1 00:00:00,000 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,850 Your support will help MIT OpenCourseWare continue to 4 00:00:06,850 --> 00:00:10,520 offer high quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,150 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,150 --> 00:00:20,320 ocw.mit.edu. 8 00:00:20,320 --> 00:00:24,610 PROFESSOR: Alright, so last time we started talking about 9 00:00:24,610 --> 00:00:30,000 more complicated mixtures. 10 00:00:30,000 --> 00:00:32,500 To get into colligative properties. 11 00:00:32,500 --> 00:00:41,540 And the example we gave was a binary system that has two 12 00:00:41,540 --> 00:00:43,880 components. 13 00:00:43,880 --> 00:00:49,200 And two phases. 14 00:00:49,200 --> 00:00:54,010 So we have a liquid phase on the bottom. 15 00:00:54,010 --> 00:00:59,470 A gas phase on top, and two components, A and B, where 16 00:00:59,470 --> 00:01:05,520 compositions xA and xB are in the liquid phase. 17 00:01:05,520 --> 00:01:06,810 Those are the mole fractions. 18 00:01:06,810 --> 00:01:14,955 And yA and yB in the gas phase, the mole fractions in 19 00:01:14,955 --> 00:01:17,290 the gas phase. 20 00:01:17,290 --> 00:01:22,860 And what we did last time was to begin to say how many 21 00:01:22,860 --> 00:01:25,410 degrees of freedom do we have, how many variables do we need 22 00:01:25,410 --> 00:01:28,220 to completely describe this mixture. 23 00:01:28,220 --> 00:01:31,940 And it turned out that for this mixture, we only needed 24 00:01:31,940 --> 00:01:32,830 two variables. 25 00:01:32,830 --> 00:01:34,910 Two degrees of freedom. 26 00:01:34,910 --> 00:01:36,970 We started out with four variables; the temperature, 27 00:01:36,970 --> 00:01:48,090 the pressure, and the components in the composition 28 00:01:48,090 --> 00:01:51,010 in the liquid phase and the composition in the gas phase. 29 00:01:51,010 --> 00:01:54,670 We need one of these, either xA or xB, and one of these, 30 00:01:54,670 --> 00:01:57,900 either yA or yB because they add up to 1. 31 00:01:57,900 --> 00:02:01,440 So you start with four, but because you are in a mixture, 32 00:02:01,440 --> 00:02:03,180 an equilibrium, you have constraints. 33 00:02:03,180 --> 00:02:10,950 You have the chemical potentials that have to be 34 00:02:10,950 --> 00:02:11,580 equal to each other. 35 00:02:11,580 --> 00:02:14,170 The chemical potential of A in the gas phase has to be equal 36 00:02:14,170 --> 00:02:16,870 to the chemical potential of A in the liquid phase. 37 00:02:16,870 --> 00:02:19,590 And the chemical potential of B in the gas phase has to B 38 00:02:19,590 --> 00:02:21,512 equal to the chemical potential of B 39 00:02:21,512 --> 00:02:22,160 in the liquid phase. 40 00:02:22,160 --> 00:02:25,540 So you start with four variables. 41 00:02:25,540 --> 00:02:32,090 Temperature, pressure, xA, yA, then you have the constraints 42 00:02:32,090 --> 00:02:38,890 that mu A, in the liquid phase has to be equal to mu 43 00:02:38,890 --> 00:02:40,860 A in the gas phase. 44 00:02:40,860 --> 00:02:42,430 Because you're at equilibrium. 45 00:02:42,430 --> 00:02:46,082 And mu B in the liquid phase has to be equal to mu 46 00:02:46,082 --> 00:02:48,790 B in the gas phase. 47 00:02:48,790 --> 00:02:51,580 So four minus two constraints means you have 48 00:02:51,580 --> 00:02:54,270 two degrees of freedom. 49 00:02:54,270 --> 00:02:58,440 And so if you want to know everything about this mixture, 50 00:02:58,440 --> 00:03:01,490 all you need is the temperature 51 00:03:01,490 --> 00:03:02,570 and the total pressure. 52 00:03:02,570 --> 00:03:02,960 And that's enough. 53 00:03:02,960 --> 00:03:05,017 It's kind of amazing. 54 00:03:05,017 --> 00:03:09,430 Give me the temperature and pressure and these two 55 00:03:09,430 --> 00:03:12,390 components, like water and alcohol, for instance, and I 56 00:03:12,390 --> 00:03:17,230 can tell you everything about the compositions. 57 00:03:17,230 --> 00:03:18,410 It's very powerful. 58 00:03:18,410 --> 00:03:20,950 And then we said that this turned out to be a special 59 00:03:20,950 --> 00:03:26,290 case, or sub-case of the more general Gibbs phase rule. 60 00:03:26,290 --> 00:03:37,380 where the Gibbs phase rule tells you that if you have C 61 00:03:37,380 --> 00:03:40,670 components, in this case here we have two, and the general 62 00:03:40,670 --> 00:03:45,900 case of C components with p phases, the number of degrees 63 00:03:45,900 --> 00:03:47,730 of freedom is C minus p plus two. 64 00:03:47,730 --> 00:03:54,050 This is components. 65 00:03:54,050 --> 00:03:57,100 And phases. 66 00:03:57,100 --> 00:03:59,250 And where we're going to start today is by proving this. 67 00:03:59,250 --> 00:04:04,450 And then we'll do more with this problem here. 68 00:04:04,450 --> 00:04:08,720 So the first thing is for us to prove Gibbs' phase rule. 69 00:04:08,720 --> 00:04:10,190 So Gibbs phase rule. 70 00:04:10,190 --> 00:04:13,810 We start out with all the variables that we have. 71 00:04:13,810 --> 00:04:16,750 And then we add the constraints. 72 00:04:16,750 --> 00:04:20,480 And for the Gibbs phase rule, then we start with the two 73 00:04:20,480 --> 00:04:22,950 knobs that we can turn externally. 74 00:04:22,950 --> 00:04:24,720 Which are the temperature and the pressure. 75 00:04:24,720 --> 00:04:27,090 So we start with temperature and pressure, as two variables 76 00:04:27,090 --> 00:04:28,550 that we have. 77 00:04:28,550 --> 00:04:29,770 And then we have a bunch of phases. 78 00:04:29,770 --> 00:04:30,940 We have C phases. 79 00:04:30,940 --> 00:04:34,210 And in each phase we have to describe the composition in 80 00:04:34,210 --> 00:04:36,270 that phase. 81 00:04:36,270 --> 00:04:40,850 So, for each phase alpha, we have to 82 00:04:40,850 --> 00:04:44,670 describe its mole fraction. 83 00:04:44,670 --> 00:04:48,760 So here we needed to describe xA for phase A. And yA for 84 00:04:48,760 --> 00:04:53,790 phase A. For component A in the gas phase. 85 00:04:53,790 --> 00:04:59,350 So now we have alpha components. 86 00:04:59,350 --> 00:05:05,860 And for each component we have to describe its mole fraction 87 00:05:05,860 --> 00:05:07,400 in that particular phase. 88 00:05:07,400 --> 00:05:12,870 So we have to describe x sub i. 89 00:05:12,870 --> 00:05:15,040 But we have a constraint on that. 90 00:05:15,040 --> 00:05:18,240 The constraint is that the sum of all the mole factions has 91 00:05:18,240 --> 00:05:19,140 to be equal to one. 92 00:05:19,140 --> 00:05:20,780 That's what it is, to be a mole fraction. 93 00:05:20,780 --> 00:05:25,745 So the constraint is that x sub i, i from one to alpha, is 94 00:05:25,745 --> 00:05:30,510 equal to one. 95 00:05:30,510 --> 00:05:36,530 So the constraint on the number of the components, 96 00:05:36,530 --> 00:05:42,470 instead of having, actually, i goes from one to C, because 97 00:05:42,470 --> 00:05:46,350 that's the number components. 98 00:05:46,350 --> 00:05:51,210 So instead of having C different compositions that we 99 00:05:51,210 --> 00:05:54,100 need to take care of, we only need C minus one. 100 00:05:54,100 --> 00:05:56,530 Just like here we only needed one, just xA. 101 00:05:56,530 --> 00:05:59,190 We didn't need both xA and xB, because we knew the sum was 102 00:05:59,190 --> 00:06:00,310 equal to one. 103 00:06:00,310 --> 00:06:05,320 So that means that we really need C minus one variables to 104 00:06:05,320 --> 00:06:10,510 describe the composition in one of the phases. 