1 00:00:00,030 --> 00:00:01,690 The following content is provided 2 00:00:01,690 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,770 Your support will help MIT OpenCourseWare continue 4 00:00:06,770 --> 00:00:10,510 to offer high quality educational resources for free. 5 00:00:10,510 --> 00:00:13,230 To make a donation or view additional materials 6 00:00:13,230 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:20,890 at ocw.mit.edu. 8 00:00:20,890 --> 00:00:24,240 PROFESSOR: Last time, under the tutelage of Professor Blendi, 9 00:00:24,240 --> 00:00:27,700 you started to see some of the consequences, 10 00:00:27,700 --> 00:00:31,780 the more immediate consequences, of some of the properties 11 00:00:31,780 --> 00:00:34,340 we've been discussing over the last few lectures 12 00:00:34,340 --> 00:00:36,760 of binary liquid mixtures. 13 00:00:36,760 --> 00:00:39,172 And in particular he started showing you 14 00:00:39,172 --> 00:00:41,130 some of the colligative properties by which you 15 00:00:41,130 --> 00:00:44,250 see changes in the vapor pressure and the boiling point 16 00:00:44,250 --> 00:00:47,870 and so forth, that arise because you've got liquids in mixtures 17 00:00:47,870 --> 00:00:50,140 and this affects their chemical potentials. 18 00:00:50,140 --> 00:00:52,230 And today I want to start by going 19 00:00:52,230 --> 00:00:56,460 through the fourth colligative property that's commonly seen, 20 00:00:56,460 --> 00:00:57,694 what called osmotic pressure. 21 00:00:57,694 --> 00:00:58,860 And I mentioned this before. 22 00:00:58,860 --> 00:01:01,210 Today we'll go through a quantitative calculation 23 00:01:01,210 --> 00:01:02,350 of its effects. 24 00:01:02,350 --> 00:01:06,090 And then we'll start in on a different unit of the course. 25 00:01:06,090 --> 00:01:07,930 We'll start in on statistical mechanics. 26 00:01:07,930 --> 00:01:28,370 So let's continue with the colligative properties. 27 00:01:28,370 --> 00:01:31,400 And osmotic pressure plays a super-important role, certainly 28 00:01:31,400 --> 00:01:35,100 in all of biology, because of course inside biological cells 29 00:01:35,100 --> 00:01:36,419 you have solutions. 30 00:01:36,419 --> 00:01:38,210 Water is in there, along with a whole bunch 31 00:01:38,210 --> 00:01:39,830 of other components. 32 00:01:39,830 --> 00:01:41,890 And so the chemical potential of the water 33 00:01:41,890 --> 00:01:44,240 is strongly affected by all the other constituents. 34 00:01:44,240 --> 00:01:47,660 And this has profound effects on what's 35 00:01:47,660 --> 00:01:50,620 involved in maintaining an equilibrium between what 36 00:01:50,620 --> 00:01:53,040 goes on inside and outside of the cell. 37 00:01:53,040 --> 00:01:54,850 And this is the case in all organisms. 38 00:01:54,850 --> 00:01:56,410 And it's also exploited of course 39 00:01:56,410 --> 00:01:59,350 in all sorts of other contexts. 40 00:01:59,350 --> 00:02:04,280 So what's happening with osmotic pressure is, 41 00:02:04,280 --> 00:02:09,280 you've got some situation where there's 42 00:02:09,280 --> 00:02:11,880 a semi-permeable membrane. 43 00:02:11,880 --> 00:02:13,690 It allows one of the constituents 44 00:02:13,690 --> 00:02:15,460 but not others to pass through. 45 00:02:15,460 --> 00:02:16,590 And typically that's water. 46 00:02:16,590 --> 00:02:18,550 Although, of course, it could be anything. 47 00:02:18,550 --> 00:02:24,090 So, in the lab the way this typically looks 48 00:02:24,090 --> 00:02:30,370 is something like this. 49 00:02:30,370 --> 00:02:42,080 Here's your semi-permeable membrane. 50 00:02:42,080 --> 00:02:45,940 Outside, we can imagine that there's 51 00:02:45,940 --> 00:02:49,580 pure liquid A. Could be water, could be something else. 52 00:02:49,580 --> 00:02:54,430 Inside, you've got A mixed with B, 53 00:02:54,430 --> 00:02:59,180 one or more other constituents. 54 00:02:59,180 --> 00:03:04,000 And then there's some level of the liquid 55 00:03:04,000 --> 00:03:05,480 outside the container. 