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 --> 00:00:06.850 Your support will help MIT OpenCourseWare continue to 00:00:06.850 --> 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 --> 00:00:17.150 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.150 --> 00:00:20.320 ocw.mit.edu. 00:00:20.320 --> 00:00:24.610 PROFESSOR: Alright, so last time we started talking about 00:00:24.610 --> 00:00:30.000 more complicated mixtures. 00:00:30.000 --> 00:00:32.500 To get into colligative properties. 00:00:32.500 --> 00:00:41.540 And the example we gave was a binary system that has two 00:00:41.540 --> 00:00:43.880 components. 00:00:43.880 --> 00:00:49.200 And two phases. 00:00:49.200 --> 00:00:54.010 So we have a liquid phase on the bottom. 00:00:54.010 --> 00:00:59.470 A gas phase on top, and two components, A and B, where 00:00:59.470 --> 00:01:05.520 compositions xA and xB are in the liquid phase. 00:01:05.520 --> 00:01:06.810 Those are the mole fractions. 00:01:06.810 --> 00:01:14.955 And yA and yB in the gas phase, the mole fractions in 00:01:14.955 --> 00:01:17.290 the gas phase. 00:01:17.290 --> 00:01:22.860 And what we did last time was to begin to say how many 00:01:22.860 --> 00:01:25.410 degrees of freedom do we have, how many variables do we need 00:01:25.410 --> 00:01:28.220 to completely describe this mixture. 00:01:28.220 --> 00:01:31.940 And it turned out that for this mixture, we only needed 00:01:31.940 --> 00:01:32.830 two variables. 00:01:32.830 --> 00:01:34.910 Two degrees of freedom. 00:01:34.910 --> 00:01:36.970 We started out with four variables; the temperature, 00:01:36.970 --> 00:01:48.090 the pressure, and the components in the composition 00:01:48.090 --> 00:01:51.010 in the liquid phase and the composition in the gas phase. 00:01:51.010 --> 00:01:54.670 We need one of these, either xA or xB, and one of these, 00:01:54.670 --> 00:01:57.900 either yA or yB because they add up to 1. 00:01:57.900 --> 00:02:01.440 So you start with four, but because you are in a mixture, 00:02:01.440 --> 00:02:03.180 an equilibrium, you have constraints. 00:02:03.180 --> 00:02:10.950 You have the chemical potentials that have to be 00:02:10.950 --> 00:02:11.580 equal to each other. 00:02:11.580 --> 00:02:14.170 The chemical potential of A in the gas phase has to be equal 00:02:14.170 --> 00:02:16.870 to the chemical potential of A in the liquid phase. 00:02:16.870 --> 00:02:19.590 And the chemical potential of B in the gas phase has to B 00:02:19.590 --> 00:02:21.512 equal to the chemical potential of B 00:02:21.512 --> 00:02:22.160 in the liquid phase. 00:02:22.160 --> 00:02:25.540 So you start with four variables. 00:02:25.540 --> 00:02:32.090 Temperature, pressure, xA, yA, then you have the constraints 00:02:32.090 --> 00:02:38.890 that mu A, in the liquid phase has to be equal to mu 00:02:38.890 --> 00:02:40.860 A in the gas phase. 00:02:40.860 --> 00:02:42.430 Because you're at equilibrium. 00:02:42.430 --> 00:02:46.082 And mu B in the liquid phase has to be equal to mu 00:02:46.082 --> 00:02:48.790 B in the gas phase. 00:02:48.790 --> 00:02:51.580 So four minus two constraints means you have 00:02:51.580 --> 00:02:54.270 two degrees of freedom. 00:02:54.270 --> 00:02:58.440 And so if you want to know everything about this mixture, 00:02:58.440 --> 00:03:01.490 all you need is the temperature 00:03:01.490 --> 00:03:02.570 and the total pressure. 00:03:02.570 --> 00:03:02.960 And that's enough. 00:03:02.960 --> 00:03:05.017 It's kind of amazing. 00:03:05.017 --> 00:03:09.430 Give me the temperature and pressure and these two 00:03:09.430 --> 00:03:12.390 components, like water and alcohol, for instance, and I 00:03:12.390 --> 00:03:17.230 can tell you everything about the compositions. 00:03:17.230 --> 00:03:18.410 It's very powerful. 00:03:18.410 --> 00:03:20.950 And then we said that this turned out to be a special 00:03:20.950 --> 00:03:26.290 case, or sub-case of the more general Gibbs phase rule. 00:03:26.290 --> 00:03:37.380 where the Gibbs phase rule tells you that if you have C 00:03:37.380 --> 00:03:40.670 components, in this case here we have two, and the general 00:03:40.670 --> 00:03:45.900 case of C components with p phases, the number of degrees 00:03:45.900 --> 00:03:47.730 of freedom is C minus p plus two. 00:03:47.730 --> 00:03:54.050 This is components. 00:03:54.050 --> 00:03:57.100 And phases. 00:03:57.100 --> 00:03:59.250 And where we're going to start today is by proving this. 00:03:59.250 --> 00:04:04.450 And then we'll do more with this problem here. 00:04:04.450 --> 00:04:08.720 So the first thing is for us to prove Gibbs' phase rule. 00:04:08.720 --> 00:04:10.190 So Gibbs phase rule. 00:04:10.190 --> 00:04:13.810 We start out with all the variables that we have. 00:04:13.810 --> 00:04:16.750 And then we add the constraints. 00:04:16.750 --> 00:04:20.480 And for the Gibbs phase rule, then we start with the two 00:04:20.480 --> 00:04:22.950 knobs that we can turn externally. 00:04:22.950 --> 00:04:24.720 Which are the temperature and the pressure. 00:04:24.720 --> 00:04:27.090 So we start with temperature and pressure, as two variables 00:04:27.090 --> 00:04:28.550 that we have. 00:04:28.550 --> 00:04:29.770 And then we have a bunch of phases. 00:04:29.770 --> 00:04:30.940 We have C phases. 00:04:30.940 --> 00:04:34.210 And in each phase we have to describe the composition in 00:04:34.210 --> 00:04:36.270 that phase. 00:04:36.270 --> 00:04:40.850 So, for each phase alpha, we have to 00:04:40.850 --> 00:04:44.670 describe its mole fraction. 00:04:44.670 --> 00:04:48.760 So here we needed to describe xA for phase A. And yA for 00:04:48.760 --> 00:04:53.790 phase A. For component A in the gas phase. 00:04:53.790 --> 00:04:59.350 So now we have alpha components. 00:04:59.350 --> 00:05:05.860 And for each component we have to describe its mole fraction 00:05:05.860 --> 00:05:07.400 in that particular phase. 00:05:07.400 --> 00:05:12.870 So we have to describe x sub i. 00:05:12.870 --> 00:05:15.040 But we have a constraint on that. 00:05:15.040 --> 00:05:18.240 The constraint is that the sum of all the mole factions has 00:05:18.240 --> 00:05:19.140 to be equal to one. 00:05:19.140 --> 00:05:20.780 That's what it is, to be a mole fraction. 00:05:20.780 --> 00:05:25.745 So the constraint is that x sub i, i from one to alpha, is 00:05:25.745 --> 00:05:30.510 equal to one. 00:05:30.510 --> 00:05:36.530 So the constraint on the number of the components, 00:05:36.530 --> 00:05:42.470 instead of having, actually, i goes from one to C, because 00:05:42.470 --> 00:05:46.350 that's the number components. 00:05:46.350 --> 00:05:51.210 So instead of having C different compositions that we 00:05:51.210 --> 00:05:54.100 need to take care of, we only need C minus one. 00:05:54.100 --> 00:05:56.530 Just like here we only needed one, just xA. 00:05:56.530 --> 00:05:59.190 We didn't need both xA and xB, because we knew the sum was 00:05:59.190 --> 00:06:00.310 equal to one. 00:06:00.310 --> 00:06:05.320 So that means that we really need C minus one variables to 00:06:05.320 --> 00:06:10.510 describe the composition in one of the phases. 