WEBVTT 00:00:00.000 --> 00:00:01.420 align:middle line:90% [SQUEAKING] 00:00:01.420 --> 00:00:01.920 align:middle line:90% 00:00:01.920 --> 00:00:03.820 align:middle line:90% [RUSTLING] 00:00:03.820 --> 00:00:04.320 align:middle line:90% 00:00:04.320 --> 00:00:07.680 align:middle line:90% [CLICKING] 00:00:07.680 --> 00:00:10.560 align:middle line:90% 00:00:10.560 --> 00:00:12.740 align:middle line:84% GIAN PAOLO BERETTA: OK, good morning. 00:00:12.740 --> 00:00:16.390 align:middle line:90% 00:00:16.390 --> 00:00:23.673 align:middle line:84% So we started talking about multi-component systems, 00:00:23.673 --> 00:00:24.173 align:middle line:90% mixtures. 00:00:24.173 --> 00:00:27.570 align:middle line:90% 00:00:27.570 --> 00:00:32.009 align:middle line:84% And our program is that at the beginning, 00:00:32.009 --> 00:00:35.100 align:middle line:84% we try to see when it is possible 00:00:35.100 --> 00:00:42.690 align:middle line:84% to infer the properties of a mixture as much as possible 00:00:42.690 --> 00:00:47.430 align:middle line:84% from the properties of the pure components. 00:00:47.430 --> 00:00:52.530 align:middle line:84% And so we are in the process of defining some ideal situations 00:00:52.530 --> 00:00:57.250 align:middle line:84% in which this is possible, which as we said, 00:00:57.250 --> 00:01:02.640 align:middle line:84% is a good start in our modeling efforts. 00:01:02.640 --> 00:01:06.540 align:middle line:84% It's like a reference simple situation 00:01:06.540 --> 00:01:17.370 align:middle line:84% in which we do not need to add experimental data to model 00:01:17.370 --> 00:01:18.900 align:middle line:90% our mixtures. 00:01:18.900 --> 00:01:24.480 align:middle line:84% But then we will find that in practice, the ideal situation 00:01:24.480 --> 00:01:28.830 align:middle line:84% is really only a limiting situation, which 00:01:28.830 --> 00:01:35.720 align:middle line:84% is not fulfilled by any real mixture, except in some limits. 00:01:35.720 --> 00:01:39.450 align:middle line:90% 00:01:39.450 --> 00:01:45.120 align:middle line:84% Nevertheless, we gain a lot of information 00:01:45.120 --> 00:01:47.380 align:middle line:84% by just looking at ideal situations. 00:01:47.380 --> 00:01:56.820 align:middle line:84% So we have defined these partial properties 00:01:56.820 --> 00:02:02.520 align:middle line:84% that are derivatives or functions of the chemical 00:02:02.520 --> 00:02:05.460 align:middle line:90% potentials. 00:02:05.460 --> 00:02:11.880 align:middle line:84% For example, the partial entropy of the constituent 00:02:11.880 --> 00:02:16.390 align:middle line:84% in the mixture, which is a function of T, p, and y. 00:02:16.390 --> 00:02:24.150 align:middle line:84% We will have to be careful in distinguishing well 00:02:24.150 --> 00:02:28.530 align:middle line:84% between the partial entropy of the constituent in the mixture 00:02:28.530 --> 00:02:38.410 align:middle line:84% and the s double i T p, which is the specific property, 00:02:38.410 --> 00:02:40.935 align:middle line:84% the specific entropy, of pure component 00:02:40.935 --> 00:02:45.370 align:middle line:84% i when at the same temperature and pressure of the mixture. 00:02:45.370 --> 00:02:49.450 align:middle line:84% We will see that they are somehow related 00:02:49.450 --> 00:02:54.280 align:middle line:84% and that the difference is measurable. 00:02:54.280 --> 00:02:58.850 align:middle line:84% So we have to pay a lot of attention to the subscripts. 00:02:58.850 --> 00:03:02.170 align:middle line:84% Like, so a single subscript is a partial property 00:03:02.170 --> 00:03:03.530 align:middle line:90% for the mixture. 00:03:03.530 --> 00:03:11.200 align:middle line:84% Double subscript is a specific property of the pure component. 00:03:11.200 --> 00:03:13.460 align:middle line:84% And then we define the partial volume, 00:03:13.460 --> 00:03:16.580 align:middle line:84% which is another derivative of the chemical potential, 00:03:16.580 --> 00:03:18.550 align:middle line:84% once we know the chemical potentials 00:03:18.550 --> 00:03:22.540 align:middle line:84% as a function of temperature, pressure, and composition. 00:03:22.540 --> 00:03:24.670 align:middle line:90% We define the partial energy. 00:03:24.670 --> 00:03:27.140 align:middle line:84% And also, we define the partial enthalpy. 00:03:27.140 --> 00:03:32.350 align:middle line:90% And let me just stress this one. 00:03:32.350 --> 00:03:36.340 align:middle line:84% If you have the expression for the chemical potential 00:03:36.340 --> 00:03:39.150 align:middle line:84% and you divide it by the temperature, 00:03:39.150 --> 00:03:41.970 align:middle line:84% and take the derivative not with respect to T, but with respect 00:03:41.970 --> 00:03:44.640 align:middle line:84% to the inverse of the temperature, 00:03:44.640 --> 00:03:48.060 align:middle line:84% then you get the partial enthalpy. 00:03:48.060 --> 00:03:53.590 align:middle line:84% This formula will come out a while often. 00:03:53.590 --> 00:04:04.140 align:middle line:84% And also, this one, which I better write here on the board, 00:04:04.140 --> 00:04:06.110 align:middle line:90% we can write it this way. 00:04:06.110 --> 00:04:09.090 align:middle line:90% 00:04:09.090 --> 00:04:15.260 align:middle line:90% OK, so we can say that-- 00:04:15.260 --> 00:04:18.000 align:middle line:84% we can see that from this relation here, 00:04:18.000 --> 00:04:21.410 align:middle line:84% we can see that the chemical potential can be-- 00:04:21.410 --> 00:04:24.230 align:middle line:84% of the component in the mixture can 00:04:24.230 --> 00:04:29.330 align:middle line:84% be written as this linear combination, enthalpy 00:04:29.330 --> 00:04:31.670 align:middle line:84% minus T s, which essentially is also 00:04:31.670 --> 00:04:33.660 align:middle line:90% equal to the Gibbs free energy. 00:04:33.660 --> 00:04:39.965 align:middle line:84% So this is also equal to the partial Gibbs free energy. 00:04:39.965 --> 00:04:44.750 align:middle line:90% 00:04:44.750 --> 00:04:49.700 align:middle line:84% And in fact, the derivative of G with respect to ni, 00:04:49.700 --> 00:04:54.390 align:middle line:84% at fixed T and p, and the other constituents, 00:04:54.390 --> 00:04:59.930 align:middle line:84% that's the definition of the chemical potential in terms 00:04:59.930 --> 00:05:01.050 align:middle line:90% of the Gibbs free energy. 00:05:01.050 --> 00:05:04.910 align:middle line:84% But it's also-- you could also call it by analogy 00:05:04.910 --> 00:05:07.490 align:middle line:84% with these definitions, the partial Gibbs 00:05:07.490 --> 00:05:09.510 align:middle line:84% free energy of the constituent in the mixture. 00:05:09.510 --> 00:05:14.190 align:middle line:84% So in other words, we're saying that the chemical potential 00:05:14.190 --> 00:05:17.170 align:middle line:84% can be interpreted itself as a partial property. 00:05:17.170 --> 00:05:18.660 align:middle line:84% It's the partial Gibbs free energy 00:05:18.660 --> 00:05:20.077 align:middle line:84% of the constituent in the mixture. 00:05:20.077 --> 00:05:25.750 align:middle line:90% 00:05:25.750 --> 00:05:31.870 align:middle line:84% And the reason why we call these properties partial 00:05:31.870 --> 00:05:40.810 align:middle line:84% is because they contribute to building up 00:05:40.810 --> 00:05:46.990 align:middle line:84% the property of the mixture by simple multiplication 00:05:46.990 --> 00:05:47.910 align:middle line:90% with their amounts. 00:05:47.910 --> 00:05:50.540 align:middle line:90% 00:05:50.540 --> 00:05:55.130 align:middle line:84% And each-- and you can do this for entropy, volume, energy, 00:05:55.130 --> 00:05:58.010 align:middle line:90% the free energies, the enthalpy. 00:05:58.010 --> 00:06:02.010 align:middle line:84% They are all-- they all have the same structure. 00:06:02.010 --> 00:06:05.590 align:middle line:90% 00:06:05.590 --> 00:06:10.150 align:middle line:84% Of course, now, we are-- we forget the small systems, 00:06:10.150 --> 00:06:11.990 align:middle line:90% and we focus on large systems. 00:06:11.990 --> 00:06:17.270 align:middle line:84% So this is the right-hand side, the highlighted formulas, 00:06:17.270 --> 00:06:19.990 align:middle line:84% because for small systems, we will 00:06:19.990 --> 00:06:23.110 align:middle line:84% have also to some contributions that 00:06:23.110 --> 00:06:29.730 align:middle line:84% correct this expression in terms of the Euler free energy. 00:06:29.730 --> 00:06:33.220 align:middle line:90% 00:06:33.220 --> 00:06:44.840 align:middle line:84% OK, so how do we measure these partial properties? 00:06:44.840 --> 00:06:45.850 align:middle line:90% And here's the setup. 00:06:45.850 --> 00:06:49.730 align:middle line:90% 00:06:49.730 --> 00:06:56.750 align:middle line:84% We take a mixture with a given composition, 00:06:56.750 --> 00:06:59.490 align:middle line:90% in stable equilibrium of course. 00:06:59.490 --> 00:07:03.660 align:middle line:84% I know I'm not repeating it often enough, 00:07:03.660 --> 00:07:07.640 align:middle line:84% but we are concentrating our attention now 00:07:07.640 --> 00:07:09.275 align:middle line:84% on the stable equilibrium states. 00:07:09.275 --> 00:07:11.810 align:middle line:90% 00:07:11.810 --> 00:07:16.520 align:middle line:84% And so we take a mixture with some given composition 00:07:16.520 --> 00:07:19.380 align:middle line:84% at a stable equilibrium state, at temperature T and pressure p. 00:07:19.380 --> 00:07:21.990 align:middle line:90% 00:07:21.990 --> 00:07:28.560 align:middle line:84% And we put it in mutual equilibrium 00:07:28.560 --> 00:07:32.280 align:middle line:84% with a thermal reservoir of the kind that 00:07:32.280 --> 00:07:35.560 align:middle line:90% can exchange also volume. 00:07:35.560 --> 00:07:38.670 align:middle line:84% So you have also the pressure equality 00:07:38.670 --> 00:07:41.700 align:middle line:84% as a condition for mutual equilibrium between your mixture 00:07:41.700 --> 00:07:42.790 align:middle line:90% and the reservoir. 00:07:42.790 --> 00:07:45.930 align:middle line:90% 00:07:45.930 --> 00:07:54.526 align:middle line:84% Then we take a little bit of pure component i, 00:07:54.526 --> 00:07:57.820 align:middle line:90% and we add it to the mixture. 00:07:57.820 --> 00:08:03.190 align:middle line:84% And we wait until the mixture goes again 00:08:03.190 --> 00:08:06.880 align:middle line:84% to mutual equilibrium with that thermal reservoir. 00:08:06.880 --> 00:08:12.490 align:middle line:84% OK, so this is a mixing process in which 00:08:12.490 --> 00:08:16.220 align:middle line:84% we add a small amount of one of the components. 00:08:16.220 --> 00:08:18.740 align:middle line:90% 00:08:18.740 --> 00:08:24.810 align:middle line:84% And then since we can measure properties of systems, 00:08:24.810 --> 00:08:28.170 align:middle line:84% for example, let me take the enthalpy. 00:08:28.170 --> 00:08:34.110 align:middle line:84% And so let me measure the difference between the enthalpy 00:08:34.110 --> 00:08:41.230 align:middle line:84% of the mixture in this final state 00:08:41.230 --> 00:08:45.610 align:middle line:84% and the enthalpy of the composite system of the mixture 00:08:45.610 --> 00:08:51.160 align:middle line:84% plus the pure component in the initial state. 00:08:51.160 --> 00:08:55.040 align:middle line:84% OK, so this is for the mixture in the final state. 00:08:55.040 --> 00:08:59.590 align:middle line:84% The amounts are the same, but the amount for the constituent i 00:08:59.590 --> 00:09:02.440 align:middle line:90% is n plus dn. 00:09:02.440 --> 00:09:08.260 align:middle line:84% This is the initial where in the mixture, I just have ni 00:09:08.260 --> 00:09:11.360 align:middle line:84% and the dni is separate from the mixture. 00:09:11.360 --> 00:09:15.700 align:middle line:90% So it contributes with its own-- 00:09:15.700 --> 00:09:19.690 align:middle line:84% to the enthalpy of the pure constituent 00:09:19.690 --> 00:09:24.760 align:middle line:84% through its specific property of the pure component. 00:09:24.760 --> 00:09:31.300 align:middle line:84% And notice that usually, as I said several times, enthalpy 00:09:31.300 --> 00:09:33.050 align:middle line:90% is not an additive property. 00:09:33.050 --> 00:09:40.060 align:middle line:84% But if you're looking at systems that are at the same temp-- 00:09:40.060 --> 00:09:43.280 align:middle line:84% pressure, then enthalpy is additive. 00:09:43.280 --> 00:09:48.580 align:middle line:84% And therefore, the enthalpy of the composite of these two 00:09:48.580 --> 00:09:52.250 align:middle line:84% subsystems is equal to the sum of the two enthalpies, 00:09:52.250 --> 00:09:55.680 align:middle line:84% like we do here, the sum of the enthalpy of the mixture, 00:09:55.680 --> 00:09:58.520 align:middle line:84% plus the enthalpy of the pure component. 00:09:58.520 --> 00:10:03.480 align:middle line:84% OK, now if we take this and divide it 00:10:03.480 --> 00:10:06.720 align:middle line:84% by the amount of constituent i that we 00:10:06.720 --> 00:10:11.490 align:middle line:84% have added to the mixture, that defines 00:10:11.490 --> 00:10:22.740 align:middle line:84% the change in the enthalpy due to the addition of that amount-- 00:10:22.740 --> 00:10:25.320 align:middle line:84% of a small amount of that component. 00:10:25.320 --> 00:10:29.170 align:middle line:84% And of course, the n is supposed to be small. 00:10:29.170 --> 00:10:32.356 align:middle line:84% So we're taking the limit of this, which is-- 00:10:32.356 --> 00:10:35.940 align:middle line:84% which can be viewed as an incremental product, at least 00:10:35.940 --> 00:10:39.690 align:middle line:84% the part with the capital H. It's 00:10:39.690 --> 00:10:43.590 align:middle line:84% the partial derivative of capital H with respect to ni. 00:10:43.590 --> 00:10:45.780 align:middle line:84% And then there is this additional contribution 00:10:45.780 --> 00:10:48.850 align:middle line:84% of the separate pure component that we have added, 00:10:48.850 --> 00:10:50.710 align:middle line:90% which since I divided-- 00:10:50.710 --> 00:10:55.560 align:middle line:84% I'm dividing by dni adds, but with the minus 00:10:55.560 --> 00:10:58.890 align:middle line:90% sign, a contribution here. 00:10:58.890 --> 00:11:01.570 align:middle line:84% Now, we recognize that this derivative 00:11:01.570 --> 00:11:06.010 align:middle line:84% is exactly what we defined as the partial enthalpy 00:11:06.010 --> 00:11:09.460 align:middle line:84% of the constituent in the mixture in the previous slide. 00:11:09.460 --> 00:11:15.320 align:middle line:84% So here, we have the difference between the partial enthalpy 00:11:15.320 --> 00:11:18.800 align:middle line:84% and the specific enthalpy of pure component 00:11:18.800 --> 00:11:22.310 align:middle line:84% at the same T and p of the mixture. 00:11:22.310 --> 00:11:24.650 align:middle line:84% This is what we call the enthalpy 00:11:24.650 --> 00:11:28.460 align:middle line:90% of isothermobaric mixing. 