105 00:06:10,510 --> 00:06:16,320 But we have p phases, so let me fix this. 106 00:06:16,320 --> 00:06:19,610 We had C components. 107 00:06:19,610 --> 00:06:21,000 We have p phases. 108 00:06:21,000 --> 00:06:24,170 And so for each phase, we have to describe the composition. 109 00:06:24,170 --> 00:06:27,850 So that means that we need p times C minus one variables to 110 00:06:27,850 --> 00:06:31,450 describe all the components and all the phases, 111 00:06:31,450 --> 00:06:33,820 plus these extra two. 112 00:06:33,820 --> 00:06:38,390 So the total number variables then that we start out with 113 00:06:38,390 --> 00:06:41,990 before putting the fact that we're in equilibrium, is going 114 00:06:41,990 --> 00:06:47,250 to be two plus p times C minus one. 115 00:06:47,250 --> 00:06:49,790 So that's a basic description of the system. 116 00:06:49,790 --> 00:06:51,220 Then we put in the same constraints 117 00:06:51,220 --> 00:06:52,330 that we had up here. 118 00:06:52,330 --> 00:06:58,150 In terms of the chemical potentials. 119 00:06:58,150 --> 00:07:00,570 If I take any of the components, let's take the 120 00:07:00,570 --> 00:07:03,240 alpha component, or the i'th component. 121 00:07:03,240 --> 00:07:07,970 Because it's in equilibrium, the chemical potential of that 122 00:07:07,970 --> 00:07:12,350 particular component has to be the same in all the phases. 123 00:07:12,350 --> 00:07:14,660 So let's take the i'th component. 124 00:07:14,660 --> 00:07:21,510 So yi in phase one has to be equal to yi in phase two, has 125 00:07:21,510 --> 00:07:25,720 to be equal to yi in phase three, has to be equal to yi 126 00:07:25,720 --> 00:07:33,350 in phase p. 127 00:07:33,350 --> 00:07:35,660 All the chemical potentials have to be the same across all 128 00:07:35,660 --> 00:07:37,370 the phases for that component. 129 00:07:37,370 --> 00:07:42,220 And if you count the number of equations, the number of 130 00:07:42,220 --> 00:07:47,240 equals signs here, that's p minus one. p minus one equals 131 00:07:47,240 --> 00:07:50,330 signs, constraints due to the fact that I have an 132 00:07:50,330 --> 00:07:51,220 equilibrium here. 133 00:07:51,220 --> 00:08:03,630 So this gives me p minus one constraints. 134 00:08:03,630 --> 00:08:05,640 And this is true for every single component. 135 00:08:05,640 --> 00:08:08,950 Every single component has to have its chemical potential, 136 00:08:08,950 --> 00:08:12,240 its chemical potential equal throughout the phases. 137 00:08:12,240 --> 00:08:15,480 So p minus one's constraint for one component times C 138 00:08:15,480 --> 00:08:16,000 components. 139 00:08:16,000 --> 00:08:26,100 So for a total of C times p minus one total constraints. 140 00:08:26,100 --> 00:08:29,200 My variables, my constraints. 141 00:08:29,200 --> 00:08:36,680 So the number of degrees of freedom then becomes p times C 142 00:08:36,680 --> 00:08:49,180 minus one minus C times p minus one and then plus two. 143 00:08:49,180 --> 00:08:52,520 And if you multiply this out, p times C gets rid of the p 144 00:08:52,520 --> 00:08:53,780 times C here. 145 00:08:53,780 --> 00:08:58,670 And you get the plus C minus p plus two. 146 00:08:58,670 --> 00:09:02,320 Which is the Gibbs phase rule, right? 147 00:09:02,320 --> 00:09:05,630 Which I'm going to bring back up here. 148 00:09:05,630 --> 00:09:08,020 C minus p plus two. 149 00:09:08,020 --> 00:09:12,750 So just purely in accounting, just accounting for the 150 00:09:12,750 --> 00:09:16,510 variables and the constraints. 151 00:09:16,510 --> 00:09:23,430 OK, any questions on how we did this? 152 00:09:23,430 --> 00:09:26,090 Great. 153 00:09:26,090 --> 00:09:26,710 OK. 154 00:09:26,710 --> 00:09:33,240 So let's move on, then, to these ideal solutions. 155 00:09:33,240 --> 00:09:35,110 Which is this case right here. 156 00:09:35,110 --> 00:09:38,070 And the first thing we're going to do is look at the 157 00:09:38,070 --> 00:09:47,470 case where we have A is a solvent, and it's volatile. 158 00:09:47,470 --> 00:09:50,450 And B is a solute, which is non-volatile. 159 00:09:50,450 --> 00:09:53,520 So let's take the first case. 160 00:09:53,520 --> 00:10:00,240 We have A volatile, that's a solvent. 161 00:10:00,240 --> 00:10:01,700 Could be water. 162 00:10:01,700 --> 00:10:05,440 And B is going to be the solute. 163 00:10:05,440 --> 00:10:09,880 It's non-volatile. 164 00:10:09,880 --> 00:10:12,750 And let's interchange solute or non-volatile to be 165 00:10:12,750 --> 00:10:14,600 consistent here. 166 00:10:14,600 --> 00:10:18,700 So this could be sugar. 167 00:10:18,700 --> 00:10:27,250 A sweet water. 168 00:10:27,250 --> 00:10:41,050 So, Mr. Raoult looked at such solutions, and found that they 169 00:10:41,050 --> 00:10:46,470 could be well described by the following diagram here. 170 00:10:46,470 --> 00:10:51,300 So if I plot on the x-axis here, from zero to one, I'm 171 00:10:51,300 --> 00:10:57,750 going to plot the partial molar fraction of the solute. 172 00:10:57,750 --> 00:11:00,010 Partial molar fractions of the solute, that's going to be xB. 173 00:11:02,530 --> 00:11:06,740 So when xB equals to zero, I have a pure water solution. 174 00:11:06,740 --> 00:11:11,370 When xB is equal to one, I have pure sugar. 175 00:11:11,370 --> 00:11:17,990 Then on the y axis, I'm going to plot the total pressure. 176 00:11:17,990 --> 00:11:20,410 So when I have a pure water solution, the total pressure 177 00:11:20,410 --> 00:11:26,660 is going to be the same as the vapor pressure of water at a 178 00:11:26,660 --> 00:11:28,150 particular temperature. 179 00:11:28,150 --> 00:11:36,570 Let's take temperature fixed at some value. 180 00:11:36,570 --> 00:11:39,840 Temperature is fixed at some value. 181 00:11:39,840 --> 00:11:44,320 I'm going to have the total pressure is going to be the 182 00:11:44,320 --> 00:11:45,970 partial pressure of the water. 183 00:11:45,970 --> 00:11:49,290 And clearly, when I don't have any water left, all I have is 184 00:11:49,290 --> 00:11:51,390 the sugar, there's no pressure. 185 00:11:51,390 --> 00:11:54,440 We're going to assume that it's totally non-volatile. 186 00:11:54,440 --> 00:11:56,000 Obviously there's going to be a little bit of vapor 187 00:11:56,000 --> 00:11:58,190 pressure, even at room temperature. 188 00:11:58,190 --> 00:12:01,980 Ten to the minus, I don't know, 13 torr or something. 189 00:12:01,980 --> 00:12:05,340 But anyway, it's basically zero here. 190 00:12:05,340 --> 00:12:07,680 And what Raoult said is that while these two points are 191 00:12:07,680 --> 00:12:13,430 just connected by a straight line. 192 00:12:13,430 --> 00:12:19,800 Simplest way to connect two points. 193 00:12:19,800 --> 00:12:27,370 Basically, he said that the pressure of, the vapor 194 00:12:27,370 --> 00:12:32,450 pressure of the water in this case, is equal 195 00:12:32,450 --> 00:12:38,310 to xA times pA star. 196 00:12:38,310 --> 00:12:42,680 I could also plot this with xA, going from one to zero, 197 00:12:42,680 --> 00:12:49,430 since xA plus xB is equal to one. 198 00:12:49,430 --> 00:12:56,160 Or I could write this as one minus xB times pA star. 199 00:12:56,160 --> 00:12:58,890 Where a star, the star, means pure water. 