WEBVTT 00:00:00.000 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.840 Your support will help MIT OpenCourseWare continue to 00:00:06.840 --> 00:00:10.520 offer high quality educational resources for free. 00:00:10.520 --> 00:00:13.390 To make a donation or view additional materials from 00:00:13.390 --> 00:00:17.580 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.580 --> 00:00:20.350 ocw.mit.edu. 00:00:20.350 --> 00:00:27.250 PROFESSOR: I want to remind and clarify as needed the 00:00:27.250 --> 00:00:36.420 equilibrium constant Kp for gas phase reaction was the 00:00:36.420 --> 00:00:43.060 ratios of the partial pressures referenced to some 00:00:43.060 --> 00:00:47.110 reference pressure, which we usually take as one, one bar. 00:00:47.110 --> 00:00:54.952 To the stoichiometry , pD divided by p naught, to the mu 00:00:54.952 --> 00:00:57.910 D where species C and D are products. 00:00:57.910 --> 00:01:11.540 And the reactants are on the bottom. 00:01:11.540 --> 00:01:19.350 And usually we don't write p naught, but it's important to 00:01:19.350 --> 00:01:22.380 remember that it's there. 00:01:22.380 --> 00:01:26.360 And then we can also write this in terms of the Gibbs 00:01:26.360 --> 00:01:29.170 free energy for the reaction. 00:01:29.170 --> 00:01:32.920 The standard Gibbs free energy, minus delta G naught 00:01:32.920 --> 00:01:39.570 of the reaction, divided by RT. 00:01:39.570 --> 00:01:43.860 And what this tells us is that this is a number. 00:01:43.860 --> 00:01:45.930 This is a number, there's no pre-factor 00:01:45.930 --> 00:01:48.240 here that has units. 00:01:48.240 --> 00:01:52.640 That's a unitless number. 00:01:52.640 --> 00:02:04.410 And it doesn't depend on the total pressure. 00:02:04.410 --> 00:02:05.905 And the other thing to remember is that delta G 00:02:05.905 --> 00:02:10.080 naught for the reaction is the process of taking the 00:02:10.080 --> 00:02:12.760 reactants separated in separate boxes, separate 00:02:12.760 --> 00:02:19.090 containers, and the final product, the final step, is 00:02:19.090 --> 00:02:22.350 the products separated in individual containers. 00:02:22.350 --> 00:02:24.435 That's what we write when we write delta G 00:02:24.435 --> 00:02:30.190 naught for the reaction. 00:02:30.190 --> 00:02:35.340 We also looked at K in terms of mole fractions. 00:02:35.340 --> 00:02:38.020 So if you replace all the partial pressures with the 00:02:38.020 --> 00:02:40.360 mole fraction times the total pressure, you get an 00:02:40.360 --> 00:02:47.560 expression for K sub x, which we define as the mole fraction 00:02:47.560 --> 00:02:59.140 of the products to the stoichiometric powers. 00:02:59.140 --> 00:03:02.420 Which is also unitless. 00:03:02.420 --> 00:03:04.880 This is true, it's unitless. 00:03:04.880 --> 00:03:08.940 But, if you write it in terms of K sub p, the total pressure 00:03:08.940 --> 00:03:11.880 comes in here. p total, divided by the reference 00:03:11.880 --> 00:03:17.770 pressure to the minus delta nu, where this is the change 00:03:17.770 --> 00:03:21.010 in the number of moles, in going from 00:03:21.010 --> 00:03:23.330 reactants to products. 00:03:23.330 --> 00:03:25.810 And there's K sub p sitting here. 00:03:25.810 --> 00:03:28.620 So unlike K sub p, which doesn't depend on the total 00:03:28.620 --> 00:03:33.040 pressure, K sub x does depend on the total pressure through 00:03:33.040 --> 00:03:47.340 this term right here. 00:03:47.340 --> 00:03:49.970 So when we look at problems where we change the pressure, 00:03:49.970 --> 00:03:53.300 the total pressure of the system, this is 00:03:53.300 --> 00:03:54.400 going to stay the same. 00:03:54.400 --> 00:03:56.130 Because it only cares about delta G 00:03:56.130 --> 00:03:57.880 naught for the reaction. 00:03:57.880 --> 00:04:01.586 But this K sub x will depend on the total pressure. 00:04:01.586 --> 00:04:10.330 And that's often a source of confusion in doing problems. 00:04:10.330 --> 00:04:11.700 OK, any questions? 00:04:11.700 --> 00:04:13.060 We're going to do an example where we change 00:04:13.060 --> 00:04:16.600 the pressure first. 00:04:16.600 --> 00:04:19.340 So there are examples in the notes, and I'm going to skip 00:04:19.340 --> 00:04:20.470 the first one. 00:04:20.470 --> 00:04:22.930 I'm going to go to the second one, which is the effect of 00:04:22.930 --> 00:04:24.840 the total pressure on the reaction. 00:04:24.840 --> 00:04:31.840 And Le Chatelier's principle, for pressure. 00:04:31.840 --> 00:04:34.720 And the example we're going to take is a fairly standard 00:04:34.720 --> 00:04:37.040 example, also. 00:04:37.040 --> 00:04:46.450 Which is the reaction of N2O4, which is a gas, to 2 NO2, 00:04:46.450 --> 00:04:49.670 which is a gas, the kind of reaction that happens when you 00:04:49.670 --> 00:04:59.080 have smog and, fairly common in big cities. 00:04:59.080 --> 00:05:05.130 The question we're going to ask is, what happens when we 00:05:05.130 --> 00:05:08.690 change the total pressure in this reaction here. 00:05:08.690 --> 00:05:11.600 Which way does the equilibrium go? 00:05:11.600 --> 00:05:13.700 Does it go to the right, does it go to the left? 00:05:13.700 --> 00:05:15.650 Does it go to the products or the reactants. 00:05:15.650 --> 00:05:19.270 And so, in order to answer that question, we're going to 00:05:19.270 --> 00:05:20.970 ask a slightly different question. 00:05:20.970 --> 00:05:24.280 We're going to ask what is the molar ratio, what is the 00:05:24.280 --> 00:05:31.270 fraction, what is fraction of the reactant, 00:05:31.270 --> 00:05:39.920 the N2O4 that's reacted. 00:05:39.920 --> 00:05:43.120 That has reacted. 00:05:43.120 --> 00:05:47.800 And we're going to call that alpha. 00:05:47.800 --> 00:05:56.470 It's the number of moles that have reacted divided by number 00:05:56.470 --> 00:06:03.350 of moles initially. 00:06:03.350 --> 00:06:08.110 So we're going to need to find at equilibrium what is the 00:06:08.110 --> 00:06:13.950 number of moles of N2O4 that has reacted. 00:06:13.950 --> 00:06:16.640 So we have to set up the problem. 00:06:16.640 --> 00:06:21.180 And so the way that, the standard way of setting up the 00:06:21.180 --> 00:06:30.300 problem is to write the equilibrium, 2 NO2 gas. 00:06:30.300 --> 00:06:34.650 And then on this line here, we write the initial conditions 00:06:34.650 --> 00:06:36.490 before we set up the equilibrium. 00:06:36.490 --> 00:06:38.960 And let's say that we have n moles of N2O4 00:06:38.960 --> 00:06:39.880 initially in the box. 00:06:39.880 --> 00:06:43.550 And zero moles of the NO2. 00:06:43.550 --> 00:06:44.970 So we have n moles here. 00:06:44.970 --> 00:06:47.320 And zero moles here. 00:06:47.320 --> 00:06:50.170 At equilibrium, let's write the number of moles. 00:06:50.170 --> 00:06:52.640 A certain number of moles of N2O4 will have reacted, let's 00:06:52.640 --> 00:06:53.880 call that x. 00:06:53.880 --> 00:06:55.940 So n minus x moles left. 00:06:55.940 --> 00:06:59.970 For every x moles of N2O4 that's reacted, we create two 00:06:59.970 --> 00:07:01.820 moles of NO2. 00:07:01.820 --> 00:07:04.270 So we have 2x here. 00:07:04.270 --> 00:07:06.060 And then we're going to need the total number of moles, 00:07:06.060 --> 00:07:07.060 because we're going to be doing mole 00:07:07.060 --> 00:07:09.460 ratios, mole fractions. 00:07:09.460 --> 00:07:15.020 So the total number of moles at any time is the sum of 00:07:15.020 --> 00:07:17.130 these two, n minus x plus 2x. 00:07:17.130 --> 00:07:18.900 It's n plus x. 00:07:18.900 --> 00:07:21.020 So if we're going to be writing our equilibrium 00:07:21.020 --> 00:07:23.780 constant in terms of mole fractions, we're going to need 00:07:23.780 --> 00:07:27.180 mole fractions. 