00:11:28.460 --> 00:11:33.020 align:middle line:84% Tp stands for isothermal and isobaric 00:11:33.020 --> 00:11:38.660 align:middle line:84% because we are doing this at fixed T and p imposed 00:11:38.660 --> 00:11:40.020 align:middle line:90% by the reservoir. 00:11:40.020 --> 00:11:44.350 align:middle line:90% 00:11:44.350 --> 00:11:47.680 align:middle line:90% This can be measured. 00:11:47.680 --> 00:11:54.010 align:middle line:84% Enthalpy in a process like this is essentially, 00:11:54.010 --> 00:11:56.590 align:middle line:84% if you make the energy balance, this 00:11:56.590 --> 00:11:59.470 align:middle line:84% is essentially the heat exchanged 00:11:59.470 --> 00:12:03.850 align:middle line:90% with the thermal reservoir. 00:12:03.850 --> 00:12:06.350 align:middle line:84% So you measure it with a calorimeter. 00:12:06.350 --> 00:12:09.140 align:middle line:90% 00:12:09.140 --> 00:12:14.940 align:middle line:84% So that means that since you can measure this with a calorimeter 00:12:14.940 --> 00:12:17.920 align:middle line:84% and you know the properties of the pure substances, 00:12:17.920 --> 00:12:20.400 align:middle line:84% then you have measured the partial enthalpy 00:12:20.400 --> 00:12:22.230 align:middle line:90% of the component in the mixture. 00:12:22.230 --> 00:12:27.570 align:middle line:84% And this doesn't require any particular assumptions 00:12:27.570 --> 00:12:31.640 align:middle line:84% about the mixture being ideal or non-ideal. 00:12:31.640 --> 00:12:36.640 align:middle line:84% We actually haven't even defined yet what ideal means. 00:12:36.640 --> 00:12:40.000 align:middle line:84% It's simply a tool, a method, with which people 00:12:40.000 --> 00:12:44.920 align:middle line:90% measure properties of mixtures. 00:12:44.920 --> 00:12:48.400 align:middle line:84% And what we have done here for the enthalpy 00:12:48.400 --> 00:12:51.710 align:middle line:90% can be repeated the same way. 00:12:51.710 --> 00:13:00.250 align:middle line:84% So I'm not repeating it here, but to define 00:13:00.250 --> 00:13:03.880 align:middle line:84% the volume of mixing, so it's the difference 00:13:03.880 --> 00:13:07.390 align:middle line:84% between the partial volume and the specific volume 00:13:07.390 --> 00:13:10.180 align:middle line:84% at the same T and p, the energy of mixing 00:13:10.180 --> 00:13:12.060 align:middle line:90% and the entropy of mixing. 00:13:12.060 --> 00:13:16.180 align:middle line:90% 00:13:16.180 --> 00:13:18.520 align:middle line:84% And these are all things that you can measure 00:13:18.520 --> 00:13:20.430 align:middle line:90% by energy and entropy balances. 00:13:20.430 --> 00:13:25.040 align:middle line:90% 00:13:25.040 --> 00:13:28.460 align:middle line:84% So these are ways in which you can characterize. 00:13:28.460 --> 00:13:30.230 align:middle line:90% OK, this is a-- 00:13:30.230 --> 00:13:31.970 align:middle line:90% oops, sorry. 00:13:31.970 --> 00:13:38.460 align:middle line:84% Yeah, there is an extra character there. 00:13:38.460 --> 00:13:41.210 align:middle line:90% OK. 00:13:41.210 --> 00:13:41.730 align:middle line:90% All right. 00:13:41.730 --> 00:13:45.350 align:middle line:84% So now, let's use that equation, which 00:13:45.350 --> 00:13:48.920 align:middle line:84% I've written up there on the board 00:13:48.920 --> 00:14:02.940 align:middle line:84% and substitute the expression of the partial enthalpy as hii, 00:14:02.940 --> 00:14:05.040 align:middle line:84% the specific enthalpy of the pure component, 00:14:05.040 --> 00:14:07.680 align:middle line:90% plus the enthalpy of mixing. 00:14:07.680 --> 00:14:11.350 align:middle line:84% And the same for the entropy, s partial. 00:14:11.350 --> 00:14:17.610 align:middle line:84% We can write it as s specific of the pure component 00:14:17.610 --> 00:14:21.080 align:middle line:84% plus the entropy of isothermobaric mixing. 00:14:21.080 --> 00:14:23.930 align:middle line:90% 00:14:23.930 --> 00:14:27.860 align:middle line:84% All right, so then, we recognize that also 00:14:27.860 --> 00:14:36.410 align:middle line:84% for the pure substance, h minus Ts is the chemical potential. 00:14:36.410 --> 00:14:41.773 align:middle line:84% So this gives us a relation for the chemical potential 00:14:41.773 --> 00:14:44.660 align:middle line:84% of the component in the mixture in terms 00:14:44.660 --> 00:14:46.850 align:middle line:84% of the chemical potential of the pure component 00:14:46.850 --> 00:14:49.940 align:middle line:84% at the same temperature and pressure of the mixture, 00:14:49.940 --> 00:14:54.500 align:middle line:84% plus the enthalpy of mixing minus T 00:14:54.500 --> 00:14:58.310 align:middle line:90% times the entropy of mixing. 00:14:58.310 --> 00:15:00.650 align:middle line:84% And at this stage, we should also remember that 00:15:00.650 --> 00:15:04.460 align:middle line:84% by the definition of partial pressures of components 00:15:04.460 --> 00:15:06.230 align:middle line:90% in a mixture. 00:15:06.230 --> 00:15:12.410 align:middle line:84% You recall that that is defined by the condition 00:15:12.410 --> 00:15:17.720 align:middle line:84% of mutual equilibrium across a membrane, permeable only, 00:15:17.720 --> 00:15:19.460 align:middle line:90% to constituent i. 00:15:19.460 --> 00:15:24.450 align:middle line:84% So at that mutual equilibrium across the membrane, 00:15:24.450 --> 00:15:27.950 align:middle line:84% you have equality of chemical potentials. 00:15:27.950 --> 00:15:31.370 align:middle line:84% So the chemical potential of a constituent in the mixture 00:15:31.370 --> 00:15:34.433 align:middle line:84% is also equal to the chemical potential. 00:15:34.433 --> 00:15:36.100 align:middle line:84% And here, there is also another mistake. 00:15:36.100 --> 00:15:38.810 align:middle line:90% This should be a double i. 00:15:38.810 --> 00:15:42.200 align:middle line:84% This is the chemical potential of a pure component 00:15:42.200 --> 00:15:50.060 align:middle line:84% on the other side of the semi-permeable membrane. 00:15:50.060 --> 00:15:51.870 align:middle line:90% And therefore, yes. 00:15:51.870 --> 00:15:53.970 align:middle line:84% At the same temperature of the mixture, 00:15:53.970 --> 00:15:58.595 align:middle line:84% but at a pressure which is what we call the partial pressure. 00:15:58.595 --> 00:16:02.820 align:middle line:90% 00:16:02.820 --> 00:16:07.080 align:middle line:90% So again, here, I-- 00:16:07.080 --> 00:16:12.720 align:middle line:84% cut and paste, makes errors propagate. 00:16:12.720 --> 00:16:17.130 align:middle line:84% So these should be double i and double i. 00:16:17.130 --> 00:16:25.080 align:middle line:84% OK, so we are now taking this last equality 00:16:25.080 --> 00:16:28.210 align:middle line:84% and putting in evidence the difference. 00:16:28.210 --> 00:16:34.080 align:middle line:84% And so taking essentially this term to the right-hand side. 00:16:34.080 --> 00:16:38.400 align:middle line:84% So I have the chemical potential difference 00:16:38.400 --> 00:16:41.370 align:middle line:84% evaluated between the pressure-- the partial pressure 00:16:41.370 --> 00:16:46.930 align:middle line:84% and the pressure of the mixture, pii minus p. 00:16:46.930 --> 00:16:50.330 align:middle line:90% 00:16:50.330 --> 00:17:02.330 align:middle line:84% Now here, we have already proved in a previous slide, 00:17:02.330 --> 00:17:05.240 align:middle line:84% maybe we can just remind ourselves 00:17:05.240 --> 00:17:10.670 align:middle line:90% of how they got here, that-- 00:17:10.670 --> 00:17:16.339 align:middle line:84% so we are having that mu double i at pii-- 00:17:16.339 --> 00:17:25.270 align:middle line:84% I'm sorry, at T and pii minus mu double i at T p. 00:17:25.270 --> 00:17:28.130 align:middle line:90% 00:17:28.130 --> 00:17:39.580 align:middle line:84% OK, you see, if I have a function of x, 00:17:39.580 --> 00:17:43.000 align:middle line:84% And the difference of a function evaluated 00:17:43.000 --> 00:17:45.640 align:middle line:84% at two different points, I can always 00:17:45.640 --> 00:17:50.560 align:middle line:84% write by the integral calculus equation 00:17:50.560 --> 00:17:55.120 align:middle line:84% that this is equal to the integral from x2 00:17:55.120 --> 00:18:01.030 align:middle line:84% to x1 of the derivative of that function. 00:18:01.030 --> 00:18:01.815 align:middle line:90% Let's call it-- 00:18:01.815 --> 00:18:04.920 align:middle line:90% 00:18:04.920 --> 00:18:11.810 align:middle line:84% And this x double prime is a dummy variable of integration. 00:18:11.810 --> 00:18:15.480 align:middle line:84% OK, so these, you can write for any function. 00:18:15.480 --> 00:18:20.120 align:middle line:90% So here, we apply this theorem. 00:18:20.120 --> 00:18:30.330 align:middle line:84% So this x2 is p, x1 is pii, and the derivative 00:18:30.330 --> 00:18:33.650 align:middle line:84% we have to put here is the partial derivative with respect 00:18:33.650 --> 00:18:38.771 align:middle line:90% to p of what? 00:18:38.771 --> 00:18:42.820 align:middle line:84% Of the chemical potential of the pure component 00:18:42.820 --> 00:18:46.450 align:middle line:84% while the other variables are T p-- 00:18:46.450 --> 00:18:56.920 align:middle line:84% I'm sorry, T. And this is to be evaluated 00:18:56.920 --> 00:19:03.550 align:middle line:84% at the value of the dummy variable, so at p double prime, 00:19:03.550 --> 00:19:07.720 align:middle line:84% and integrated over the dummy variable. 00:19:07.720 --> 00:19:14.050 align:middle line:84% But if you look up the many equations 00:19:14.050 --> 00:19:17.980 align:middle line:84% that we have derived also for the pure substance, 00:19:17.980 --> 00:19:23.025 align:middle line:84% this derivative is the specific volume. 00:19:23.025 --> 00:19:26.260 align:middle line:90% 00:19:26.260 --> 00:19:33.410 align:middle line:84% So this becomes the specific volume of the pure component 00:19:33.410 --> 00:19:37.790 align:middle line:84% evaluated at temperature T and at the dummy pressure p 00:19:37.790 --> 00:19:41.430 align:middle line:90% of integration p double prime. 00:19:41.430 --> 00:19:44.070 align:middle line:90% OK, all right. 00:19:44.070 --> 00:19:49.235 align:middle line:84% So that explains this integral here. 00:19:49.235 --> 00:19:52.210 align:middle line:90% 00:19:52.210 --> 00:19:55.810 align:middle line:84% But above, we've seen that this difference in chemical 00:19:55.810 --> 00:20:04.480 align:middle line:84% potentials is equal to the delta h mix minus T s delta s mix. 00:20:04.480 --> 00:20:12.970 align:middle line:90% OK, now look at this volume. 00:20:12.970 --> 00:20:15.875 align:middle line:90% 00:20:15.875 --> 00:20:22.030 align:middle line:84% If I have an ideal gas, and if I can-- 00:20:22.030 --> 00:20:27.970 align:middle line:84% my substance i behaves according to the ideal gas relation, 00:20:27.970 --> 00:20:35.230 align:middle line:84% the specific volume can be written as RT over p. 00:20:35.230 --> 00:20:39.980 align:middle line:84% And the pressure that we should evaluate is p that is here. 00:20:39.980 --> 00:20:43.480 align:middle line:90% So it's p double prime. 00:20:43.480 --> 00:20:50.710 align:middle line:84% And so when I do this integral, RT goes outside 00:20:50.710 --> 00:20:53.890 align:middle line:84% because I'm integrating over p, p double prime. 00:20:53.890 --> 00:20:59.440 align:middle line:84% And the integral of 1 over p is the logarithm. 00:20:59.440 --> 00:21:10.330 align:middle line:84% So that gives me the formula RT log of pii over p. 00:21:10.330 --> 00:21:12.340 align:middle line:90% And that's for the gas. 00:21:12.340 --> 00:21:15.250 align:middle line:90% 00:21:15.250 --> 00:21:20.860 align:middle line:84% If instead I have a liquid and I can 00:21:20.860 --> 00:21:26.020 align:middle line:84% use the incompressible liquid model for which 00:21:26.020 --> 00:21:33.860 align:middle line:84% the specific volume vii is approximately constant, 00:21:33.860 --> 00:21:38.590 align:middle line:84% meaning independent of temperature and pressure, 00:21:38.590 --> 00:21:41.810 align:middle line:84% well, if it is a constant, it goes out of the integral. 00:21:41.810 --> 00:21:48.820 align:middle line:84% And then this becomes the specific volume times 00:21:48.820 --> 00:21:53.830 align:middle line:90% pii minus p. 00:21:53.830 --> 00:21:59.380 align:middle line:84% And here in the slide, I just switched 00:21:59.380 --> 00:22:03.100 align:middle line:84% because the pressure is usually greater 00:22:03.100 --> 00:22:04.250 align:middle line:90% than the partial pressure. 00:22:04.250 --> 00:22:07.660 align:middle line:84% So I put this difference in parentheses 00:22:07.660 --> 00:22:11.290 align:middle line:84% and then I pick up a minus sign on the other side. 00:22:11.290 --> 00:22:18.105 align:middle line:84% OK, so this formula will be useful in a moment. 00:22:18.105 --> 00:22:23.220 align:middle line:90% 00:22:23.220 --> 00:22:27.210 align:middle line:84% And again, we see that there is a relation 00:22:27.210 --> 00:22:37.745 align:middle line:84% between the partial pressures, very explicit relations in terms 00:22:37.745 --> 00:22:42.530 align:middle line:84% of the enthalpy and the entropy of isothermobaric mixing. 00:22:42.530 --> 00:22:48.250 align:middle line:84% So this is a way in which you can measure partial pressures 00:22:48.250 --> 00:22:50.770 align:middle line:84% because you can measure the properties 00:22:50.770 --> 00:22:52.095 align:middle line:90% of isothermobaric mixing. 00:22:52.095 --> 00:23:01.230 align:middle line:90% 00:23:01.230 --> 00:23:12.000 align:middle line:84% For example, here, I want to point you to a couple 00:23:12.000 --> 00:23:20.300 align:middle line:84% of references, which if you go and take a look, and they'll-- 00:23:20.300 --> 00:23:23.275 align:middle line:90% 00:23:23.275 --> 00:23:30.840 align:middle line:84% a huge literature, so a huge community 00:23:30.840 --> 00:23:36.840 align:middle line:84% that over many decades, or perhaps 00:23:36.840 --> 00:23:41.010 align:middle line:84% over more than a century, have been 00:23:41.010 --> 00:23:44.310 align:middle line:90% measuring properties this way. 00:23:44.310 --> 00:23:47.640 align:middle line:84% And they're still doing it because the applications 00:23:47.640 --> 00:23:49.123 align:middle line:90% are growing. 00:23:49.123 --> 00:23:52.820 align:middle line:90% 00:23:52.820 --> 00:23:58.250 align:middle line:84% For example, this is an experimental paper 00:23:58.250 --> 00:24:01.370 align:middle line:90% where they collect-- 00:24:01.370 --> 00:24:05.570 align:middle line:84% they measure the enthalpy of isothermobaric mixing 00:24:05.570 --> 00:24:07.970 align:middle line:90% of liquids. 00:24:07.970 --> 00:24:16.330 align:middle line:84% And these liquids are molten salts that, as you may know, 00:24:16.330 --> 00:24:20.150 align:middle line:84% are used in various applications, 00:24:20.150 --> 00:24:30.170 align:middle line:84% including solar thermal applications in those collectors 00:24:30.170 --> 00:24:37.350 align:middle line:84% with mirrors that want to achieve high temperatures. 