56 00:03:05,480 --> 00:03:10,350 Of course, A can pass between, pass through, the membrane. 57 00:03:10,350 --> 00:03:13,340 And so what ends up happening is, if you just 58 00:03:13,340 --> 00:03:15,470 look at the situation, what you immediately 59 00:03:15,470 --> 00:03:19,580 notice is that the liquid in here 60 00:03:19,580 --> 00:03:27,852 goes up higher than the liquid around it. 61 00:03:27,852 --> 00:03:28,810 It's at a higher level. 62 00:03:28,810 --> 00:03:30,270 It seems counterintuitive at first. 63 00:03:30,270 --> 00:03:32,880 And in fact it can be enormously higher. 64 00:03:32,880 --> 00:03:35,030 But that's what happens. 65 00:03:35,030 --> 00:03:47,690 So there's extra height, h, of liquid inside the container 66 00:03:47,690 --> 00:03:50,910 relative to what's outside it. 67 00:03:50,910 --> 00:03:52,970 And what that means is the pressure inside here 68 00:03:52,970 --> 00:03:54,100 must be higher, right? 69 00:03:54,100 --> 00:03:56,160 Of course, the weight of this liquid 70 00:03:56,160 --> 00:03:59,730 is pushing down on what's down here. 71 00:03:59,730 --> 00:04:02,210 So the pressure inside here is higher 72 00:04:02,210 --> 00:04:03,820 than the pressure outside. 73 00:04:03,820 --> 00:04:05,910 But of course, from what you know already 74 00:04:05,910 --> 00:04:09,280 about liquid mixtures, this should in fact make sense. 75 00:04:09,280 --> 00:04:12,690 Because what's happening is that if these things are 76 00:04:12,690 --> 00:04:15,670 at the same pressure, you know that the chemical potential 77 00:04:15,670 --> 00:04:18,250 of A is going to be lower in the mixture. 78 00:04:18,250 --> 00:04:22,050 It's always lower in the mixture than it is in pure A. 79 00:04:22,050 --> 00:04:25,620 So there's nothing to stop A from rushing in. 80 00:04:25,620 --> 00:04:27,170 So A does rush in. 81 00:04:27,170 --> 00:04:31,830 If the liquid level goes up, it raises the pressure down here. 82 00:04:31,830 --> 00:04:36,470 But as the pressure rises, now the chemical potential 83 00:04:36,470 --> 00:04:38,520 changes and goes up. 84 00:04:38,520 --> 00:04:41,290 So eventually, equilibrium is reestablished. 85 00:04:41,290 --> 00:04:43,040 Because these aren't at the same pressure. 86 00:04:43,040 --> 00:04:45,890 The additional pressure means the chemical potential of A 87 00:04:45,890 --> 00:04:46,680 goes up. 88 00:04:46,680 --> 00:04:49,570 Of course, it goes down by virtue of being in the mixture. 89 00:04:49,570 --> 00:04:52,426 And those two things will eventually reach an equilibrium 90 00:04:52,426 --> 00:04:53,800 where the chemical potential of A 91 00:04:53,800 --> 00:04:57,550 is the same everywhere, as it has to be. 92 00:04:57,550 --> 00:04:59,930 So what we'll calculate now is, OK, 93 00:04:59,930 --> 00:05:02,210 how much does the pressure go up. 94 00:05:02,210 --> 00:05:04,140 When that happens. 95 00:05:04,140 --> 00:05:05,467 So let's take a look. 96 00:05:05,467 --> 00:05:07,550 And so of course, there are all kinds of examples. 97 00:05:07,550 --> 00:05:09,340 In a typical one, A would be water. 98 00:05:09,340 --> 00:05:11,310 And B would be one or more sugars. 99 00:05:11,310 --> 00:05:14,790 All sorts of other potential constituents also. 100 00:05:14,790 --> 00:05:18,820 OK, so let's just consider the pressure 101 00:05:18,820 --> 00:05:21,090 at two places that are the same level. 102 00:05:21,090 --> 00:05:22,500 The same height. 103 00:05:22,500 --> 00:05:25,640 We'll label this one point alpha. 104 00:05:25,640 --> 00:05:41,710 Let's put some color in here. 105 00:05:41,710 --> 00:05:46,690 We'll have points alpha and beta. 106 00:05:46,690 --> 00:05:54,450 And we'll make this distance l. 107 00:05:54,450 --> 00:05:55,970 So now let's just first calculate 108 00:05:55,970 --> 00:05:57,764 what the pressure is at both points. 109 00:05:57,764 --> 00:05:59,430 It's going to be higher here, because of 110 00:05:59,430 --> 00:06:01,340 this additional liquid on top. 111 00:06:01,340 --> 00:06:11,130 So, if we look at point alpha, well, 112 00:06:11,130 --> 00:06:12,630 let's start at point beta, actually. 