200 00:12:58,890 --> 00:13:06,660 So pure A. And pA is the vapor pressure of the mixture. 201 00:13:06,660 --> 00:13:09,000 Which in this case here is the total pressure, because I 202 00:13:09,000 --> 00:13:12,630 don't have anything else that's volatile. 203 00:13:12,630 --> 00:13:29,240 And this is Raoult's law. 204 00:13:29,240 --> 00:13:32,930 And the first thing that you can use this for is to look at 205 00:13:32,930 --> 00:13:35,780 your first colligative property which is vapor 206 00:13:35,780 --> 00:13:45,900 pressure lowering. 207 00:13:45,900 --> 00:13:48,120 And let me point out another thing first. 208 00:13:48,120 --> 00:13:51,940 So if a solution obeys that property here, it's called an 209 00:13:51,940 --> 00:13:53,230 ideal solution. 210 00:13:53,230 --> 00:13:56,470 So if the water-sugar solution really did behave like a 211 00:13:56,470 --> 00:13:58,290 straight line like this, as you increase the amount of 212 00:13:58,290 --> 00:14:00,190 sugar, then it would be an ideal solution. 213 00:14:00,190 --> 00:14:02,270 Many things are very close to it. 214 00:14:02,270 --> 00:14:07,230 It's not such a bad approximation. 215 00:14:07,230 --> 00:14:09,490 Especially in that region up here. 216 00:14:09,490 --> 00:14:12,015 Around zero, when the amount of sugar is pretty small. 217 00:14:12,015 --> 00:14:19,070 You expect to obey an ideal solution behavior. 218 00:14:19,070 --> 00:14:24,320 Alright, so let's look now at a glass of water that has 219 00:14:24,320 --> 00:14:26,150 sugar in it. 220 00:14:26,150 --> 00:14:28,320 We've got my glass of water, H20. 221 00:14:28,320 --> 00:14:34,640 With some sugar dissolved in it. 222 00:14:34,640 --> 00:14:42,260 And it's got a certain vapor pressure on top. pA, or pH2O. 223 00:14:42,260 --> 00:14:43,590 And I'm going to ask the question, what's the 224 00:14:43,590 --> 00:14:48,950 difference between the vapor pressure what it would be 225 00:14:48,950 --> 00:14:51,370 without the sugar in it. 226 00:14:51,370 --> 00:14:59,670 What is pA star minus pA? 227 00:14:59,670 --> 00:15:07,230 Obeys Raoult's law. pA star, pA is xA pA star, 228 00:15:07,230 --> 00:15:09,230 plug that in here. 229 00:15:09,230 --> 00:15:15,900 That's one minus xA pA star, one minus xA is xB. 230 00:15:15,900 --> 00:15:18,410 xB pA star. 231 00:15:18,410 --> 00:15:21,010 Well, the first thing I know is that this is a positive 232 00:15:21,010 --> 00:15:23,573 number. xB is a positive number. pA star 233 00:15:23,573 --> 00:15:25,020 is a positive number. 234 00:15:25,020 --> 00:15:26,500 This is greater than zero. 235 00:15:26,500 --> 00:15:31,050 That means that the act of putting some solute in my 236 00:15:31,050 --> 00:15:37,540 water decreases the vapor pressure of the 237 00:15:37,540 --> 00:15:40,450 water in the gas phase. 238 00:15:40,450 --> 00:15:41,880 Pretty simple. 239 00:15:41,880 --> 00:15:47,130 It's called vapor pressure law. 240 00:15:47,130 --> 00:15:49,550 And we can actually take this one step further. 241 00:15:49,550 --> 00:15:57,510 We can draw a phase diagram. 242 00:15:57,510 --> 00:15:59,990 So our usual phase diagram here with temperature on this 243 00:15:59,990 --> 00:16:04,470 axis here and pressure on this axis here, and we've got our 244 00:16:04,470 --> 00:16:09,390 triple point sitting here. 245 00:16:09,390 --> 00:16:13,280 And our gas-liquid line, with a critical point here. 246 00:16:13,280 --> 00:16:16,770 So this is gas here, there's a liquid here. 247 00:16:16,770 --> 00:16:21,430 And we've got our, this is going to be water. 248 00:16:21,430 --> 00:16:23,690 So it's got a negative slope. 249 00:16:23,690 --> 00:16:26,550 I think I've got it right this time. 250 00:16:26,550 --> 00:16:29,400 And there's the solid phase here. 251 00:16:29,400 --> 00:16:31,420 OK, this is for the pure stuff. 252 00:16:31,420 --> 00:16:40,770 So now I can draw a similar diagram for sugared water. 253 00:16:40,770 --> 00:16:43,500 And I just looked at this, assuming it 254 00:16:43,500 --> 00:16:44,480 obeys Raoult's law. 255 00:16:44,480 --> 00:16:50,870 What we looked at here was the gas-liquid coexistence. 256 00:16:50,870 --> 00:16:55,350 And we found that the gas-liquid coexistence point, 257 00:16:55,350 --> 00:16:58,290 the vapor pressure of the water, was less 258 00:16:58,290 --> 00:16:59,740 than if it were pure. 259 00:16:59,740 --> 00:17:02,670 So that means that if I looked at the gas-liquid coexistence, 260 00:17:02,670 --> 00:17:05,110 which is this line right here. 261 00:17:05,110 --> 00:17:08,710 In the presence of sugar, that whole line is 262 00:17:08,710 --> 00:17:11,620 going to get lowered. 263 00:17:11,620 --> 00:17:14,140 The pressure, the vapor pressure, of the water is 264 00:17:14,140 --> 00:17:17,400 going to be lower when the sugar is there. 265 00:17:17,400 --> 00:17:26,770 So the whole thing is going to get lower. 266 00:17:26,770 --> 00:17:32,325 Now, if I have a solid, if the water is a solid, it's going 267 00:17:32,325 --> 00:17:34,730 to crystallize the water, the sugar is going to be separated 268 00:17:34,730 --> 00:17:36,080 from the water. 269 00:17:36,080 --> 00:17:38,250 The sugar and the water are not going to really know about 270 00:17:38,250 --> 00:17:40,700 each other. 271 00:17:40,700 --> 00:17:44,840 They're just going to have big chunks of pure water, with 272 00:17:44,840 --> 00:17:49,430 every now and then a molecule of sugar that's in there. 273 00:17:49,430 --> 00:17:53,680 So you don't really expect to see any difference between the 274 00:17:53,680 --> 00:17:58,980 pure solid water, in terms of the gas-solid interface, and 275 00:17:58,980 --> 00:18:02,290 the pure liquid water. 276 00:18:02,290 --> 00:18:06,120 There's no sugar in the gas phase. 277 00:18:06,120 --> 00:18:11,130 And as far as the gas phase is concerned, that solid phase is 278 00:18:11,130 --> 00:18:11,820 pure water. 279 00:18:11,820 --> 00:18:14,230 It's just seeing pure ice crystals on the surface. 280 00:18:14,230 --> 00:18:17,400 Every now and then there's a molecule of sugar. 281 00:18:17,400 --> 00:18:22,970 So you expect this to be pretty close here. 282 00:18:22,970 --> 00:18:24,850 So you expect the triple point to go down. 283 00:18:24,850 --> 00:18:30,246 And if I now, I go up to the solid-liquid coexistence, and 284 00:18:30,246 --> 00:18:34,470 go up like this, I can make a prediction about solid-liquid 285 00:18:34,470 --> 00:18:36,950 coexistence. 286 00:18:36,950 --> 00:18:42,380 I can predict that if I'm looking at, let's put one bar 287 00:18:42,380 --> 00:18:49,530 here somewhere. 288 00:18:49,530 --> 00:18:52,050 One bar. 289 00:18:52,050 --> 00:18:53,570 Room temperature, sitting here somewhere. 290 00:18:53,570 --> 00:18:57,360 One bar, room temperature, water is all liquid. 291 00:18:57,360 --> 00:19:03,720 This is what it would usually, solid, liquid, that would be 292 00:19:03,720 --> 00:19:06,510 273 degrees Kelvin. 293 00:19:06,510 --> 00:19:10,570 That's the melting point of water. 294 00:19:10,570 --> 00:19:14,090 So now if I put sugar in the water, and I follow my diagram 295 00:19:14,090 --> 00:19:21,140 here, what I find is that the new melting point of water is 296 00:19:21,140 --> 00:19:29,850 some new temperature T, T1, where T1 is less than 273 297 00:19:29,850 --> 00:19:30,094 degrees Kelvin. 