00:06:10.510 --> 00:06:16.320 But we have p phases, so let me fix this. 00:06:16.320 --> 00:06:19.610 We had C components. 00:06:19.610 --> 00:06:21.000 We have p phases. 00:06:21.000 --> 00:06:24.170 And so for each phase, we have to describe the composition. 00:06:24.170 --> 00:06:27.850 So that means that we need p times C minus one variables to 00:06:27.850 --> 00:06:31.450 describe all the components and all the phases, 00:06:31.450 --> 00:06:33.820 plus these extra two. 00:06:33.820 --> 00:06:38.390 So the total number variables then that we start out with 00:06:38.390 --> 00:06:41.990 before putting the fact that we're in equilibrium, is going 00:06:41.990 --> 00:06:47.250 to be two plus p times C minus one. 00:06:47.250 --> 00:06:49.790 So that's a basic description of the system. 00:06:49.790 --> 00:06:51.220 Then we put in the same constraints 00:06:51.220 --> 00:06:52.330 that we had up here. 00:06:52.330 --> 00:06:58.150 In terms of the chemical potentials. 00:06:58.150 --> 00:07:00.570 If I take any of the components, let's take the 00:07:00.570 --> 00:07:03.240 alpha component, or the i'th component. 00:07:03.240 --> 00:07:07.970 Because it's in equilibrium, the chemical potential of that 00:07:07.970 --> 00:07:12.350 particular component has to be the same in all the phases. 00:07:12.350 --> 00:07:14.660 So let's take the i'th component. 00:07:14.660 --> 00:07:21.510 So yi in phase one has to be equal to yi in phase two, has 00:07:21.510 --> 00:07:25.720 to be equal to yi in phase three, has to be equal to yi 00:07:25.720 --> 00:07:33.350 in phase p. 00:07:33.350 --> 00:07:35.660 All the chemical potentials have to be the same across all 00:07:35.660 --> 00:07:37.370 the phases for that component. 00:07:37.370 --> 00:07:42.220 And if you count the number of equations, the number of 00:07:42.220 --> 00:07:47.240 equals signs here, that's p minus one. p minus one equals 00:07:47.240 --> 00:07:50.330 signs, constraints due to the fact that I have an 00:07:50.330 --> 00:07:51.220 equilibrium here. 00:07:51.220 --> 00:08:03.630 So this gives me p minus one constraints. 00:08:03.630 --> 00:08:05.640 And this is true for every single component. 00:08:05.640 --> 00:08:08.950 Every single component has to have its chemical potential, 00:08:08.950 --> 00:08:12.240 its chemical potential equal throughout the phases. 00:08:12.240 --> 00:08:15.480 So p minus one's constraint for one component times C 00:08:15.480 --> 00:08:16.000 components. 00:08:16.000 --> 00:08:26.100 So for a total of C times p minus one total constraints. 00:08:26.100 --> 00:08:29.200 My variables, my constraints. 00:08:29.200 --> 00:08:36.680 So the number of degrees of freedom then becomes p times C 00:08:36.680 --> 00:08:49.180 minus one minus C times p minus one and then plus two. 00:08:49.180 --> 00:08:52.520 And if you multiply this out, p times C gets rid of the p 00:08:52.520 --> 00:08:53.780 times C here. 00:08:53.780 --> 00:08:58.670 And you get the plus C minus p plus two. 00:08:58.670 --> 00:09:02.320 Which is the Gibbs phase rule, right? 00:09:02.320 --> 00:09:05.630 Which I'm going to bring back up here. 00:09:05.630 --> 00:09:08.020 C minus p plus two. 00:09:08.020 --> 00:09:12.750 So just purely in accounting, just accounting for the 00:09:12.750 --> 00:09:16.510 variables and the constraints. 00:09:16.510 --> 00:09:23.430 OK, any questions on how we did this? 00:09:23.430 --> 00:09:26.090 Great. 00:09:26.090 --> 00:09:26.710 OK. 00:09:26.710 --> 00:09:33.240 So let's move on, then, to these ideal solutions. 00:09:33.240 --> 00:09:35.110 Which is this case right here. 00:09:35.110 --> 00:09:38.070 And the first thing we're going to do is look at the 00:09:38.070 --> 00:09:47.470 case where we have A is a solvent, and it's volatile. 00:09:47.470 --> 00:09:50.450 And B is a solute, which is non-volatile. 00:09:50.450 --> 00:09:53.520 So let's take the first case. 00:09:53.520 --> 00:10:00.240 We have A volatile, that's a solvent. 00:10:00.240 --> 00:10:01.700 Could be water. 00:10:01.700 --> 00:10:05.440 And B is going to be the solute. 00:10:05.440 --> 00:10:09.880 It's non-volatile. 00:10:09.880 --> 00:10:12.750 And let's interchange solute or non-volatile to be 00:10:12.750 --> 00:10:14.600 consistent here. 00:10:14.600 --> 00:10:18.700 So this could be sugar. 00:10:18.700 --> 00:10:27.250 A sweet water. 00:10:27.250 --> 00:10:41.050 So, Mr. Raoult looked at such solutions, and found that they 00:10:41.050 --> 00:10:46.470 could be well described by the following diagram here. 00:10:46.470 --> 00:10:51.300 So if I plot on the x-axis here, from zero to one, I'm 00:10:51.300 --> 00:10:57.750 going to plot the partial molar fraction of the solute. 00:10:57.750 --> 00:11:00.010 Partial molar fractions of the solute, that's going to be xB. 00:11:02.530 --> 00:11:06.740 So when xB equals to zero, I have a pure water solution. 00:11:06.740 --> 00:11:11.370 When xB is equal to one, I have pure sugar. 00:11:11.370 --> 00:11:17.990 Then on the y axis, I'm going to plot the total pressure. 00:11:17.990 --> 00:11:20.410 So when I have a pure water solution, the total pressure 00:11:20.410 --> 00:11:26.660 is going to be the same as the vapor pressure of water at a 00:11:26.660 --> 00:11:28.150 particular temperature. 00:11:28.150 --> 00:11:36.570 Let's take temperature fixed at some value. 00:11:36.570 --> 00:11:39.840 Temperature is fixed at some value. 00:11:39.840 --> 00:11:44.320 I'm going to have the total pressure is going to be the 00:11:44.320 --> 00:11:45.970 partial pressure of the water. 00:11:45.970 --> 00:11:49.290 And clearly, when I don't have any water left, all I have is 00:11:49.290 --> 00:11:51.390 the sugar, there's no pressure. 00:11:51.390 --> 00:11:54.440 We're going to assume that it's totally non-volatile. 00:11:54.440 --> 00:11:56.000 Obviously there's going to be a little bit of vapor 00:11:56.000 --> 00:11:58.190 pressure, even at room temperature. 00:11:58.190 --> 00:12:01.980 Ten to the minus, I don't know, 13 torr or something. 00:12:01.980 --> 00:12:05.340 But anyway, it's basically zero here. 00:12:05.340 --> 00:12:07.680 And what Raoult said is that while these two points are 00:12:07.680 --> 00:12:13.430 just connected by a straight line. 00:12:13.430 --> 00:12:19.800 Simplest way to connect two points. 00:12:19.800 --> 00:12:27.370 Basically, he said that the pressure of, the vapor 00:12:27.370 --> 00:12:32.450 pressure of the water in this case, is equal 00:12:32.450 --> 00:12:38.310 to xA times pA star. 00:12:38.310 --> 00:12:42.680 I could also plot this with xA, going from one to zero, 00:12:42.680 --> 00:12:49.430 since xA plus xB is equal to one. 00:12:49.430 --> 00:12:56.160 Or I could write this as one minus xB times pA star. 00:12:56.160 --> 00:12:58.890 Where a star, the star, means pure water. 