00:24:37.350 --> 00:24:41.660 align:middle line:84% The molten salts allow-- are stable 00:24:41.660 --> 00:24:43.440 align:middle line:90% also at very high temperatures. 00:24:43.440 --> 00:24:52.910 align:middle line:84% And therefore, using these fluids, 00:24:52.910 --> 00:24:58.400 align:middle line:84% you can collect solar energy at high temperatures 00:24:58.400 --> 00:25:00.730 align:middle line:90% up to maybe 600 Celsius. 00:25:00.730 --> 00:25:05.450 align:middle line:90% 00:25:05.450 --> 00:25:11.360 align:middle line:84% But of course, molten salts are used also in nuclear engineering 00:25:11.360 --> 00:25:21.170 align:middle line:84% and in processing industries for producing materials 00:25:21.170 --> 00:25:25.100 align:middle line:84% that are useful for all sorts of applications, 00:25:25.100 --> 00:25:30.310 align:middle line:90% from electronics to magnets. 00:25:30.310 --> 00:25:33.200 align:middle line:90% 00:25:33.200 --> 00:25:36.860 align:middle line:84% So this is a reference in which you 00:25:36.860 --> 00:25:42.560 align:middle line:84% can see how people measure and then try to correlate. 00:25:42.560 --> 00:25:46.430 align:middle line:84% So there is also a number of correlations, 00:25:46.430 --> 00:25:52.910 align:middle line:84% which go under the names of the inventors of the correlations, 00:25:52.910 --> 00:26:01.670 align:middle line:84% that do good jobs in determining these effects. 00:26:01.670 --> 00:26:09.980 align:middle line:84% You see, here, the gray molecule, cesium ion, 00:26:09.980 --> 00:26:14.030 align:middle line:90% is sort of the solvent. 00:26:14.030 --> 00:26:20.900 align:middle line:84% It's like the prevailing, or the common substance 00:26:20.900 --> 00:26:23.190 align:middle line:90% of these mixtures. 00:26:23.190 --> 00:26:27.450 align:middle line:84% And then they keep changing the other substance. 00:26:27.450 --> 00:26:31.710 align:middle line:90% 00:26:31.710 --> 00:26:40.380 align:middle line:84% So it's actually not only for dilute, 00:26:40.380 --> 00:26:45.690 align:middle line:84% it's for over the full range of compositions from 0 to 1. 00:26:45.690 --> 00:26:52.220 align:middle line:84% And you can see that you have a behavior in which the enthalpy 00:26:52.220 --> 00:26:55.076 align:middle line:90% of mixing is negative. 00:26:55.076 --> 00:27:01.710 align:middle line:90% 00:27:01.710 --> 00:27:06.210 align:middle line:90% Meaning that the substance-- 00:27:06.210 --> 00:27:12.430 align:middle line:84% that when you mix the substance, you have an exothermic reaction. 00:27:12.430 --> 00:27:17.590 align:middle line:84% It's not really a reaction, but it's a physical chemical effect, 00:27:17.590 --> 00:27:19.780 align:middle line:90% which releases heat. 00:27:19.780 --> 00:27:21.390 align:middle line:90% So in other words, you have to-- 00:27:21.390 --> 00:27:24.670 align:middle line:84% if you want to return to mutual equilibrium with the reservoir, 00:27:24.670 --> 00:27:27.870 align:middle line:84% you have to cool down after the mixing. 00:27:27.870 --> 00:27:30.410 align:middle line:84% You have to cool down by this amount. 00:27:30.410 --> 00:27:34.410 align:middle line:90% 00:27:34.410 --> 00:27:38.070 align:middle line:84% And that cooling measures is exactly-- 00:27:38.070 --> 00:27:40.620 align:middle line:84% so we-- that's how with a calorimeter, 00:27:40.620 --> 00:27:49.470 align:middle line:84% you can measure that cooling, and therefore this delta H. 00:27:49.470 --> 00:27:54.410 align:middle line:84% You can also do similar measurements 00:27:54.410 --> 00:27:58.800 align:middle line:90% for the volume of mixing. 00:27:58.800 --> 00:28:07.320 align:middle line:90% And here, I sketched the-- 00:28:07.320 --> 00:28:12.120 align:middle line:84% yeah, I think this line should belong a little lower here 00:28:12.120 --> 00:28:14.250 align:middle line:90% to the 0 level. 00:28:14.250 --> 00:28:18.870 align:middle line:84% And I sketched-- this is just to show 00:28:18.870 --> 00:28:24.280 align:middle line:84% that it's the volume of mixing is not always positive. 00:28:24.280 --> 00:28:27.750 align:middle line:84% It can be positive, it can be negative, 00:28:27.750 --> 00:28:32.670 align:middle line:84% it can be positive in one range and become negative 00:28:32.670 --> 00:28:38.300 align:middle line:84% in the other range, depending on the composition. 00:28:38.300 --> 00:28:41.870 align:middle line:84% And it all depends on what kind of molecules you have, 00:28:41.870 --> 00:28:48.680 align:middle line:84% and how the molecules interact with each other. 00:28:48.680 --> 00:28:52.640 align:middle line:84% And as you can see, these two references 00:28:52.640 --> 00:28:57.920 align:middle line:84% where you may want to take a look to see experimental data, 00:28:57.920 --> 00:29:03.655 align:middle line:84% and how they're treated, date back to at least 40 years ago. 00:29:03.655 --> 00:29:11.130 align:middle line:90% 00:29:11.130 --> 00:29:19.600 align:middle line:84% OK, now, this slide is also just a parenthetical. 00:29:19.600 --> 00:29:22.475 align:middle line:90% It just says, all right. 00:29:22.475 --> 00:29:25.230 align:middle line:90% 00:29:25.230 --> 00:29:27.480 align:middle line:84% So we have seen that in practice, 00:29:27.480 --> 00:29:33.660 align:middle line:84% with those measurements, we can infer the values of the chemical 00:29:33.660 --> 00:29:37.245 align:middle line:84% potentials, and therefore the partial properties. 00:29:37.245 --> 00:29:45.750 align:middle line:84% So the question that we address in this slide is, can we-- 00:29:45.750 --> 00:29:48.360 align:middle line:84% is the meaning of partial that we 00:29:48.360 --> 00:29:54.570 align:middle line:84% use in partial properties the same as the meaning of partial 00:29:54.570 --> 00:29:57.330 align:middle line:90% as we use in partial pressures? 00:29:57.330 --> 00:30:00.550 align:middle line:84% And the answer is no because the definition is different. 00:30:00.550 --> 00:30:05.100 align:middle line:84% However, you can somehow relate partial properties 00:30:05.100 --> 00:30:07.290 align:middle line:90% in terms of partial pressures. 00:30:07.290 --> 00:30:14.850 align:middle line:84% And here, here is the way of doing it. 00:30:14.850 --> 00:30:18.520 align:middle line:90% 00:30:18.520 --> 00:30:22.160 align:middle line:84% It's just a matter of writing the chemical potential, 00:30:22.160 --> 00:30:24.520 align:middle line:90% as we see here. 00:30:24.520 --> 00:30:28.130 align:middle line:84% Again, as a function, as it is of the partial pressure. 00:30:28.130 --> 00:30:33.310 align:middle line:84% So if you take the differential of this relation 00:30:33.310 --> 00:30:36.760 align:middle line:84% here, which defines the partial pressure, 00:30:36.760 --> 00:30:41.230 align:middle line:84% and then remember that the derivatives of the chemical 00:30:41.230 --> 00:30:43.130 align:middle line:84% potential are the partial properties, 00:30:43.130 --> 00:30:47.273 align:middle line:84% then that is where that relation comes about. 00:30:47.273 --> 00:30:50.670 align:middle line:90% 00:30:50.670 --> 00:30:52.000 align:middle line:90% And here are the details. 00:30:52.000 --> 00:30:54.900 align:middle line:84% So if you like to take derivatives, 00:30:54.900 --> 00:30:58.800 align:middle line:90% this is how you can practice it. 00:30:58.800 --> 00:31:05.330 align:middle line:84% So I'm not going to explain the details here. 00:31:05.330 --> 00:31:13.459 align:middle line:90% 00:31:13.459 --> 00:31:19.100 align:middle line:84% At this stage, we need to recall something 00:31:19.100 --> 00:31:25.220 align:middle line:84% about what happens when molecules 00:31:25.220 --> 00:31:27.595 align:middle line:90% come close to one another. 00:31:27.595 --> 00:31:31.090 align:middle line:90% 00:31:31.090 --> 00:31:40.730 align:middle line:84% And the Lennard-Jones model gives you 00:31:40.730 --> 00:31:44.360 align:middle line:84% the potential energy of the interaction 00:31:44.360 --> 00:31:45.630 align:middle line:90% between two molecules. 00:31:45.630 --> 00:31:49.850 align:middle line:84% So suppose I have one molecule and I stay in-- 00:31:49.850 --> 00:31:52.580 align:middle line:84% and I sit on that molecule on the reference frame that 00:31:52.580 --> 00:31:55.850 align:middle line:84% moves with that molecule, and I see 00:31:55.850 --> 00:31:57.980 align:middle line:90% another molecule approaching. 00:31:57.980 --> 00:32:04.560 align:middle line:84% The Lennard-Jones model is a pair-interaction kind of model. 00:32:04.560 --> 00:32:08.540 align:middle line:84% It looks just what happens between one molecule 00:32:08.540 --> 00:32:11.910 align:middle line:90% and the other, not the fact-- 00:32:11.910 --> 00:32:16.490 align:middle line:84% it doesn't apply well if you have to consider also the fact 00:32:16.490 --> 00:32:19.070 align:middle line:84% that this molecule may also, at the same time, 00:32:19.070 --> 00:32:27.005 align:middle line:84% be interacting with a third and maybe a fourth molecule, 00:32:27.005 --> 00:32:30.240 align:middle line:84% which of course, is important in liquids. 00:32:30.240 --> 00:32:32.850 align:middle line:90% But OK, so here's the idea. 00:32:32.850 --> 00:32:34.580 align:middle line:84% I'm sitting on that molecule, and I see 00:32:34.580 --> 00:32:39.290 align:middle line:90% another molecule approaching. 00:32:39.290 --> 00:32:43.760 align:middle line:84% Now, because of the electronic structure 00:32:43.760 --> 00:32:48.170 align:middle line:84% of atoms and molecules, the electrons 00:32:48.170 --> 00:32:51.570 align:middle line:84% that are negatively charged, yes, 00:32:51.570 --> 00:32:55.940 align:middle line:84% are attracted by the nucleus or the nuclei of the molecule, 00:32:55.940 --> 00:32:58.310 align:middle line:90% or the atom. 00:32:58.310 --> 00:33:01.560 align:middle line:84% But as they approach other molecules, 00:33:01.560 --> 00:33:08.270 align:middle line:84% they are also a bit attracted by the nucleus of the other atom. 00:33:08.270 --> 00:33:14.630 align:middle line:84% And that produces these so-called van 00:33:14.630 --> 00:33:16.800 align:middle line:90% der Waals attractive forces. 00:33:16.800 --> 00:33:20.760 align:middle line:84% So the potential going negative-- 00:33:20.760 --> 00:33:23.120 align:middle line:84% so if this is the potential, remember 00:33:23.120 --> 00:33:28.040 align:middle line:84% that I can calculate the force by taking 00:33:28.040 --> 00:33:30.260 align:middle line:84% the derivative of the potential with respect 00:33:30.260 --> 00:33:33.560 align:middle line:90% to distance minus sign. 00:33:33.560 --> 00:33:36.140 align:middle line:90% So this force is a vector. 00:33:36.140 --> 00:33:44.100 align:middle line:84% And ir is the vector that points in the direction outward, 00:33:44.100 --> 00:33:48.770 align:middle line:84% so further from the molecule I am sitting on. 00:33:48.770 --> 00:33:53.570 align:middle line:84% So it's like on this axis, i sub r 00:33:53.570 --> 00:33:56.030 align:middle line:84% is the unit vector in the r direction, 00:33:56.030 --> 00:33:58.170 align:middle line:90% in the right direction. 00:33:58.170 --> 00:34:01.660 align:middle line:90% 00:34:01.660 --> 00:34:03.910 align:middle line:84% So the usefulness of having a potential 00:34:03.910 --> 00:34:05.980 align:middle line:84% is that then, you can calculate forces 00:34:05.980 --> 00:34:08.270 align:middle line:84% by taking the derivative of the potential. 00:34:08.270 --> 00:34:10.360 align:middle line:84% And as you see, here is the force-- 00:34:10.360 --> 00:34:12.896 align:middle line:90% if the force is negative. 00:34:12.896 --> 00:34:16.860 align:middle line:90% 00:34:16.860 --> 00:34:23.969 align:middle line:84% it means that its direction is in the opposite sign of the unit 00:34:23.969 --> 00:34:26.050 align:middle line:90% vector in the r direction. 00:34:26.050 --> 00:34:27.780 align:middle line:90% So it points left. 00:34:27.780 --> 00:34:31.469 align:middle line:90% So it's attractive. 00:34:31.469 --> 00:34:47.540 align:middle line:84% Up to this point, which is about here, the derivative of V 00:34:47.540 --> 00:34:48.739 align:middle line:90% is positive. 00:34:48.739 --> 00:34:50.750 align:middle line:90% So the force is negative. 00:34:50.750 --> 00:34:57.850 align:middle line:90% And yeah. 00:34:57.850 --> 00:35:01.620 align:middle line:84% Yeah, OK, up to this point, where it is 0, 00:35:01.620 --> 00:35:07.100 align:middle line:84% we get to the minimum, it's attractive. 00:35:07.100 --> 00:35:10.850 align:middle line:84% And from this point on, it becomes repulsive. 00:35:10.850 --> 00:35:13.230 align:middle line:90% The force becomes positive. 00:35:13.230 --> 00:35:16.280 align:middle line:90% 00:35:16.280 --> 00:35:18.890 align:middle line:84% That repulsion is due to the fact 00:35:18.890 --> 00:35:24.410 align:middle line:84% that now, the nuclei, which are positively charged, 00:35:24.410 --> 00:35:28.330 align:middle line:84% begin to fill the positive charge of the nuclei 00:35:28.330 --> 00:35:29.990 align:middle line:90% of the particle I'm sitting on. 00:35:29.990 --> 00:35:31.493 align:middle line:90% And therefore, that repels. 00:35:31.493 --> 00:35:38.140 align:middle line:90% 00:35:38.140 --> 00:35:48.530 align:middle line:84% OK, so most of the times, molecules 00:35:48.530 --> 00:35:55.160 align:middle line:84% are running around in their Brownian motion. 00:35:55.160 --> 00:36:01.440 align:middle line:84% So in which they have some kinetic energy 00:36:01.440 --> 00:36:04.050 align:middle line:90% due to the velocity they have. 00:36:04.050 --> 00:36:06.520 align:middle line:84% And they have also this potential energy. 00:36:06.520 --> 00:36:12.480 align:middle line:84% Some of that kinetic energy gets transformed in potential energy. 00:36:12.480 --> 00:36:15.990 align:middle line:84% Or actually when the molecule approaches and gets 00:36:15.990 --> 00:36:21.520 align:middle line:84% an attractive force, it gains kinetic energy. 00:36:21.520 --> 00:36:25.330 align:middle line:84% So it's accelerated towards the particle. 00:36:25.330 --> 00:36:28.800 align:middle line:84% So if the molecule is coming from the right 00:36:28.800 --> 00:36:36.540 align:middle line:84% and it's approaching, it's accelerated towards the molecule 00:36:36.540 --> 00:36:38.950 align:middle line:90% I'm sitting in. 00:36:38.950 --> 00:36:44.620 align:middle line:84% But then, all of a sudden, it finds also a repulsive force. 00:36:44.620 --> 00:36:47.710 align:middle line:90% So that's decelerating. 00:36:47.710 --> 00:36:53.950 align:middle line:84% Of course, if the kinetic energy is enough, 00:36:53.950 --> 00:36:58.310 align:middle line:90% this molecule can go and hit. 00:36:58.310 --> 00:37:00.910 align:middle line:90% So make a collision. 00:37:00.910 --> 00:37:03.710 align:middle line:84% Eventually, it will make a collision 00:37:03.710 --> 00:37:07.370 align:middle line:84% because the force is going to become very large. 