00:07:27.180 --> 00:07:31.010 So the mole fraction at any time is n minus x divided by 00:07:31.010 --> 00:07:33.290 the total number of moles, which we just 00:07:33.290 --> 00:07:36.110 calculated as n plus x. 00:07:36.110 --> 00:07:40.210 And this is 2x divided by the total number of 00:07:40.210 --> 00:07:43.610 moles, n plus x. 00:07:43.610 --> 00:07:49.640 And what we want is this ratio here. 00:07:49.640 --> 00:07:55.070 We want the ratio of the number of moles reacted, which 00:07:55.070 --> 00:08:01.010 is x, that's the number of moles that's gone. 00:08:01.010 --> 00:08:03.560 That have reacted. 00:08:03.560 --> 00:08:09.130 Divided by the number of moles initially, which is n. 00:08:09.130 --> 00:08:10.120 That's what we want. 00:08:10.120 --> 00:08:15.740 We want to see how that is going to change with pressure. 00:08:15.740 --> 00:08:18.160 So we're going to deal first with Kp, because Kp doesn't 00:08:18.160 --> 00:08:20.990 depend on total pressure. 00:08:20.990 --> 00:08:21.870 We're going to write that down. 00:08:21.870 --> 00:08:24.850 Then we're going to go to Kx, somehow. 00:08:24.850 --> 00:08:26.040 And that's going to depend on pressure. 00:08:26.040 --> 00:08:37.160 So let's see what Kp is here. 00:08:37.160 --> 00:08:41.020 So K sub p, you've got the products on top. 00:08:41.020 --> 00:08:45.490 So it's the partial pressure of NO2 to the second power, 00:08:45.490 --> 00:08:47.900 divided by the partial pressure of 00:08:47.900 --> 00:08:50.130 N2O4 to the one power. 00:08:50.130 --> 00:08:51.600 And everything is referenced to one bar, 00:08:51.600 --> 00:08:54.180 everything's in bar. 00:08:54.180 --> 00:08:57.060 And in terms of the molar fractions, it's the total 00:08:57.060 --> 00:09:04.380 pressure squared, times the mole fraction of NO2 squared, 00:09:04.380 --> 00:09:07.290 divided by the total pressure to the first power. 00:09:07.290 --> 00:09:09.680 So the square root on top gets divided by 00:09:09.680 --> 00:09:11.350 one factor of pressure. 00:09:11.350 --> 00:09:12.830 So we have total pressure in front. 00:09:12.830 --> 00:09:25.340 Divided by x to the N2O4, and that's p times Kx. 00:09:25.340 --> 00:09:27.950 So let's plug in what these mole fractions are from our 00:09:27.950 --> 00:09:30.950 table here. 00:09:30.950 --> 00:09:35.930 The mole fraction of NO2 is 2x divided by n plus x. 00:09:35.930 --> 00:09:39.040 2x divided by n plus x to the square power. 00:09:39.040 --> 00:09:43.650 Mole fraction of N2O4, n minus x over n plus x. n minus x 00:09:43.650 --> 00:09:46.560 over n plus x. 00:09:46.560 --> 00:09:50.270 Multiply, square the top, 4x squared. 00:09:50.270 --> 00:09:52.220 Divided by n plus x squared. 00:09:52.220 --> 00:09:54.900 Things sort of cancel out here. 00:09:54.900 --> 00:09:56.770 Rearrange. 00:09:56.770 --> 00:10:05.830 4x squared divided by n squared minus x squared. 00:10:05.830 --> 00:10:08.220 What we're really interested in is, we're not 00:10:08.220 --> 00:10:09.550 interested in x. 00:10:09.550 --> 00:10:12.690 We're interested in x divided by n. 00:10:12.690 --> 00:10:17.610 So let's divide both the top and the bottom by n squared. 00:10:17.610 --> 00:10:20.120 And we're going to get alpha come up. 00:10:20.120 --> 00:10:28.200 So this is then p times 4 alpha squared divided by one 00:10:28.200 --> 00:10:32.350 minus alpha squared. 00:10:32.350 --> 00:10:35.390 This is not, there's no total pressure here. 00:10:35.390 --> 00:10:39.310 The only way, the only place, where the total pressure comes 00:10:39.310 --> 00:10:42.500 in, is right here. 00:10:42.500 --> 00:10:44.655 This is just a number that doesn't care what the total 00:10:44.655 --> 00:10:45.250 pressure is. 00:10:45.250 --> 00:10:47.260 Which is why we're using it. 00:10:47.260 --> 00:10:51.810 And not K sub x, which cares what the total pressure is. 00:10:51.810 --> 00:10:54.590 So now we can solve for alpha. 00:10:54.590 --> 00:11:03.870 We can solve for alpha by rearranging this equation. 00:11:03.870 --> 00:11:07.140 This is just a number. 00:11:07.140 --> 00:11:11.290 And this is where the total pressure comes in. 00:11:11.290 --> 00:11:15.020 So you rearrange that, and you get alpha is equal to 1 plus 00:11:15.020 --> 00:11:22.050 4p divided by Kp, to the minus 1/2 power. 00:11:22.050 --> 00:11:25.080 And if I rewrite that slightly to make it a little bit easier 00:11:25.080 --> 00:11:28.880 to see what's going to happen, when I change the pressure, 1 00:11:28.880 --> 00:11:37.120 plus 4p divided by Kp, to the 1/2 power. 00:11:37.120 --> 00:11:41.530 And that's what I'm after. 00:11:41.530 --> 00:11:48.910 This tells me what happens at equilibrium to the amount of 00:11:48.910 --> 00:11:53.250 NO2 as I change the total pressure. 00:11:53.250 --> 00:11:54.780 This is the only place where it comes in. 00:11:54.780 --> 00:12:03.020 So now I can see that if I raise the pressure in my 00:12:03.020 --> 00:12:09.200 container, raise the pressure, this is in the denominator, so 00:12:09.200 --> 00:12:12.030 this fraction gets smaller. 00:12:12.030 --> 00:12:16.550 Alpha gets smaller. 00:12:16.550 --> 00:12:22.250 I raise the pressure, the fraction of material that 00:12:22.250 --> 00:12:24.170 reacts gets smaller. 00:12:24.170 --> 00:12:30.480 Therefore, the reaction goes towards the reactants. 00:12:30.480 --> 00:12:38.160 If I decrease the pressure, this is a smaller number here. 00:12:38.160 --> 00:12:40.500 The fraction gets bigger. 00:12:40.500 --> 00:12:40.990 Alpha goes up. 00:12:40.990 --> 00:12:47.210 If I decrease the pressure and I compare what happens to the 00:12:47.210 --> 00:12:51.320 number of moles of reactants that react, more of it reacts. 00:12:51.320 --> 00:12:55.140 Equilibrium shifts towards the product. 00:12:55.140 --> 00:13:02.680 And this is Le Chatelier that you already know. 00:13:02.680 --> 00:13:10.480 Le Chatelier's principle, for pressure. 00:13:10.480 --> 00:13:14.860 The way it works is that Le Chatelier's principle states 00:13:14.860 --> 00:13:23.050 that, this chemical system wants to stay as close to what 00:13:23.050 --> 00:13:23.820 it was before. 00:13:23.820 --> 00:13:24.620 It doesn't like change. 00:13:24.620 --> 00:13:28.030 It doesn't want to have any change happen. 00:13:28.030 --> 00:13:34.280 So if you increase the pressure, the chemical system 00:13:34.280 --> 00:13:37.320 says, hey, you know I'm not so happy that you're increasing 00:13:37.320 --> 00:13:38.860 the pressure on me. 00:13:38.860 --> 00:13:43.030 I'd like to go back to a smaller pressure. 00:13:43.030 --> 00:13:44.020 It doesn't like change. 00:13:44.020 --> 00:13:46.780 Very conservative. 00:13:46.780 --> 00:13:50.760 And the way to decrease the pressure is to decrease the 00:13:50.760 --> 00:13:53.230 number of moles in the container. 00:13:53.230 --> 00:13:55.140 How does it decrease the number of moles? 00:13:55.140 --> 00:13:56.590 Goes back to where there are fewer moles. 00:13:56.590 --> 00:14:03.870 And that's on the reactant side. 00:14:03.870 --> 00:14:06.450 How many of you know Lenz's law? 00:14:06.450 --> 00:14:07.980 In magnetism. 00:14:07.980 --> 00:14:11.070 The diamagnetic materials. 00:14:11.070 --> 00:14:11.830 Right. 00:14:11.830 --> 00:14:13.760 At least one person knows it. 00:14:13.760 --> 00:14:17.640 It's the same idea. 00:14:17.640 --> 00:14:20.880 You take a diamagnetic material in the absence of a 00:14:20.880 --> 00:14:24.820 magnetic field, and you slowly move it into a place where 00:14:24.820 --> 00:14:27.920 there's high magnetic field, what does the diamagnetic 00:14:27.920 --> 00:14:30.110 material do? 00:14:30.110 --> 00:14:33.680 It orients its magnetic moment to reverse the field. 00:14:33.680 --> 00:14:35.160 So that there's no field inside. 