:10.010 align:middle line:84% But let's try to rewrite this quadratic form 01:23:10.010 --> 01:23:15.000 align:middle line:84% into a form in which you have something squared 01:23:15.000 --> 01:23:18.020 align:middle line:90% and dy squared. 01:23:18.020 --> 01:23:22.340 align:middle line:84% So a combination of dx and dy, if you 01:23:22.340 --> 01:23:25.880 align:middle line:84% choose the proper combination of dx and dy-- 01:23:25.880 --> 01:23:27.680 align:middle line:84% and this is the proper combination. 01:23:27.680 --> 01:23:29.520 align:middle line:84% I've already built in the answer. 01:23:29.520 --> 01:23:33.070 align:middle line:84% So if you want to check it, you just substitute. 01:23:33.070 --> 01:23:37.510 align:middle line:84% If you take this expression and work it out, 01:23:37.510 --> 01:23:41.025 align:middle line:84% you will prove to yourself that it's equivalent to that one. 01:23:41.025 --> 01:23:44.960 align:middle line:90% 01:23:44.960 --> 01:23:46.910 align:middle line:84% And this ratio, this coefficient here 01:23:46.910 --> 01:23:51.140 align:middle line:84% is the ratio of the second derivative 01:23:51.140 --> 01:23:54.170 align:middle line:84% with respect to xy divided by the second derivative 01:23:54.170 --> 01:23:54.980 align:middle line:90% with respect to xx. 01:23:54.980 --> 01:23:59.730 align:middle line:90% 01:23:59.730 --> 01:24:03.790 align:middle line:84% And then if you substitute, by comparison 01:24:03.790 --> 01:24:07.140 align:middle line:84% you find that what you need to put in front of the dy squared 01:24:07.140 --> 01:24:10.470 align:middle line:90% is this lambda, which is not-- 01:24:10.470 --> 01:24:12.550 align:middle line:90% sorry that I used lambda here. 01:24:12.550 --> 01:24:13.686 align:middle line:90% It's not the eigenvalues. 01:24:13.686 --> 01:24:17.660 align:middle line:90% 01:24:17.660 --> 01:24:23.400 align:middle line:84% But this lambda is given by this expression. 01:24:23.400 --> 01:24:27.450 align:middle line:90% Again, in terms of the original. 01:24:27.450 --> 01:24:31.890 align:middle line:84% You can do it in this way, where in your summation, 01:24:31.890 --> 01:24:33.780 align:middle line:90% you keep the square of-- 01:24:33.780 --> 01:24:37.660 align:middle line:84% dy squared, or you can do it also in this other way. 01:24:37.660 --> 01:24:41.370 align:middle line:84% This is another canonical form in which you keep dx squared. 01:24:41.370 --> 01:24:45.910 align:middle line:90% They are two equivalent ways. 01:24:45.910 --> 01:24:47.830 align:middle line:84% Of course, here the lambda is different 01:24:47.830 --> 01:24:49.785 align:middle line:84% and the lambda x is given by this one. 01:24:49.785 --> 01:24:52.760 align:middle line:90% 01:24:52.760 --> 01:24:54.444 align:middle line:90% OK. 01:24:54.444 --> 01:25:00.520 align:middle line:90% Now do I have more space? 01:25:00.520 --> 01:25:07.490 align:middle line:90% 01:25:07.490 --> 01:25:12.230 align:middle line:90% Now let's look at-- 01:25:12.230 --> 01:25:16.790 align:middle line:84% suppose that our function A is such 01:25:16.790 --> 01:25:23.040 align:middle line:84% that its second differential must be positive. 01:25:23.040 --> 01:25:26.020 align:middle line:90% OK. 01:25:26.020 --> 01:25:34.810 align:middle line:84% So positive means that this matrix is positive definite 01:25:34.810 --> 01:25:38.500 align:middle line:90% or semidefinite. 01:25:38.500 --> 01:25:40.960 align:middle line:90% And as we remember-- 01:25:40.960 --> 01:25:42.220 align:middle line:90% yeah. 01:25:42.220 --> 01:25:44.290 align:middle line:90% I took away the wrong. 