00:37:07.370 --> 00:37:09.170 align:middle line:84% The repulsive force becomes very large. 00:37:09.170 --> 00:37:13.350 align:middle line:84% And therefore, the particle is pushed away back. 00:37:13.350 --> 00:37:14.970 align:middle line:90% That's a collision. 00:37:14.970 --> 00:37:17.730 align:middle line:84% It's like a soft collision, but it's a collision. 00:37:17.730 --> 00:37:22.880 align:middle line:90% 00:37:22.880 --> 00:37:30.370 align:middle line:84% An even simpler model of this would be the so-called hard 00:37:30.370 --> 00:37:37.840 align:middle line:84% sphere model, in which the potential is-- would be like-- 00:37:37.840 --> 00:37:49.820 align:middle line:90% 00:37:49.820 --> 00:37:52.510 align:middle line:90% will be like this. 00:37:52.510 --> 00:37:56.260 align:middle line:84% A function, so V, is a function which 00:37:56.260 --> 00:38:03.760 align:middle line:84% goes to infinite at this point, which is 1. 00:38:03.760 --> 00:38:08.650 align:middle line:84% It's the distance-- 1 is because we're normalizing these 00:38:08.650 --> 00:38:17.050 align:middle line:84% to the diameter of these molecules, of these spheres, 00:38:17.050 --> 00:38:19.480 align:middle line:90% assuming they're equal. 00:38:19.480 --> 00:38:22.330 align:middle line:84% Otherwise, it would be like the-- have to normalize 00:38:22.330 --> 00:38:23.666 align:middle line:90% to the average-- 00:38:23.666 --> 00:38:27.260 align:middle line:90% 00:38:27.260 --> 00:38:30.740 align:middle line:84% the sum of the radii of the two molecules. 00:38:30.740 --> 00:38:32.670 align:middle line:90% And it's 0 here. 00:38:32.670 --> 00:38:37.940 align:middle line:84% So the hard sphere model assumes no attractive force 00:38:37.940 --> 00:38:47.330 align:middle line:84% until the sphere hits like in a billiard ball situation. 00:38:47.330 --> 00:38:51.050 align:middle line:84% It hits the other molecule and it gets repelled 00:38:51.050 --> 00:38:56.146 align:middle line:90% by some elastic collision. 00:38:56.146 --> 00:38:59.940 align:middle line:90% 00:38:59.940 --> 00:39:09.250 align:middle line:84% OK, let's return to this case here and now consider 00:39:09.250 --> 00:39:10.955 align:middle line:90% two important regimes. 00:39:10.955 --> 00:39:13.480 align:middle line:90% 00:39:13.480 --> 00:39:20.980 align:middle line:84% One regime is the ideal gas, the gas regime. 00:39:20.980 --> 00:39:27.100 align:middle line:84% So it's true that the molecules of the gas 00:39:27.100 --> 00:39:40.200 align:middle line:84% very often, billions of collisions per second, we-- 00:39:40.200 --> 00:39:47.580 align:middle line:84% maybe we'll have a slide later on that does that calculation, 00:39:47.580 --> 00:39:50.130 align:middle line:90% do have experienced collisions. 00:39:50.130 --> 00:39:58.290 align:middle line:84% So these molecules do get to feel this potential when 00:39:58.290 --> 00:40:03.210 align:middle line:84% they approach the other molecules and have a collision. 00:40:03.210 --> 00:40:05.550 align:middle line:90% That happens very often. 00:40:05.550 --> 00:40:09.090 align:middle line:84% But if you look up, if you make an average, 00:40:09.090 --> 00:40:17.580 align:middle line:84% a time average, of how much time this molecule spends 00:40:17.580 --> 00:40:23.970 align:middle line:84% in this area, where the Lennard-Jones potential is 00:40:23.970 --> 00:40:28.440 align:middle line:90% negative, is very small. 00:40:28.440 --> 00:40:35.710 align:middle line:84% Most of the times, on average, the particles are far apart. 00:40:35.710 --> 00:40:45.241 align:middle line:84% And therefore, on average, they feel zero potential. 00:40:45.241 --> 00:40:50.960 align:middle line:84% See, this is why when you try to measure the properties, 00:40:50.960 --> 00:40:55.730 align:middle line:84% you're actually measuring the properties on average, 00:40:55.730 --> 00:40:59.460 align:middle line:84% because remember that somehow, the state of your system 00:40:59.460 --> 00:41:04.820 align:middle line:84% is the probability distribution of things. 00:41:04.820 --> 00:41:08.690 align:middle line:84% For example, here is the probability distribution 00:41:08.690 --> 00:41:13.160 align:middle line:84% of finding the particle at these various distances. 00:41:13.160 --> 00:41:20.130 align:middle line:84% And that probability for a gas is-- 00:41:20.130 --> 00:41:25.620 align:middle line:84% has this very small tail here, and it peaks away. 00:41:25.620 --> 00:41:28.920 align:middle line:84% So most of the times, the particles 00:41:28.920 --> 00:41:32.220 align:middle line:84% are far away from each other even though they 00:41:32.220 --> 00:41:34.090 align:middle line:90% experience a lot of collisions. 00:41:34.090 --> 00:41:39.260 align:middle line:90% 00:41:39.260 --> 00:41:41.660 align:middle line:84% But that means that when you're calculating 00:41:41.660 --> 00:41:43.820 align:middle line:84% the stable equilibrium properties, therefore, 00:41:43.820 --> 00:41:45.170 align:middle line:90% the average-- 00:41:45.170 --> 00:41:52.690 align:middle line:84% time average properties, this potential is irrelevant. 00:41:52.690 --> 00:41:59.930 align:middle line:84% And that is what allows you to, for example, 00:41:59.930 --> 00:42:03.400 align:middle line:84% use the hard sphere model because that's 00:42:03.400 --> 00:42:06.340 align:middle line:84% totally equivalent if the particles live far 00:42:06.340 --> 00:42:13.193 align:middle line:84% away on average and the details of the potential 00:42:13.193 --> 00:42:14.860 align:middle line:84% of their interaction when they are close 00:42:14.860 --> 00:42:20.890 align:middle line:84% is not important, well, those details, you can forget. 00:42:20.890 --> 00:42:22.420 align:middle line:84% They don't enter into determining 00:42:22.420 --> 00:42:25.300 align:middle line:90% the properties of the mixtures. 00:42:25.300 --> 00:42:29.860 align:middle line:84% So much that in the ideal gas, and also, 00:42:29.860 --> 00:42:40.500 align:middle line:84% we will hopefully have a slide also on this, the ideal gas, 00:42:40.500 --> 00:42:46.050 align:middle line:84% this relation here that the volume is RT over p, 00:42:46.050 --> 00:42:51.210 align:middle line:84% you can derive for a particle, a single particle 00:42:51.210 --> 00:42:59.900 align:middle line:84% in a box, which is a nice illustration also of-- 00:42:59.900 --> 00:43:02.050 align:middle line:84% that's why I will have a slide on this. 00:43:02.050 --> 00:43:05.330 align:middle line:90% 00:43:05.330 --> 00:43:08.750 align:middle line:84% But in other words, this-- here in this context 00:43:08.750 --> 00:43:17.015 align:middle line:84% is just to say that the properties of an ideal gas, so 00:43:17.015 --> 00:43:23.465 align:middle line:84% the origin of this ideal gas equation of state 00:43:23.465 --> 00:43:27.230 align:middle line:84% comes from the single particle because the properties 00:43:27.230 --> 00:43:31.331 align:middle line:84% at stable equilibrium in this regime 00:43:31.331 --> 00:43:35.900 align:middle line:84% are those of particles that leave on average by themselves. 00:43:35.900 --> 00:43:40.080 align:middle line:84% They don't feel the effects of the other particles. 00:43:40.080 --> 00:43:44.590 align:middle line:90% 00:43:44.590 --> 00:43:49.340 align:middle line:84% There is another case also, which is the liquid case. 00:43:49.340 --> 00:43:54.940 align:middle line:84% We will return to the liquid in a while, maybe today. 00:43:54.940 --> 00:44:01.870 align:middle line:84% But what's important is that instead, the liquid particles 00:44:01.870 --> 00:44:03.890 align:middle line:90% live very close to one another. 00:44:03.890 --> 00:44:05.670 align:middle line:90% They live near this minimum. 00:44:05.670 --> 00:44:08.220 align:middle line:90% 00:44:08.220 --> 00:44:14.470 align:middle line:84% So for them, this potential is more important. 00:44:14.470 --> 00:44:31.760 align:middle line:90% 00:44:31.760 --> 00:44:40.610 align:middle line:84% This-- the Lennard-Jones potential formula can be 00:44:40.610 --> 00:44:41.150 align:middle line:90% written-- 00:44:41.150 --> 00:44:44.010 align:middle line:84% is written in two different ways here. 00:44:44.010 --> 00:44:48.920 align:middle line:84% One is in terms of the actual distance, 00:44:48.920 --> 00:44:53.030 align:middle line:84% and one is in terms of the distance normalized 00:44:53.030 --> 00:45:00.590 align:middle line:84% to this minimum, to the distance at which the minimum occurs. 00:45:00.590 --> 00:45:05.450 align:middle line:90% Sigma, as I already said, is-- 00:45:05.450 --> 00:45:07.650 align:middle line:90% r is normalized here to sigma. 00:45:07.650 --> 00:45:14.720 align:middle line:84% So it's 1 at the distance where it 00:45:14.720 --> 00:45:17.070 align:middle line:84% is equal to the diameter of the two particles. 00:45:17.070 --> 00:45:21.530 align:middle line:84% So where the particles are colliding. 00:45:21.530 --> 00:45:27.650 align:middle line:84% And here, we also in the plot normalized the potential 00:45:27.650 --> 00:45:28.700 align:middle line:90% to epsilon. 00:45:28.700 --> 00:45:36.450 align:middle line:84% And epsilon is the depth of this well here. 00:45:36.450 --> 00:45:38.110 align:middle line:90% So it goes to minus 1. 00:45:38.110 --> 00:45:41.480 align:middle line:90% 00:45:41.480 --> 00:45:46.730 align:middle line:84% So stronger interactions will have-- 00:45:46.730 --> 00:45:50.455 align:middle line:84% would be characterized by stronger values of this epsilon. 00:45:50.455 --> 00:45:55.630 align:middle line:90% 00:45:55.630 --> 00:46:03.430 align:middle line:84% OK, so with this introduction, let's 00:46:03.430 --> 00:46:10.660 align:middle line:84% consider a first ideal behavior, which 00:46:10.660 --> 00:46:14.950 align:middle line:84% we will call Gibbs-Dalton ideal behavior, 00:46:14.950 --> 00:46:21.580 align:middle line:84% and which will be a very good approximation for gases 00:46:21.580 --> 00:46:28.540 align:middle line:84% in the gas phase, because in practice, we 00:46:28.540 --> 00:46:33.820 align:middle line:84% will assume that not only particles of the same kind 00:46:33.820 --> 00:46:36.830 align:middle line:84% like you would have in a pure substance, 00:46:36.830 --> 00:46:43.450 align:middle line:84% but also particles of different kind like we have in a mixture, 00:46:43.450 --> 00:46:48.200 align:middle line:84% they also live on average far away from one another. 00:46:48.200 --> 00:46:51.430 align:middle line:84% And therefore, we can neglect the effect 00:46:51.430 --> 00:46:54.680 align:middle line:84% of their intermolecular interactions. 00:46:54.680 --> 00:47:03.050 align:middle line:90% 00:47:03.050 --> 00:47:03.550 align:middle line:90% All right. 00:47:03.550 --> 00:47:08.800 align:middle line:84% Formally, this is how the story goes. 00:47:08.800 --> 00:47:12.330 align:middle line:84% We will first define ideal behavior. 00:47:12.330 --> 00:47:16.950 align:middle line:84% And just so this will be our-- this will be 00:47:16.950 --> 00:47:18.830 align:middle line:90% the final result of this slide. 00:47:18.830 --> 00:47:21.930 align:middle line:90% 00:47:21.930 --> 00:47:25.680 align:middle line:90% And the motivation is that-- 00:47:25.680 --> 00:47:29.610 align:middle line:84% of this definition that we are going to do here-- 00:47:29.610 --> 00:47:35.610 align:middle line:84% is that we will find that under this approximation, 00:47:35.610 --> 00:47:37.530 align:middle line:90% or in this limit-- 00:47:37.530 --> 00:47:43.650 align:middle line:84% when these assumptions are valid, 00:47:43.650 --> 00:47:46.800 align:middle line:84% then we obtain a famous result, which 00:47:46.800 --> 00:47:50.610 align:middle line:84% is the Dalton law of additive partial pressures. 00:47:50.610 --> 00:47:54.870 align:middle line:84% So the pressure of the mixture is 00:47:54.870 --> 00:47:57.420 align:middle line:84% going to be equal to the sum of the partial pressures 00:47:57.420 --> 00:48:02.320 align:middle line:84% of the constituents of the mixture. 00:48:02.320 --> 00:48:04.310 align:middle line:84% But in general, that is not true. 00:48:04.310 --> 00:48:07.770 align:middle line:84% Not-- I mean, this is not true for a non-ideal mixture. 00:48:07.770 --> 00:48:12.810 align:middle line:90% 00:48:12.810 --> 00:48:14.680 align:middle line:84% So the derivation goes as follows. 00:48:14.680 --> 00:48:19.940 align:middle line:84% We start from the Euler relation written for the mixture. 00:48:19.940 --> 00:48:22.880 align:middle line:90% 00:48:22.880 --> 00:48:28.816 align:middle line:84% We write also the Euler relation for the pure components. 00:48:28.816 --> 00:48:32.070 align:middle line:90% 00:48:32.070 --> 00:48:35.130 align:middle line:84% And when-- and here, I'm going to write it 00:48:35.130 --> 00:48:39.570 align:middle line:84% for the pure components at just on the other side 00:48:39.570 --> 00:48:44.370 align:middle line:84% of the semi-permeable membrane so they 00:48:44.370 --> 00:48:46.990 align:middle line:84% are at the same temperature of the mixture, 00:48:46.990 --> 00:48:49.680 align:middle line:90% but at the partial pressure. 00:48:49.680 --> 00:48:56.860 align:middle line:84% So these uii here, I'm not writing the explicit dependence, 00:48:56.860 --> 00:49:03.960 align:middle line:84% but I'm doing it here, are to be evaluated at T 00:49:03.960 --> 00:49:08.070 align:middle line:90% and at the partial pressure. 00:49:08.070 --> 00:49:16.050 align:middle line:84% Then I take these individual pure constituent 00:49:16.050 --> 00:49:18.220 align:middle line:90% Euler relations, we multiply-- 00:49:18.220 --> 00:49:24.660 align:middle line:84% I multiply them by the amount ni with which that constituent is 00:49:24.660 --> 00:49:29.010 align:middle line:84% present in the mixture and sum over all components. 00:49:29.010 --> 00:49:32.230 align:middle line:84% And then I take this summation, subtract it 00:49:32.230 --> 00:49:39.190 align:middle line:84% from the Euler relation for the mixture, and here is the result. 00:49:39.190 --> 00:49:43.030 align:middle line:84% So you can still identify here the Euler relation 00:49:43.030 --> 00:49:45.340 align:middle line:84% for the mixture is given by this term. 00:49:45.340 --> 00:49:52.210 align:middle line:90% This TS minus pV plus mui ni. 00:49:52.210 --> 00:49:57.190 align:middle line:84% And what I subtracted is the summation of the ni's 00:49:57.190 --> 00:49:58.285 align:middle line:90% with which-- 00:49:58.285 --> 00:50:01.150 align:middle line:84% that are the amounts in the mixture 00:50:01.150 --> 00:50:08.080 align:middle line:84% times the single component, pure component, Euler relation. 00:50:08.080 --> 00:50:20.850 align:middle line:84% uii minus T sii plus pii vii and plus the mu's. 00:50:20.850 --> 00:50:29.170 align:middle line:90% 00:50:29.170 --> 00:50:34.400 align:middle line:84% Actually, there's a mistake here because there 00:50:34.400 --> 00:50:35.730 align:middle line:90% shouldn't be a summation here. 00:50:35.730 --> 00:50:37.430 align:middle line:84% This is, again, a cut-and-paste problem. 