298 00:19:30,094 --> 00:19:31,090 You all know this. 299 00:19:31,090 --> 00:19:34,640 When you put something, like an impurity, a salt or sugar 300 00:19:34,640 --> 00:19:37,830 in the water, the melting point gets depressed. 301 00:19:37,830 --> 00:19:43,110 It goes down. 302 00:19:43,110 --> 00:19:44,030 Straight from here. 303 00:19:44,030 --> 00:19:46,820 You can build it up straight from here. 304 00:19:46,820 --> 00:19:51,850 When we get to the colligative properties, we'll attack it 305 00:19:51,850 --> 00:19:55,740 from the point of chemical potentials. 306 00:19:55,740 --> 00:19:57,620 We'll equate chemical potentials and we'll look at 307 00:19:57,620 --> 00:19:59,990 the change in the chemical potential in the water, and 308 00:19:59,990 --> 00:20:02,090 the presence of the sugar and we'll do it rigorously. 309 00:20:02,090 --> 00:20:05,010 But this is sort of the first indication that these 310 00:20:05,010 --> 00:20:08,460 colligative properties are all connected to each other. 311 00:20:08,460 --> 00:20:13,270 They're connected to this diagram here. 312 00:20:13,270 --> 00:20:14,360 OK. 313 00:20:14,360 --> 00:20:19,940 So when I was a kid, and I knew that this was the case. 314 00:20:19,940 --> 00:20:21,380 Everybody knows this is the case, right? 315 00:20:21,380 --> 00:20:24,536 You put salt in the water and you expect all sorts 316 00:20:24,536 --> 00:20:25,180 of things to happen. 317 00:20:25,180 --> 00:20:27,720 You expect the vapor pressure to go down. 318 00:20:27,720 --> 00:20:31,273 And certainly you know that you add salt to the roads and 319 00:20:31,273 --> 00:20:33,620 the water melts, et cetera. 320 00:20:33,620 --> 00:20:36,850 And my mother always told me that the reason why you added 321 00:20:36,850 --> 00:20:42,360 salt to the water was so that the temperature of the water 322 00:20:42,360 --> 00:20:46,050 would get higher when you try to boil pasta. 323 00:20:46,050 --> 00:20:50,770 And for, I don't know, fifteen years I believed her. 324 00:20:50,770 --> 00:20:53,110 Until I taught thermodynamics. 325 00:20:53,110 --> 00:20:58,010 And I actually calculated the change in the temperature by 326 00:20:58,010 --> 00:21:01,020 adding a few pinches of salt to the water. 327 00:21:01,020 --> 00:21:04,730 And you know how small that temperature change is? 328 00:21:04,730 --> 00:21:05,960 You should calculate it. 329 00:21:05,960 --> 00:21:07,880 It's really tiny. 330 00:21:07,880 --> 00:21:09,920 It won't make any difference for the water. 331 00:21:09,920 --> 00:21:12,370 Probably the difference in the temperature of the water will 332 00:21:12,370 --> 00:21:16,040 be the same as if I cooked here versus halfway up Mount 333 00:21:16,040 --> 00:21:18,460 Washington in New Hampshire or something. 334 00:21:18,460 --> 00:21:19,220 Probably even less. 335 00:21:19,220 --> 00:21:20,875 There's no difference. 336 00:21:20,875 --> 00:21:23,820 The reason why you add salt is so that it tastes better. 337 00:21:23,820 --> 00:21:24,530 That's the only reason. 338 00:21:24,530 --> 00:21:28,610 It has nothing to do with thermodynamics. 339 00:21:28,610 --> 00:21:35,730 Alright, any questions on this? 340 00:21:35,730 --> 00:21:37,240 So we going to make it a little bit more complicated 341 00:21:37,240 --> 00:21:38,090 now, if there's no questions. 342 00:21:38,090 --> 00:21:43,070 We're going to now have, instead of having one 343 00:21:43,070 --> 00:21:47,080 non-volatile solute, use we're going to have two, we're going 344 00:21:47,080 --> 00:21:47,730 to have this mixture. 345 00:21:47,730 --> 00:21:58,580 Where both components are volatile. 346 00:21:58,580 --> 00:22:03,230 Can we erase this? 347 00:22:03,230 --> 00:22:03,950 Actually, I may not erase this. 348 00:22:03,950 --> 00:22:12,140 Because I want to keep Raoult's law here. 349 00:22:12,140 --> 00:22:16,000 OK, we're going to have a mixture of two volatile 350 00:22:16,000 --> 00:22:16,550 components. 351 00:22:16,550 --> 00:22:19,960 For instance, water and alcohol. 352 00:22:19,960 --> 00:22:25,510 Your your typical martini type situation. 353 00:22:25,510 --> 00:22:26,660 Or margarita. 354 00:22:26,660 --> 00:22:40,370 Whatever's your favorite drink. 355 00:22:40,370 --> 00:22:47,840 So now we have xA, we have xB, we have yA, we have yB. 356 00:22:47,840 --> 00:23:00,430 And we're going to assume that both obey Raoult's law with 357 00:23:00,430 --> 00:23:02,590 respect to each other. 358 00:23:02,590 --> 00:23:08,100 So now if I draw a diagram, and I just focus on, so 359 00:23:08,100 --> 00:23:12,080 there's the total pressure sitting here. 360 00:23:12,080 --> 00:23:14,500 And I just focus on one component 361 00:23:14,500 --> 00:23:15,400 and ignore the other. 362 00:23:15,400 --> 00:23:20,800 Suppose I focus on A. And I know that A in the presence of 363 00:23:20,800 --> 00:23:25,590 B, the partial pressure of A is going to obey Raoult's law. 364 00:23:25,590 --> 00:23:28,563 And let's take A to be the more volatile component in 365 00:23:28,563 --> 00:23:30,240 this case here. 366 00:23:30,240 --> 00:23:35,770 So pA star is bigger than pB star. 367 00:23:35,770 --> 00:23:39,820 The partial pressure, pure A, had a particular temperature, 368 00:23:39,820 --> 00:23:46,090 T fixed is bigger than pB star, it's the more volatile 369 00:23:46,090 --> 00:23:48,490 of the two. 370 00:23:48,490 --> 00:23:56,980 So if I were to plot pA as a function of xB, going from 371 00:23:56,980 --> 00:23:59,820 zero to one, so it'll be Raoult's law, I'm going to 372 00:23:59,820 --> 00:24:04,360 start at pA star. 373 00:24:04,360 --> 00:24:09,680 And I'm going to go down linearly to xB equals one. 374 00:24:09,680 --> 00:24:12,445 Now I can do the same thing for the partial pressure of B. 375 00:24:12,445 --> 00:24:19,100 And assume that in the presence of A, just looking at 376 00:24:19,100 --> 00:24:21,200 this component B, now, and I'm looking at the partial 377 00:24:21,200 --> 00:24:24,530 pressure of B, and it's got its impurity in it, A. B obeys 378 00:24:24,530 --> 00:24:27,450 Raoult's law with A in there. 379 00:24:27,450 --> 00:24:34,080 But now if I want to keep the same x axis here, I can look 380 00:24:34,080 --> 00:24:36,920 at xA here, going from zero to one. 381 00:24:36,920 --> 00:24:40,310 So I just basically flip the thing over. 382 00:24:40,310 --> 00:24:44,950 So at xA equals zero that's pure B. pB star is 383 00:24:44,950 --> 00:24:45,960 less than pA star. 384 00:24:45,960 --> 00:24:49,780 I'm starting here, on that side here. pB star, and I'm 385 00:24:49,780 --> 00:25:00,246 going down in a straight line to zero. pB is equal to xB pB 386 00:25:00,246 --> 00:25:02,110 star, straight line all the way to zero when 387 00:25:02,110 --> 00:25:05,210 xB is equal to zero. 388 00:25:05,210 --> 00:25:08,180 So that's pB. 389 00:25:08,180 --> 00:25:10,860 And that's pA. 390 00:25:10,860 --> 00:25:17,250 Now, the total pressure is the sum of the two. pA plus pB. 391 00:25:17,250 --> 00:25:21,570 So the total pressure in this system here, that I measure 392 00:25:21,570 --> 00:25:23,740 here, is just the sum of these two straight lines. 