01:25:44.290 --> 01:25:53.650 align:middle line:84% So if we remember, positive definite for a matrix 01:25:53.650 --> 01:25:59.460 align:middle line:84% implies all these things are positive or non-negative. 01:25:59.460 --> 01:26:07.620 align:middle line:90% 01:26:07.620 --> 01:26:14.110 align:middle line:84% So we have to have that A xx is greater than zero. 01:26:14.110 --> 01:26:16.280 align:middle line:84% And in fact, here is just what I have written. 01:26:16.280 --> 01:26:20.200 align:middle line:90% A xx is greater than 0-- 01:26:20.200 --> 01:26:22.990 align:middle line:84% is greater than something else and greater than zero. 01:26:22.990 --> 01:26:25.575 align:middle line:84% And also A yy is greater than zero. 01:26:25.575 --> 01:26:29.330 align:middle line:90% 01:26:29.330 --> 01:26:35.450 align:middle line:84% But here we have written also something more strong than that. 01:26:35.450 --> 01:26:43.400 align:middle line:84% We have written that A xx or A yy is greater than lambda. 01:26:43.400 --> 01:26:44.190 align:middle line:90% Let's see. 01:26:44.190 --> 01:26:47.730 align:middle line:90% 01:26:47.730 --> 01:27:00.550 align:middle line:84% The lambda-- for example, lambda y-- 01:27:00.550 --> 01:27:04.950 align:middle line:90% 01:27:04.950 --> 01:27:07.830 align:middle line:90% let me rewrite it this way. 01:27:07.830 --> 01:27:22.650 align:middle line:84% It has at the denominator A xx squared and at the numerator 01:27:22.650 --> 01:27:34.290 align:middle line:84% A xx times A yy minus A xy squared. 01:27:34.290 --> 01:27:43.350 align:middle line:84% That is the determinant of the Hess matrix, that numerator. 01:27:43.350 --> 01:27:51.820 align:middle line:90% 01:27:51.820 --> 01:27:58.160 align:middle line:84% So since the matrix is positive, the numerator is positive, 01:27:58.160 --> 01:28:03.120 align:middle line:84% the denominator is positive, so this is also positive. 01:28:03.120 --> 01:28:03.620 align:middle line:90% OK. 01:28:03.620 --> 01:28:06.870 align:middle line:90% 01:28:06.870 --> 01:28:08.680 align:middle line:90% So this explains why lambda-- 01:28:08.680 --> 01:28:12.140 align:middle line:90% 01:28:12.140 --> 01:28:16.710 align:middle line:84% you can do that for y, but also for x. 01:28:16.710 --> 01:28:22.310 align:middle line:84% The lambdas are also non-negative. 01:28:22.310 --> 01:28:28.010 align:middle line:84% And now you can also see here why lambda-- 01:28:28.010 --> 01:28:35.970 align:middle line:84% for example, lambda y is smaller than A yy because look here. 01:28:35.970 --> 01:28:36.720 align:middle line:90% This is lambda. 01:28:36.720 --> 01:28:40.820 align:middle line:84% This is A. And then you subtract something positive. 01:28:40.820 --> 01:28:44.310 align:middle line:84% So if from A you subtract something positive, 01:28:44.310 --> 01:28:50.940 align:middle line:84% it means that lambda is smaller than A yy. 01:28:50.940 --> 01:28:53.680 align:middle line:90% OK. 01:28:53.680 --> 01:28:54.180 align:middle line:90% All right. 01:28:54.180 --> 01:29:05.540 align:middle line:84% So this is the mathematics that stands behind the Le 01:29:05.540 --> 01:29:09.410 align:middle line:84% Chatelier-Braun principle that we are going to do in a second. 01:29:09.410 --> 01:29:11.810 align:middle line:84% Of course, you can do exactly the same 01:29:11.810 --> 01:29:16.360 align:middle line:84% that I've done here for positive semidefinite. 01:29:16.360 --> 01:29:19.740 align:middle line:84% You can also do it for negative semidefinite. 