00:50:37.430 --> 00:50:41.450 align:middle line:84% This should be-- I hope I'll remember all these things 00:50:41.450 --> 00:50:44.210 align:middle line:90% to correct in the viewgraph. 00:50:44.210 --> 00:50:48.330 align:middle line:84% Yeah, so for the pure component, there's only one. 00:50:48.330 --> 00:50:49.650 align:middle line:90% And so there is no summation. 00:50:49.650 --> 00:50:53.620 align:middle line:90% 00:50:53.620 --> 00:51:02.590 align:middle line:84% OK, now, look at these elements of the last summation. 00:51:02.590 --> 00:51:05.410 align:middle line:84% They are identically 0 because of the condition 00:51:05.410 --> 00:51:08.620 align:middle line:84% for mutual equilibrium that the chemical potential 00:51:08.620 --> 00:51:12.550 align:middle line:84% of component i in the mixture is equal to the chemical potential 00:51:12.550 --> 00:51:14.830 align:middle line:84% of the pure component when evaluated 00:51:14.830 --> 00:51:16.370 align:middle line:90% at the partial pressure. 00:51:16.370 --> 00:51:19.270 align:middle line:90% So these are 0. 00:51:19.270 --> 00:51:23.080 align:middle line:84% And then I get a relation here, which contains the pressure 00:51:23.080 --> 00:51:25.030 align:middle line:84% and contains the partial pressures, 00:51:25.030 --> 00:51:27.910 align:middle line:84% because after all, what we wanted to see here is, 00:51:27.910 --> 00:51:31.880 align:middle line:84% is there a relation between pressure and partial pressures? 00:51:31.880 --> 00:51:34.520 align:middle line:84% And here is the result. So I put in evidence, 00:51:34.520 --> 00:51:38.630 align:middle line:84% I solve this expression for the total pressure, 00:51:38.630 --> 00:51:42.670 align:middle line:84% and here is how it depends on the partial pressures. 00:51:42.670 --> 00:51:45.490 align:middle line:90% It's all over. 00:51:45.490 --> 00:51:57.470 align:middle line:84% However, if I make these three assumptions, you see this? 00:51:57.470 --> 00:52:03.380 align:middle line:84% The first assumption kills these terms in parentheses. 00:52:03.380 --> 00:52:07.580 align:middle line:84% The second assumption kills this other term. 00:52:07.580 --> 00:52:13.970 align:middle line:84% The third assumption, and note here that this is not a mistake, 00:52:13.970 --> 00:52:16.300 align:middle line:84% here, there is no summation sign. 00:52:16.300 --> 00:52:20.730 align:middle line:90% 00:52:20.730 --> 00:52:25.210 align:middle line:84% So if this holds, then we'll discuss what it means. 00:52:25.210 --> 00:52:30.810 align:middle line:84% But if this holds, it means that inside this summation, 00:52:30.810 --> 00:52:39.490 align:middle line:84% every time I see ni times vii, I can put capital V, 00:52:39.490 --> 00:52:41.960 align:middle line:90% which doesn't depend on i. 00:52:41.960 --> 00:52:43.390 align:middle line:90% It doesn't have a subscript. 00:52:43.390 --> 00:52:46.270 align:middle line:84% So it goes outside of the summation 00:52:46.270 --> 00:52:49.720 align:middle line:84% and it cancels with the V in the denominator. 00:52:49.720 --> 00:52:52.540 align:middle line:84% And then what I'm left with in the summation 00:52:52.540 --> 00:52:54.670 align:middle line:90% is just the partial pressures. 00:52:54.670 --> 00:52:59.680 align:middle line:84% So I finally get that the pressure, if these three 00:52:59.680 --> 00:53:06.670 align:middle line:84% assumptions are together, all holding, the pressure 00:53:06.670 --> 00:53:08.660 align:middle line:84% is equal to the sum of the partial pressures. 00:53:08.660 --> 00:53:10.310 align:middle line:90% And this is Dalton's law. 00:53:10.310 --> 00:53:17.430 align:middle line:90% 00:53:17.430 --> 00:53:27.950 align:middle line:84% Let's see how we could interpret those conditions that 00:53:27.950 --> 00:53:29.935 align:middle line:84% define the ideal Gibbs-Dalton behavior. 00:53:29.935 --> 00:53:32.570 align:middle line:90% 00:53:32.570 --> 00:53:36.420 align:middle line:84% Essentially, it says that I can see the mixture, 00:53:36.420 --> 00:53:40.880 align:middle line:84% or at least I can model the properties of the mixture, 00:53:40.880 --> 00:53:47.540 align:middle line:84% as the model, as the properties of a composite system in which I 00:53:47.540 --> 00:53:54.950 align:middle line:84% have the pure components living each one in the same volume 00:53:54.950 --> 00:53:59.570 align:middle line:84% of the mixture, so boxes of the same size, 00:53:59.570 --> 00:54:05.755 align:middle line:84% at the same temperature and at the partial pressure. 00:54:05.755 --> 00:54:08.890 align:middle line:90% 00:54:08.890 --> 00:54:18.020 align:middle line:84% OK, then in these conditions, this is the specific-- 00:54:18.020 --> 00:54:20.480 align:middle line:84% they have a specific energy since they are pure. 00:54:20.480 --> 00:54:21.980 align:middle line:84% For example, this is component, say, 00:54:21.980 --> 00:54:25.790 align:middle line:84% component i has a specific energy, which 00:54:25.790 --> 00:54:28.850 align:middle line:84% is u double i evaluated at T and pii 00:54:28.850 --> 00:54:33.230 align:middle line:84% because here it is at the partial pressure. 00:54:33.230 --> 00:54:35.750 align:middle line:90% The same for the entropy. 00:54:35.750 --> 00:54:49.400 align:middle line:84% And now, if I use the fact that energy and entropy are 00:54:49.400 --> 00:55:00.553 align:middle line:84% additive properties, the energy of this one is n1 times u11. 00:55:00.553 --> 00:55:03.770 align:middle line:90% 00:55:03.770 --> 00:55:08.270 align:middle line:90% And so I sum up ni times uii. 00:55:08.270 --> 00:55:13.940 align:middle line:84% And that gives me the properties, 00:55:13.940 --> 00:55:15.300 align:middle line:90% the property of the mixture. 00:55:15.300 --> 00:55:19.400 align:middle line:90% So sum of ni times uii. 00:55:19.400 --> 00:55:26.570 align:middle line:84% So that's-- this last equation is the defining equation 00:55:26.570 --> 00:55:29.900 align:middle line:90% for ideal Gibbs-Dalton behavior. 00:55:29.900 --> 00:55:32.510 align:middle line:84% The first one here I have added also 00:55:32.510 --> 00:55:36.780 align:middle line:84% the general expression, which is valid always, 00:55:36.780 --> 00:55:43.880 align:middle line:84% which gives the property in terms of the partial properties. 00:55:43.880 --> 00:55:45.220 align:middle line:90% It's the same for the entropy. 00:55:45.220 --> 00:55:46.450 align:middle line:90% So it's additive. 00:55:46.450 --> 00:55:48.750 align:middle line:90% So I can view this. 00:55:48.750 --> 00:55:52.575 align:middle line:84% And instead-- and here, I've used the fact that each-- 00:55:52.575 --> 00:55:55.320 align:middle line:90% 00:55:55.320 --> 00:55:58.440 align:middle line:84% you see here, I could also write the specific volume 00:55:58.440 --> 00:56:03.780 align:middle line:84% of the constituent being at the partial pressure. 00:56:03.780 --> 00:56:06.430 align:middle line:90% And the fact that it-- 00:56:06.430 --> 00:56:09.130 align:middle line:84% that in the same amount of the mixture, it occupies, 00:56:09.130 --> 00:56:15.480 align:middle line:84% the same volume of the mixture is the sort of interpretation 00:56:15.480 --> 00:56:16.520 align:middle line:90% of this formula here. 00:56:16.520 --> 00:56:20.080 align:middle line:90% 00:56:20.080 --> 00:56:29.080 align:middle line:84% It also goes together with the idea 00:56:29.080 --> 00:56:33.130 align:middle line:84% that in the gas limit, when molecules live 00:56:33.130 --> 00:56:39.700 align:middle line:84% far apart from one another in either this or the Lennard-Jones 00:56:39.700 --> 00:56:43.783 align:middle line:84% kind of potential, if they're on average, they live here. 00:56:43.783 --> 00:56:50.080 align:middle line:90% 00:56:50.080 --> 00:56:56.320 align:middle line:84% Essentially, they stay in-- they occupy the volume of the mixture 00:56:56.320 --> 00:57:00.060 align:middle line:84% as if they were by themselves, each molecule by itself. 00:57:00.060 --> 00:57:04.360 align:middle line:90% 00:57:04.360 --> 00:57:08.350 align:middle line:90% And that's why we have-- 00:57:08.350 --> 00:57:10.990 align:middle line:84% they obey the ideal gas law, which 00:57:10.990 --> 00:57:16.380 align:middle line:84% is valid for the single molecule in a box 00:57:16.380 --> 00:57:18.430 align:middle line:90% even if I have many box-- 00:57:18.430 --> 00:57:20.260 align:middle line:90% many particles in that box. 00:57:20.260 --> 00:57:25.060 align:middle line:84% But it is as if each one uses the box by itself. 00:57:25.060 --> 00:57:27.970 align:middle line:84% It doesn't recognize on average, the presence of the other, 00:57:27.970 --> 00:57:33.210 align:middle line:84% even though it does continuously collide with the other. 00:57:33.210 --> 00:57:38.130 align:middle line:90% But the time that it takes-- 00:57:38.130 --> 00:57:43.050 align:middle line:84% that the collision-- and so that-- 00:57:43.050 --> 00:57:44.550 align:middle line:90% it lasts. 00:57:44.550 --> 00:57:46.680 align:middle line:84% The time that the collision lasts 00:57:46.680 --> 00:57:49.740 align:middle line:84% is very short with respect to the rest of the time 00:57:49.740 --> 00:57:53.940 align:middle line:84% that the particle moves freely in space 00:57:53.940 --> 00:57:58.600 align:middle line:84% before encountering the other molecule for the next collision. 00:57:58.600 --> 00:58:01.330 align:middle line:90% 00:58:01.330 --> 00:58:07.780 align:middle line:84% So that's the ideal Gibbs-Dalton behavior. 00:58:07.780 --> 00:58:10.510 align:middle line:84% And if we apply it now, we haven't 00:58:10.510 --> 00:58:11.970 align:middle line:90% assumed yet the ideal gas. 00:58:11.970 --> 00:58:14.670 align:middle line:90% 00:58:14.670 --> 00:58:16.990 align:middle line:84% So now, if we in addition to those, 00:58:16.990 --> 00:58:26.090 align:middle line:84% we also assume that each pure substance behaves 00:58:26.090 --> 00:58:29.630 align:middle line:84% as an ideal gas, so it obeys the equation 00:58:29.630 --> 00:58:34.640 align:middle line:84% of state of an ideal gas in the full range of pressures 00:58:34.640 --> 00:58:39.740 align:middle line:84% from the partial pressure to the total pressure of the mixture, 00:58:39.740 --> 00:58:42.670 align:middle line:84% I shouldn't call it total, to the pressure of the mixture. 00:58:42.670 --> 00:58:45.440 align:middle line:90% 00:58:45.440 --> 00:58:48.590 align:middle line:84% So I can write the equation of state 00:58:48.590 --> 00:58:54.650 align:middle line:84% for the gas pV equals RT both at partial pressure 00:58:54.650 --> 00:58:56.380 align:middle line:84% and at the pressure of the mixture. 00:58:56.380 --> 00:59:00.910 align:middle line:90% 00:59:00.910 --> 00:59:09.035 align:middle line:84% Then you see from assumption three, which is this, 00:59:09.035 --> 00:59:17.180 align:middle line:84% I substitute this, obtain this result. 00:59:17.180 --> 00:59:20.005 align:middle line:84% So I obtain the partial pressure written this way. 00:59:20.005 --> 00:59:23.440 align:middle line:90% 00:59:23.440 --> 00:59:29.890 align:middle line:84% Now, I'm using what we proved in the Gibbs-Dalton theorem, 00:59:29.890 --> 00:59:36.460 align:middle line:84% about the Dalton law that the pressure of the mixture 00:59:36.460 --> 00:59:38.720 align:middle line:84% is the sum of the partial pressures. 00:59:38.720 --> 00:59:42.480 align:middle line:84% So take the sum of these ones, see 00:59:42.480 --> 00:59:47.270 align:middle line:84% RT and V do not have subscripts, so they go out of the summation. 00:59:47.270 --> 00:59:49.390 align:middle line:84% I'm left with the sum of the n's, 00:59:49.390 --> 00:59:55.450 align:middle line:84% which is just the total n of the particles in the mixture. 00:59:55.450 --> 00:59:58.600 align:middle line:90% So here, what do I get? 00:59:58.600 --> 01:00:01.700 align:middle line:90% That pV equals nRT. 01:00:01.700 --> 01:00:09.640 align:middle line:84% So it means that also the mixture behaves as an ideal gas. 01:00:09.640 --> 01:00:14.630 align:middle line:84% So what this proves is that an ideal Gibbs-Dalton mixture 01:00:14.630 --> 01:00:19.490 align:middle line:84% of ideal gases obeys the equation 01:00:19.490 --> 01:00:21.380 align:middle line:90% of state of an ideal gas. 01:00:21.380 --> 01:00:23.065 align:middle line:90% So it is an ideal gas itself. 01:00:23.065 --> 01:00:25.910 align:middle line:90% 01:00:25.910 --> 01:00:37.400 align:middle line:84% And the other important result is that if you extract RT over V 01:00:37.400 --> 01:00:39.770 align:middle line:84% from here, you see that you can write it 01:00:39.770 --> 01:00:43.760 align:middle line:84% as partial pressure over the amount. 01:00:43.760 --> 01:00:47.810 align:middle line:84% And RT over V from here, it's the pressure 01:00:47.810 --> 01:00:49.970 align:middle line:90% over the total amount. 01:00:49.970 --> 01:00:56.900 align:middle line:84% So that's-- essentially you combine these two and you obtain 01:00:56.900 --> 01:01:01.100 align:middle line:84% that the ratio of the partial pressure to the total pressure 01:01:01.100 --> 01:01:06.410 align:middle line:84% is equal to the ratio of the individual amount with the total 01:01:06.410 --> 01:01:10.300 align:middle line:84% amount, which is the mole fraction. 01:01:10.300 --> 01:01:13.300 align:middle line:84% In other words, the partial pressure 01:01:13.300 --> 01:01:17.860 align:middle line:84% is given by the mole fraction of that constituent 01:01:17.860 --> 01:01:22.325 align:middle line:84% times the pressure of the mixture. 01:01:22.325 --> 01:01:26.720 align:middle line:90% 01:01:26.720 --> 01:01:37.160 align:middle line:84% And this relation, if we put it in this relation 01:01:37.160 --> 01:01:42.560 align:middle line:84% that we have derived before, so pii over p 01:01:42.560 --> 01:01:51.320 align:middle line:84% becomes in the limit of a Gibbs-Dalton mixture. 01:01:51.320 --> 01:01:54.450 align:middle line:90% That is the mole fraction. 01:01:54.450 --> 01:01:59.050 align:middle line:84% So this is RT the logarithm of the mole fraction. 01:01:59.050 --> 01:02:06.210 align:middle line:90% 01:02:06.210 --> 01:02:08.150 align:middle line:84% And this is a very important relation. 01:02:08.150 --> 01:02:12.230 align:middle line:90% 01:02:12.230 --> 01:02:15.185 align:middle line:84% It gives the chemical potential of one constituent 01:02:15.185 --> 01:02:16.640 align:middle line:90% in the mixture. 01:02:16.640 --> 01:02:21.690 align:middle line:84% In terms of the chemical potential that it has, 01:02:21.690 --> 01:02:24.920 align:middle line:84% when it is a pure substance, at the same temperature 01:02:24.920 --> 01:02:27.530 align:middle line:90% and pressure of the mixture. 