00:12:58.890 --> 00:13:06.660 So pure A. And pA is the vapor pressure of the mixture. 00:13:06.660 --> 00:13:09.000 Which in this case here is the total pressure, because I 00:13:09.000 --> 00:13:12.630 don't have anything else that's volatile. 00:13:12.630 --> 00:13:29.240 And this is Raoult's law. 00:13:29.240 --> 00:13:32.930 And the first thing that you can use this for is to look at 00:13:32.930 --> 00:13:35.780 your first colligative property which is vapor 00:13:35.780 --> 00:13:45.900 pressure lowering. 00:13:45.900 --> 00:13:48.120 And let me point out another thing first. 00:13:48.120 --> 00:13:51.940 So if a solution obeys that property here, it's called an 00:13:51.940 --> 00:13:53.230 ideal solution. 00:13:53.230 --> 00:13:56.470 So if the water-sugar solution really did behave like a 00:13:56.470 --> 00:13:58.290 straight line like this, as you increase the amount of 00:13:58.290 --> 00:14:00.190 sugar, then it would be an ideal solution. 00:14:00.190 --> 00:14:02.270 Many things are very close to it. 00:14:02.270 --> 00:14:07.230 It's not such a bad approximation. 00:14:07.230 --> 00:14:09.490 Especially in that region up here. 00:14:09.490 --> 00:14:12.015 Around zero, when the amount of sugar is pretty small. 00:14:12.015 --> 00:14:19.070 You expect to obey an ideal solution behavior. 00:14:19.070 --> 00:14:24.320 Alright, so let's look now at a glass of water that has 00:14:24.320 --> 00:14:26.150 sugar in it. 00:14:26.150 --> 00:14:28.320 We've got my glass of water, H20. 00:14:28.320 --> 00:14:34.640 With some sugar dissolved in it. 00:14:34.640 --> 00:14:42.260 And it's got a certain vapor pressure on top. pA, or pH2O. 00:14:42.260 --> 00:14:43.590 And I'm going to ask the question, what's the 00:14:43.590 --> 00:14:48.950 difference between the vapor pressure what it would be 00:14:48.950 --> 00:14:51.370 without the sugar in it. 00:14:51.370 --> 00:14:59.670 What is pA star minus pA? 00:14:59.670 --> 00:15:07.230 Obeys Raoult's law. pA star, pA is xA pA star, 00:15:07.230 --> 00:15:09.230 plug that in here. 00:15:09.230 --> 00:15:15.900 That's one minus xA pA star, one minus xA is xB. 00:15:15.900 --> 00:15:18.410 xB pA star. 00:15:18.410 --> 00:15:21.010 Well, the first thing I know is that this is a positive 00:15:21.010 --> 00:15:23.573 number. xB is a positive number. pA star 00:15:23.573 --> 00:15:25.020 is a positive number. 00:15:25.020 --> 00:15:26.500 This is greater than zero. 00:15:26.500 --> 00:15:31.050 That means that the act of putting some solute in my 00:15:31.050 --> 00:15:37.540 water decreases the vapor pressure of the 00:15:37.540 --> 00:15:40.450 water in the gas phase. 00:15:40.450 --> 00:15:41.880 Pretty simple. 00:15:41.880 --> 00:15:47.130 It's called vapor pressure law. 00:15:47.130 --> 00:15:49.550 And we can actually take this one step further. 00:15:49.550 --> 00:15:57.510 We can draw a phase diagram. 00:15:57.510 --> 00:15:59.990 So our usual phase diagram here with temperature on this 00:15:59.990 --> 00:16:04.470 axis here and pressure on this axis here, and we've got our 00:16:04.470 --> 00:16:09.390 triple point sitting here. 00:16:09.390 --> 00:16:13.280 And our gas-liquid line, with a critical point here. 00:16:13.280 --> 00:16:16.770 So this is gas here, there's a liquid here. 00:16:16.770 --> 00:16:21.430 And we've got our, this is going to be water. 00:16:21.430 --> 00:16:23.690 So it's got a negative slope. 00:16:23.690 --> 00:16:26.550 I think I've got it right this time. 00:16:26.550 --> 00:16:29.400 And there's the solid phase here. 00:16:29.400 --> 00:16:31.420 OK, this is for the pure stuff. 00:16:31.420 --> 00:16:40.770 So now I can draw a similar diagram for sugared water. 00:16:40.770 --> 00:16:43.500 And I just looked at this, assuming it 00:16:43.500 --> 00:16:44.480 obeys Raoult's law. 00:16:44.480 --> 00:16:50.870 What we looked at here was the gas-liquid coexistence. 00:16:50.870 --> 00:16:55.350 And we found that the gas-liquid coexistence point, 00:16:55.350 --> 00:16:58.290 the vapor pressure of the water, was less 00:16:58.290 --> 00:16:59.740 than if it were pure. 00:16:59.740 --> 00:17:02.670 So that means that if I looked at the gas-liquid coexistence, 00:17:02.670 --> 00:17:05.110 which is this line right here. 00:17:05.110 --> 00:17:08.710 In the presence of sugar, that whole line is 00:17:08.710 --> 00:17:11.620 going to get lowered. 00:17:11.620 --> 00:17:14.140 The pressure, the vapor pressure, of the water is 00:17:14.140 --> 00:17:17.400 going to be lower when the sugar is there. 00:17:17.400 --> 00:17:26.770 So the whole thing is going to get lower. 00:17:26.770 --> 00:17:32.325 Now, if I have a solid, if the water is a solid, it's going 00:17:32.325 --> 00:17:34.730 to crystallize the water, the sugar is going to be separated 00:17:34.730 --> 00:17:36.080 from the water. 00:17:36.080 --> 00:17:38.250 The sugar and the water are not going to really know about 00:17:38.250 --> 00:17:40.700 each other. 00:17:40.700 --> 00:17:44.840 They're just going to have big chunks of pure water, with 00:17:44.840 --> 00:17:49.430 every now and then a molecule of sugar that's in there. 00:17:49.430 --> 00:17:53.680 So you don't really expect to see any difference between the 00:17:53.680 --> 00:17:58.980 pure solid water, in terms of the gas-solid interface, and 00:17:58.980 --> 00:18:02.290 the pure liquid water. 00:18:02.290 --> 00:18:06.120 There's no sugar in the gas phase. 00:18:06.120 --> 00:18:11.130 And as far as the gas phase is concerned, that solid phase is 00:18:11.130 --> 00:18:11.820 pure water. 00:18:11.820 --> 00:18:14.230 It's just seeing pure ice crystals on the surface. 00:18:14.230 --> 00:18:17.400 Every now and then there's a molecule of sugar. 00:18:17.400 --> 00:18:22.970 So you expect this to be pretty close here. 00:18:22.970 --> 00:18:24.850 So you expect the triple point to go down. 00:18:24.850 --> 00:18:30.246 And if I now, I go up to the solid-liquid coexistence, and 00:18:30.246 --> 00:18:34.470 go up like this, I can make a prediction about solid-liquid 00:18:34.470 --> 00:18:36.950 coexistence. 00:18:36.950 --> 00:18:42.380 I can predict that if I'm looking at, let's put one bar 00:18:42.380 --> 00:18:49.530 here somewhere. 00:18:49.530 --> 00:18:52.050 One bar. 00:18:52.050 --> 00:18:53.570 Room temperature, sitting here somewhere. 00:18:53.570 --> 00:18:57.360 One bar, room temperature, water is all liquid. 00:18:57.360 --> 00:19:03.720 This is what it would usually, solid, liquid, that would be 00:19:03.720 --> 00:19:06.510 273 degrees Kelvin. 00:19:06.510 --> 00:19:10.570 That's the melting point of water. 