01:29:19.740 --> 01:29:23.660 align:middle line:90% 01:29:23.660 --> 01:29:24.160 align:middle line:90% OK. 01:29:24.160 --> 01:29:46.336 align:middle line:90% 01:29:46.336 --> 01:29:56.140 align:middle line:84% These steps help understanding the various steps 01:29:56.140 --> 01:29:58.830 align:middle line:90% on the previous slide. 01:29:58.830 --> 01:30:02.850 align:middle line:90% 01:30:02.850 --> 01:30:14.130 align:middle line:84% And the bottom line is here, you can rewrite that A double x-- 01:30:14.130 --> 01:30:17.280 align:middle line:90% 01:30:17.280 --> 01:30:20.990 align:middle line:84% this is equal to the partial of A (with respect to) 01:30:20.990 --> 01:30:27.960 align:middle line:84% x, which is the conjugate potential to variable x. 01:30:27.960 --> 01:30:33.490 align:middle line:90% So yeah. 01:30:33.490 --> 01:30:45.070 align:middle line:84% So suppose our function is S. And S is a function of energy. 01:30:45.070 --> 01:30:48.880 align:middle line:84% And so S EE, which is the equivalent-- 01:30:48.880 --> 01:30:53.440 align:middle line:84% so our function is the entropy and the x variable 01:30:53.440 --> 01:30:54.610 align:middle line:90% is the energy. 01:30:54.610 --> 01:31:00.940 align:middle line:84% So S EE means partial of S with respect to E twice. 01:31:00.940 --> 01:31:11.820 align:middle line:84% But since the partial of S with respect to E is 1 over T, 01:31:11.820 --> 01:31:13.740 align:middle line:90% this is equivalent also to-- 01:31:13.740 --> 01:31:20.040 align:middle line:84% you can write it as S comma E, which 01:31:20.040 --> 01:31:29.970 align:middle line:84% is the inverse of the temperature, again comma E. 01:31:29.970 --> 01:31:35.060 align:middle line:84% And of course, you can do also the other like that. 01:31:35.060 --> 01:31:39.980 align:middle line:84% So this is the meaning of partial of A comma x, again 01:31:39.980 --> 01:31:42.705 align:middle line:90% comma x, done at constant y. 01:31:42.705 --> 01:31:46.710 align:middle line:90% 01:31:46.710 --> 01:31:57.740 align:middle line:84% The point is that, by doing this math here that I want to skip, 01:31:57.740 --> 01:32:02.040 align:middle line:84% I prove that the lambda also, the lambdas can 01:32:02.040 --> 01:32:08.640 align:middle line:84% be written as partials of these potentials-- 01:32:08.640 --> 01:32:10.320 align:middle line:84% so of the first derivative with respect 01:32:10.320 --> 01:32:14.580 align:middle line:84% to x but not while keeping constant y 01:32:14.580 --> 01:32:22.790 align:middle line:84% but while keeping constant the other potential conjugated to y. 01:32:22.790 --> 01:32:29.600 align:middle line:84% So what changes in these two terms is what is kept constant. 01:32:29.600 --> 01:32:31.350 align:middle line:90% Here I keep constant y. 01:32:31.350 --> 01:32:37.400 align:middle line:84% Here I keep constant the potential conjugated to y. 01:32:37.400 --> 01:32:52.130 align:middle line:84% And that's the key to the practical consequences 01:32:52.130 --> 01:32:54.200 align:middle line:90% of this mathematics. 01:32:54.200 --> 01:33:00.130 align:middle line:84% So for example, let's change once more variables 01:33:00.130 --> 01:33:02.940 align:middle line:84% so that we get even more confused. 01:33:02.940 --> 01:33:07.260 align:middle line:84% But we just have five minutes so it's OK. 01:33:07.260 --> 01:33:13.390 align:middle line:84% So we start from the function E from this fundamental relation, 01:33:13.390 --> 01:33:16.710 align:middle line:84% this form of the fundamental relation. 