01:02:27.530 --> 01:02:32.660 align:middle line:84% Plus RT, the log of its mole fraction, 01:02:32.660 --> 01:02:36.565 align:middle line:84% of the mole fraction with which it appears in the mixture. 01:02:36.565 --> 01:02:41.330 align:middle line:90% 01:02:41.330 --> 01:02:46.200 align:middle line:84% So it doesn't depend on other things of the mixture. 01:02:46.200 --> 01:02:54.070 align:middle line:84% For example, what are the other components, or how many they 01:02:54.070 --> 01:03:00.070 align:middle line:84% are except that all we need is to know what fraction 01:03:00.070 --> 01:03:03.650 align:middle line:84% our component occupies in the mixture. 01:03:03.650 --> 01:03:11.300 align:middle line:90% 01:03:11.300 --> 01:03:28.250 align:middle line:84% OK, the first few lines of this slide 01:03:28.250 --> 01:03:34.970 align:middle line:84% are just a cut and paste from the previous. 01:03:34.970 --> 01:03:38.930 align:middle line:84% But here, we add some other items. 01:03:38.930 --> 01:03:42.330 align:middle line:84% For example, we compute the entropy-- 01:03:42.330 --> 01:03:48.950 align:middle line:84% the partial entropy of the component in the mixture, 01:03:48.950 --> 01:03:53.000 align:middle line:84% as you remember, is this derivative. 01:03:53.000 --> 01:04:04.330 align:middle line:84% It is equal to the pure component entropy 01:04:04.330 --> 01:04:09.130 align:middle line:90% plus a term that in-- 01:04:09.130 --> 01:04:13.300 align:middle line:84% when we define the entropy of mixing at the beginning today, 01:04:13.300 --> 01:04:18.250 align:middle line:90% this is delta si mix. 01:04:18.250 --> 01:04:24.790 align:middle line:84% But if we-- since now, for this ideal gas mixture of ideal 01:04:24.790 --> 01:04:27.430 align:middle line:84% gases, we have the expression for the chemical potential 01:04:27.430 --> 01:04:29.470 align:middle line:90% in terms of-- 01:04:29.470 --> 01:04:34.060 align:middle line:84% explicitly here, taking the derivative with respect to T, 01:04:34.060 --> 01:04:39.640 align:middle line:84% we see leaves just R times the log of y. 01:04:39.640 --> 01:04:44.290 align:middle line:84% And since there is a minus sign here, this is minus R log of y. 01:04:44.290 --> 01:04:47.720 align:middle line:84% Essentially, this-- and we'll return to this later, 01:04:47.720 --> 01:04:56.890 align:middle line:84% but this says that the entropy of isothermobaric mixing 01:04:56.890 --> 01:05:03.740 align:middle line:84% of that molecule is equal to minus R log of y. 01:05:03.740 --> 01:05:12.610 align:middle line:90% 01:05:12.610 --> 01:05:24.320 align:middle line:84% And this is also equal to the pure component entropy evaluated 01:05:24.320 --> 01:05:27.230 align:middle line:90% at pii. 01:05:27.230 --> 01:05:45.360 align:middle line:84% This is because you remember for a gas, 01:05:45.360 --> 01:05:54.540 align:middle line:84% for an ideal gas, if I evaluate the difference in the entropy 01:05:54.540 --> 01:06:02.150 align:middle line:84% at different temperature and pressure, OK, this is equal to-- 01:06:02.150 --> 01:06:04.700 align:middle line:90% 01:06:04.700 --> 01:06:05.900 align:middle line:90% and this is the perfect-- 01:06:05.900 --> 01:06:08.540 align:middle line:84% let's suppose it's a perfect gas. 01:06:08.540 --> 01:06:16.790 align:middle line:84% It's T2 over T1 minus R log p2 over p1. 01:06:16.790 --> 01:06:26.180 align:middle line:84% So in this relation here, we have s evaluated 01:06:26.180 --> 01:06:48.920 align:middle line:90% at pii minus s evaluated at p. 01:06:48.920 --> 01:06:50.190 align:middle line:90% Same temperature. 01:06:50.190 --> 01:06:53.190 align:middle line:84% So this doesn't give any contribution. 01:06:53.190 --> 01:06:59.000 align:middle line:84% So I'm left with minus R log of pii over p. 01:06:59.000 --> 01:07:06.490 align:middle line:84% But that, in our limit of ideal mixture, is the mole fraction. 01:07:06.490 --> 01:07:09.320 align:middle line:90% 01:07:09.320 --> 01:07:14.000 align:middle line:84% So hopefully, what I've written on the board 01:07:14.000 --> 01:07:16.340 align:middle line:90% is this relation here. 01:07:16.340 --> 01:07:19.195 align:middle line:84% That's to justify this equal sign. 01:07:19.195 --> 01:07:23.090 align:middle line:90% 01:07:23.090 --> 01:07:28.400 align:middle line:84% We can also compute the partial volume, 01:07:28.400 --> 01:07:33.515 align:middle line:84% taking the derivative of this with respect to p 01:07:33.515 --> 01:07:36.920 align:middle line:84% and you see that since this last term doesn't have pressure, 01:07:36.920 --> 01:07:40.880 align:middle line:84% I just pick up something from the derivative 01:07:40.880 --> 01:07:42.920 align:middle line:84% of the chemical potential of the pure component. 01:07:42.920 --> 01:07:46.240 align:middle line:90% And that's the specific volume. 01:07:46.240 --> 01:07:49.440 align:middle line:90% 01:07:49.440 --> 01:07:53.400 align:middle line:84% So here, we see that there is no volume of mixing. 01:07:53.400 --> 01:07:55.290 align:middle line:90% There is no iso-- 01:07:55.290 --> 01:08:04.230 align:middle line:84% the delta-- essentially, we are proving that delta vi mix 01:08:04.230 --> 01:08:07.080 align:middle line:84% for a Gibbs-Dalton mixture of ideal gases 01:08:07.080 --> 01:08:10.810 align:middle line:90% and so on is equal to 0. 01:08:10.810 --> 01:08:16.180 align:middle line:84% Up here, we proved that this is equal to delta si 01:08:16.180 --> 01:08:20.400 align:middle line:84% mix for a Gibbs-Dalton mixture of ideal gases. 01:08:20.400 --> 01:08:23.390 align:middle line:90% 01:08:23.390 --> 01:08:25.606 align:middle line:84% I should write it because otherwise-- 01:08:25.606 --> 01:08:40.174 align:middle line:90% 01:08:40.174 --> 01:08:43.160 align:middle line:90% The same for the enthalpy. 01:08:43.160 --> 01:08:50.210 align:middle line:84% Remember that derivative that I emphasized this morning earlier, 01:08:50.210 --> 01:08:53.960 align:middle line:84% if I take the expression for the chemical potential divided by T 01:08:53.960 --> 01:08:59.790 align:middle line:84% and take the derivative with respect to 1 over t, 01:08:59.790 --> 01:09:02.910 align:middle line:84% OK, well, if I divide this by T, it 01:09:02.910 --> 01:09:04.899 align:middle line:84% becomes independent of temperature. 01:09:04.899 --> 01:09:06.899 align:middle line:84% So when I take the derivative with respect 01:09:06.899 --> 01:09:10.170 align:middle line:84% to any function of temperature, it doesn't contribute anything. 01:09:10.170 --> 01:09:17.910 align:middle line:84% So that means that the partial enthalpy is related just 01:09:17.910 --> 01:09:26.270 align:middle line:84% to the specific enthalpy of the gas, which here also, I 01:09:26.270 --> 01:09:30.740 align:middle line:84% removed the pressure dependence, because for an ideal gas, 01:09:30.740 --> 01:09:35.250 align:middle line:84% the enthalpy depends only on temperature. 01:09:35.250 --> 01:09:38.410 align:middle line:90% And similarly for the-- 01:09:38.410 --> 01:09:47.020 align:middle line:84% so that also proves that also delta h mix equals 0. 01:09:47.020 --> 01:09:53.100 align:middle line:90% 01:09:53.100 --> 01:09:56.290 align:middle line:84% And remember what these properties are. 01:09:56.290 --> 01:09:57.870 align:middle line:84% These are the things that I measure 01:09:57.870 --> 01:10:01.430 align:middle line:84% when I add a little bit of a substance to my mixture. 01:10:01.430 --> 01:10:08.795 align:middle line:84% If I do it with an ideal mixture of ideal gases, those are 0. 01:10:08.795 --> 01:10:13.810 align:middle line:90% 01:10:13.810 --> 01:10:21.640 align:middle line:84% We can also prove for a gas, for this kind of mixture, 01:10:21.640 --> 01:10:31.440 align:middle line:84% the so-called Amagat law of additive volumes, which 01:10:31.440 --> 01:10:34.830 align:middle line:90% can be interpreted this way. 01:10:34.830 --> 01:10:46.532 align:middle line:84% Suppose I take my pure components 01:10:46.532 --> 01:10:51.860 align:middle line:84% that you remember from when we interpreted the mixture before, 01:10:51.860 --> 01:10:55.420 align:middle line:84% if they are at the partial pressure, 01:10:55.420 --> 01:10:59.200 align:middle line:84% they occupy the entire volume of the mixture. 01:10:59.200 --> 01:11:05.950 align:middle line:84% But if I squeeze them from the partial pressure to the pressure 01:11:05.950 --> 01:11:09.880 align:middle line:84% of the mixture, they are going to occupy a volume, which 01:11:09.880 --> 01:11:17.290 align:middle line:84% is called the proper volume, V prime, which see, this is not-- 01:11:17.290 --> 01:11:20.630 align:middle line:84% So this is the volume they occupy 01:11:20.630 --> 01:11:23.540 align:middle line:84% at the same temperature and pressure of the mixture 01:11:23.540 --> 01:11:24.475 align:middle line:90% when pure. 01:11:24.475 --> 01:11:28.581 align:middle line:90% 01:11:28.581 --> 01:11:31.070 align:middle line:84% And I do the same for all of them. 01:11:31.070 --> 01:11:34.900 align:middle line:84% So they are all squeezed at the same pressure of the mixture. 01:11:34.900 --> 01:11:38.860 align:middle line:84% Well, it turns out that the sum of these volumes 01:11:38.860 --> 01:11:42.195 align:middle line:84% is equal to the volume of the mixture. 01:11:42.195 --> 01:11:49.051 align:middle line:90% 01:11:49.051 --> 01:11:52.785 align:middle line:84% And that is Amagat's law of additive volumes. 01:11:52.785 --> 01:12:05.680 align:middle line:90% 01:12:05.680 --> 01:12:11.990 align:middle line:84% OK, so now, let's apply for example, 01:12:11.990 --> 01:12:17.060 align:middle line:84% for an important example, what we have seen. 01:12:17.060 --> 01:12:21.790 align:middle line:84% So we want to compute the so-called-- 01:12:21.790 --> 01:12:24.490 align:middle line:84% the bottom line here is a formula 01:12:24.490 --> 01:12:31.580 align:middle line:84% that we call the entropy of spontaneous mixing, which 01:12:31.580 --> 01:12:36.380 align:middle line:84% is actually going to be an isothermobaric mixing. 01:12:36.380 --> 01:12:38.930 align:middle line:84% So the idea here is that we start 01:12:38.930 --> 01:12:44.720 align:middle line:84% with a container, which is isolated 01:12:44.720 --> 01:12:46.585 align:middle line:90% so it doesn't experience any-- 01:12:46.585 --> 01:12:49.250 align:middle line:90% 01:12:49.250 --> 01:12:54.500 align:middle line:84% there is no heat bath, no thermal reservoir, no contact 01:12:54.500 --> 01:12:56.010 align:middle line:90% with thermal reservoirs. 01:12:56.010 --> 01:12:58.280 align:middle line:90% It's just isolated. 01:12:58.280 --> 01:13:04.250 align:middle line:84% And I have my r constituents all in mutual equilibrium 01:13:04.250 --> 01:13:05.960 align:middle line:84% at the same temperature and pressure 01:13:05.960 --> 01:13:10.970 align:middle line:84% because these partitions that separate them are movable, 01:13:10.970 --> 01:13:15.770 align:middle line:84% and therefore allow exchanges of volume and energy. 01:13:15.770 --> 01:13:17.810 align:middle line:84% So the conditions for mutual equilibrium 01:13:17.810 --> 01:13:23.960 align:middle line:84% is equality of temperature and equality of pressure. 01:13:23.960 --> 01:13:32.420 align:middle line:84% Then, I make those partitions, those moving partitions, 01:13:32.420 --> 01:13:33.166 align:middle line:90% disappear. 01:13:33.166 --> 01:13:38.700 align:middle line:90% 01:13:38.700 --> 01:13:44.340 align:middle line:84% As you know, since we are within the simple-system model, 01:13:44.340 --> 01:13:53.100 align:middle line:84% taking the partition away doesn't change the properties 01:13:53.100 --> 01:13:56.850 align:middle line:90% of any of the properties. 01:13:56.850 --> 01:14:00.360 align:middle line:84% And the moment I do that removal, 01:14:00.360 --> 01:14:03.986 align:middle line:84% the particles are still separated. 01:14:03.986 --> 01:14:07.040 align:middle line:90% 01:14:07.040 --> 01:14:24.330 align:middle line:84% So the particles-- which start from here. 01:14:24.330 --> 01:14:27.166 align:middle line:84% And my system is the system with the partitions. 01:14:27.166 --> 01:14:30.560 align:middle line:90% 01:14:30.560 --> 01:14:36.720 align:middle line:84% When I remove the partitions, the stable equilibrium state, 01:14:36.720 --> 01:14:40.320 align:middle line:84% and the new system-- there is a new system because I've 01:14:40.320 --> 01:14:44.070 align:middle line:84% changed the internal structure, it 01:14:44.070 --> 01:14:50.060 align:middle line:84% has a new curve for the stable equilibrium states. 01:14:50.060 --> 01:14:57.590 align:middle line:90% 01:14:57.590 --> 01:15:01.670 align:middle line:84% And the moment I have removed the partitions 01:15:01.670 --> 01:15:06.200 align:middle line:84% and the properties haven't changed, this state here, 01:15:06.200 --> 01:15:07.976 align:middle line:90% which is-- 01:15:07.976 --> 01:15:10.920 align:middle line:84% which was a stable equilibrium state with the partitions, 01:15:10.920 --> 01:15:14.300 align:middle line:84% becomes a non-equilibrium state with respect to the system 01:15:14.300 --> 01:15:16.190 align:middle line:90% without partitions. 01:15:16.190 --> 01:15:22.070 align:middle line:84% And the system will tend to evolve spontaneously 01:15:22.070 --> 01:15:26.140 align:middle line:84% towards a new equilibrium, which is going to be the mixture. 01:15:26.140 --> 01:15:29.860 align:middle line:90% 01:15:29.860 --> 01:15:31.750 align:middle line:90% The particles will mix. 01:15:31.750 --> 01:15:36.610 align:middle line:84% And each one of them, if you think 01:15:36.610 --> 01:15:39.320 align:middle line:84% in terms of the single particle contribution, 01:15:39.320 --> 01:15:47.190 align:middle line:84% each particle here was squeezed to a small volume. 01:15:47.190 --> 01:15:50.860 align:middle line:84% And now, it sees the entire volume available. 01:15:50.860 --> 01:15:53.340 align:middle line:90% So it expands. 01:15:53.340 --> 01:15:55.780 align:middle line:84% And the same for the particles here, and the same for 01:15:55.780 --> 01:15:56.710 align:middle line:90% particles there. 01:15:56.710 --> 01:16:00.690 align:middle line:84% So each one experiences a spontaneous expansion. 01:16:00.690 --> 01:16:07.620 align:middle line:84% It's like if you take the single particle view, 01:16:07.620 --> 01:16:16.150 align:middle line:84% so say particle one, and here, I have vacuum, 01:16:16.150 --> 01:16:21.730 align:middle line:84% OK, I remove the partition and the particle 01:16:21.730 --> 01:16:25.410 align:middle line:84% now fills the entire volume and fills the entire volume. 01:16:25.410 --> 01:16:28.530 align:middle line:84% There is no more vacuum, because at the beginning, 01:16:28.530 --> 01:16:33.810 align:middle line:84% I had zero probability of finding my particles here, 01:16:33.810 --> 01:16:38.460 align:middle line:84% and probability 1 of finding it there. 