00:19:10.570 --> 00:19:14.090 So now if I put sugar in the water, and I follow my diagram 00:19:14.090 --> 00:19:21.140 here, what I find is that the new melting point of water is 00:19:21.140 --> 00:19:29.850 some new temperature T, T1, where T1 is less than 273 00:19:29.850 --> 00:19:30.094 degrees Kelvin. 00:19:30.094 --> 00:19:31.090 You all know this. 00:19:31.090 --> 00:19:34.640 When you put something, like an impurity, a salt or sugar 00:19:34.640 --> 00:19:37.830 in the water, the melting point gets depressed. 00:19:37.830 --> 00:19:43.110 It goes down. 00:19:43.110 --> 00:19:44.030 Straight from here. 00:19:44.030 --> 00:19:46.820 You can build it up straight from here. 00:19:46.820 --> 00:19:51.850 When we get to the colligative properties, we'll attack it 00:19:51.850 --> 00:19:55.740 from the point of chemical potentials. 00:19:55.740 --> 00:19:57.620 We'll equate chemical potentials and we'll look at 00:19:57.620 --> 00:19:59.990 the change in the chemical potential in the water, and 00:19:59.990 --> 00:20:02.090 the presence of the sugar and we'll do it rigorously. 00:20:02.090 --> 00:20:05.010 But this is sort of the first indication that these 00:20:05.010 --> 00:20:08.460 colligative properties are all connected to each other. 00:20:08.460 --> 00:20:13.270 They're connected to this diagram here. 00:20:13.270 --> 00:20:14.360 OK. 00:20:14.360 --> 00:20:19.940 So when I was a kid, and I knew that this was the case. 00:20:19.940 --> 00:20:21.380 Everybody knows this is the case, right? 00:20:21.380 --> 00:20:24.536 You put salt in the water and you expect all sorts 00:20:24.536 --> 00:20:25.180 of things to happen. 00:20:25.180 --> 00:20:27.720 You expect the vapor pressure to go down. 00:20:27.720 --> 00:20:31.273 And certainly you know that you add salt to the roads and 00:20:31.273 --> 00:20:33.620 the water melts, et cetera. 00:20:33.620 --> 00:20:36.850 And my mother always told me that the reason why you added 00:20:36.850 --> 00:20:42.360 salt to the water was so that the temperature of the water 00:20:42.360 --> 00:20:46.050 would get higher when you try to boil pasta. 00:20:46.050 --> 00:20:50.770 And for, I don't know, fifteen years I believed her. 00:20:50.770 --> 00:20:53.110 Until I taught thermodynamics. 00:20:53.110 --> 00:20:58.010 And I actually calculated the change in the temperature by 00:20:58.010 --> 00:21:01.020 adding a few pinches of salt to the water. 00:21:01.020 --> 00:21:04.730 And you know how small that temperature change is? 00:21:04.730 --> 00:21:05.960 You should calculate it. 00:21:05.960 --> 00:21:07.880 It's really tiny. 00:21:07.880 --> 00:21:09.920 It won't make any difference for the water. 00:21:09.920 --> 00:21:12.370 Probably the difference in the temperature of the water will 00:21:12.370 --> 00:21:16.040 be the same as if I cooked here versus halfway up Mount 00:21:16.040 --> 00:21:18.460 Washington in New Hampshire or something. 00:21:18.460 --> 00:21:19.220 Probably even less. 00:21:19.220 --> 00:21:20.875 There's no difference. 00:21:20.875 --> 00:21:23.820 The reason why you add salt is so that it tastes better. 00:21:23.820 --> 00:21:24.530 That's the only reason. 00:21:24.530 --> 00:21:28.610 It has nothing to do with thermodynamics. 00:21:28.610 --> 00:21:35.730 Alright, any questions on this? 00:21:35.730 --> 00:21:37.240 So we going to make it a little bit more complicated 00:21:37.240 --> 00:21:38.090 now, if there's no questions. 00:21:38.090 --> 00:21:43.070 We're going to now have, instead of having one 00:21:43.070 --> 00:21:47.080 non-volatile solute, use we're going to have two, we're going 00:21:47.080 --> 00:21:47.730 to have this mixture. 00:21:47.730 --> 00:21:58.580 Where both components are volatile. 00:21:58.580 --> 00:22:03.230 Can we erase this? 00:22:03.230 --> 00:22:03.950 Actually, I may not erase this. 00:22:03.950 --> 00:22:12.140 Because I want to keep Raoult's law here. 00:22:12.140 --> 00:22:16.000 OK, we're going to have a mixture of two volatile 00:22:16.000 --> 00:22:16.550 components. 00:22:16.550 --> 00:22:19.960 For instance, water and alcohol. 00:22:19.960 --> 00:22:25.510 Your your typical martini type situation. 00:22:25.510 --> 00:22:26.660 Or margarita. 00:22:26.660 --> 00:22:40.370 Whatever's your favorite drink. 00:22:40.370 --> 00:22:47.840 So now we have xA, we have xB, we have yA, we have yB. 00:22:47.840 --> 00:23:00.430 And we're going to assume that both obey Raoult's law with 00:23:00.430 --> 00:23:02.590 respect to each other. 00:23:02.590 --> 00:23:08.100 So now if I draw a diagram, and I just focus on, so 00:23:08.100 --> 00:23:12.080 there's the total pressure sitting here. 00:23:12.080 --> 00:23:14.500 And I just focus on one component 00:23:14.500 --> 00:23:15.400 and ignore the other. 00:23:15.400 --> 00:23:20.800 Suppose I focus on A. And I know that A in the presence of 00:23:20.800 --> 00:23:25.590 B, the partial pressure of A is going to obey Raoult's law. 00:23:25.590 --> 00:23:28.563 And let's take A to be the more volatile component in 00:23:28.563 --> 00:23:30.240 this case here. 00:23:30.240 --> 00:23:35.770 So pA star is bigger than pB star. 00:23:35.770 --> 00:23:39.820 The partial pressure, pure A, had a particular temperature, 00:23:39.820 --> 00:23:46.090 T fixed is bigger than pB star, it's the more volatile 00:23:46.090 --> 00:23:48.490 of the two. 00:23:48.490 --> 00:23:56.980 So if I were to plot pA as a function of xB, going from 00:23:56.980 --> 00:23:59.820 zero to one, so it'll be Raoult's law, I'm going to 00:23:59.820 --> 00:24:04.360 start at pA star. 00:24:04.360 --> 00:24:09.680 And I'm going to go down linearly to xB equals one. 00:24:09.680 --> 00:24:12.445 Now I can do the same thing for the partial pressure of B. 00:24:12.445 --> 00:24:19.100 And assume that in the presence of A, just looking at 00:24:19.100 --> 00:24:21.200 this component B, now, and I'm looking at the partial 00:24:21.200 --> 00:24:24.530 pressure of B, and it's got its impurity in it, A. B obeys 00:24:24.530 --> 00:24:27.450 Raoult's law with A in there. 00:24:27.450 --> 00:24:34.080 But now if I want to keep the same x axis here, I can look 00:24:34.080 --> 00:24:36.920 at xA here, going from zero to one. 00:24:36.920 --> 00:24:40.310 So I just basically flip the thing over. 00:24:40.310 --> 00:24:44.950 So at xA equals zero that's pure B. pB star is 00:24:44.950 --> 00:24:45.960 less than pA star. 00:24:45.960 --> 00:24:49.780 I'm starting here, on that side here. pB star, and I'm 00:24:49.780 --> 00:25:00.246 going down in a straight line to zero. pB is equal to xB pB 00:25:00.246 --> 00:25:02.110 star, straight line all the way to zero when 00:25:02.110 --> 00:25:05.210 xB is equal to zero. 00:25:05.210 --> 00:25:08.180 So that's pB. 00:25:08.180 --> 00:25:10.860 And that's pA. 00:25:10.860 --> 00:25:17.250 Now, the total pressure is the sum of the two. pA plus pB. 00:25:17.250 --> 00:25:21.570 So the total pressure in this system here, that I measure 00:25:21.570 --> 00:25:23.740 here, is just the sum of these two straight lines. 