00:14:35.160 --> 00:14:36.740 It starts out with no field. 00:14:36.740 --> 00:14:38.640 Doesn't like change. 00:14:38.640 --> 00:14:42.590 So Lenz's law says it's going to do whatever it can so that 00:14:42.590 --> 00:14:44.450 it retains no field inside. 00:14:44.450 --> 00:14:47.790 Le Chatelier's principle is basically the same thing. 00:14:47.790 --> 00:14:50.100 Equilibrium systems are very unhappy if you 00:14:50.100 --> 00:14:52.330 try to change them. 00:14:52.330 --> 00:14:56.950 And that's what happens for Le Chatelier here with pressure. 00:14:56.950 --> 00:15:01.790 Any questions? 00:15:01.790 --> 00:15:04.660 So before we go to Le Chatelier's with temperature, 00:15:04.660 --> 00:15:06.570 and the van 't Hoff equation. 00:15:06.570 --> 00:15:11.290 Let's do a little detour here and talk about equilibrium in 00:15:11.290 --> 00:15:14.960 solution, which is really as important, or if not important 00:15:14.960 --> 00:15:20.430 for a lot of you, then gas phase equilibrium. 00:15:20.430 --> 00:15:23.320 Although gas phase equilibrium was where everything started. 00:15:23.320 --> 00:15:26.860 And still a huge deal. 00:15:26.860 --> 00:15:32.330 OK, so in equilibrium now, when we talk about equilibrium 00:15:32.330 --> 00:15:35.290 in solution, we still have to, still going to be 00:15:35.290 --> 00:15:36.870 the chemical potential. 00:15:36.870 --> 00:15:39.680 It's still going to be looking at how chemical potential 00:15:39.680 --> 00:15:41.530 likes to go downhill. 00:15:41.530 --> 00:15:46.730 And we're going to have to write chemical potential for a 00:15:46.730 --> 00:15:51.140 species, A, let's say, which is in solution. 00:15:51.140 --> 00:15:54.920 At some concentration c sub A in solution. 00:15:54.920 --> 00:16:03.860 And the concentration could be given in moles per liter. 00:16:03.860 --> 00:16:08.510 Or it could be in grams per liter. 00:16:08.510 --> 00:16:13.910 Or it could be in grams per 1000 grams. 00:16:13.910 --> 00:16:16.860 Whatever your favorite unit of concentration is. 00:16:16.860 --> 00:16:17.740 Use it. 00:16:17.740 --> 00:16:20.210 Stick to it. 00:16:20.210 --> 00:16:22.260 And in order to do equilibrium, we're going to 00:16:22.260 --> 00:16:28.360 have to reference it to, so this would be the species at 00:16:28.360 --> 00:16:30.280 some arbitrary concentration. 00:16:30.280 --> 00:16:33.110 We're going to have to reference it to some reference 00:16:33.110 --> 00:16:33.730 concentration. 00:16:33.730 --> 00:16:37.210 Just like we referenced everything to one bar before, 00:16:37.210 --> 00:16:39.920 as our standard pressure. 00:16:39.920 --> 00:16:41.860 And we're going to take, usually you take one mole per 00:16:41.860 --> 00:16:45.690 liter, or one gram per liter, or one whatever. 00:16:45.690 --> 00:16:47.750 One as your reference concentration. 00:16:47.750 --> 00:16:49.220 And the reference concentration is going to 00:16:49.220 --> 00:16:50.460 disappear from the equation. 00:16:50.460 --> 00:16:52.140 It's just like the reference pressure. 00:16:52.140 --> 00:16:54.100 Disappear from the equation. 00:16:54.100 --> 00:16:59.320 So we're going to reference this to some standard state 00:16:59.320 --> 00:17:01.580 chemical potential. 00:17:01.580 --> 00:17:06.390 Where the naught refers now to the standard concentration. 00:17:06.390 --> 00:17:12.245 And instead of having RT log p, now we're going 00:17:12.245 --> 00:17:16.500 to have RT log cA. 00:17:16.500 --> 00:17:19.390 It's kind of like considering the molecules in the solution 00:17:19.390 --> 00:17:26.240 to act like an ideal gas. 00:17:26.240 --> 00:17:28.730 Knowing fully well that behind, that underneath the 00:17:28.730 --> 00:17:33.280 cA, is this reference concentration of one. 00:17:33.280 --> 00:17:36.960 One whatever is your favorite units. 00:17:36.960 --> 00:17:39.230 Now, it's a little bit more complicated than 00:17:39.230 --> 00:17:41.420 for the ideal gas. 00:17:41.420 --> 00:17:48.550 Because your solution may contain other things than your 00:17:48.550 --> 00:17:50.740 reactants and your products. 00:17:50.740 --> 00:17:52.920 Especially if you're doing biology. 00:17:52.920 --> 00:17:54.430 It could be a buffer. 00:17:54.430 --> 00:17:56.390 It could be a buffer, it could have salt. 00:17:56.390 --> 00:18:00.160 It could have a pH that's not equal to seven, whatever. 00:18:00.160 --> 00:18:04.060 And so this reference, chemical potential, now needs 00:18:04.060 --> 00:18:12.380 to be referenced to a particular pH or salt 00:18:12.380 --> 00:18:13.120 concentration. 00:18:13.120 --> 00:18:20.280 Or whatever the properties of your solvent are. 00:18:20.280 --> 00:18:23.240 Or your solution are. 00:18:23.240 --> 00:18:25.660 And that's the big difference. 00:18:25.660 --> 00:18:26.980 In an ideal gas, it's 00:18:26.980 --> 00:18:28.870 reference to vacuum, basically. 00:18:28.870 --> 00:18:30.090 There's nothing there. 00:18:30.090 --> 00:18:32.340 In here, in solution you have all these molecules of 00:18:32.340 --> 00:18:38.000 solvent, molecules of salt, molecules of acid, or 00:18:38.000 --> 00:18:42.260 whatever, that are going to be around to buffer the pH. 00:18:42.260 --> 00:18:45.320 And that's going to change what the chemical potential of 00:18:45.320 --> 00:18:46.370 a species is. 00:18:46.370 --> 00:18:50.160 And if I change the pH and I've got a molecule that I'm 00:18:50.160 --> 00:18:54.500 interested in, it may not have an acidic moiety on it, but it 00:18:54.500 --> 00:18:59.660 could still care what the pH is, slightly. 00:18:59.660 --> 00:19:01.990 And that would change what the reference potential is, 00:19:01.990 --> 00:19:03.920 chemical potential is. 00:19:03.920 --> 00:19:05.540 So this is really important to remember. 00:19:05.540 --> 00:19:07.640 And there are textbooks that are written on how to do this 00:19:07.640 --> 00:19:09.090 the right way. 00:19:09.090 --> 00:19:10.110 We're not going to do that here. 00:19:10.110 --> 00:19:13.020 We're just going to remember this is, we're going to assume 00:19:13.020 --> 00:19:15.290 that this is done correctly. 00:19:15.290 --> 00:19:20.360 Once you take that as a given, that you have a way to have a 00:19:20.360 --> 00:19:23.630 reference chemical potential at a properly referenced pH 00:19:23.630 --> 00:19:25.520 and salt concentration, then you can go through the same 00:19:25.520 --> 00:19:29.740 analysis that we went to for partial pressures. 00:19:29.740 --> 00:19:34.590 This looks just like the ideal gas, where the concentration 00:19:34.590 --> 00:19:38.130 replaces the partial pressure. 00:19:38.130 --> 00:19:43.360 Or the pressure of chemical A. And you can go through, then 00:19:43.360 --> 00:19:52.440 the same argument, where you take your reaction. 00:19:52.440 --> 00:20:00.370 And you initially have some delta G for the reactants. 00:20:00.370 --> 00:20:05.420 And you have some delta G for the products. 00:20:05.420 --> 00:20:07.830 And the difference is the delta G for the reaction, 00:20:07.830 --> 00:20:17.110 delta G naught for the reaction, and then on this 00:20:17.110 --> 00:20:26.510 side here, you have a solution of A, so the reaction would be 00:20:26.510 --> 00:20:30.390 nu A times A, which is in a solution. 00:20:30.390 --> 00:20:31.980 Temperature and pressure. 00:20:31.980 --> 00:20:36.910 Plus nu B of reactant B, in a solution, 00:20:36.910 --> 00:20:38.340 temperature and pressure. 00:20:38.340 --> 00:20:44.330 Going to a nu C, C solution, temperature and pressure plus 00:20:44.330 --> 00:20:48.560 nu D, D in a solution, constant 00:20:48.560 --> 00:20:50.230 temperature and pressure. 00:20:50.230 --> 00:20:54.420 So this is taking a solution of A, in one container, a 00:20:54.420 --> 00:20:56.150 solution of B in another the container. 