01:33:16.710 --> 01:33:19.440 align:middle line:84% Keep the n fixed and look at what 01:33:19.440 --> 01:33:21.995 align:middle line:84% happens by applying what we've just seen. 01:33:21.995 --> 01:33:26.910 align:middle line:90% 01:33:26.910 --> 01:33:30.210 align:middle line:84% This is the train of inequalities 01:33:30.210 --> 01:33:35.300 align:middle line:84% that is implied by one of the forms, by this form, 01:33:35.300 --> 01:33:41.440 align:middle line:84% and by the second form is written here. 01:33:41.440 --> 01:33:44.480 align:middle line:90% You can do the substitutions. 01:33:44.480 --> 01:33:47.110 align:middle line:84% You see I put here what the variable x and y 01:33:47.110 --> 01:33:51.460 align:middle line:84% is, what A sub x, A sub y, the temperature and the pressure 01:33:51.460 --> 01:33:52.990 align:middle line:90% in our case. 01:33:52.990 --> 01:33:58.390 align:middle line:84% And here is the result. Now, by the way, 01:33:58.390 --> 01:34:02.430 align:middle line:84% by looking at these derivatives, you find old friends. 01:34:02.430 --> 01:34:06.180 align:middle line:84% And you can see that, for example, you can rewrite this 01:34:06.180 --> 01:34:11.070 align:middle line:84% as CP is greater than or equal to CV, which 01:34:11.070 --> 01:34:17.600 align:middle line:84% is greater than or equal to 0, or that one, the compressibility 01:34:17.600 --> 01:34:20.060 align:middle line:84% at constant temperature is greater than the compressibility 01:34:20.060 --> 01:34:22.940 align:middle line:90% at constant entropy. 01:34:22.940 --> 01:34:29.745 align:middle line:84% So what's the reading that Le Chatelier-Brown does? 01:34:29.745 --> 01:34:32.880 align:middle line:90% 01:34:32.880 --> 01:34:38.370 align:middle line:84% What's the difference here between these two derivatives? 01:34:38.370 --> 01:34:43.080 align:middle line:90% It is what is kept constant. 01:34:43.080 --> 01:34:49.830 align:middle line:84% I change the entropy and measure what temperature change occurs 01:34:49.830 --> 01:34:54.740 align:middle line:84% when I keep volume or when I keep pressure fixed. 01:34:54.740 --> 01:34:59.260 align:middle line:90% 01:34:59.260 --> 01:35:06.170 align:middle line:84% You see, volume is the original variable. 01:35:06.170 --> 01:35:11.330 align:middle line:84% Pressure is the conjugated potential. 01:35:11.330 --> 01:35:17.130 align:middle line:84% It has to do with the tendency to escape, to gain volume. 01:35:17.130 --> 01:35:19.375 align:middle line:84% So minus pressure is the tendency to yield volume. 01:35:19.375 --> 01:35:23.270 align:middle line:90% 01:35:23.270 --> 01:35:26.870 align:middle line:84% So the idea is that the interpretation 01:35:26.870 --> 01:35:28.700 align:middle line:84% of these derivatives being positive 01:35:28.700 --> 01:35:34.340 align:middle line:84% is that the system reacts if you move it away 01:35:34.340 --> 01:35:38.500 align:middle line:84% from the mutual equilibrium state of the environment. 01:35:38.500 --> 01:35:41.160 align:middle line:84% If you add some entropy, for example, 01:35:41.160 --> 01:35:44.150 align:middle line:84% it will increase its temperature-- 01:35:44.150 --> 01:35:53.860 align:middle line:84% this is positive-- so as to increase its tendency 01:35:53.860 --> 01:36:00.520 align:middle line:84% to return energy to contrast your change by giving energy 01:36:00.520 --> 01:36:02.090 align:middle line:90% away to your environment. 