00:25:23.740 --> 00:25:25.220 Let me do it in blue. 00:25:25.220 --> 00:25:30.510 The total pressure, if you add up these two straight lines 00:25:30.510 --> 00:25:32.320 together, you got another straight line. 00:25:32.320 --> 00:25:38.460 Two straight lines make a straight line. p is pA plus pB 00:25:38.460 --> 00:25:48.030 is equal to xA pA star plus xB pB star. 00:25:48.030 --> 00:25:54.200 Now, xA and xB are related. xB is one minus xA. 00:25:54.200 --> 00:25:57.080 And so really, this is only a function of one variable here. 00:25:57.080 --> 00:26:02.820 Linearly, with respect to one variable. 00:26:02.820 --> 00:26:04.190 OK, what does this diagram tell me? 00:26:04.190 --> 00:26:13.020 So now, let's ignore how we built it. 00:26:13.020 --> 00:26:19.490 And let's take a look at it. 00:26:19.490 --> 00:26:23.760 So I have, going to take xB on the bottom here. 00:26:23.760 --> 00:26:25.140 From zero to one. 00:26:25.140 --> 00:26:29.120 We could choose either one, and I've got the straight line 00:26:29.120 --> 00:26:36.370 for the total pressure going from pA star to pB star. 00:26:36.370 --> 00:26:38.890 T is fixed. 00:26:38.890 --> 00:26:42.360 I need to tell you that T is fixed because I have two 00:26:42.360 --> 00:26:43.920 degrees of freedom, right? 00:26:43.920 --> 00:26:45.700 I have two degrees of freedom. 00:26:45.700 --> 00:26:48.500 My degrees of freedom are the temperature, which I'm fixing. 00:26:48.500 --> 00:26:50.880 And the composition in the liquid 00:26:50.880 --> 00:26:52.500 phase, which I'm fixing. 00:26:52.500 --> 00:26:59.850 So my two variables are xB and T. Those are my 00:26:59.850 --> 00:27:00.820 two degrees of freedom. 00:27:00.820 --> 00:27:03.330 And I'm telling you what the total pressure is. 00:27:03.330 --> 00:27:08.190 I'm giving you total pressure as a function of xB and T. By 00:27:08.190 --> 00:27:11.510 fixing T, I'm really telling you what the pressure is of 00:27:11.510 --> 00:27:13.190 the function of xB. 00:27:13.190 --> 00:27:16.250 So the Gibbs phase rule then tells me that the pressure 00:27:16.250 --> 00:27:20.300 should be only a function of xB if T is fixed. 00:27:20.300 --> 00:27:21.600 Should be a line. 00:27:21.600 --> 00:27:22.980 Not necessarily straight, but it should be a 00:27:22.980 --> 00:27:26.030 line in this diagram. 00:27:26.030 --> 00:27:28.070 To have coexistence. 00:27:28.070 --> 00:27:32.330 So what this line is, then, this line is the line of 00:27:32.330 --> 00:27:37.230 points that tells me when I have coexistence between the 00:27:37.230 --> 00:27:41.350 gas phase and the liquid phase. 00:27:41.350 --> 00:27:41.516 The coexistence line. 00:27:41.516 --> 00:27:44.960 This is the gas-liquids coexistence 00:27:44.960 --> 00:27:47.370 line for the mixture. 00:27:47.370 --> 00:27:55.440 It's not so different than this coexistence line up here. 00:27:55.440 --> 00:27:59.250 So you can think of this as a phase diagram, kind of like 00:27:59.250 --> 00:28:02.810 you thought, you know this as a phase diagram. 00:28:02.810 --> 00:28:05.360 It's got a coexistence line. 00:28:05.360 --> 00:28:08.400 And then if I'm above this line, if I'm at a certain 00:28:08.400 --> 00:28:13.810 pressure here, I'm at this point here, this pressure 00:28:13.810 --> 00:28:17.620 well, let's go to a slightly higher 00:28:17.620 --> 00:28:18.730 different pressure here. 00:28:18.730 --> 00:28:22.890 And this composition here. 00:28:22.890 --> 00:28:24.540 I'm not on the line. 00:28:24.540 --> 00:28:27.230 That means that I don't have coexistence 00:28:27.230 --> 00:28:29.380 between liquid and gas. 00:28:29.380 --> 00:28:30.260 I'm above the line. 00:28:30.260 --> 00:28:35.215 I'm at a pressure higher then where I would get coexistence. 00:28:35.215 --> 00:28:38.060 The pressure is higher, means I probably don't 00:28:38.060 --> 00:28:42.170 have a gas any more. 00:28:42.170 --> 00:28:45.320 Probably means that the total pressure pushing down on my 00:28:45.320 --> 00:28:48.730 mixture is too high to have a gas phase around. 00:28:48.730 --> 00:28:55.700 It means that the top part here is a liquid phase. 00:28:55.700 --> 00:29:00.760 So I'm in the liquid phase if I'm above this line and I have 00:29:00.760 --> 00:29:03.590 a coexistence between the gas and the liquid 00:29:03.590 --> 00:29:06.310 if I'm on the line. 00:29:06.310 --> 00:29:13.210 Now, if I'm below this line here, well, I get in trouble. 00:29:13.210 --> 00:29:17.740 I get in trouble because my first instinct would be, well 00:29:17.740 --> 00:29:19.120 this is just gas. 00:29:19.120 --> 00:29:22.400 I go from liquid, and then I have coexistence. 00:29:22.400 --> 00:29:24.180 Then I go into the gas phase. 00:29:24.180 --> 00:29:27.090 But my x-axis here is the composition 00:29:27.090 --> 00:29:28.670 in the liquid phase. 00:29:28.670 --> 00:29:31.990 There's no liquid around. 00:29:31.990 --> 00:29:35.130 So I really can't say anything here. 00:29:35.130 --> 00:29:37.300 This diagram is a little bit meaningless down here. 00:29:37.300 --> 00:29:40.500 Because my x-axis is describing a phase which 00:29:40.500 --> 00:29:45.150 doesn't exist on the diagram. 00:29:45.150 --> 00:29:48.650 On this diagram, here. 00:29:48.650 --> 00:29:51.430 So when you see it, this diagram here with Raoult's law 00:29:51.430 --> 00:29:54.810 on it, you've got to remember that the part that's really 00:29:54.810 --> 00:30:00.090 interesting is above the line and at the line itself. 00:30:00.090 --> 00:30:14.100 So you could do an experiment, then, on this line here. 00:30:14.100 --> 00:30:14.330 T fixed. 00:30:14.330 --> 00:30:20.170 This is the liquid phase. 00:30:20.170 --> 00:30:22.420 So suppose I start with some high 00:30:22.420 --> 00:30:23.620 pressure up here somewhere. 00:30:23.620 --> 00:30:29.220 At this point, let's call this point one. 00:30:29.220 --> 00:30:36.680 We've got my container, my piston here. p is equal to p1. 00:30:36.680 --> 00:30:38.676 Again, I put the point in an awkward place. 00:30:38.676 --> 00:30:51.400 Let's put it right there. p1, and composition xB. 00:30:51.400 --> 00:30:56.800 p1 and xB here. 00:30:56.800 --> 00:31:02.510 And I'm going to slowly decrease the pressure. 00:31:02.510 --> 00:31:03.770 I start in this composition. 00:31:03.770 --> 00:31:05.720 The composition doesn't change. 00:31:05.720 --> 00:31:07.070 I'm just decreasing the pressure. 00:31:07.070 --> 00:31:08.450 Decrease the pressure, decrease the pressure, 00:31:08.450 --> 00:31:09.060 decrease the pressure. 00:31:09.060 --> 00:31:14.530 And at some point, I get to that line. 00:31:14.530 --> 00:31:18.540 Get to that line, what happens when I get to that line? 00:31:18.540 --> 00:31:21.260 I start to see bubbles coming out of the liquid. 00:31:21.260 --> 00:31:22.530 Vapor starts to form, because it wants to be in the 00:31:22.530 --> 00:31:24.330 coexistence line. 00:31:24.330 --> 00:31:25.490 So I get to that line. 00:31:25.490 --> 00:31:29.100 And I get a little bit of bubbles forming. 