01:16:38.460 --> 01:16:43.050 align:middle line:84% When I remove the partition, the particle is still one. 01:16:43.050 --> 01:16:45.540 align:middle line:84% But now, I have an even probability 01:16:45.540 --> 01:16:47.020 align:middle line:90% of finding it everywhere. 01:16:47.020 --> 01:16:50.670 align:middle line:90% 01:16:50.670 --> 01:17:00.360 align:middle line:84% OK, so for the moment, we want to describe 01:17:00.360 --> 01:17:04.110 align:middle line:90% that spontaneous process. 01:17:04.110 --> 01:17:08.590 align:middle line:84% And so we want to analyze our process here. 01:17:08.590 --> 01:17:11.290 align:middle line:84% We have a final state and an initial state. 01:17:11.290 --> 01:17:13.890 align:middle line:84% And the tools are, as usual, the energy balance 01:17:13.890 --> 01:17:15.310 align:middle line:90% and the entropy balance. 01:17:15.310 --> 01:17:17.437 align:middle line:84% Plus the auxiliary balances that are 01:17:17.437 --> 01:17:23.370 align:middle line:84% the balance for the volume, the balance for the amounts 01:17:23.370 --> 01:17:24.760 align:middle line:90% of the constituents. 01:17:24.760 --> 01:17:28.140 align:middle line:90% 01:17:28.140 --> 01:17:32.310 align:middle line:84% So I need to write the various ingredients of those balance 01:17:32.310 --> 01:17:36.580 align:middle line:84% equations, which are the expressions for the energy, 01:17:36.580 --> 01:17:40.590 align:middle line:84% the volume, and the entropy in the initial state, 01:17:40.590 --> 01:17:42.730 align:middle line:90% and in the final state. 01:17:42.730 --> 01:17:47.880 align:middle line:84% I've called this final state B prime. 01:17:47.880 --> 01:17:51.360 align:middle line:84% And I call that final temperature T prime and pressure 01:17:51.360 --> 01:17:51.860 align:middle line:90% p prime. 01:17:51.860 --> 01:17:55.670 align:middle line:90% 01:17:55.670 --> 01:17:58.430 align:middle line:84% For the initial state, things are 01:17:58.430 --> 01:18:03.480 align:middle line:84% easy because the components are separate. 01:18:03.480 --> 01:18:08.930 align:middle line:84% So these are-- this is just a composite 01:18:08.930 --> 01:18:13.670 align:middle line:90% of separate substances. 01:18:13.670 --> 01:18:16.850 align:middle line:84% We know how to compute properties of pure substances. 01:18:16.850 --> 01:18:23.690 align:middle line:84% And here in particular, they are ideal gases, even simpler. 01:18:23.690 --> 01:18:27.440 align:middle line:84% And so we simply have to use additivity 01:18:27.440 --> 01:18:35.050 align:middle line:84% for energy, for volume, and for entropy 01:18:35.050 --> 01:18:39.610 align:middle line:84% and sum up the energy of each one of them. 01:18:39.610 --> 01:18:44.590 align:middle line:84% Since u is the specific energy, the energy of this system 01:18:44.590 --> 01:18:48.190 align:middle line:84% is obtained by the specific energy times the amount. 01:18:48.190 --> 01:18:51.588 align:middle line:84% And then I sum, and that's the additivity. 01:18:51.588 --> 01:18:53.505 align:middle line:84% And the same for volume, the same for entropy. 01:18:53.505 --> 01:18:57.180 align:middle line:90% 01:18:57.180 --> 01:19:09.150 align:middle line:84% In the final state, in the final state, 01:19:09.150 --> 01:19:12.770 align:middle line:84% things are not as simple because it's a mixture. 01:19:12.770 --> 01:19:15.300 align:middle line:90% 01:19:15.300 --> 01:19:19.710 align:middle line:84% So we have to take a model for mixture behavior. 01:19:19.710 --> 01:19:23.850 align:middle line:84% So here, we are going to adopt the Gibbs-Dalton-- 01:19:23.850 --> 01:19:27.190 align:middle line:84% ideal Gibbs-Dalton mixture model for this state. 01:19:27.190 --> 01:19:31.970 align:middle line:90% 01:19:31.970 --> 01:19:35.330 align:middle line:84% Therefore, the energy for the mixture 01:19:35.330 --> 01:19:41.660 align:middle line:84% is equal to the sum weighted by the amounts 01:19:41.660 --> 01:19:51.050 align:middle line:84% of the specific properties of the pure components evaluated 01:19:51.050 --> 01:19:55.280 align:middle line:84% at the temperature and the partial pressure. 01:19:55.280 --> 01:20:00.410 align:middle line:84% So here in principle, I should have written comma pii. 01:20:00.410 --> 01:20:06.600 align:middle line:84% But I didn't because I remember that for an ideal gas, 01:20:06.600 --> 01:20:09.750 align:middle line:84% the energy is a function only of temperature. 01:20:09.750 --> 01:20:15.860 align:middle line:84% So that dependence isn't there because if you remember, 01:20:15.860 --> 01:20:20.270 align:middle line:90% the u is equal to cv dT. 01:20:20.270 --> 01:20:25.650 align:middle line:84% And for an ideal gas, there is nothing, no pressure dependence. 01:20:25.650 --> 01:20:28.580 align:middle line:84% And here, we are talking about something 01:20:28.580 --> 01:20:31.150 align:middle line:84% that holds for each of the pure substances. 01:20:31.150 --> 01:20:38.950 align:middle line:90% 01:20:38.950 --> 01:20:46.230 align:middle line:84% Then, we can use the relation for-- 01:20:46.230 --> 01:20:50.410 align:middle line:84% this is Amagat's relation, or you 01:20:50.410 --> 01:20:52.935 align:middle line:84% can use, in any case, the equation of state. 01:20:52.935 --> 01:20:57.290 align:middle line:90% 01:20:57.290 --> 01:21:01.730 align:middle line:84% And we already proved that V is equal to-- 01:21:01.730 --> 01:21:07.940 align:middle line:84% that the mixture of ideal gases under the Gibbs-Dalton model 01:21:07.940 --> 01:21:09.780 align:middle line:90% behaves itself as an ideal gas. 01:21:09.780 --> 01:21:15.050 align:middle line:84% So there is the relation, equation of state pV 01:21:15.050 --> 01:21:19.680 align:middle line:84% equals nRT for the entire mixture. 01:21:19.680 --> 01:21:22.220 align:middle line:90% And that's what we use here. 01:21:22.220 --> 01:21:26.780 align:middle line:84% And then for the entropy, the entropy 01:21:26.780 --> 01:21:31.130 align:middle line:84% is the sum of the entropies of the specific entropies evaluated 01:21:31.130 --> 01:21:34.460 align:middle line:90% at the partial pressures. 01:21:34.460 --> 01:21:40.410 align:middle line:84% Now, the entropy does depend on pressure. 01:21:40.410 --> 01:21:46.330 align:middle line:84% So I did leave the partial pressure 01:21:46.330 --> 01:21:52.210 align:middle line:84% that the components have in the final state. 01:21:52.210 --> 01:21:53.650 align:middle line:84% So these are just the ingredients 01:21:53.650 --> 01:21:54.820 align:middle line:90% of the balance equations. 01:21:54.820 --> 01:21:56.860 align:middle line:84% Now, I make the balanced equation. 01:21:56.860 --> 01:22:02.980 align:middle line:84% The balanced equation, when the system is completely isolated 01:22:02.980 --> 01:22:05.590 align:middle line:90% is that the energy is conserved. 01:22:05.590 --> 01:22:11.720 align:middle line:84% So U final minus U initial is equal to 0. 01:22:11.720 --> 01:22:17.630 align:middle line:84% So I have U final, I have U initial, I subtract them. 01:22:17.630 --> 01:22:22.800 align:middle line:84% I can put this under the same summation. 01:22:22.800 --> 01:22:25.970 align:middle line:84% Then, I remember this equation here. 01:22:25.970 --> 01:22:28.950 align:middle line:90% 01:22:28.950 --> 01:22:43.610 align:middle line:84% And so for each of the constituents, I can write that-- 01:22:43.610 --> 01:22:49.400 align:middle line:84% integrate from T to T prime, this difference, 01:22:49.400 --> 01:22:52.040 align:middle line:84% so this is the integral from T to T prime of cv. 01:22:52.040 --> 01:22:53.735 align:middle line:84% And here, there should be a cvii. 01:22:53.735 --> 01:22:59.100 align:middle line:90% 01:22:59.100 --> 01:23:00.900 align:middle line:90% OK, OK. 01:23:00.900 --> 01:23:05.020 align:middle line:84% So this integral has to be equal to 0. 01:23:05.020 --> 01:23:08.530 align:middle line:84% Now, when can that be equal to 0? 01:23:08.530 --> 01:23:11.020 align:middle line:84% Now, here, we have to remember a few things, 01:23:11.020 --> 01:23:19.860 align:middle line:84% that for each pure substance, the specific heats are 01:23:19.860 --> 01:23:23.430 align:middle line:84% non-negative, are greater than 0. 01:23:23.430 --> 01:23:26.910 align:middle line:84% We proved it because they are related to the second derivative 01:23:26.910 --> 01:23:28.330 align:middle line:90% of the fundamental relations. 01:23:28.330 --> 01:23:32.180 align:middle line:84% They are related to this concavity here. 01:23:32.180 --> 01:23:35.830 align:middle line:90% 01:23:35.830 --> 01:23:41.050 align:middle line:84% So we have an integral of positive quantities, 01:23:41.050 --> 01:23:44.710 align:middle line:84% of a positive function, weighted and then 01:23:44.710 --> 01:23:47.620 align:middle line:84% multiplied by positive numbers because the amounts can only 01:23:47.620 --> 01:23:48.710 align:middle line:90% be positive, of course. 01:23:48.710 --> 01:23:52.750 align:middle line:90% It cannot be a negative amount. 01:23:52.750 --> 01:23:55.270 align:middle line:84% So you can view this as an integral 01:23:55.270 --> 01:24:00.430 align:middle line:84% of a function, an integral from, say, 01:24:00.430 --> 01:24:09.220 align:middle line:84% x1 to x2 of a function that has to be equal to 0, 01:24:09.220 --> 01:24:15.660 align:middle line:90% while this function is positive. 01:24:15.660 --> 01:24:21.780 align:middle line:84% So here is a function, positive, and the integral 01:24:21.780 --> 01:24:27.570 align:middle line:84% is equal to this area from x1 to x2. 01:24:27.570 --> 01:24:31.240 align:middle line:84% So if the function is positive and I want the integral to be 0, 01:24:31.240 --> 01:24:35.790 align:middle line:84% the only way is that the interval of integration 01:24:35.790 --> 01:24:37.620 align:middle line:90% must be 0. 01:24:37.620 --> 01:24:42.630 align:middle line:84% So that implies that this integral is equal to 0 only 01:24:42.630 --> 01:24:46.170 align:middle line:90% when x2 is equal to x1. 01:24:46.170 --> 01:24:50.490 align:middle line:84% So that proves that the final temperature 01:24:50.490 --> 01:24:52.600 align:middle line:84% must be equal to the initial temperature. 01:24:52.600 --> 01:24:55.500 align:middle line:90% 01:24:55.500 --> 01:24:59.610 align:middle line:84% We've no need to make any assumptions of what 01:24:59.610 --> 01:25:02.070 align:middle line:84% these functions are, whether they 01:25:02.070 --> 01:25:05.950 align:middle line:84% are functions of whether they are constant or not. 01:25:05.950 --> 01:25:09.970 align:middle line:84% They can be functions, arbitrary functions of temperature, 01:25:09.970 --> 01:25:17.200 align:middle line:90% still get the same result. 01:25:17.200 --> 01:25:24.220 align:middle line:84% Then, if I make the volume balance equation, here 01:25:24.220 --> 01:25:30.820 align:middle line:84% is another mistake, this is a V, not a U. 01:25:30.820 --> 01:25:35.900 align:middle line:84% So the final volume, since we are in the same overall box, 01:25:35.900 --> 01:25:38.180 align:middle line:84% final volume is equal to initial volume. 01:25:38.180 --> 01:25:40.810 align:middle line:90% So V-- these two are equal. 01:25:40.810 --> 01:25:43.960 align:middle line:84% If I look at this relation here, it 01:25:43.960 --> 01:25:48.430 align:middle line:84% says that if I put them equal, it says that T over p 01:25:48.430 --> 01:25:51.610 align:middle line:84% is equal to T prime over p prime. 01:25:51.610 --> 01:25:53.620 align:middle line:84% So that means that if we just proved 01:25:53.620 --> 01:25:56.200 align:middle line:84% that T prime is equal to T, so the final temperature is 01:25:56.200 --> 01:25:59.050 align:middle line:84% equal to the initial, the volume balance 01:25:59.050 --> 01:26:02.020 align:middle line:84% implies that also the final pressure is 01:26:02.020 --> 01:26:02.910 align:middle line:90% equal to the initial. 01:26:02.910 --> 01:26:08.250 align:middle line:90% 01:26:08.250 --> 01:26:14.180 align:middle line:84% So spontaneous mixing with these conditions is isothermobaric. 01:26:14.180 --> 01:26:19.910 align:middle line:90% 01:26:19.910 --> 01:26:22.435 align:middle line:84% Different story is for the entropy balance. 01:26:22.435 --> 01:26:27.510 align:middle line:90% 01:26:27.510 --> 01:26:33.000 align:middle line:84% The entropy balance is written here with another little mistake 01:26:33.000 --> 01:26:37.170 align:middle line:84% because this should be a capital S. So the final entropy 01:26:37.170 --> 01:26:39.023 align:middle line:90% minus the initial entropy. 01:26:39.023 --> 01:26:42.630 align:middle line:90% 01:26:42.630 --> 01:26:44.390 align:middle line:90% There are no interactions. 01:26:44.390 --> 01:26:47.250 align:middle line:90% So there is no entropy transfer. 01:26:47.250 --> 01:26:49.860 align:middle line:90% There are no heat interactions. 01:26:49.860 --> 01:26:54.370 align:middle line:84% But there can be entropy produced by irreversibility. 01:26:54.370 --> 01:27:01.160 align:middle line:90% 01:27:01.160 --> 01:27:03.230 align:middle line:90% So we rewrite this. 01:27:03.230 --> 01:27:06.995 align:middle line:84% So S here is equal to SB minus SA. 01:27:06.995 --> 01:27:10.900 align:middle line:90% And I have the terms here. 01:27:10.900 --> 01:27:14.290 align:middle line:90% So I substitute. 01:27:14.290 --> 01:27:20.290 align:middle line:84% Now, I realize that this we can also write-- 01:27:20.290 --> 01:27:23.145 align:middle line:84% well, actually, it's this expression here. 01:27:23.145 --> 01:27:26.320 align:middle line:90% 01:27:26.320 --> 01:27:28.580 align:middle line:90% I don't have to re-prove it. 01:27:28.580 --> 01:27:30.550 align:middle line:90% It's written here. 01:27:30.550 --> 01:27:33.280 align:middle line:84% This difference is equal to minus R log 01:27:33.280 --> 01:27:36.580 align:middle line:84% y, which is also-- is interpreted 01:27:36.580 --> 01:27:41.290 align:middle line:84% as the entropy of isothermobaric mixing. 01:27:41.290 --> 01:27:48.190 align:middle line:84% And therefore, the-- if I take R, the minus sign, outside 01:27:48.190 --> 01:27:51.070 align:middle line:84% of the summation, I take also the total amount 01:27:51.070 --> 01:27:52.630 align:middle line:90% out of the summation. 01:27:52.630 --> 01:27:57.880 align:middle line:84% Then, I'm left inside with the mole fraction times the log 01:27:57.880 --> 01:27:58.755 align:middle line:90% of the mole fraction. 01:27:58.755 --> 01:28:03.040 align:middle line:90% 01:28:03.040 --> 01:28:11.260 align:middle line:84% Now, at first sight, one would be worried about the minus sign 01:28:11.260 --> 01:28:19.090 align:middle line:84% because we know that the entropy produced by irreversibility 01:28:19.090 --> 01:28:21.570 align:middle line:84% cannot be negative, must be positive. 01:28:21.570 --> 01:28:25.240 align:middle line:84% But if you look at this function y log 01:28:25.240 --> 01:28:33.880 align:middle line:84% y, the function in the interval between 0 and 1, 01:28:33.880 --> 01:28:37.840 align:middle line:84% the function y log y does something like that. 01:28:37.840 --> 01:28:43.186 align:middle line:90% 01:28:43.186 --> 01:28:45.620 align:middle line:90% Versus y. 01:28:45.620 --> 01:28:47.690 align:middle line:90% So it's negative. 