00:20:56.150 --> 00:20:57.360 That's the initial point. 00:20:57.360 --> 00:20:57.970 Mix them together. 00:20:57.970 --> 00:20:59.690 Let them react. 00:20:59.690 --> 00:21:01.580 Then you take the product, you put them in separate 00:21:01.580 --> 00:21:02.640 containers. 00:21:02.640 --> 00:21:06.770 And that gets you the stuff that you have the reaction. 00:21:06.770 --> 00:21:09.980 So when you mix A and B, you're going to have the same 00:21:09.980 --> 00:21:10.980 entropy of mixing. 00:21:10.980 --> 00:21:14.100 You're going to lower the delta G. Of the solution. 00:21:14.100 --> 00:21:16.640 And you're going to have the same curve that 00:21:16.640 --> 00:21:18.530 goes down like this. 00:21:18.530 --> 00:21:21.890 To the mixture of products. 00:21:21.890 --> 00:21:24.630 And just like for the gases, where we wanted to know what 00:21:24.630 --> 00:21:26.360 is the bottom of this curve which gives us the 00:21:26.360 --> 00:21:29.480 equilibrium, we can do the same thing. 00:21:29.480 --> 00:21:34.110 Exactly the same thing, for solutions. 00:21:34.110 --> 00:21:37.050 And so we start out with a mixture of the A and B in 00:21:37.050 --> 00:21:39.470 solution and C and D, reactants 00:21:39.470 --> 00:21:41.070 and products together. 00:21:41.070 --> 00:21:45.650 And we let the reaction proceed a little bit. 00:21:45.650 --> 00:21:53.030 And we look at the change in delta G, going from, say this 00:21:53.030 --> 00:21:56.130 point here through that point here. 00:21:56.130 --> 00:21:58.570 This will be delta G of epsilon. 00:21:58.570 --> 00:22:01.600 We ask, is this positive, negative, or zero. 00:22:01.600 --> 00:22:04.100 And if it's zero, that means that we're in equilibrium, 00:22:04.100 --> 00:22:07.150 that we're actually sitting down here. 00:22:07.150 --> 00:22:09.420 And that gives us the equilibrium constant. 00:22:09.420 --> 00:22:13.280 So, just for the sake of completeness, let me just 00:22:13.280 --> 00:22:15.730 write down what we would do. 00:22:15.730 --> 00:22:20.240 We would react it for a small amount. 00:22:20.240 --> 00:22:24.530 And then we'd end up with nu C. 00:22:24.530 --> 00:22:28.050 So you would have the chemical potentials of the products 00:22:28.050 --> 00:22:33.480 minus the chemical potentials of the reactants, nu C mu C, 00:22:33.480 --> 00:22:44.160 plus nu D mu D, minus nu A mu A, minus nu B mu B. And 00:22:44.160 --> 00:22:46.680 instead of these chemical potentials, you would write 00:22:46.680 --> 00:22:51.130 them in terms of the pure chemical potentials times 00:22:51.130 --> 00:22:55.630 their concentrations. 00:22:55.630 --> 00:23:01.470 And then you'd end up with epsilon times delta G naught 00:23:01.470 --> 00:23:10.360 of the reaction, plus RT log, and then the concentrations. 00:23:10.360 --> 00:23:12.450 And then you write them in a different way. 00:23:12.450 --> 00:23:15.380 So if it's moles per liter, you usually write that with 00:23:15.380 --> 00:23:18.240 these brackets here. 00:23:18.240 --> 00:23:21.940 That means concentration of A in moles per liter, I'd say. 00:23:21.940 --> 00:23:29.420 So C to the nu C power, D to the nu D power, A to the nu A 00:23:29.420 --> 00:23:34.110 power, and B to the nu B power, in these 00:23:34.110 --> 00:23:37.370 concentrations. 00:23:37.370 --> 00:23:41.420 Where this ratio of logs comes from expanding out the 00:23:41.420 --> 00:23:44.380 chemical potential here. 00:23:44.380 --> 00:23:46.750 And there's the log term here. 00:23:46.750 --> 00:23:48.670 Just like an ideal gas. 00:23:48.670 --> 00:23:51.070 Then at equilibrium, this is equal to zero, you're at the 00:23:51.070 --> 00:23:52.620 bottom of that curve. 00:23:52.620 --> 00:23:57.350 And you set these two things equal to each other. 00:23:57.350 --> 00:24:04.880 And you get the chemical, you get your equation 00:24:04.880 --> 00:24:06.200 that you know well. 00:24:06.200 --> 00:24:11.550 For the equilibrium constant. 00:24:11.550 --> 00:24:16.050 And this time it's not K sub p, it's just K. And that's 00:24:16.050 --> 00:24:19.700 what you know from doing solution equilibrium. 00:24:19.700 --> 00:24:23.390 And it's just like V. 00:24:23.390 --> 00:24:28.810 So the thing to remember, which is the slightly more 00:24:28.810 --> 00:24:32.520 advanced part, which you'll have to worry about at some 00:24:32.520 --> 00:24:37.820 point if you stay in some sort of biochemistry oriented 00:24:37.820 --> 00:24:41.060 field, is that you've got to reference your initial 00:24:41.060 --> 00:24:43.340 solution properly. 00:24:43.340 --> 00:24:48.150 To get to the right equilibrium constant. 00:24:48.150 --> 00:24:53.960 OK, any questions? 00:24:53.960 --> 00:24:57.260 Alright, then now we can do the temperature dependence of 00:24:57.260 --> 00:25:02.900 the equilibrium constant in a general way. 00:25:02.900 --> 00:25:05.260 Whether it be a gas or a solution. 00:25:05.260 --> 00:25:06.160 It doesn't matter. 00:25:06.160 --> 00:25:13.970 It's going to be the same thing. 00:25:13.970 --> 00:25:18.080 So the question that we ask now is, suppose that I change 00:25:18.080 --> 00:25:21.320 the temperature of my equilibrium, which way is the 00:25:21.320 --> 00:25:23.990 equilibrium going to shift? 00:25:23.990 --> 00:25:28.580 And you all know the answer already, probably. 00:25:28.580 --> 00:25:31.010 But let's derive it out. 00:25:31.010 --> 00:25:34.010 So we're going to want to know basically, we want to know 00:25:34.010 --> 00:25:35.260 what is dKp/dT. 00:25:38.060 --> 00:25:41.850 How does equilibrium constant, or dK/dT, if you're doing 00:25:41.850 --> 00:25:45.150 solution, how does the equilibrium change with 00:25:45.150 --> 00:25:46.230 temperature? 00:25:46.230 --> 00:25:46.880 What's the slope? 00:25:46.880 --> 00:25:49.090 Is it positive, negative? 00:25:49.090 --> 00:25:51.350 If we have this, we can integrate it out. 00:25:51.350 --> 00:25:53.560 We can do an integral over temperature. 00:25:53.560 --> 00:25:56.380 And get an actual change. 00:25:56.380 --> 00:25:58.540 So that's our goal. 00:25:58.540 --> 00:26:00.670 To find how the equilibrium constant changes with 00:26:00.670 --> 00:26:03.350 temperature. 00:26:03.350 --> 00:26:04.220 What do we know? 00:26:04.220 --> 00:26:11.640 Well, we know how Kp depends on temperature, through the 00:26:11.640 --> 00:26:14.740 Gibbs free energy of the reaction. 00:26:14.740 --> 00:26:16.780 The Gibbs free energy, delta G naught, has a temperature 00:26:16.780 --> 00:26:18.060 dependence. 00:26:18.060 --> 00:26:19.700 And then there's an RT sitting on the bottom. 00:26:19.700 --> 00:26:23.890 There's another temperature dependence here. 00:26:23.890 --> 00:26:27.000 Well, dKp/dT is sort of like, we could also ask 00:26:27.000 --> 00:26:28.990 what's d log Kp dT. 00:26:28.990 --> 00:26:30.500 That might be an easier question. 00:26:30.500 --> 00:26:32.810 It's basically the same question. 00:26:32.810 --> 00:26:35.010 Especially since we have something which is log K, is 00:26:35.010 --> 00:26:35.830 equal to something. 00:26:35.830 --> 00:26:38.320 So let's ask this question instead. 00:26:38.320 --> 00:26:42.580 Let's ask, what is d log Kp dT? 00:26:42.580 --> 00:26:48.380 Alright, so let's differentiate both sides. 00:26:48.380 --> 00:26:54.880 d/dT, d/dT here. 00:26:54.880 --> 00:26:56.950 Got to use the chain rule now. 00:26:56.950 --> 00:26:58.490 Because we've got temperature as part of 00:26:58.490 --> 00:27:04.270 delta G Write it out. 00:27:04.270 --> 00:27:05.930 So let's take the derivative with respect to the 00:27:05.930 --> 00:27:07.180 temperature on the bottom first. 00:27:07.180 --> 00:27:10.710 We have delta G naught, which is a function of temperature, 00:27:10.710 --> 00:27:14.100 divided by RT squared. 