00:31:29.100 --> 00:31:32.060 And a little bit of vapor. 00:31:32.060 --> 00:31:33.980 That's the liquid here. 00:31:33.980 --> 00:31:36.090 Little bit of gas. 00:31:36.090 --> 00:31:44.870 And that's why this line here is called the bubble line. 00:31:44.870 --> 00:31:48.250 So I've gone down, decreased the pressure. 00:31:48.250 --> 00:31:50.740 So p has gone down now. 00:31:50.740 --> 00:31:52.600 Decreased the pressure. 00:31:52.600 --> 00:31:58.740 And if I keep on decreasing pressure, well, I'm not going 00:31:58.740 --> 00:32:00.450 to jump into this area of the diagram. 00:32:00.450 --> 00:32:02.690 Because this area of the diagram means there's no 00:32:02.690 --> 00:32:05.410 liquid, it's a meaningless area. 00:32:05.410 --> 00:32:08.416 If I keep decreasing the pressure, I'm still going to 00:32:08.416 --> 00:32:09.190 be in coexistence. 00:32:09.190 --> 00:32:12.170 I'm just going to make more gas. 00:32:12.170 --> 00:32:13.170 More vapor. 00:32:13.170 --> 00:32:15.610 I'm going to transfer more of the liquid 00:32:15.610 --> 00:32:17.070 into the vapor phase. 00:32:17.070 --> 00:32:21.120 I'm going to ride down that line. 00:32:21.120 --> 00:32:22.370 I'm going to ride down the line. 00:32:22.370 --> 00:32:23.320 I'm decreasing the pressure. 00:32:23.320 --> 00:32:24.510 I'm not going to skip into here. 00:32:24.510 --> 00:32:29.150 I'm going to ride down the line here. 00:32:29.150 --> 00:32:33.690 We're going to keep decreasing the pressure even more. p goes 00:32:33.690 --> 00:32:34.060 down even more. 00:32:34.060 --> 00:32:37.820 Now, I make even more vapor. 00:32:37.820 --> 00:32:41.620 There's my piston right there. 00:32:41.620 --> 00:32:43.850 There's the gas phase. 00:32:43.850 --> 00:32:46.240 There's the liquid phase. 00:32:46.240 --> 00:32:51.400 Now I've transferred material from the liquid 00:32:51.400 --> 00:32:53.730 phase to the gas phase. 00:32:53.730 --> 00:32:55.810 But I also changed the composition 00:32:55.810 --> 00:32:57.500 of my liquid phase. 00:32:57.500 --> 00:33:02.420 And if I read my new composition, this is what I 00:33:02.420 --> 00:33:05.080 started out with. xB. 00:33:05.080 --> 00:33:09.990 My new composition in the liquid phase is going to be, 00:33:09.990 --> 00:33:15.220 let's call it xB prime. 00:33:15.220 --> 00:33:17.340 There's going to be more solute in there. 00:33:17.340 --> 00:33:24.310 Or more B in there, than what I started out with. 00:33:24.310 --> 00:33:25.800 Now which one was the more volatile one? 00:33:25.800 --> 00:33:27.950 A or B? 00:33:27.950 --> 00:33:29.320 A was more volatile, right? 00:33:29.320 --> 00:33:34.610 So as I decreased the pressure and I started bubbling off the 00:33:34.610 --> 00:33:38.410 material, which one's going to bubble off preferentially? 00:33:38.410 --> 00:33:39.910 A is going to. 00:33:39.910 --> 00:33:47.350 So it makes sense that I would concentrate B in there. 00:33:47.350 --> 00:33:49.040 Decrease the pressure. 00:33:49.040 --> 00:33:51.820 Bubbles that are rich in A come out. 00:33:51.820 --> 00:33:52.940 Both A and B come out. 00:33:52.940 --> 00:33:56.940 But more A comes out than B. And what's left over is rich 00:33:56.940 --> 00:34:01.080 and then more than the less volatile material which is B. 00:34:01.080 --> 00:34:05.920 So we're beginning to see how distillation comes about. 00:34:05.920 --> 00:34:07.730 This is the first step in it. 00:34:07.730 --> 00:34:18.220 In a distillation process. 00:34:18.220 --> 00:34:21.410 So let's make this a little bit more complicated now. 00:34:21.410 --> 00:34:30.380 Any questions, first? 00:34:30.380 --> 00:34:32.490 Alright, now, suppose I wanted to know. 00:34:32.490 --> 00:34:36.280 So I found out where the composition in the liquid 00:34:36.280 --> 00:34:38.080 phase is here. 00:34:38.080 --> 00:34:41.505 And I know that should tell me immediately whether the 00:34:41.505 --> 00:34:43.970 composition in the gas phase is here. 00:34:43.970 --> 00:34:47.860 So I have all, I have everything I need to know to 00:34:47.860 --> 00:34:52.030 calculate what the composition in the gas phase is. 00:34:52.030 --> 00:34:55.220 So I have everything I need, then, to calculate and draw a 00:34:55.220 --> 00:34:58.510 diagram that looks just like this. 00:34:58.510 --> 00:35:04.520 Except where my x-axis is the y's instead of the x's. 00:35:04.520 --> 00:35:06.660 Going to make the same diagram, except now I'm going 00:35:06.660 --> 00:35:10.380 to use the gas phase as my reference point. 00:35:10.380 --> 00:35:15.590 The composition in the gas phase. 00:35:15.590 --> 00:35:22.190 So I need to get, so here I got p as a function of xB. 00:35:22.190 --> 00:35:30.250 I want p, total pressure, as a function of yB. 00:35:30.250 --> 00:35:33.240 So I can draw a diagram that looks just like this except 00:35:33.240 --> 00:35:41.310 now with the x-axis being the gas phase composition. 00:35:41.310 --> 00:35:41.980 Let's turn the crank. 00:35:41.980 --> 00:35:45.880 What do I know? 00:35:45.880 --> 00:35:55.400 I know Dalton's law. pA is yA times total pressure. 00:35:55.400 --> 00:35:59.810 I'm trying mix up the partial pressures and the total 00:35:59.810 --> 00:36:02.950 pressure in all ways that I know how to write it. 00:36:02.950 --> 00:36:05.595 And then hope that something's going to come out that's going 00:36:05.595 --> 00:36:06.540 to be helpful. 00:36:06.540 --> 00:36:10.700 So that's Dalton's law. 00:36:10.700 --> 00:36:17.960 I know Raoult's law. pA is xA star times pA star. 00:36:17.960 --> 00:36:22.740 And that's Raoult. 00:36:22.740 --> 00:36:25.400 And I can write Raoult's law in a different way. 00:36:25.400 --> 00:36:31.120 I can write as pB is equal to xB pB star, that's just 00:36:31.120 --> 00:36:41.310 Raoult's law for B. One minus xA pB star. 00:36:41.310 --> 00:36:45.200 And I'm looking for this. 00:36:45.200 --> 00:36:48.340 I'm looking for the composition in the gas phase. 00:36:48.340 --> 00:36:50.687 Or the total pressure as a function of the composition in 00:36:50.687 --> 00:36:51.010 the gas phase. 00:36:51.010 --> 00:36:54.830 Let's start by finding the composition in the gas phase 00:36:54.830 --> 00:36:56.610 as a function of the composition 00:36:56.610 --> 00:36:58.450 in the liquid phase. 00:36:58.450 --> 00:37:01.990 And the total pressure. 00:37:01.990 --> 00:37:08.940 Rewrite yA is equal to pA over P. Total 00:37:08.940 --> 00:37:11.380 pressure is pA plus pB. 00:37:11.380 --> 00:37:16.320 Sum of the two partial pressures. pA over pA plus pB. 00:37:16.320 --> 00:37:19.290 And I'm trying to get yA in terms of the xA's. 00:37:19.290 --> 00:37:21.000 Or xB's, in terms of composition 00:37:21.000 --> 00:37:22.680 in the liquid phase. 00:37:22.680 --> 00:37:25.860 And I know how to relate that to a constant. 