01:36:02.090 --> 01:36:07.030 align:middle line:84% So the system responds by trying to spontaneously reestablish 01:36:07.030 --> 01:36:08.940 align:middle line:90% equilibrium. 01:36:08.940 --> 01:36:16.150 align:middle line:84% But it does more so if you do it at constant volume 01:36:16.150 --> 01:36:18.740 align:middle line:84% than if you do it at constant pressure. 01:36:18.740 --> 01:36:26.198 align:middle line:84% So the reaction of the system is stronger if what? 01:36:26.198 --> 01:36:27.240 align:middle line:90% So what's the difference? 01:36:27.240 --> 01:36:31.970 align:middle line:84% The difference is that, here, the system has a smaller 01:36:31.970 --> 01:36:35.820 align:middle line:90% reaction if you keep-- 01:36:35.820 --> 01:36:41.670 align:middle line:84% keeping the pressure constant means keeping your system 01:36:41.670 --> 01:36:43.600 align:middle line:84% in equilibrium with your reservoir 01:36:43.600 --> 01:36:48.630 align:middle line:84% at least as for the condition that requires pressure equality. 01:36:48.630 --> 01:36:50.610 align:middle line:84% So you're not touching that condition 01:36:50.610 --> 01:36:51.570 align:middle line:90% for mutual equilibrium. 01:36:51.570 --> 01:36:53.530 align:middle line:84% You're touching only something else. 01:36:53.530 --> 01:36:56.830 align:middle line:84% So if you do it there, the system says, 01:36:56.830 --> 01:37:00.780 align:middle line:84% all right, well, after all, you're moving me away but not so 01:37:00.780 --> 01:37:03.660 align:middle line:90% much so its reaction is softer. 01:37:03.660 --> 01:37:07.500 align:middle line:84% If you move it away so that also its pressure changes 01:37:07.500 --> 01:37:11.520 align:middle line:84% and therefore like you do when you keep the volume constant, 01:37:11.520 --> 01:37:18.990 align:middle line:90% then it will increase not only-- 01:37:18.990 --> 01:37:22.830 align:middle line:84% so essentially, you are touching two conditions 01:37:22.830 --> 01:37:25.800 align:middle line:84% of mutual equilibrium, touching equality of temperature 01:37:25.800 --> 01:37:27.760 align:middle line:84% and touching equality of pressure. 01:37:27.760 --> 01:37:31.820 align:middle line:84% And therefore the system reacts stronger. 01:37:31.820 --> 01:37:35.490 align:middle line:84% Wants more to come back to equilibrium. 01:37:35.490 --> 01:37:38.270 align:middle line:90% 01:37:38.270 --> 01:37:44.830 align:middle line:90% This is the-- this is the idea. 01:37:44.830 --> 01:37:48.120 align:middle line:90% And since I wanted to-- 01:37:48.120 --> 01:37:50.010 align:middle line:90% give me two more minutes. 01:37:50.010 --> 01:37:56.330 align:middle line:84% I wanted to show you the application here. 01:37:56.330 --> 01:37:59.100 align:middle line:90% This is an air spring. 01:37:59.100 --> 01:38:02.920 align:middle line:84% So it's a piston with air inside. 01:38:02.920 --> 01:38:06.940 align:middle line:84% And it acts like a standard mechanical spring. 01:38:06.940 --> 01:38:08.745 align:middle line:84% You can compute the spring constant. 01:38:08.745 --> 01:38:11.940 align:middle line:90% 01:38:11.940 --> 01:38:13.090 align:middle line:90% Here are the details. 01:38:13.090 --> 01:38:17.735 align:middle line:84% It's kind of easy mechanical engineering, so to speak. 01:38:17.735 --> 01:38:22.350 align:middle line:90% 01:38:22.350 --> 01:38:28.740 align:middle line:84% This k is the spring constant that relates, 01:38:28.740 --> 01:38:34.650 align:middle line:84% the increasing reaction or force of your spring 01:38:34.650 --> 01:38:39.710 align:middle line:90% to a change in its compression. 