393 00:25:23,740 --> 00:25:25,220 Let me do it in blue. 394 00:25:25,220 --> 00:25:30,510 The total pressure, if you add up these two straight lines 395 00:25:30,510 --> 00:25:32,320 together, you got another straight line. 396 00:25:32,320 --> 00:25:38,460 Two straight lines make a straight line. p is pA plus pB 397 00:25:38,460 --> 00:25:48,030 is equal to xA pA star plus xB pB star. 398 00:25:48,030 --> 00:25:54,200 Now, xA and xB are related. xB is one minus xA. 399 00:25:54,200 --> 00:25:57,080 And so really, this is only a function of one variable here. 400 00:25:57,080 --> 00:26:02,820 Linearly, with respect to one variable. 401 00:26:02,820 --> 00:26:04,190 OK, what does this diagram tell me? 402 00:26:04,190 --> 00:26:13,020 So now, let's ignore how we built it. 403 00:26:13,020 --> 00:26:19,490 And let's take a look at it. 404 00:26:19,490 --> 00:26:23,760 So I have, going to take xB on the bottom here. 405 00:26:23,760 --> 00:26:25,140 From zero to one. 406 00:26:25,140 --> 00:26:29,120 We could choose either one, and I've got the straight line 407 00:26:29,120 --> 00:26:36,370 for the total pressure going from pA star to pB star. 408 00:26:36,370 --> 00:26:38,890 T is fixed. 409 00:26:38,890 --> 00:26:42,360 I need to tell you that T is fixed because I have two 410 00:26:42,360 --> 00:26:43,920 degrees of freedom, right? 411 00:26:43,920 --> 00:26:45,700 I have two degrees of freedom. 412 00:26:45,700 --> 00:26:48,500 My degrees of freedom are the temperature, which I'm fixing. 413 00:26:48,500 --> 00:26:50,880 And the composition in the liquid 414 00:26:50,880 --> 00:26:52,500 phase, which I'm fixing. 415 00:26:52,500 --> 00:26:59,850 So my two variables are xB and T. Those are my 416 00:26:59,850 --> 00:27:00,820 two degrees of freedom. 417 00:27:00,820 --> 00:27:03,330 And I'm telling you what the total pressure is. 418 00:27:03,330 --> 00:27:08,190 I'm giving you total pressure as a function of xB and T. By 419 00:27:08,190 --> 00:27:11,510 fixing T, I'm really telling you what the pressure is of 420 00:27:11,510 --> 00:27:13,190 the function of xB. 421 00:27:13,190 --> 00:27:16,250 So the Gibbs phase rule then tells me that the pressure 422 00:27:16,250 --> 00:27:20,300 should be only a function of xB if T is fixed. 423 00:27:20,300 --> 00:27:21,600 Should be a line. 424 00:27:21,600 --> 00:27:22,980 Not necessarily straight, but it should be a 425 00:27:22,980 --> 00:27:26,030 line in this diagram. 426 00:27:26,030 --> 00:27:28,070 To have coexistence. 427 00:27:28,070 --> 00:27:32,330 So what this line is, then, this line is the line of 428 00:27:32,330 --> 00:27:37,230 points that tells me when I have coexistence between the 429 00:27:37,230 --> 00:27:41,350 gas phase and the liquid phase. 430 00:27:41,350 --> 00:27:41,516 The coexistence line. 431 00:27:41,516 --> 00:27:44,960 This is the gas-liquids coexistence 432 00:27:44,960 --> 00:27:47,370 line for the mixture. 433 00:27:47,370 --> 00:27:55,440 It's not so different than this coexistence line up here. 434 00:27:55,440 --> 00:27:59,250 So you can think of this as a phase diagram, kind of like 435 00:27:59,250 --> 00:28:02,810 you thought, you know this as a phase diagram. 436 00:28:02,810 --> 00:28:05,360 It's got a coexistence line. 437 00:28:05,360 --> 00:28:08,400 And then if I'm above this line, if I'm at a certain 438 00:28:08,400 --> 00:28:13,810 pressure here, I'm at this point here, this pressure 439 00:28:13,810 --> 00:28:17,620 well, let's go to a slightly higher 440 00:28:17,620 --> 00:28:18,730 different pressure here. 441 00:28:18,730 --> 00:28:22,890 And this composition here. 442 00:28:22,890 --> 00:28:24,540 I'm not on the line. 443 00:28:24,540 --> 00:28:27,230 That means that I don't have coexistence 444 00:28:27,230 --> 00:28:29,380 between liquid and gas. 445 00:28:29,380 --> 00:28:30,260 I'm above the line. 446 00:28:30,260 --> 00:28:35,215 I'm at a pressure higher then where I would get coexistence. 447 00:28:35,215 --> 00:28:38,060 The pressure is higher, means I probably don't 448 00:28:38,060 --> 00:28:42,170 have a gas any more. 449 00:28:42,170 --> 00:28:45,320 Probably means that the total pressure pushing down on my 450 00:28:45,320 --> 00:28:48,730 mixture is too high to have a gas phase around. 451 00:28:48,730 --> 00:28:55,700 It means that the top part here is a liquid phase. 452 00:28:55,700 --> 00:29:00,760 So I'm in the liquid phase if I'm above this line and I have 453 00:29:00,760 --> 00:29:03,590 a coexistence between the gas and the liquid 454 00:29:03,590 --> 00:29:06,310 if I'm on the line. 455 00:29:06,310 --> 00:29:13,210 Now, if I'm below this line here, well, I get in trouble. 456 00:29:13,210 --> 00:29:17,740 I get in trouble because my first instinct would be, well 457 00:29:17,740 --> 00:29:19,120 this is just gas. 458 00:29:19,120 --> 00:29:22,400 I go from liquid, and then I have coexistence. 459 00:29:22,400 --> 00:29:24,180 Then I go into the gas phase. 460 00:29:24,180 --> 00:29:27,090 But my x-axis here is the composition 461 00:29:27,090 --> 00:29:28,670 in the liquid phase. 462 00:29:28,670 --> 00:29:31,990 There's no liquid around. 463 00:29:31,990 --> 00:29:35,130 So I really can't say anything here. 464 00:29:35,130 --> 00:29:37,300 This diagram is a little bit meaningless down here. 465 00:29:37,300 --> 00:29:40,500 Because my x-axis is describing a phase which 466 00:29:40,500 --> 00:29:45,150 doesn't exist on the diagram. 467 00:29:45,150 --> 00:29:48,650 On this diagram, here. 468 00:29:48,650 --> 00:29:51,430 So when you see it, this diagram here with Raoult's law 469 00:29:51,430 --> 00:29:54,810 on it, you've got to remember that the part that's really 470 00:29:54,810 --> 00:30:00,090 interesting is above the line and at the line itself. 471 00:30:00,090 --> 00:30:14,100 So you could do an experiment, then, on this line here. 472 00:30:14,100 --> 00:30:14,330 T fixed. 473 00:30:14,330 --> 00:30:20,170 This is the liquid phase. 474 00:30:20,170 --> 00:30:22,420 So suppose I start with some high 475 00:30:22,420 --> 00:30:23,620 pressure up here somewhere. 476 00:30:23,620 --> 00:30:29,220 At this point, let's call this point one. 477 00:30:29,220 --> 00:30:36,680 We've got my container, my piston here. p is equal to p1. 478 00:30:36,680 --> 00:30:38,676 Again, I put the point in an awkward place. 479 00:30:38,676 --> 00:30:51,400 Let's put it right there. p1, and composition xB. 480 00:30:51,400 --> 00:30:56,800 p1 and xB here. 481 00:30:56,800 --> 00:31:02,510 And I'm going to slowly decrease the pressure. 482 00:31:02,510 --> 00:31:03,770 I start in this composition. 483 00:31:03,770 --> 00:31:05,720 The composition doesn't change. 484 00:31:05,720 --> 00:31:07,070 I'm just decreasing the pressure. 485 00:31:07,070 --> 00:31:08,450 Decrease the pressure, decrease the pressure, 486 00:31:08,450 --> 00:31:09,060 decrease the pressure. 487 00:31:09,060 --> 00:31:14,530 And at some point, I get to that line. 488 00:31:14,530 --> 00:31:18,540 Get to that line, what happens when I get to that line? 489 00:31:18,540 --> 00:31:21,260 I start to see bubbles coming out of the liquid. 490 00:31:21,260 --> 00:31:22,530 Vapor starts to form, because it wants to be in the 491 00:31:22,530 --> 00:31:24,330 coexistence line. 492 00:31:24,330 --> 00:31:25,490 So I get to that line. 493 00:31:25,490 --> 00:31:29,100 And I get a little bit of bubbles forming. 