00:37:25.860 --> 00:37:29.850 Which is the vapor pressure of the pure material times the, 00:37:29.850 --> 00:37:33.280 and the composition of the liquid phase, Raoult's law. 00:37:33.280 --> 00:37:44.490 This is xA pA star divided by xA pA star plus xB pB star. 00:37:44.490 --> 00:37:46.750 And I know that xB is one minus xA. 00:37:46.750 --> 00:37:59.320 So this is xA pA star times pB star plus pA star minus pB 00:37:59.320 --> 00:38:02.900 star times xA. 00:38:02.900 --> 00:38:07.330 Where all I did was replace xB with one minus xA. 00:38:07.330 --> 00:38:10.070 And rearrange my equation on the bottom. 00:38:10.070 --> 00:38:15.310 So I've gotten the composition in the gas phase in terms of 00:38:15.310 --> 00:38:17.960 the composition in the liquid phase. 00:38:17.960 --> 00:38:18.830 They're not independent. 00:38:18.830 --> 00:38:20.430 I knew they weren't independent. 00:38:20.430 --> 00:38:24.220 And this just shows me that math works. 00:38:24.220 --> 00:38:24.490 OK. 00:38:24.490 --> 00:38:25.500 So I've got that. 00:38:25.500 --> 00:38:28.620 And I can invert this to get the composition in the liquid 00:38:28.620 --> 00:38:31.080 phase in terms of the composition in the gas phase. 00:38:31.080 --> 00:38:35.200 It's not so straightforward, but you can get xA as a 00:38:35.200 --> 00:38:37.720 function of yA, as well. 00:38:37.720 --> 00:38:40.300 You've got xA, if you reverse this you get xA is 00:38:40.300 --> 00:38:44.605 equal to yA pB star. 00:38:44.605 --> 00:38:54.420 It looks kind of the same. pB star plus pA star plus pB star 00:38:54.420 --> 00:39:01.310 minus pA star times yA. xA as a function of yA. 00:39:05.560 --> 00:39:05.880 Alright. 00:39:05.880 --> 00:39:11.690 So now you can actually put everything together, starting 00:39:11.690 --> 00:39:13.200 from Dalton's law. 00:39:13.200 --> 00:39:15.920 Because really what we want is the total pressure as a 00:39:15.920 --> 00:39:19.070 function of the composition in the gas phase. 00:39:19.070 --> 00:39:20.500 So there's our total pressure here. 00:39:20.500 --> 00:39:22.740 There's the composition in the gas phase. 00:39:22.740 --> 00:39:24.520 There's pA. 00:39:24.520 --> 00:39:26.960 We've found composition in the gas phase in terms of 00:39:26.960 --> 00:39:30.930 composition in the liquid phase. 00:39:30.930 --> 00:39:38.620 P is equal to yA over pA. 00:39:38.620 --> 00:39:45.050 Which is equal to yA over xA pA star. 00:39:45.050 --> 00:39:48.030 So if we could get rid of this here, in terms of 00:39:48.030 --> 00:39:49.200 yA, we'd be all set. 00:39:49.200 --> 00:39:52.800 We'd get the total pressure as a function of the composition 00:39:52.800 --> 00:39:54.240 in the gas phase. 00:39:54.240 --> 00:39:57.860 And there's what we need. 00:39:57.860 --> 00:40:00.420 So we plug that in here. 00:40:00.420 --> 00:40:06.320 And now we have the total pressure in terms of constants 00:40:06.320 --> 00:40:12.310 like these pure vapor pressures and the composition 00:40:12.310 --> 00:40:18.380 of A in the gas phase. 00:40:18.380 --> 00:40:21.110 So let me find a way to write that somewhere. 00:40:21.110 --> 00:40:24.870 Let me write it right here in this little box here. 00:40:24.870 --> 00:40:29.150 Plug it in, you get something that's not a straight line. 00:40:29.150 --> 00:40:31.010 Unfortunately. 00:40:31.010 --> 00:40:37.330 Doesn't look like Raoult's law. pA star pB star. pA star 00:40:37.330 --> 00:40:48.790 plus pB star minus pA star times yA. 00:40:48.790 --> 00:40:49.950 It's not a straight line. 00:40:49.950 --> 00:40:57.390 It's a more complicated function. 00:40:57.390 --> 00:41:16.860 What does it look like? 00:41:16.860 --> 00:41:17.430 Looks like this. 00:41:17.430 --> 00:41:23.050 I'm going to use the same sort of plot. 00:41:23.050 --> 00:41:25.040 I going to plot the total pressure on this axis here, on 00:41:25.040 --> 00:41:25.920 the y axis. 00:41:25.920 --> 00:41:28.120 I'm going to keep the T fixed. 00:41:28.120 --> 00:41:31.210 Temperature is fixed. 00:41:31.210 --> 00:41:34.220 So I'm going to find the total pressure as a function of the 00:41:34.220 --> 00:41:35.270 composition in the gas phase. 00:41:35.270 --> 00:41:42.870 I'm going to put yB here. yB going from zero to one. 00:41:42.870 --> 00:41:50.720 So it's kind of looking like what I had before. 00:41:50.720 --> 00:41:51.990 Total pressure is going to look the same. 00:41:51.990 --> 00:41:54.450 If you want to do it in terms of B, you just have everywhere 00:41:54.450 --> 00:41:56.240 you see B replace it with A. And everywhere you see A 00:41:56.240 --> 00:42:02.130 replace it with B. You just need to change the two. 00:42:02.130 --> 00:42:06.730 You know that if yB is equal to zero, that you have pure A. 00:42:06.730 --> 00:42:12.190 So you know that this is going to be pA star here. 00:42:12.190 --> 00:42:14.630 You know if you have yB equals to one in the gas phase, you 00:42:14.630 --> 00:42:16.080 have pure B in the gas phase. 00:42:16.080 --> 00:42:20.320 You know that what you have is the vapor pressure of the pure 00:42:20.320 --> 00:42:26.540 B, in this case the pure water. pB star, OK? 00:42:26.540 --> 00:42:32.300 Where A is more volatile then B. A would be the ethanol and 00:42:32.300 --> 00:42:35.830 B would be the water. 00:42:35.830 --> 00:42:37.330 And it's not a straight line. 00:42:37.330 --> 00:42:39.414 If it were Raoult, he would say just connect these two 00:42:39.414 --> 00:42:40.280 with a straight line. 00:42:40.280 --> 00:42:41.310 But it's not. 00:42:41.310 --> 00:42:44.280 It's not, we just figured out with an equation here. 00:42:44.280 --> 00:42:49.050 And it turns out to be a line that looks like this. 00:42:49.050 --> 00:42:55.150 It's a curved line that looks a bit like this. 00:42:55.150 --> 00:42:57.020 OK, what does this line mean? 00:42:57.020 --> 00:43:01.540 This line is also a coexistence line. 00:43:01.540 --> 00:43:02.510 That's what we started out with. 00:43:02.510 --> 00:43:05.410 We started out with Raoult. 00:43:05.410 --> 00:43:07.560 That was part of our derivation. 00:43:07.560 --> 00:43:15.120 Raoult's law tells you the composition at coexistence. 00:43:15.120 --> 00:43:18.260 Tells you the pressure of, the total pressure, or partial 00:43:18.260 --> 00:43:21.200 pressure at coexistence and the composition. 00:43:21.200 --> 00:43:23.820 So this is a coexistence line between the 00:43:23.820 --> 00:43:26.390 liquid and the gas. 00:43:26.390 --> 00:43:28.120 Coexistence between liquid and gas. 00:43:28.120 --> 00:43:30.382 But now, it's not as a function of the composition at 00:43:30.382 --> 00:43:30.990 the liquid phase. 00:43:30.990 --> 00:43:34.500 It's a function of the composition in the gas phase. 00:43:34.500 --> 00:43:38.460 Which was missing before. 00:43:38.460 --> 00:43:43.890 So now if I do my experiment, and I'm, so 00:43:43.890 --> 00:43:44.950 what does this mean? 00:43:44.950 --> 00:43:48.540 So if I have a point, if I have a pressure which is lower 00:43:48.540 --> 00:43:51.110 than the line here, then I have pure gas. 00:43:51.110 --> 00:43:52.820 Pressure, low pressure, I have pure gas. 00:43:52.820 --> 00:43:53.810 Don't have any liquids. 