00:27:14.100 --> 00:27:17.250 The minus sign here disappears when you take the derivative 00:27:17.250 --> 00:27:18.960 on the bottom. 00:27:18.960 --> 00:27:30.160 Minus one over RT, d/dT of delta G naught. 00:27:30.160 --> 00:27:36.840 So this is a derivative of delta G, where zero 00:27:36.840 --> 00:27:38.370 means here one bar. 00:27:38.370 --> 00:27:39.210 Fixed at one bar. 00:27:39.210 --> 00:27:46.490 So really, d/dT, with delta G naught fixed on one bar, is 00:27:46.490 --> 00:27:50.130 the same thing as the partial derivative of delta G with 00:27:50.130 --> 00:27:53.605 respect to temperature, keeping p is equal to constant 00:27:53.605 --> 00:27:55.650 at one bar. 00:27:55.650 --> 00:28:01.620 It's the same thing, just different notation. 00:28:01.620 --> 00:28:05.530 And we know what this is. 00:28:05.530 --> 00:28:12.540 In terms of other things that we can find in books, like 00:28:12.540 --> 00:28:16.960 delta H, or delta S. Because we can go to 00:28:16.960 --> 00:28:18.900 the fundamental equations. 00:28:18.900 --> 00:28:27.370 To find out how delta G depends on temperature. 00:28:27.370 --> 00:28:31.170 And our goal is to get rid of delta G, which clearly has a 00:28:31.170 --> 00:28:33.790 nice temperature dependence through the entropy term. 00:28:33.790 --> 00:28:36.510 And to replace delta G with delta H, if we can. 00:28:36.510 --> 00:28:38.750 Because delta h is going to be much less sensitive to 00:28:38.750 --> 00:28:43.390 temperature and it's, delta H is going to be over small 00:28:43.390 --> 00:28:44.700 temperature ranges, is going to be independent of 00:28:44.700 --> 00:28:46.100 temperature. 00:28:46.100 --> 00:28:49.380 And we know the temperature dependence of delta H, because 00:28:49.380 --> 00:28:51.020 it's through the heat capacities. 00:28:51.020 --> 00:28:55.150 So our goal is to get rid of this delta G here. 00:28:55.150 --> 00:28:59.430 And to try to replace it with delta H, if at all possible. 00:28:59.430 --> 00:29:04.260 So we go to the fundamental equation for G, dG is equal to 00:29:04.260 --> 00:29:10.100 minus S dT plus V dp. 00:29:10.100 --> 00:29:17.800 And sitting right here is dG/dT at constant p. 00:29:17.800 --> 00:29:19.940 Which is what we have here. 00:29:19.940 --> 00:29:25.110 So we get rid of this derivative of G. And replace 00:29:25.110 --> 00:29:35.890 it with S. So now, now we have d log Kp dT, and I mentioned 00:29:35.890 --> 00:29:38.890 already that I want to get rid of G. Because it has a strong 00:29:38.890 --> 00:29:39.680 temperature dependence. 00:29:39.680 --> 00:29:42.490 And I want to somehow get H in there, which is not going to 00:29:42.490 --> 00:29:44.130 have a strong temperature dependence. 00:29:44.130 --> 00:29:47.625 And delta G naught, I can write in terms of H and S, and 00:29:47.625 --> 00:29:56.410 T. Delta H naught minus T delta S naught 00:29:56.410 --> 00:29:59.630 divided by RT squared. 00:29:59.630 --> 00:30:04.800 And then my derivative here, I have minus one over RT. 00:30:04.800 --> 00:30:08.890 Partial of G with respect to T. p is equal to one bar. 00:30:08.890 --> 00:30:13.080 Well, that's just delta S. p is equal to one bar, well, 00:30:13.080 --> 00:30:14.260 that's just delta S naught. 00:30:14.260 --> 00:30:20.660 Times delta S naught. 00:30:20.660 --> 00:30:23.800 And that's great, because now there's minus T 00:30:23.800 --> 00:30:24.920 divided by RT squared. 00:30:24.920 --> 00:30:27.080 That's one over RT. 00:30:27.080 --> 00:30:31.340 And somewhere I've lost a sign. 00:30:31.340 --> 00:30:33.300 And there's my sign that I lost, right there. 00:30:33.300 --> 00:30:40.580 This minus sign here. d/dT of delta G naught is minus S. It 00:30:40.580 --> 00:30:43.000 actually includes this minus sign right here. 00:30:43.000 --> 00:30:45.650 Which is great, because now things work out, because this 00:30:45.650 --> 00:30:49.090 becomes a plus sign. 00:30:49.090 --> 00:30:54.460 And this and this cancel out. 00:30:54.460 --> 00:31:03.380 And this becomes delta H naught over RT squared. 00:31:03.380 --> 00:31:04.300 Great, so we have what we want. 00:31:04.300 --> 00:31:07.320 We have how the equilibrium constant depends on 00:31:07.320 --> 00:31:10.050 temperature in a way which is very clear. 00:31:10.050 --> 00:31:13.110 Where the top part is only very weakly dependent on 00:31:13.110 --> 00:31:14.740 temperature, usually. 00:31:14.740 --> 00:31:23.160 And this is called the van 't Hoff equation. 00:31:23.160 --> 00:31:26.900 And this will tell us what happens to equilibrium when we 00:31:26.900 --> 00:31:30.340 change the temperature. 00:31:30.340 --> 00:31:34.710 So if you want to do a finite temperature change, now what 00:31:34.710 --> 00:31:52.700 you need to do is, you need to integrate. 00:31:52.700 --> 00:31:59.320 You integrate both sides here. 00:31:59.320 --> 00:32:02.520 From some T1 to T2. 00:32:02.520 --> 00:32:09.880 From T1 to T2, and that tells you, then, that the log of the 00:32:09.880 --> 00:32:13.480 equilibrium constant at the new temperature is equal to 00:32:13.480 --> 00:32:15.650 the log of the equilibrium constant of the old 00:32:15.650 --> 00:32:21.310 temperature, T1, plus the integral from T1 to T2 of 00:32:21.310 --> 00:32:28.180 delta H over RT delta H naught, over RT squared. 00:32:28.180 --> 00:32:31.690 And this could be slightly temperature dependent. dT. 00:32:31.690 --> 00:32:34.810 And this is the integrated van 't Hoff equation. 00:32:34.810 --> 00:32:39.480 And if you're going to be designing a chemical plant 00:32:39.480 --> 00:32:43.810 where you have high temperatures and high 00:32:43.810 --> 00:32:48.070 pressures around, you better use that. 00:32:48.070 --> 00:32:50.070 Because there is some temperature dependence in 00:32:50.070 --> 00:32:53.150 delta H, through the heat capacities of the reactants 00:32:53.150 --> 00:32:54.460 and the products. 00:32:54.460 --> 00:32:57.880 And that could make the difference between your plant 00:32:57.880 --> 00:33:00.750 running nice and smoothly or your plant exploding. 00:33:00.750 --> 00:33:04.210 And you don't want to have exploding plants around. 00:33:04.210 --> 00:33:10.160 So for heavy-duty uses of this equation, you've got to do the 00:33:10.160 --> 00:33:11.800 integral properly. 00:33:11.800 --> 00:33:14.830 But for most normal applications, like if you're 00:33:14.830 --> 00:33:17.030 doing biology, where the temperature changes by a few 00:33:17.030 --> 00:33:19.730 degrees, like today I have a little bit of a cold. 00:33:19.730 --> 00:33:22.500 I don't have a fever, but I could have a fever. 00:33:22.500 --> 00:33:25.050 So my biochemistry would change if I had a fever. 00:33:25.050 --> 00:33:27.420 The equilibrium constant of all my reactions would change 00:33:27.420 --> 00:33:28.820 a little bit. 00:33:28.820 --> 00:33:31.350 It's a small change in temperature. 00:33:31.350 --> 00:33:34.640 I'm not going to explode. 00:33:34.640 --> 00:33:37.330 And so you can then take the approximation. 00:33:37.330 --> 00:33:47.140 In that case, the delta H naught, is independent of T. 00:33:47.140 --> 00:33:53.960 And this is fine over small temperature ranges. 00:33:53.960 --> 00:33:55.940 And that's the one that you're most used to. 00:33:55.940 --> 00:33:57.770 Is this approximation here, this 00:33:57.770 --> 00:34:00.020 approximate van 't Hoff equation. 00:34:00.020 --> 00:34:02.860 Which is really fine for most cases that you're going to be 00:34:02.860 --> 00:34:05.870 dealing with. 00:34:05.870 --> 00:34:09.190 So then, if that's the case then you can take your delta 00:34:09.190 --> 00:34:12.830 H. Ignore the temperature dependence and take it outside 00:34:12.830 --> 00:34:14.210 of the integral. 