494 00:31:29,100 --> 00:31:32,060 And a little bit of vapor. 495 00:31:32,060 --> 00:31:33,980 That's the liquid here. 496 00:31:33,980 --> 00:31:36,090 Little bit of gas. 497 00:31:36,090 --> 00:31:44,870 And that's why this line here is called the bubble line. 498 00:31:44,870 --> 00:31:48,250 So I've gone down, decreased the pressure. 499 00:31:48,250 --> 00:31:50,740 So p has gone down now. 500 00:31:50,740 --> 00:31:52,600 Decreased the pressure. 501 00:31:52,600 --> 00:31:58,740 And if I keep on decreasing pressure, well, I'm not going 502 00:31:58,740 --> 00:32:00,450 to jump into this area of the diagram. 503 00:32:00,450 --> 00:32:02,690 Because this area of the diagram means there's no 504 00:32:02,690 --> 00:32:05,410 liquid, it's a meaningless area. 505 00:32:05,410 --> 00:32:08,416 If I keep decreasing the pressure, I'm still going to 506 00:32:08,416 --> 00:32:09,190 be in coexistence. 507 00:32:09,190 --> 00:32:12,170 I'm just going to make more gas. 508 00:32:12,170 --> 00:32:13,170 More vapor. 509 00:32:13,170 --> 00:32:15,610 I'm going to transfer more of the liquid 510 00:32:15,610 --> 00:32:17,070 into the vapor phase. 511 00:32:17,070 --> 00:32:21,120 I'm going to ride down that line. 512 00:32:21,120 --> 00:32:22,370 I'm going to ride down the line. 513 00:32:22,370 --> 00:32:23,320 I'm decreasing the pressure. 514 00:32:23,320 --> 00:32:24,510 I'm not going to skip into here. 515 00:32:24,510 --> 00:32:29,150 I'm going to ride down the line here. 516 00:32:29,150 --> 00:32:33,690 We're going to keep decreasing the pressure even more. p goes 517 00:32:33,690 --> 00:32:34,060 down even more. 518 00:32:34,060 --> 00:32:37,820 Now, I make even more vapor. 519 00:32:37,820 --> 00:32:41,620 There's my piston right there. 520 00:32:41,620 --> 00:32:43,850 There's the gas phase. 521 00:32:43,850 --> 00:32:46,240 There's the liquid phase. 522 00:32:46,240 --> 00:32:51,400 Now I've transferred material from the liquid 523 00:32:51,400 --> 00:32:53,730 phase to the gas phase. 524 00:32:53,730 --> 00:32:55,810 But I also changed the composition 525 00:32:55,810 --> 00:32:57,500 of my liquid phase. 526 00:32:57,500 --> 00:33:02,420 And if I read my new composition, this is what I 527 00:33:02,420 --> 00:33:05,080 started out with. xB. 528 00:33:05,080 --> 00:33:09,990 My new composition in the liquid phase is going to be, 529 00:33:09,990 --> 00:33:15,220 let's call it xB prime. 530 00:33:15,220 --> 00:33:17,340 There's going to be more solute in there. 531 00:33:17,340 --> 00:33:24,310 Or more B in there, than what I started out with. 532 00:33:24,310 --> 00:33:25,800 Now which one was the more volatile one? 533 00:33:25,800 --> 00:33:27,950 A or B? 534 00:33:27,950 --> 00:33:29,320 A was more volatile, right? 535 00:33:29,320 --> 00:33:34,610 So as I decreased the pressure and I started bubbling off the 536 00:33:34,610 --> 00:33:38,410 material, which one's going to bubble off preferentially? 537 00:33:38,410 --> 00:33:39,910 A is going to. 538 00:33:39,910 --> 00:33:47,350 So it makes sense that I would concentrate B in there. 539 00:33:47,350 --> 00:33:49,040 Decrease the pressure. 540 00:33:49,040 --> 00:33:51,820 Bubbles that are rich in A come out. 541 00:33:51,820 --> 00:33:52,940 Both A and B come out. 542 00:33:52,940 --> 00:33:56,940 But more A comes out than B. And what's left over is rich 543 00:33:56,940 --> 00:34:01,080 and then more than the less volatile material which is B. 544 00:34:01,080 --> 00:34:05,920 So we're beginning to see how distillation comes about. 545 00:34:05,920 --> 00:34:07,730 This is the first step in it. 546 00:34:07,730 --> 00:34:18,220 In a distillation process. 547 00:34:18,220 --> 00:34:21,410 So let's make this a little bit more complicated now. 548 00:34:21,410 --> 00:34:30,380 Any questions, first? 549 00:34:30,380 --> 00:34:32,490 Alright, now, suppose I wanted to know. 550 00:34:32,490 --> 00:34:36,280 So I found out where the composition in the liquid 551 00:34:36,280 --> 00:34:38,080 phase is here. 552 00:34:38,080 --> 00:34:41,505 And I know that should tell me immediately whether the 553 00:34:41,505 --> 00:34:43,970 composition in the gas phase is here. 554 00:34:43,970 --> 00:34:47,860 So I have all, I have everything I need to know to 555 00:34:47,860 --> 00:34:52,030 calculate what the composition in the gas phase is. 556 00:34:52,030 --> 00:34:55,220 So I have everything I need, then, to calculate and draw a 557 00:34:55,220 --> 00:34:58,510 diagram that looks just like this. 558 00:34:58,510 --> 00:35:04,520 Except where my x-axis is the y's instead of the x's. 559 00:35:04,520 --> 00:35:06,660 Going to make the same diagram, except now I'm going 560 00:35:06,660 --> 00:35:10,380 to use the gas phase as my reference point. 561 00:35:10,380 --> 00:35:15,590 The composition in the gas phase. 562 00:35:15,590 --> 00:35:22,190 So I need to get, so here I got p as a function of xB. 563 00:35:22,190 --> 00:35:30,250 I want p, total pressure, as a function of yB. 564 00:35:30,250 --> 00:35:33,240 So I can draw a diagram that looks just like this except 565 00:35:33,240 --> 00:35:41,310 now with the x-axis being the gas phase composition. 566 00:35:41,310 --> 00:35:41,980 Let's turn the crank. 567 00:35:41,980 --> 00:35:45,880 What do I know? 568 00:35:45,880 --> 00:35:55,400 I know Dalton's law. pA is yA times total pressure. 569 00:35:55,400 --> 00:35:59,810 I'm trying mix up the partial pressures and the total 570 00:35:59,810 --> 00:36:02,950 pressure in all ways that I know how to write it. 571 00:36:02,950 --> 00:36:05,595 And then hope that something's going to come out that's going 572 00:36:05,595 --> 00:36:06,540 to be helpful. 573 00:36:06,540 --> 00:36:10,700 So that's Dalton's law. 574 00:36:10,700 --> 00:36:17,960 I know Raoult's law. pA is xA star times pA star. 575 00:36:17,960 --> 00:36:22,740 And that's Raoult. 576 00:36:22,740 --> 00:36:25,400 And I can write Raoult's law in a different way. 577 00:36:25,400 --> 00:36:31,120 I can write as pB is equal to xB pB star, that's just 578 00:36:31,120 --> 00:36:41,310 Raoult's law for B. One minus xA pB star. 579 00:36:41,310 --> 00:36:45,200 And I'm looking for this. 580 00:36:45,200 --> 00:36:48,340 I'm looking for the composition in the gas phase. 581 00:36:48,340 --> 00:36:50,687 Or the total pressure as a function of the composition in 582 00:36:50,687 --> 00:36:51,010 the gas phase. 583 00:36:51,010 --> 00:36:54,830 Let's start by finding the composition in the gas phase 584 00:36:54,830 --> 00:36:56,610 as a function of the composition 585 00:36:56,610 --> 00:36:58,450 in the liquid phase. 586 00:36:58,450 --> 00:37:01,990 And the total pressure. 587 00:37:01,990 --> 00:37:08,940 Rewrite yA is equal to pA over P. Total 588 00:37:08,940 --> 00:37:11,380 pressure is pA plus pB. 589 00:37:11,380 --> 00:37:16,320 Sum of the two partial pressures. pA over pA plus pB. 590 00:37:16,320 --> 00:37:19,290 And I'm trying to get yA in terms of the xA's. 591 00:37:19,290 --> 00:37:21,000 Or xB's, in terms of composition 592 00:37:21,000 --> 00:37:22,680 in the liquid phase. 593 00:37:22,680 --> 00:37:25,860 And I know how to relate that to a constant. 