00:43:53.810 --> 00:43:56.950 And that's fine, because I can, I know 00:43:56.950 --> 00:43:57.600 what that point is. 00:43:57.600 --> 00:44:02.870 If I'm sitting below the line here somewhere, 00:44:02.870 --> 00:44:06.640 so I'm sitting here. 00:44:06.640 --> 00:44:09.630 Some composition in the gas phase, a certain pressure. 00:44:09.630 --> 00:44:11.300 That's fine. 00:44:11.300 --> 00:44:15.280 It's a meaningful point. 00:44:15.280 --> 00:44:18.870 So my container here has a piston on it. 00:44:18.870 --> 00:44:26.120 And I'm starting out with this pressure, p1 here. 00:44:26.120 --> 00:44:27.490 p1 pressing down. 00:44:27.490 --> 00:44:30.240 And I got yB in this here. 00:44:30.240 --> 00:44:33.930 And I increase the pressure, pressure goes up. 00:44:33.930 --> 00:44:35.340 Composition doesn't change. 00:44:35.340 --> 00:44:38.250 I'm just increasing the pressure. 00:44:38.250 --> 00:44:39.640 Increasing the pressure, increasing the pressure. 00:44:39.640 --> 00:44:42.380 Now, at some point the pressure becomes big enough 00:44:42.380 --> 00:44:44.720 that I start to see liquid forming. 00:44:44.720 --> 00:44:54.970 Little drops of liquid forming in my container. 00:44:54.970 --> 00:44:58.255 So now I have little drops of liquid that are forming in my 00:44:58.255 --> 00:45:01.750 container, that are going to start to rain down and form a 00:45:01.750 --> 00:45:04.290 little bit of liquid pooling at the 00:45:04.290 --> 00:45:05.640 bottom of the container. 00:45:05.640 --> 00:45:10.420 And that's called the dew line. 00:45:10.420 --> 00:45:12.860 The other one was called the bubble line, this is called 00:45:12.860 --> 00:45:15.520 the dew line. 00:45:15.520 --> 00:45:19.180 So when I reach the dew line, I start forming liquid. 00:45:19.180 --> 00:45:21.210 And again, just like in the previous case, if I keep 00:45:21.210 --> 00:45:24.360 increasing the pressure, I don't go into this area here. 00:45:24.360 --> 00:45:25.770 First of all, this area's meaningless for 00:45:25.770 --> 00:45:27.620 this diagram here. 00:45:27.620 --> 00:45:31.550 Because this area's pure liquid. 00:45:31.550 --> 00:45:34.070 But the x-axis here doesn't tell me about the composition 00:45:34.070 --> 00:45:34.530 of pure liquid. 00:45:34.530 --> 00:45:35.680 It's all yB. 00:45:35.680 --> 00:45:38.130 It's all about the composition in the gas -- yes. 00:45:38.130 --> 00:45:40.980 STUDENT: [INAUDIBLE] 00:45:40.980 --> 00:45:43.130 PROFESSOR: Between the dew line and the bubble line? 00:45:43.130 --> 00:45:43.410 STUDENT: Yeah. 00:45:43.410 --> 00:45:44.330 PROFESSOR: This never, never world. 00:45:44.330 --> 00:45:47.970 Where you're not allowed to inhabit. 00:45:47.970 --> 00:45:54.760 We're going to go into that next time. 00:45:54.760 --> 00:45:57.920 On how to use those two curves together. 00:45:57.920 --> 00:45:59.220 OK, but the same thing happens here. 00:45:59.220 --> 00:46:01.270 If I increase the pressure, I'm going to ride the 00:46:01.270 --> 00:46:01.680 coexistence line up. 00:46:01.680 --> 00:46:12.770 I ride it up, and keep riding it up. 00:46:12.770 --> 00:46:14.780 And I get liquid forming in here. 00:46:14.780 --> 00:46:16.325 And I can find out the composition in 00:46:16.325 --> 00:46:17.100 the gas phase now. 00:46:17.100 --> 00:46:20.490 The composition in the gas phase, as I increase the 00:46:20.490 --> 00:46:25.620 pressure, I get yB prime, there's yB that I 00:46:25.620 --> 00:46:26.920 started out with with. 00:46:26.920 --> 00:46:30.510 With yB prime is less than yB. 00:46:30.510 --> 00:46:37.320 And remember, B was the less volatile of the two. 00:46:37.320 --> 00:46:42.200 As I increase the pressure, I am forming liquid. 00:46:42.200 --> 00:46:46.030 The composition in the gas phase changes. 00:46:46.030 --> 00:46:50.750 You decrease the mole fraction of the less volatile material 00:46:50.750 --> 00:46:51.440 in the gas phase. 00:46:51.440 --> 00:46:55.300 The more volatile A becomes more 00:46:55.300 --> 00:46:57.010 prevalent in the gas phase. 00:46:57.010 --> 00:47:01.250 And what's coming down is mostly, then, it's enriched in 00:47:01.250 --> 00:47:07.190 B. You've done an enrichment of the less volatile material 00:47:07.190 --> 00:47:10.220 down in here. 00:47:10.220 --> 00:47:12.750 And now you can put the two together, as the 00:47:12.750 --> 00:47:19.420 question was asked. 00:47:19.420 --> 00:47:21.970 Now we can have, we mix the two together. 00:47:21.970 --> 00:47:27.610 We mix our Raoult diagram with our dew line diagram. 00:47:27.610 --> 00:47:31.050 We mix the bubble line and the dew line together. 00:47:31.050 --> 00:47:32.500 In one diagram, which is going to be a 00:47:32.500 --> 00:47:48.730 powerful diagram to use. 00:47:48.730 --> 00:47:51.040 Total pressure. 00:47:51.040 --> 00:47:54.510 Temperature fixed. 00:47:54.510 --> 00:48:02.600 You've got two materials, pA star is bigger than pB star. 00:48:02.600 --> 00:48:08.830 So I'm going to have pA star here, pB star down here. 00:48:08.830 --> 00:48:13.590 And if I choose, as my x-axis, to have xB going from zero to 00:48:13.590 --> 00:48:18.700 one, I can draw a coexistence line on this axis. 00:48:18.700 --> 00:48:20.520 On this diagram. 00:48:20.520 --> 00:48:24.770 That's the coexistence line in terms of xB. 00:48:24.770 --> 00:48:29.910 And any point up here is well-described by this xB 00:48:29.910 --> 00:48:32.870 value, and the p value. 00:48:32.870 --> 00:48:36.770 And as long as I have xB here, I'm not allowed to touch the 00:48:36.770 --> 00:48:38.600 bottom part of this diagram. 00:48:38.600 --> 00:48:43.220 But now if I add another x-axis, I'm going to add yB. 00:48:46.680 --> 00:48:50.550 From zero to one. 00:48:50.550 --> 00:48:53.230 I draw a coexistence line in terms of yB. 00:48:55.830 --> 00:48:56.500 Like this. 00:48:56.500 --> 00:48:59.550 So this line is p as a function of 00:48:59.550 --> 00:49:01.850 yB coexistence line. 00:49:01.850 --> 00:49:08.030 This line is total pressure of function of xB. 00:49:08.030 --> 00:49:11.460 And as long as I'm on this line or below this line, then 00:49:11.460 --> 00:49:15.900 this axis makes sense. 00:49:15.900 --> 00:49:17.320 That's how we build this diagram here. 00:49:17.320 --> 00:49:18.880 It's got a bubble line on top, and the 00:49:18.880 --> 00:49:21.520 dew line on the bottom. 00:49:21.520 --> 00:49:26.060 So I'm up here going down, I'm going to hit the bubble line. 00:49:26.060 --> 00:49:28.590 If I'm down here and go up in pressure, I'm going 00:49:28.590 --> 00:49:29.460 to hit the dew line. 00:49:29.460 --> 00:49:32.610 And next time -- well, next time we have an exam. 00:49:32.610 --> 00:49:38.020 But on Friday, we'll figure out how to use this diagram.