00:34:14.210 --> 00:34:17.370 And now you can do the integral fine. 00:34:17.370 --> 00:34:21.190 And then you have an analytic expression for the change in 00:34:21.190 --> 00:34:23.600 the equilibrium constant with temperature. 00:34:23.600 --> 00:34:29.170 Log Kp at a new temperature, T2, is log Kp temperature T1. 00:34:29.170 --> 00:34:31.610 And I'm carrying this little p around everywhere. 00:34:31.610 --> 00:34:33.410 But really, it doesn't have to be there. 00:34:33.410 --> 00:34:35.180 This could be solution. 00:34:35.180 --> 00:34:38.450 I shouldn't really have written this for the specific 00:34:38.450 --> 00:34:40.240 case of partial pressures. 00:34:40.240 --> 00:34:43.090 But it's equally valid for solutions. 00:34:43.090 --> 00:34:46.970 Then we have delta H naught over R. And then we have the 00:34:46.970 --> 00:34:51.430 integral from T1 to T2, over one of RT squared. 00:34:51.430 --> 00:34:53.520 And if you do that the right way, you get T2 00:34:53.520 --> 00:34:57.610 minus T1 over T1 T2. 00:34:57.610 --> 00:35:00.020 And that gives you the 00:35:00.020 --> 00:35:04.280 approximate van 't Hoff equation. 00:35:04.280 --> 00:35:06.480 Which is fine. 00:35:06.480 --> 00:35:09.080 And you'll know that it's fine in problem sets or exam, 00:35:09.080 --> 00:35:11.450 because we'll say assume that delta H is temperature - yes. 00:35:11.450 --> 00:35:18.940 STUDENT: You said that the [INAUDIBLE] 00:35:18.940 --> 00:35:20.740 PROFESSOR: For K, if K is solution K. 00:35:20.740 --> 00:35:23.260 STUDENT: Right. 00:35:23.260 --> 00:35:23.770 PROFESSOR: Yeah. 00:35:23.770 --> 00:35:26.930 STUDENT: So then, can you also use Kx? 00:35:26.930 --> 00:35:28.450 PROFESSOR: Can you use Kx? 00:35:28.450 --> 00:35:33.140 Well, as long as you keep the pressure, the total pressure, 00:35:33.140 --> 00:35:38.720 constant, then you should be able to use Kx. 00:35:38.720 --> 00:35:43.920 Let me think about this. 00:35:43.920 --> 00:35:47.100 Yeah, here you would have p, you have a log c so, you can 00:35:47.100 --> 00:35:47.840 use Kx, fine. 00:35:47.840 --> 00:35:50.560 Because then you would have log p to the minus delta nu 00:35:50.560 --> 00:35:53.890 times Kx, log p to the delta nu minus Kx. 00:35:53.890 --> 00:35:56.770 And the log of the multiplication is 00:35:56.770 --> 00:35:58.380 the sum of the logs. 00:35:58.380 --> 00:35:59.440 And the logs will just fall out. 00:35:59.440 --> 00:36:01.460 So it could be any K that you want. 00:36:01.460 --> 00:36:04.080 Doesn't matter. 00:36:04.080 --> 00:36:06.190 As long as you keep the pressure constant. 00:36:06.190 --> 00:36:10.630 If you change the pressure, then you're in trouble. 00:36:10.630 --> 00:36:14.660 So now we can see what happens when you do change the 00:36:14.660 --> 00:36:18.280 temperature. 00:36:18.280 --> 00:36:25.420 If I have some equilibrium, and it's all going to depend 00:36:25.420 --> 00:36:28.020 on the sign of delta H. Whether the reaction is 00:36:28.020 --> 00:36:33.200 exothermic or endothermic. 00:36:33.200 --> 00:36:36.340 And it's the same thing as Le Chatelier's for pressure, or 00:36:36.340 --> 00:36:37.090 Lenz's law. 00:36:37.090 --> 00:36:40.300 The system doesn't want to have change happening. 00:36:40.300 --> 00:36:43.430 So if you have something that's, delta H is less than 00:36:43.430 --> 00:36:46.810 zero, it's exothermic. 00:36:46.810 --> 00:36:51.750 Exothermic, that means that it's putting out heat. it 00:36:51.750 --> 00:36:55.300 wants to heat up its environment. 00:36:55.300 --> 00:36:59.990 And if I take temperature and I raise the temperature, the 00:36:59.990 --> 00:37:02.130 system's not going to like that very much. 00:37:02.130 --> 00:37:06.540 It doesn't want to get hotter, and going from reactants to 00:37:06.540 --> 00:37:08.550 products makes things hotter. 00:37:08.550 --> 00:37:09.520 And if you go from products to 00:37:09.520 --> 00:37:10.980 reactants, that's the opposite. 00:37:10.980 --> 00:37:13.840 Go from reactants to products, that becomes endothermic. 00:37:13.840 --> 00:37:15.420 It sucks in heat. 00:37:15.420 --> 00:37:16.610 You raise the temperature. 00:37:16.610 --> 00:37:19.210 The system said no, no, no, I'm happy where I am at my 00:37:19.210 --> 00:37:20.450 original temperature. 00:37:20.450 --> 00:37:22.960 I'm going to start sucking in heat, to try to get the 00:37:22.960 --> 00:37:24.810 temperature down. 00:37:24.810 --> 00:37:26.630 And it's going to try to make more product. 00:37:26.630 --> 00:37:29.250 More reactants. 00:37:29.250 --> 00:37:40.170 So, equilibrium K is going to go down. and the reaction goes 00:37:40.170 --> 00:37:43.810 towards the reactants. 00:37:43.810 --> 00:37:46.090 And the opposite if you have something that's endothermic 00:37:46.090 --> 00:37:49.670 to begin with. 00:37:49.670 --> 00:37:51.230 OK, delta H is positive here. 00:37:51.230 --> 00:37:54.650 Delta H naught is positive. 00:37:54.650 --> 00:37:57.850 You raise the temperature, delta H naught is positive, 00:37:57.850 --> 00:37:58.590 T2's bigger. 00:37:58.590 --> 00:38:00.620 This is a positive number. 00:38:00.620 --> 00:38:06.910 K becomes larger at higher temperature. 00:38:06.910 --> 00:38:08.850 K goes up. 00:38:08.850 --> 00:38:10.900 The reaction goes to products. 00:38:10.900 --> 00:38:15.210 So if you think of it in terms of the system, the system is 00:38:15.210 --> 00:38:16.960 at some temperature. 00:38:16.960 --> 00:38:18.820 You raise the temperature. 00:38:18.820 --> 00:38:20.500 System doesn't like it. 00:38:20.500 --> 00:38:21.910 Says, I want to go back to my original 00:38:21.910 --> 00:38:24.520 temperature, what can I do. 00:38:24.520 --> 00:38:27.690 I can try to suck in heat that you're trying to put in the 00:38:27.690 --> 00:38:28.970 environment. 00:38:28.970 --> 00:38:31.630 That's great because if I make more products, that's 00:38:31.630 --> 00:38:32.730 endothermic. 00:38:32.730 --> 00:38:34.810 And I'm just going to make more products until I try to 00:38:34.810 --> 00:38:35.740 lower my temperature. 00:38:35.740 --> 00:38:41.740 So I move the equilibrium to the products. 00:38:41.740 --> 00:38:56.130 OK, Le Chatelier for temperature. 00:38:56.130 --> 00:38:59.910 Any questions? 00:38:59.910 --> 00:39:08.300 On equilibrium. 00:39:08.300 --> 00:39:09.970 OK, let's do a quick example. 00:39:09.970 --> 00:39:13.410 Because this was the example that we started out with, 00:39:13.410 --> 00:39:14.760 talking about, the Haber process. 00:39:14.760 --> 00:39:21.570 This important industrial reaction that started the 00:39:21.570 --> 00:39:23.670 chemical industry, essentially. 00:39:23.670 --> 00:39:27.860 That uses up 1%, or close to 1%, of the world's energy. 00:39:27.860 --> 00:39:29.740 If you think about it, that's an amazing number. 00:39:29.740 --> 00:39:31.960 1% of all energy produced in the 00:39:31.960 --> 00:39:35.110 world goes to one reaction. 00:39:35.110 --> 00:39:37.050 One industrial reaction. 00:39:37.050 --> 00:39:45.430 Just shows how important it is. 00:39:45.430 --> 00:39:47.210 OK, why does it take so much energy? 00:39:47.210 --> 00:39:48.260 We're going to find out. 00:39:48.260 --> 00:39:55.310 We're going to find out why it takes so much energy to run 00:39:55.310 --> 00:39:57.500 this here reaction. 00:39:57.500 --> 00:40:00.100 Alright, let's look at this Haber process. 00:40:00.100 --> 00:40:04.990 Take some nitrogen gas. 00:40:04.990 --> 00:40:11.270 Plus some hydrogen gas. 00:40:11.270 --> 00:40:14.380 And this is usually done over catalysts, like an iron oxide 00:40:14.380 --> 00:40:15.850 catalyst or something. 00:40:15.850 --> 00:40:17.090 To try to speed it up. 00:40:17.090 --> 00:40:18.330 It doesn't change the thermodynamics. 