594 00:37:25,860 --> 00:37:29,850 Which is the vapor pressure of the pure material times the, 595 00:37:29,850 --> 00:37:33,280 and the composition of the liquid phase, Raoult's law. 596 00:37:33,280 --> 00:37:44,490 This is xA pA star divided by xA pA star plus xB pB star. 597 00:37:44,490 --> 00:37:46,750 And I know that xB is one minus xA. 598 00:37:46,750 --> 00:37:59,320 So this is xA pA star times pB star plus pA star minus pB 599 00:37:59,320 --> 00:38:02,900 star times xA. 600 00:38:02,900 --> 00:38:07,330 Where all I did was replace xB with one minus xA. 601 00:38:07,330 --> 00:38:10,070 And rearrange my equation on the bottom. 602 00:38:10,070 --> 00:38:15,310 So I've gotten the composition in the gas phase in terms of 603 00:38:15,310 --> 00:38:17,960 the composition in the liquid phase. 604 00:38:17,960 --> 00:38:18,830 They're not independent. 605 00:38:18,830 --> 00:38:20,430 I knew they weren't independent. 606 00:38:20,430 --> 00:38:24,220 And this just shows me that math works. 607 00:38:24,220 --> 00:38:24,490 OK. 608 00:38:24,490 --> 00:38:25,500 So I've got that. 609 00:38:25,500 --> 00:38:28,620 And I can invert this to get the composition in the liquid 610 00:38:28,620 --> 00:38:31,080 phase in terms of the composition in the gas phase. 611 00:38:31,080 --> 00:38:35,200 It's not so straightforward, but you can get xA as a 612 00:38:35,200 --> 00:38:37,720 function of yA, as well. 613 00:38:37,720 --> 00:38:40,300 You've got xA, if you reverse this you get xA is 614 00:38:40,300 --> 00:38:44,605 equal to yA pB star. 615 00:38:44,605 --> 00:38:54,420 It looks kind of the same. pB star plus pA star plus pB star 616 00:38:54,420 --> 00:39:01,310 minus pA star times yA. xA as a function of yA. 617 00:39:05,560 --> 00:39:05,880 Alright. 618 00:39:05,880 --> 00:39:11,690 So now you can actually put everything together, starting 619 00:39:11,690 --> 00:39:13,200 from Dalton's law. 620 00:39:13,200 --> 00:39:15,920 Because really what we want is the total pressure as a 621 00:39:15,920 --> 00:39:19,070 function of the composition in the gas phase. 622 00:39:19,070 --> 00:39:20,500 So there's our total pressure here. 623 00:39:20,500 --> 00:39:22,740 There's the composition in the gas phase. 624 00:39:22,740 --> 00:39:24,520 There's pA. 625 00:39:24,520 --> 00:39:26,960 We've found composition in the gas phase in terms of 626 00:39:26,960 --> 00:39:30,930 composition in the liquid phase. 627 00:39:30,930 --> 00:39:38,620 P is equal to yA over pA. 628 00:39:38,620 --> 00:39:45,050 Which is equal to yA over xA pA star. 629 00:39:45,050 --> 00:39:48,030 So if we could get rid of this here, in terms of 630 00:39:48,030 --> 00:39:49,200 yA, we'd be all set. 631 00:39:49,200 --> 00:39:52,800 We'd get the total pressure as a function of the composition 632 00:39:52,800 --> 00:39:54,240 in the gas phase. 633 00:39:54,240 --> 00:39:57,860 And there's what we need. 634 00:39:57,860 --> 00:40:00,420 So we plug that in here. 635 00:40:00,420 --> 00:40:06,320 And now we have the total pressure in terms of constants 636 00:40:06,320 --> 00:40:12,310 like these pure vapor pressures and the composition 637 00:40:12,310 --> 00:40:18,380 of A in the gas phase. 638 00:40:18,380 --> 00:40:21,110 So let me find a way to write that somewhere. 639 00:40:21,110 --> 00:40:24,870 Let me write it right here in this little box here. 640 00:40:24,870 --> 00:40:29,150 Plug it in, you get something that's not a straight line. 641 00:40:29,150 --> 00:40:31,010 Unfortunately. 642 00:40:31,010 --> 00:40:37,330 Doesn't look like Raoult's law. pA star pB star. pA star 643 00:40:37,330 --> 00:40:48,790 plus pB star minus pA star times yA. 644 00:40:48,790 --> 00:40:49,950 It's not a straight line. 645 00:40:49,950 --> 00:40:57,390 It's a more complicated function. 646 00:40:57,390 --> 00:41:16,860 What does it look like? 647 00:41:16,860 --> 00:41:17,430 Looks like this. 648 00:41:17,430 --> 00:41:23,050 I'm going to use the same sort of plot. 649 00:41:23,050 --> 00:41:25,040 I going to plot the total pressure on this axis here, on 650 00:41:25,040 --> 00:41:25,920 the y axis. 651 00:41:25,920 --> 00:41:28,120 I'm going to keep the T fixed. 652 00:41:28,120 --> 00:41:31,210 Temperature is fixed. 653 00:41:31,210 --> 00:41:34,220 So I'm going to find the total pressure as a function of the 654 00:41:34,220 --> 00:41:35,270 composition in the gas phase. 655 00:41:35,270 --> 00:41:42,870 I'm going to put yB here. yB going from zero to one. 656 00:41:42,870 --> 00:41:50,720 So it's kind of looking like what I had before. 657 00:41:50,720 --> 00:41:51,990 Total pressure is going to look the same. 658 00:41:51,990 --> 00:41:54,450 If you want to do it in terms of B, you just have everywhere 659 00:41:54,450 --> 00:41:56,240 you see B replace it with A. And everywhere you see A 660 00:41:56,240 --> 00:42:02,130 replace it with B. You just need to change the two. 661 00:42:02,130 --> 00:42:06,730 You know that if yB is equal to zero, that you have pure A. 662 00:42:06,730 --> 00:42:12,190 So you know that this is going to be pA star here. 663 00:42:12,190 --> 00:42:14,630 You know if you have yB equals to one in the gas phase, you 664 00:42:14,630 --> 00:42:16,080 have pure B in the gas phase. 665 00:42:16,080 --> 00:42:20,320 You know that what you have is the vapor pressure of the pure 666 00:42:20,320 --> 00:42:26,540 B, in this case the pure water. pB star, OK? 667 00:42:26,540 --> 00:42:32,300 Where A is more volatile then B. A would be the ethanol and 668 00:42:32,300 --> 00:42:35,830 B would be the water. 669 00:42:35,830 --> 00:42:37,330 And it's not a straight line. 670 00:42:37,330 --> 00:42:39,414 If it were Raoult, he would say just connect these two 671 00:42:39,414 --> 00:42:40,280 with a straight line. 672 00:42:40,280 --> 00:42:41,310 But it's not. 673 00:42:41,310 --> 00:42:44,280 It's not, we just figured out with an equation here. 674 00:42:44,280 --> 00:42:49,050 And it turns out to be a line that looks like this. 675 00:42:49,050 --> 00:42:55,150 It's a curved line that looks a bit like this. 676 00:42:55,150 --> 00:42:57,020 OK, what does this line mean? 677 00:42:57,020 --> 00:43:01,540 This line is also a coexistence line. 678 00:43:01,540 --> 00:43:02,510 That's what we started out with. 679 00:43:02,510 --> 00:43:05,410 We started out with Raoult. 680 00:43:05,410 --> 00:43:07,560 That was part of our derivation. 681 00:43:07,560 --> 00:43:15,120 Raoult's law tells you the composition at coexistence. 682 00:43:15,120 --> 00:43:18,260 Tells you the pressure of, the total pressure, or partial 683 00:43:18,260 --> 00:43:21,200 pressure at coexistence and the composition. 684 00:43:21,200 --> 00:43:23,820 So this is a coexistence line between the 685 00:43:23,820 --> 00:43:26,390 liquid and the gas. 686 00:43:26,390 --> 00:43:28,120 Coexistence between liquid and gas. 687 00:43:28,120 --> 00:43:30,382 But now, it's not as a function of the composition at 688 00:43:30,382 --> 00:43:30,990 the liquid phase. 689 00:43:30,990 --> 00:43:34,500 It's a function of the composition in the gas phase. 690 00:43:34,500 --> 00:43:38,460 Which was missing before. 691 00:43:38,460 --> 00:43:43,890 So now if I do my experiment, and I'm, so 692 00:43:43,890 --> 00:43:44,950 what does this mean? 693 00:43:44,950 --> 00:43:48,540 So if I have a point, if I have a pressure which is lower 694 00:43:48,540 --> 00:43:51,110 than the line here, then I have pure gas. 695 00:43:51,110 --> 00:43:52,820 Pressure, low pressure, I have pure gas. 696 00:43:52,820 --> 00:43:53,810 Don't have any liquids. 