01:28:47.690 --> 01:28:52.250 align:middle line:90% So minus y log y is positive. 01:28:52.250 --> 01:28:53.380 align:middle line:90% And we are safe. 01:28:53.380 --> 01:29:02.210 align:middle line:84% That means for any reasonable, in other words, possible values 01:29:02.210 --> 01:29:05.870 align:middle line:84% of the mole fractions, which of course can go only 01:29:05.870 --> 01:29:12.210 align:middle line:84% between 0 and 1, that function is non-negative. 01:29:12.210 --> 01:29:17.710 align:middle line:84% But it also says that there is entropy production. 01:29:17.710 --> 01:29:20.730 align:middle line:84% So the process is irreversible as we 01:29:20.730 --> 01:29:25.140 align:middle line:84% have anticipated by looking at the energy 01:29:25.140 --> 01:29:26.783 align:middle line:90% versus entropy graphs. 01:29:26.783 --> 01:29:34.790 align:middle line:90% 01:29:34.790 --> 01:29:36.985 align:middle line:84% And this is called the entropy of mixing. 01:29:36.985 --> 01:29:44.220 align:middle line:90% 01:29:44.220 --> 01:29:51.610 align:middle line:84% Now, so if I look at that graph, immediately, I also 01:29:51.610 --> 01:29:58.480 align:middle line:84% realize that I could have done better. 01:29:58.480 --> 01:30:03.300 align:middle line:84% In other words, we could have done the mixing reversibly 01:30:03.300 --> 01:30:15.300 align:middle line:84% because if instead of going this way, I go this way, 01:30:15.300 --> 01:30:19.270 align:middle line:90% I do not generate entropy. 01:30:19.270 --> 01:30:20.870 align:middle line:90% And I extract some work. 01:30:20.870 --> 01:30:23.390 align:middle line:90% 01:30:23.390 --> 01:30:26.470 align:middle line:84% So this is what we call the adiabatic availability 01:30:26.470 --> 01:30:29.470 align:middle line:84% of the state that is achieved the moment I 01:30:29.470 --> 01:30:30.760 align:middle line:90% remove the partitions. 01:30:30.760 --> 01:30:33.730 align:middle line:84% That non-equilibrium state has adiabatic availability 01:30:33.730 --> 01:30:36.220 align:middle line:84% if I am quick enough, or smart enough, 01:30:36.220 --> 01:30:39.830 align:middle line:90% to do whatever needs to be done. 01:30:39.830 --> 01:30:44.590 align:middle line:84% And we'll see how we could do it in a moment. 01:30:44.590 --> 01:30:46.065 align:middle line:90% We could extract some work. 01:30:46.065 --> 01:30:52.400 align:middle line:90% 01:30:52.400 --> 01:30:58.170 align:middle line:84% So here is the calculation of that adiabatic availability. 01:30:58.170 --> 01:31:04.010 align:middle line:90% 01:31:04.010 --> 01:31:13.460 align:middle line:84% Now, the final state, OK, what we just derived 01:31:13.460 --> 01:31:17.630 align:middle line:84% is that if we mix this way, the final state 01:31:17.630 --> 01:31:22.760 align:middle line:84% has the same temperature and pressure. 01:31:22.760 --> 01:31:27.740 align:middle line:84% Clearly, if we go down and we have extracted some energy, 01:31:27.740 --> 01:31:33.530 align:middle line:84% the final state here will have a smaller temperature, which 01:31:33.530 --> 01:31:34.945 align:middle line:90% we call it T double prime. 01:31:34.945 --> 01:31:42.000 align:middle line:90% 01:31:42.000 --> 01:31:43.360 align:middle line:90% And how do we measure it? 01:31:43.360 --> 01:31:46.830 align:middle line:90% How do we do the computation? 01:31:46.830 --> 01:31:50.650 align:middle line:84% Well, we start-- the previous computation, 01:31:50.650 --> 01:31:52.290 align:middle line:84% we started from the energy balance 01:31:52.290 --> 01:31:55.890 align:middle line:84% setting that final energy is equal to initial energy. 01:31:55.890 --> 01:31:58.080 align:middle line:90% That determines the temperature. 01:31:58.080 --> 01:32:01.080 align:middle line:84% Now, we start from the entropy balance 01:32:01.080 --> 01:32:03.720 align:middle line:84% because for a reversible process because we 01:32:03.720 --> 01:32:05.980 align:middle line:84% want to extract the adiabatic availability, 01:32:05.980 --> 01:32:08.140 align:middle line:84% so that's a reversible weight process. 01:32:08.140 --> 01:32:10.800 align:middle line:90% The entropy stays constant. 01:32:10.800 --> 01:32:14.380 align:middle line:84% We apply equality of final and initial entropy. 01:32:14.380 --> 01:32:16.295 align:middle line:84% And so we start from the entropy balance. 01:32:16.295 --> 01:32:31.170 align:middle line:90% 01:32:31.170 --> 01:32:33.820 align:middle line:84% Yeah, here is the entropy balance. 01:32:33.820 --> 01:32:36.815 align:middle line:84% S final minus S initial equals 0. 01:32:36.815 --> 01:32:39.420 align:middle line:90% 01:32:39.420 --> 01:32:44.300 align:middle line:84% So we write it this way with the double primes. 01:32:44.300 --> 01:32:47.670 align:middle line:90% 01:32:47.670 --> 01:32:55.725 align:middle line:84% And then this difference here, we can-- 01:32:55.725 --> 01:32:59.980 align:middle line:90% 01:32:59.980 --> 01:33:08.920 align:middle line:84% yeah, it's evaluated in this line because, OK, T double prime 01:33:08.920 --> 01:33:12.610 align:middle line:84% is the final temperature that we have. 01:33:12.610 --> 01:33:16.840 align:middle line:84% p double prime double i is the partial pressure 01:33:16.840 --> 01:33:20.890 align:middle line:84% that the components will have in that new final state. 01:33:20.890 --> 01:33:23.650 align:middle line:84% So that partial pressure is going 01:33:23.650 --> 01:33:28.090 align:middle line:84% to be equal to the mole fraction. 01:33:28.090 --> 01:33:31.660 align:middle line:84% So that partial pressure will be equal to the mole fraction 01:33:31.660 --> 01:33:35.670 align:middle line:84% times the final pressure p double prime. 01:33:35.670 --> 01:33:38.180 align:middle line:90% 01:33:38.180 --> 01:33:49.930 align:middle line:84% But if we remember, if we remember the question of-- 01:33:49.930 --> 01:33:56.380 align:middle line:84% if we look at the volume balance, which is written here, 01:33:56.380 --> 01:34:02.035 align:middle line:84% p double prime over T double prime is equal to p over T. 01:34:02.035 --> 01:34:10.490 align:middle line:84% So I can think of p double prime, p double prime, 01:34:10.490 --> 01:34:13.850 align:middle line:90% as what is it? 01:34:13.850 --> 01:34:17.500 align:middle line:90% 01:34:17.500 --> 01:34:27.416 align:middle line:84% As p times T double prime over T. 01:34:27.416 --> 01:34:35.710 align:middle line:84% And that's why I am writing this here. 01:34:35.710 --> 01:34:44.660 align:middle line:84% Then I apply the relation for how the entropy depends 01:34:44.660 --> 01:34:47.770 align:middle line:90% on temperature and pressure. 01:34:47.770 --> 01:34:50.360 align:middle line:84% Here, we also have a temperature dependence. 01:34:50.360 --> 01:34:53.470 align:middle line:84% So we also have the cp log contribution 01:34:53.470 --> 01:34:57.760 align:middle line:84% of the ratio of temperatures minus R, the log 01:34:57.760 --> 01:34:59.960 align:middle line:90% of the ratio of the pressures. 01:34:59.960 --> 01:35:02.600 align:middle line:90% 01:35:02.600 --> 01:35:07.160 align:middle line:84% OK, then I can split this logarithm 01:35:07.160 --> 01:35:08.885 align:middle line:84% as being the logarithm of the product, 01:35:08.885 --> 01:35:11.790 align:middle line:84% so it's going to be the sum of the logarithms. 01:35:11.790 --> 01:35:15.570 align:middle line:84% So I take the minus R log y here. 01:35:15.570 --> 01:35:18.740 align:middle line:84% And then I combine the log of T double prime 01:35:18.740 --> 01:35:22.940 align:middle line:84% that appears from this part and from that part. 01:35:22.940 --> 01:35:26.370 align:middle line:84% It gets multiplied by cv minus R, 01:35:26.370 --> 01:35:28.675 align:middle line:84% but by the Mayer relation that's cv. 01:35:28.675 --> 01:35:34.680 align:middle line:90% 01:35:34.680 --> 01:35:37.870 align:middle line:84% OK, so we finally have this expression. 01:35:37.870 --> 01:35:39.995 align:middle line:84% We can put it in the entropy balance. 01:35:39.995 --> 01:35:46.760 align:middle line:90% 01:35:46.760 --> 01:35:57.560 align:middle line:84% Now, I see here that I could define an average-specific heat 01:35:57.560 --> 01:36:03.560 align:middle line:84% at constant volume of the mixture as being-- 01:36:03.560 --> 01:36:08.815 align:middle line:84% the weighted as being this summation here. 01:36:08.815 --> 01:36:12.462 align:middle line:90% 01:36:12.462 --> 01:36:18.990 align:middle line:84% And then I recognize this final term, which is very close. 01:36:18.990 --> 01:36:23.790 align:middle line:84% I mean, it's exactly what we called the entropy of mixing. 01:36:23.790 --> 01:36:25.980 align:middle line:90% But in any case, by setting-- 01:36:25.980 --> 01:36:30.090 align:middle line:84% since this has to be equal to 0 by the entropy balance, 01:36:30.090 --> 01:36:33.090 align:middle line:84% this gives me an expression-- a way 01:36:33.090 --> 01:36:38.660 align:middle line:84% to calculate T double prime, the final temperature. 01:36:38.660 --> 01:36:46.940 align:middle line:84% And this is the result. Now, the fact that these functions are 01:36:46.940 --> 01:36:54.000 align:middle line:84% negative, so e to a negative number is less than 1, 01:36:54.000 --> 01:36:57.600 align:middle line:84% shows that, of course, as we expected, 01:36:57.600 --> 01:36:59.550 align:middle line:84% the final temperature will be less 01:36:59.550 --> 01:37:01.210 align:middle line:90% than the initial temperature. 01:37:01.210 --> 01:37:03.910 align:middle line:90% 01:37:03.910 --> 01:37:07.190 align:middle line:84% And now that we have the final temperature, 01:37:07.190 --> 01:37:10.100 align:middle line:84% we can perform the energy balance. 01:37:10.100 --> 01:37:13.600 align:middle line:84% Now, the energy balance here, since the system is not isolated 01:37:13.600 --> 01:37:17.980 align:middle line:84% anymore, because we have lifted the weight in a weight process, 01:37:17.980 --> 01:37:21.280 align:middle line:84% we have extracted the adiabatic availability. 01:37:21.280 --> 01:37:25.450 align:middle line:84% So the energy given to the weight 01:37:25.450 --> 01:37:27.580 align:middle line:84% is equal to the difference between the initial 01:37:27.580 --> 01:37:30.045 align:middle line:90% and the final energy. 01:37:30.045 --> 01:37:33.110 align:middle line:90% 01:37:33.110 --> 01:37:36.650 align:middle line:84% So by substituting these ingredients that we calculated 01:37:36.650 --> 01:37:43.490 align:middle line:84% up there and using the same-- the specific heat 01:37:43.490 --> 01:37:46.280 align:middle line:84% at constant volume, and then the fact that the summation 01:37:46.280 --> 01:37:56.180 align:middle line:84% we defined as sort of an average mixing mixture, specific heat 01:37:56.180 --> 01:37:57.055 align:middle line:90% at constant volume. 01:37:57.055 --> 01:37:59.640 align:middle line:90% 01:37:59.640 --> 01:38:01.500 align:middle line:84% And we substitute T double prime. 01:38:01.500 --> 01:38:05.100 align:middle line:84% And you compute the value for the adiabatic availability. 01:38:05.100 --> 01:38:11.150 align:middle line:84% So we see that this represents this work. 01:38:11.150 --> 01:38:16.230 align:middle line:90% 01:38:16.230 --> 01:38:21.910 align:middle line:84% Just to show you how we could actually do it with a picture, 01:38:21.910 --> 01:38:23.590 align:middle line:84% so we conclude, we have one minute. 01:38:23.590 --> 01:38:26.340 align:middle line:90% 01:38:26.340 --> 01:38:27.790 align:middle line:90% How can we do that? 01:38:27.790 --> 01:38:30.200 align:middle line:84% Can we actually separate a mixture? 01:38:30.200 --> 01:38:35.080 align:middle line:90% 01:38:35.080 --> 01:38:42.910 align:middle line:84% Remember, the problem here with the single particle, 01:38:42.910 --> 01:38:47.840 align:middle line:84% if I have a single particle there, and I have vacuum, 01:38:47.840 --> 01:38:51.100 align:middle line:84% and I have this-- and instead of removing the piston, 01:38:51.100 --> 01:39:03.250 align:middle line:84% I can just have it move rightward while connected 01:39:03.250 --> 01:39:05.740 align:middle line:90% through a pulley to a weight. 01:39:05.740 --> 01:39:10.080 align:middle line:84% And the pressure of the particle will push it to the right. 01:39:10.080 --> 01:39:13.350 align:middle line:84% Therefore, the particle will do work and lift. 01:39:13.350 --> 01:39:16.550 align:middle line:90% 01:39:16.550 --> 01:39:19.850 align:middle line:84% And similarly, if I have this other particle 01:39:19.850 --> 01:39:23.540 align:middle line:84% on the other different particle on the other side, 01:39:23.540 --> 01:39:28.640 align:middle line:84% say, suppose I have a particle of different kind 01:39:28.640 --> 01:39:31.500 align:middle line:84% that is confined, that's separated, 01:39:31.500 --> 01:39:33.590 align:middle line:90% and it's on the other side. 01:39:33.590 --> 01:39:38.240 align:middle line:84% Also here, I can do the same and have 01:39:38.240 --> 01:39:41.840 align:middle line:84% the pressure of the particle, or the particles. 01:39:41.840 --> 01:39:46.160 align:middle line:84% Doesn't really matter because each one acts by itself. 01:39:46.160 --> 01:39:50.120 align:middle line:90% So that pulls this this way. 01:39:50.120 --> 01:39:53.880 align:middle line:84% And when you have the presence of both particles, 01:39:53.880 --> 01:39:55.290 align:middle line:84% you have to do it more carefully. 01:39:55.290 --> 01:40:00.120 align:middle line:84% Instead of having an impermeable piston, 01:40:00.120 --> 01:40:05.270 align:middle line:84% you have to use semi-permeable membranes so that when this one 01:40:05.270 --> 01:40:09.145 align:middle line:90% moves to the right, it will-- 01:40:09.145 --> 01:40:12.230 align:middle line:90% 01:40:12.230 --> 01:40:17.870 align:middle line:84% so this one allows the black particles 01:40:17.870 --> 01:40:21.290 align:middle line:90% to freely move in and out. 01:40:21.290 --> 01:40:26.480 align:middle line:84% And then it feels only the pressure of the gray particles 01:40:26.480 --> 01:40:28.260 align:middle line:90% until you move to the right. 01:40:28.260 --> 01:40:32.450 align:middle line:84% And this other membrane, instead, is-- 01:40:32.450 --> 01:40:36.200 align:middle line:84% allows the gray particles through, and instead, 01:40:36.200 --> 01:40:39.110 align:middle line:84% feels the pressure of the black particles. 01:40:39.110 --> 01:40:44.430 align:middle line:84% OK, so when you're done, and if you've done it reversibly, 01:40:44.430 --> 01:40:49.680 align:middle line:84% you have extracted this adiabatic availability. 01:40:49.680 --> 01:40:53.020 align:middle line:84% So I think we have to stop here today. 01:40:53.020 --> 01:40:58.000 align:middle line:90%