00:40:18.330 --> 00:40:22.070 As you know, and you'll hear again in this class, catalysts 00:40:22.070 --> 00:40:23.140 just affect the kinetics. 00:40:23.140 --> 00:40:25.330 They don't change the thermodynamics. 00:40:25.330 --> 00:40:27.950 So this is usually done over some catalyst to try to 00:40:27.950 --> 00:40:29.220 speed things up. 00:40:29.220 --> 00:40:30.450 To make ammonia. 00:40:30.450 --> 00:40:35.200 And ammonia becomes the feedstock for fertilizers, for 00:40:35.200 --> 00:40:37.310 almost anything that contains an amine in it, or 00:40:37.310 --> 00:40:38.680 a nitrogen in it. 00:40:38.680 --> 00:40:42.090 If you're going to make proteins or whatever. 00:40:42.090 --> 00:40:45.740 You've got to have ammonia somewhere in the process. 00:40:45.740 --> 00:40:48.470 OK, delta H naught of the reaction, we're given all 00:40:48.470 --> 00:40:50.210 these numbers. 00:40:50.210 --> 00:40:54.630 At 298 degrees Kelvin. 00:40:54.630 --> 00:40:57.080 And they're in your notes, so I'm not going to go through 00:40:57.080 --> 00:40:58.390 them in detail. 00:40:58.390 --> 00:40:59.890 Delta G naught for the reaction, 00:40:59.890 --> 00:41:01.940 we're given that number. 00:41:01.940 --> 00:41:06.810 At 298 degrees Kelvin, that's minus 16, roughly minus 16 00:41:06.810 --> 00:41:12.340 kilojoules per mole. 00:41:12.340 --> 00:41:16.500 And we want to know, what is the equilibrium constant. 00:41:16.500 --> 00:41:17.430 Room temperature. 00:41:17.430 --> 00:41:18.880 So you know how to calculate that. 00:41:18.880 --> 00:41:22.760 Minus RT log Kp, log K is equal to minus RT. 00:41:25.450 --> 00:41:28.110 Minus delta G naught over RT. 00:41:28.110 --> 00:41:30.340 So you put that in there. 00:41:30.340 --> 00:41:33.530 You get Kp is equal to 860. 00:41:33.530 --> 00:41:35.170 A number, no units. 00:41:35.170 --> 00:41:36.820 It's a big number. 00:41:36.820 --> 00:41:38.800 It's a big number, you've got the products 00:41:38.800 --> 00:41:40.500 divided by the reactants. 00:41:40.500 --> 00:41:42.450 It means that the products are favored. 00:41:42.450 --> 00:41:45.160 This is great. 00:41:45.160 --> 00:41:48.760 What a wonderful reaction. 00:41:48.760 --> 00:41:51.340 Shouldn't take energy to for us to do that, right? 00:41:51.340 --> 00:41:53.090 It's a room temperature reaction. 00:41:53.090 --> 00:41:55.560 Thermodynamics is great. 00:41:55.560 --> 00:41:59.570 But even over a catalyst, this is a really, really, really 00:41:59.570 --> 00:42:01.230 slow reaction. 00:42:01.230 --> 00:42:04.590 We'd still be waiting here for Mr. Haber to produce his first 00:42:04.590 --> 00:42:08.190 mole of amine, if you were doing it, or ammonia if we 00:42:08.190 --> 00:42:10.040 were doing it at room temperature. 00:42:10.040 --> 00:42:13.770 It's just so slow. 00:42:13.770 --> 00:42:16.380 Thus, not at all practical. 00:42:16.380 --> 00:42:18.620 We're not going to run the world on room temperature 00:42:18.620 --> 00:42:23.840 Haber process. 00:42:23.840 --> 00:42:27.710 But it turns out, if you raise the temperature, kinetics is 00:42:27.710 --> 00:42:29.240 wonderful in terms of the temperature dependence. 00:42:29.240 --> 00:42:31.180 It's exponential. 00:42:31.180 --> 00:42:32.620 Arrhenius rate law. 00:42:32.620 --> 00:42:34.970 Great thing, you raise the temperature by a little bit. 00:42:34.970 --> 00:42:37.190 Rates speed up, things go faster. 00:42:37.190 --> 00:42:39.750 So if you were to raise the temperature from 298 degrees 00:42:39.750 --> 00:42:49.050 Kelvin to 800 degrees Kelvin, the rate speeds up. 00:42:49.050 --> 00:42:50.540 You're going to need some energy. 00:42:50.540 --> 00:42:53.620 As input here, to feed that. 00:42:53.620 --> 00:42:57.510 Hence the 1% energy use. 00:42:57.510 --> 00:42:58.750 Rate speeds up, that's great. 00:42:58.750 --> 00:43:00.030 Things happen faster. 00:43:00.030 --> 00:43:00.930 It becomes practical. 00:43:00.930 --> 00:43:04.600 But, what happens if you raise the temperature? 00:43:04.600 --> 00:43:04.910 Let's see. 00:43:04.910 --> 00:43:09.380 This is an endothermic, or exothermic, negative sign. 00:43:09.380 --> 00:43:10.740 And negative sign's exothermic. 00:43:10.740 --> 00:43:12.780 I raise the temperature, K goes down. 00:43:12.780 --> 00:43:14.590 I know how to calculate it here. 00:43:14.590 --> 00:43:17.090 And if I want to be super careful, because it's a fairly 00:43:17.090 --> 00:43:20.420 large temperature range, I can even use the exact form of the 00:43:20.420 --> 00:43:21.280 van 't Hoff equation. 00:43:21.280 --> 00:43:27.110 And what I find, if I do that, and putting the heat 00:43:27.110 --> 00:43:30.190 capacities for all these gases, I find 00:43:30.190 --> 00:43:32.150 that Kp does go down. 00:43:32.150 --> 00:43:34.310 In fact, it goes down quite a bit. 00:43:34.310 --> 00:43:39.050 It becomes 0.0007. 00:43:39.050 --> 00:43:40.580 Two zero's. 00:43:40.580 --> 00:43:43.350 Still really small. 00:43:43.350 --> 00:43:45.560 That's not practical. 00:43:45.560 --> 00:43:48.510 Not practical at all. 00:43:48.510 --> 00:43:49.440 No good. 00:43:49.440 --> 00:43:53.810 We can't run an industrial plant with this kind of yield. 00:43:53.810 --> 00:43:56.290 There's just no way it's going to work. 00:43:56.290 --> 00:43:59.190 So, you're a chemical engineer. 00:43:59.190 --> 00:44:01.350 Or a chemist, like Haber and Bosch were. 00:44:01.350 --> 00:44:05.510 And you're trying, you know by Le Chatelier, you know that it 00:44:05.510 --> 00:44:07.310 went in the wrong direction for you here. 00:44:07.310 --> 00:44:10.070 And then you look at your reaction and you say, how many 00:44:10.070 --> 00:44:12.180 moles of reactants do I have? 00:44:12.180 --> 00:44:15.860 3/2 plus 1/2, that's two moles of reactants. 00:44:15.860 --> 00:44:18.040 And I've got one mole of product. 00:44:18.040 --> 00:44:20.600 Two moles reactants, one mole of product. 00:44:20.600 --> 00:44:21.720 Two moles reactant... 00:44:21.720 --> 00:44:25.060 What happens if I change the pressure? 00:44:25.060 --> 00:44:27.040 If I change the pressure, if I increase the pressure, the 00:44:27.040 --> 00:44:29.480 system is going to say, no, I don't want 00:44:29.480 --> 00:44:31.050 the pressure increased. 00:44:31.050 --> 00:44:33.520 It's going to go to where there's less moles. 00:44:33.520 --> 00:44:36.070 And the less moles in the product area. 00:44:36.070 --> 00:44:37.420 It's going to go to my product. 00:44:37.420 --> 00:44:38.020 That's great. 00:44:38.020 --> 00:44:39.710 I've got to increase the pressure. 00:44:39.710 --> 00:44:40.220 Wonderful. 00:44:40.220 --> 00:44:41.710 Let's start increasing the pressure. 00:44:41.710 --> 00:44:43.710 Again, we need some energy to do that. 00:44:43.710 --> 00:44:46.380 We're going to go from one bar to some higher pressure. 00:44:46.380 --> 00:44:48.020 It's going to make our lives more complicated. 00:44:48.020 --> 00:44:49.390 The plant might explode now. 00:44:49.390 --> 00:44:50.890 If the pressure's too high. 00:44:50.890 --> 00:44:53.230 All sorts of problems going to come into play. 00:44:53.230 --> 00:44:56.660 But, let's do it. 00:44:56.660 --> 00:44:59.790 Let's increase the pressure. 00:44:59.790 --> 00:45:10.280 So, you increase the pressure from one bar to 100 bar. 00:45:10.280 --> 00:45:18.160 And you calculate Kx. 00:45:18.160 --> 00:45:20.420 Which is really what you want. 00:45:20.420 --> 00:45:24.730 So Kx, in this case here, is equal to p times Kp. 00:45:24.730 --> 00:45:25.770 Kp doesn't change. 00:45:25.770 --> 00:45:27.590 Kp doesn't care what the total pressure is. 00:45:27.590 --> 00:45:29.190 It's Kx that cares. 00:45:29.190 --> 00:45:32.800 At one bar, Kx is equal Kp. 00:45:32.800 --> 00:45:35.050 Kx is p to the minus delta nu. 00:45:35.050 --> 00:45:40.130 Number of moles of products minus the number of reactants. 