01:38:39.710 --> 01:38:42.410 align:middle line:90% 01:38:42.410 --> 01:38:45.650 align:middle line:84% And this is also a similar situation. 01:38:45.650 --> 01:38:48.480 align:middle line:90% This acts as a spring. 01:38:48.480 --> 01:38:53.530 align:middle line:84% And you can compute the dF over dx. 01:38:53.530 --> 01:38:59.170 align:middle line:84% And if you go down here, you see it's related to how much 01:38:59.170 --> 01:39:03.950 align:middle line:84% the area, little a, of the piston, the volume. 01:39:03.950 --> 01:39:05.840 align:middle line:84% And this thing at the denominator 01:39:05.840 --> 01:39:07.380 align:middle line:90% is the compressibility. 01:39:07.380 --> 01:39:10.500 align:middle line:90% 01:39:10.500 --> 01:39:12.240 align:middle line:90% So you could have-- 01:39:12.240 --> 01:39:17.590 align:middle line:84% so if you act your spring while keeping 01:39:17.590 --> 01:39:21.790 align:middle line:84% the temperature constant-- so if your spring is 01:39:21.790 --> 01:39:26.630 align:middle line:84% in contact with the thermal reservoir and therefore, 01:39:26.630 --> 01:39:28.360 align:middle line:84% it has the time to exchange energy 01:39:28.360 --> 01:39:33.650 align:middle line:84% to remain in mutual equilibrium, which typically is when you 01:39:33.650 --> 01:39:37.790 align:middle line:84% push it, but push it slow enough so that the system has time 01:39:37.790 --> 01:39:41.150 align:middle line:84% to exchange heat with the environment 01:39:41.150 --> 01:39:49.310 align:middle line:84% and remain in equilibrium, then you get one spring constant. 01:39:49.310 --> 01:39:52.800 align:middle line:84% If instead you push the spring fast, 01:39:52.800 --> 01:40:01.160 align:middle line:84% like when the car goes on a bump, 01:40:01.160 --> 01:40:06.270 align:middle line:84% the system doesn't have time to exchange energy or heat 01:40:06.270 --> 01:40:09.730 align:middle line:84% and the entropy remains constant. 01:40:09.730 --> 01:40:12.850 align:middle line:84% So that's the isothermal compressibility-- 01:40:12.850 --> 01:40:16.880 align:middle line:84% I'm sorry, the isentropic compressibility that appears. 01:40:16.880 --> 01:40:24.405 align:middle line:84% And so you see that the ratio of these two spring constants 01:40:24.405 --> 01:40:34.220 align:middle line:84% is equal to CP over CV, which for air is 1.4. 01:40:34.220 --> 01:40:36.520 align:middle line:90% We just reviewed. 01:40:36.520 --> 01:40:41.380 align:middle line:84% So that means that if you act on the spring fast, 01:40:41.380 --> 01:40:46.060 align:middle line:84% you get the spring constant that is 40% higher than if you 01:40:46.060 --> 01:40:48.540 align:middle line:90% act on the spring slow. 01:40:48.540 --> 01:40:53.460 align:middle line:84% And these devices, if you look up air springs, 01:40:53.460 --> 01:40:56.810 align:middle line:84% you can see that it's an enormous business 01:40:56.810 --> 01:41:04.880 align:middle line:84% because, as I said, trains, buses, bicycles, and cars use 01:41:04.880 --> 01:41:09.110 align:middle line:84% spring constants combined also with the damping system 01:41:09.110 --> 01:41:14.480 align:middle line:84% to give you suspension systems that 01:41:14.480 --> 01:41:22.620 align:middle line:84% are allowed to absorb vibrations and make it more comfortable. 01:41:22.620 --> 01:41:23.170 align:middle line:90% OK. 01:41:23.170 --> 01:41:27.910 align:middle line:84% So sorry if I went a little bit late. 01:41:27.910 --> 01:41:32.000 align:middle line:90%