697 00:43:53,810 --> 00:43:56,950 And that's fine, because I can, I know 698 00:43:56,950 --> 00:43:57,600 what that point is. 699 00:43:57,600 --> 00:44:02,870 If I'm sitting below the line here somewhere, 700 00:44:02,870 --> 00:44:06,640 so I'm sitting here. 701 00:44:06,640 --> 00:44:09,630 Some composition in the gas phase, a certain pressure. 702 00:44:09,630 --> 00:44:11,300 That's fine. 703 00:44:11,300 --> 00:44:15,280 It's a meaningful point. 704 00:44:15,280 --> 00:44:18,870 So my container here has a piston on it. 705 00:44:18,870 --> 00:44:26,120 And I'm starting out with this pressure, p1 here. 706 00:44:26,120 --> 00:44:27,490 p1 pressing down. 707 00:44:27,490 --> 00:44:30,240 And I got yB in this here. 708 00:44:30,240 --> 00:44:33,930 And I increase the pressure, pressure goes up. 709 00:44:33,930 --> 00:44:35,340 Composition doesn't change. 710 00:44:35,340 --> 00:44:38,250 I'm just increasing the pressure. 711 00:44:38,250 --> 00:44:39,640 Increasing the pressure, increasing the pressure. 712 00:44:39,640 --> 00:44:42,380 Now, at some point the pressure becomes big enough 713 00:44:42,380 --> 00:44:44,720 that I start to see liquid forming. 714 00:44:44,720 --> 00:44:54,970 Little drops of liquid forming in my container. 715 00:44:54,970 --> 00:44:58,255 So now I have little drops of liquid that are forming in my 716 00:44:58,255 --> 00:45:01,750 container, that are going to start to rain down and form a 717 00:45:01,750 --> 00:45:04,290 little bit of liquid pooling at the 718 00:45:04,290 --> 00:45:05,640 bottom of the container. 719 00:45:05,640 --> 00:45:10,420 And that's called the dew line. 720 00:45:10,420 --> 00:45:12,860 The other one was called the bubble line, this is called 721 00:45:12,860 --> 00:45:15,520 the dew line. 722 00:45:15,520 --> 00:45:19,180 So when I reach the dew line, I start forming liquid. 723 00:45:19,180 --> 00:45:21,210 And again, just like in the previous case, if I keep 724 00:45:21,210 --> 00:45:24,360 increasing the pressure, I don't go into this area here. 725 00:45:24,360 --> 00:45:25,770 First of all, this area's meaningless for 726 00:45:25,770 --> 00:45:27,620 this diagram here. 727 00:45:27,620 --> 00:45:31,550 Because this area's pure liquid. 728 00:45:31,550 --> 00:45:34,070 But the x-axis here doesn't tell me about the composition 729 00:45:34,070 --> 00:45:34,530 of pure liquid. 730 00:45:34,530 --> 00:45:35,680 It's all yB. 731 00:45:35,680 --> 00:45:38,130 It's all about the composition in the gas -- yes. 732 00:45:38,130 --> 00:45:40,980 STUDENT: [INAUDIBLE] 733 00:45:40,980 --> 00:45:43,130 PROFESSOR: Between the dew line and the bubble line? 734 00:45:43,130 --> 00:45:43,410 STUDENT: Yeah. 735 00:45:43,410 --> 00:45:44,330 PROFESSOR: This never, never world. 736 00:45:44,330 --> 00:45:47,970 Where you're not allowed to inhabit. 737 00:45:47,970 --> 00:45:54,760 We're going to go into that next time. 738 00:45:54,760 --> 00:45:57,920 On how to use those two curves together. 739 00:45:57,920 --> 00:45:59,220 OK, but the same thing happens here. 740 00:45:59,220 --> 00:46:01,270 If I increase the pressure, I'm going to ride the 741 00:46:01,270 --> 00:46:01,680 coexistence line up. 742 00:46:01,680 --> 00:46:12,770 I ride it up, and keep riding it up. 743 00:46:12,770 --> 00:46:14,780 And I get liquid forming in here. 744 00:46:14,780 --> 00:46:16,325 And I can find out the composition in 745 00:46:16,325 --> 00:46:17,100 the gas phase now. 746 00:46:17,100 --> 00:46:20,490 The composition in the gas phase, as I increase the 747 00:46:20,490 --> 00:46:25,620 pressure, I get yB prime, there's yB that I 748 00:46:25,620 --> 00:46:26,920 started out with with. 749 00:46:26,920 --> 00:46:30,510 With yB prime is less than yB. 750 00:46:30,510 --> 00:46:37,320 And remember, B was the less volatile of the two. 751 00:46:37,320 --> 00:46:42,200 As I increase the pressure, I am forming liquid. 752 00:46:42,200 --> 00:46:46,030 The composition in the gas phase changes. 753 00:46:46,030 --> 00:46:50,750 You decrease the mole fraction of the less volatile material 754 00:46:50,750 --> 00:46:51,440 in the gas phase. 755 00:46:51,440 --> 00:46:55,300 The more volatile A becomes more 756 00:46:55,300 --> 00:46:57,010 prevalent in the gas phase. 757 00:46:57,010 --> 00:47:01,250 And what's coming down is mostly, then, it's enriched in 758 00:47:01,250 --> 00:47:07,190 B. You've done an enrichment of the less volatile material 759 00:47:07,190 --> 00:47:10,220 down in here. 760 00:47:10,220 --> 00:47:12,750 And now you can put the two together, as the 761 00:47:12,750 --> 00:47:19,420 question was asked. 762 00:47:19,420 --> 00:47:21,970 Now we can have, we mix the two together. 763 00:47:21,970 --> 00:47:27,610 We mix our Raoult diagram with our dew line diagram. 764 00:47:27,610 --> 00:47:31,050 We mix the bubble line and the dew line together. 765 00:47:31,050 --> 00:47:32,500 In one diagram, which is going to be a 766 00:47:32,500 --> 00:47:48,730 powerful diagram to use. 767 00:47:48,730 --> 00:47:51,040 Total pressure. 768 00:47:51,040 --> 00:47:54,510 Temperature fixed. 769 00:47:54,510 --> 00:48:02,600 You've got two materials, pA star is bigger than pB star. 770 00:48:02,600 --> 00:48:08,830 So I'm going to have pA star here, pB star down here. 771 00:48:08,830 --> 00:48:13,590 And if I choose, as my x-axis, to have xB going from zero to 772 00:48:13,590 --> 00:48:18,700 one, I can draw a coexistence line on this axis. 773 00:48:18,700 --> 00:48:20,520 On this diagram. 774 00:48:20,520 --> 00:48:24,770 That's the coexistence line in terms of xB. 775 00:48:24,770 --> 00:48:29,910 And any point up here is well-described by this xB 776 00:48:29,910 --> 00:48:32,870 value, and the p value. 777 00:48:32,870 --> 00:48:36,770 And as long as I have xB here, I'm not allowed to touch the 778 00:48:36,770 --> 00:48:38,600 bottom part of this diagram. 779 00:48:38,600 --> 00:48:43,220 But now if I add another x-axis, I'm going to add yB. 780 00:48:46,680 --> 00:48:50,550 From zero to one. 781 00:48:50,550 --> 00:48:53,230 I draw a coexistence line in terms of yB. 782 00:48:55,830 --> 00:48:56,500 Like this. 783 00:48:56,500 --> 00:48:59,550 So this line is p as a function of 784 00:48:59,550 --> 00:49:01,850 yB coexistence line. 785 00:49:01,850 --> 00:49:08,030 This line is total pressure of function of xB. 786 00:49:08,030 --> 00:49:11,460 And as long as I'm on this line or below this line, then 787 00:49:11,460 --> 00:49:15,900 this axis makes sense. 788 00:49:15,900 --> 00:49:17,320 That's how we build this diagram here. 789 00:49:17,320 --> 00:49:18,880 It's got a bubble line on top, and the 790 00:49:18,880 --> 00:49:21,520 dew line on the bottom. 791 00:49:21,520 --> 00:49:26,060 So I'm up here going down, I'm going to hit the bubble line. 792 00:49:26,060 --> 00:49:28,590 If I'm down here and go up in pressure, I'm going 793 00:49:28,590 --> 00:49:29,460 to hit the dew line. 794 00:49:29,460 --> 00:49:32,610 And next time -- well, next time we have an exam. 795 00:49:32,610 --> 00:49:38,020 But on Friday, we'll figure out how to use this diagram.