00:45:40.130 --> 00:45:47.950 If I go from one bar to 100 bars, Kx goes from 0.007 to 00:45:47.950 --> 00:45:53.320 100 times 0.007, which is equal to 0.7. 00:45:53.320 --> 00:45:55.970 That's a lot better. 00:45:55.970 --> 00:45:58.340 Kx is the mole fraction or the, of 00:45:58.340 --> 00:46:00.860 products divided by reactants. 00:46:00.860 --> 00:46:09.220 And if I go to p is equal to 300 bars, then Kx goes to 2.1. 00:46:09.220 --> 00:46:11.960 three times 0.7. 00:46:11.960 --> 00:46:12.870 This is great. 00:46:12.870 --> 00:46:14.630 Now I'm really starting to make good products. 00:46:14.630 --> 00:46:16.670 But I've got to go to 300 bar. 00:46:16.670 --> 00:46:22.300 I've got to go to 300 bar, and 800 degrees Kelvin. 00:46:22.300 --> 00:46:25.500 That is incredibly energy-intensive. 00:46:25.500 --> 00:46:27.330 But it works. 00:46:27.330 --> 00:46:29.170 That's why Haber and Bosch made this work. 00:46:29.170 --> 00:46:32.500 And why Germany stayed in the war longer than after 1916. 00:46:32.500 --> 00:46:35.180 The first world war. 00:46:35.180 --> 00:46:39.840 Ended in 1918. 00:46:39.840 --> 00:46:41.240 Nobel Prizes. 00:46:41.240 --> 00:46:44.910 Merck, Bayer, all these german companies. 00:46:44.910 --> 00:46:48.770 Because they figured how to do this at high pressure and high 00:46:48.770 --> 00:46:49.510 temperature. 00:46:49.510 --> 00:46:54.450 Without blowing everything up. 00:46:54.450 --> 00:46:55.910 OK. 00:46:55.910 --> 00:47:01.710 Any questions? 00:47:01.710 --> 00:47:02.000 Alright. 00:47:02.000 --> 00:47:07.560 The last topic is, so far we've seen equilibria where 00:47:07.560 --> 00:47:09.770 you had things that were well mixed. 00:47:09.770 --> 00:47:11.470 equilibria of ideal gases. 00:47:11.470 --> 00:47:14.730 Or in solutions, where your solutes, your solute molecules 00:47:14.730 --> 00:47:17.440 are mixing around. 00:47:17.440 --> 00:47:20.590 And the entropy of mixing was really super important. 00:47:20.590 --> 00:47:23.260 Our curve, our going down for delta G, was all because of 00:47:23.260 --> 00:47:24.440 the entropy of mixing. 00:47:24.440 --> 00:47:28.610 Now, suppose that I have a heterogeneous mixture. 00:47:28.610 --> 00:47:32.230 I've got some solids or some pure liquids that are refusing 00:47:32.230 --> 00:47:35.930 to share their environment. 00:47:35.930 --> 00:47:38.680 And staying as pure materials. 00:47:38.680 --> 00:47:43.870 So, for instance, if I have a beaker with some solid 00:47:43.870 --> 00:47:45.770 reactant on the bottom here. 00:47:45.770 --> 00:47:55.080 And the products are in solution. 00:47:55.080 --> 00:47:57.440 How do I deal with that equilibrium? 00:47:57.440 --> 00:48:00.030 Well, you know the answer, but let's just do it out again. 00:48:00.030 --> 00:48:04.470 So we're going to have nu A moles of A, of solid. 00:48:04.470 --> 00:48:06.270 Not mixed in the solution. 00:48:06.270 --> 00:48:08.400 Let's say we have multiple phases here. 00:48:08.400 --> 00:48:10.980 Nu B moles of B, which is a gas. 00:48:10.980 --> 00:48:13.760 Instead of a solution, let's do a gas phase reaction. 00:48:13.760 --> 00:48:18.540 Nu C moles of C, which is a pure liquid. 00:48:18.540 --> 00:48:22.020 And nu D moles of D, which is a gas. 00:48:22.020 --> 00:48:23.280 So these two gases can mix. 00:48:23.280 --> 00:48:26.580 But the pure solid and the pure liquid can't mix. 00:48:26.580 --> 00:48:28.650 So let's think again, where does the equilibrium 00:48:28.650 --> 00:48:29.460 constant come from? 00:48:29.460 --> 00:48:34.310 It comes from looking at this delta G of the mixture and 00:48:34.310 --> 00:48:36.210 letting it react a little bit more. 00:48:36.210 --> 00:48:39.905 And taking out these chemical potentials for the species and 00:48:39.905 --> 00:48:40.750 the mixture. 00:48:40.750 --> 00:48:46.460 Expanding it out in terms of log p or log concentration. 00:48:46.460 --> 00:48:48.360 So we need to have this delta G in. 00:48:48.360 --> 00:48:51.100 Let's take epsilon equal to one, to make our 00:48:51.100 --> 00:48:51.800 life simpler here. 00:48:51.800 --> 00:48:57.740 So now we have nu C mu C of the pure. 00:48:57.740 --> 00:48:59.350 The pure solid. 00:48:59.350 --> 00:49:04.740 Plus nu D mu C of the gas. 00:49:04.740 --> 00:49:07.150 Which is in the mixture. 00:49:07.150 --> 00:49:14.220 Minus nu A mu A of the pure liquid, minus nu 00:49:14.220 --> 00:49:17.860 B mu B of the gas. 00:49:17.860 --> 00:49:20.200 Which is in the mixture. 00:49:20.200 --> 00:49:24.190 That's what this delta G is, when we allow the reaction to 00:49:24.190 --> 00:49:27.380 proceed for a little bit more. 00:49:27.380 --> 00:49:29.300 We add a little bit of chemical 00:49:29.300 --> 00:49:30.890 potentials from the products. 00:49:30.890 --> 00:49:32.210 Subtract a little bit of chemical 00:49:32.210 --> 00:49:34.370 potential from the reactants. 00:49:34.370 --> 00:49:37.330 And then we expand it out in terms of the standard chemical 00:49:37.330 --> 00:49:39.950 potentials for everything being pure. 00:49:39.950 --> 00:49:41.720 Entropy of mixing comes in here. 00:49:41.720 --> 00:49:42.610 Comes in here. 00:49:42.610 --> 00:49:46.300 Delta G of mixing comes in here. 00:49:46.300 --> 00:49:54.860 And we end up with something that looks like nu C mu C 00:49:54.860 --> 00:50:02.760 naught, plus nu D mu D naught, minus nu A mu A naught, minus 00:50:02.760 --> 00:50:06.670 nu B mu B naught. 00:50:06.670 --> 00:50:11.730 Plus RT log, and the only place where we have these log 00:50:11.730 --> 00:50:14.590 p's, or log concentration coming in, is for those 00:50:14.590 --> 00:50:17.640 species that were not pure. 00:50:17.640 --> 00:50:19.930 And those are only these two guys here. 00:50:19.930 --> 00:50:22.340 The ones that are in the gas phase. 00:50:22.340 --> 00:50:26.820 D and B. So we end up with partial pressure of D to the 00:50:26.820 --> 00:50:27.770 nu D power. 00:50:27.770 --> 00:50:32.500 Partial pressure of B, to the nu B power. 00:50:32.500 --> 00:50:34.560 The other two species don't come in there. 00:50:34.560 --> 00:50:36.630 Because they started out as pure. 00:50:36.630 --> 00:50:39.360 There's no mixing going on. 00:50:39.360 --> 00:50:44.060 And there's no expansion of the log for them. 00:50:44.060 --> 00:50:49.985 And so now, when we look at the Q for the reaction, the 00:50:49.985 --> 00:50:52.010 reaction quotient, it doesn't contain 00:50:52.010 --> 00:50:53.050 any of the pure species. 00:50:53.050 --> 00:50:56.950 It only contains those species that are allowed to mix. 00:50:56.950 --> 00:51:00.440 Those that are in the gas phase or in solution. 00:51:00.440 --> 00:51:08.350 And so K, then, for this reaction only takes in those 00:51:08.350 --> 00:51:14.780 products like D. Or reactants like B, which 00:51:14.780 --> 00:51:16.380 are in the gas phase. 00:51:16.380 --> 00:51:18.170 The pure solids or pure liquids don't come in. 00:51:18.170 --> 00:51:19.640 They come in for delta G naught. 00:51:19.640 --> 00:51:23.090 There's delta G naught sitting right here. 00:51:23.090 --> 00:51:26.850 Delta G naught for the reaction is 00:51:26.850 --> 00:51:28.650 sitting right there. 00:51:28.650 --> 00:51:32.365 So when you write your log K is equal to minus delta G 00:51:32.365 --> 00:51:36.940 naught over RT, the delta G naught has everything in it. 00:51:36.940 --> 00:51:40.710 The pure stuff, the solution stuff, the gas phase stuff. 00:51:40.710 --> 00:51:44.570 But the K only has the gas phase and solution stuff. 00:51:44.570 --> 00:51:49.560 Alright, any questions? 00:51:49.560 --> 00:51:49.760 Good. 00:51:49.760 --> 00:51:51.900 Next time we'll do an example. 00:51:51.900 --> 00:51:54.130 And then we'll go on phase transitions.