113 00:06:12,630 --> 00:06:14,254 That's the simpler one since it doesn't 114 00:06:14,254 --> 00:06:17,800 have the additional liquid. 115 00:06:17,800 --> 00:06:22,470 And first of all, there's just atmospheric pressure. 116 00:06:22,470 --> 00:06:26,140 So there's an external pressure. 117 00:06:26,140 --> 00:06:28,960 Certainly, typically, it would just be atmosphere pressure. 118 00:06:28,960 --> 00:06:30,880 It's pushing down here. 119 00:06:30,880 --> 00:06:36,560 It's also there. 120 00:06:36,560 --> 00:06:44,090 So that has to factor in. 121 00:06:44,090 --> 00:06:48,060 So there's our external pressure. 122 00:06:48,060 --> 00:06:53,410 And then there is the product of this height 123 00:06:53,410 --> 00:06:58,370 times the density of the solution. 124 00:06:58,370 --> 00:07:02,740 Times the gravitational force constant. 125 00:07:02,740 --> 00:07:05,070 And I'll work through that momentarily 126 00:07:05,070 --> 00:07:07,690 so you see that that indeed is the pressure. 127 00:07:07,690 --> 00:07:10,770 That's exerted by the amount of liquid 128 00:07:10,770 --> 00:07:14,360 that's up above this level. 129 00:07:14,360 --> 00:07:22,040 But before I do that, let's put the pressure at point alpha. 130 00:07:22,040 --> 00:07:23,690 So p alpha. 131 00:07:23,690 --> 00:07:29,840 And that's p external plus l rho g. 132 00:07:29,840 --> 00:07:35,910 But now, there's this additional height on top of that. 133 00:07:35,910 --> 00:07:40,960 So there's also h rho g. 134 00:07:40,960 --> 00:07:44,730 There's an extra pressure. 135 00:07:44,730 --> 00:07:55,510 We'll label this pi. 136 00:07:55,510 --> 00:08:05,080 So we can write p alpha is equals to p beta plus h rho g. 137 00:08:05,080 --> 00:08:15,280 Which we'll say is equal to p beta plus pi. 138 00:08:15,280 --> 00:08:18,290 Now, just to make sure we're all on the same page 139 00:08:18,290 --> 00:08:21,310 about these quantities being the pressure that's 140 00:08:21,310 --> 00:08:23,690 exerted by the liquid, let's just look at that. 141 00:08:23,690 --> 00:08:38,800 So l times rho times g. 142 00:08:38,800 --> 00:08:45,280 It should be a pressure which is a force per unit area. 143 00:08:45,280 --> 00:08:56,200 And in particular, this is going to be a weight per unit area. 144 00:08:56,200 --> 00:09:02,100 Remember the weight is the force that a mass exerts. 145 00:09:02,100 --> 00:09:05,820 And it exerts it because of the gravitational constant. 146 00:09:05,820 --> 00:09:07,600 So what happens? 147 00:09:07,600 --> 00:09:17,490 Well, this is just a length. 148 00:09:17,490 --> 00:09:22,600 Rho is a density, right? 149 00:09:22,600 --> 00:09:36,630 So it's mass per unit volume. 150 00:09:36,630 --> 00:09:47,690 And of course, g is just our gravitational force constant. 151 00:09:47,690 --> 00:09:53,110 So it'll be in units of meters per second squared. 152 00:09:53,110 --> 00:09:55,100 So what's going to happen? 153 00:09:55,100 --> 00:09:58,710 If we multiply l times rho together, 154 00:09:58,710 --> 00:10:00,210 this is units of length. 155 00:10:00,210 --> 00:10:01,890 This is units of volume. 156 00:10:01,890 --> 00:10:06,005 What we're going to get then is mass per unit area. 157 00:10:06,005 --> 00:10:07,380 And that's what it is, of course. 158 00:10:07,380 --> 00:10:11,160 It's distributing across the area of that surface 159 00:10:11,160 --> 00:10:13,310 where point alpha is. 160 00:10:13,310 --> 00:10:20,350 So these two together, l times rho, 161 00:10:20,350 --> 00:10:29,270 give us mass per unit area. 162 00:10:29,270 --> 00:10:34,970 So it could be in units of kilograms. 163 00:10:34,970 --> 00:10:38,370 Per unit area, per meters squared. 164 00:10:38,370 --> 00:10:41,550 And now we're going to multiply that by the gravitational force 165 00:10:41,550 --> 00:10:42,460 constant. 166 00:10:42,460 --> 00:10:47,790 So now we're going to have all together 167 00:10:47,790 --> 00:10:55,240 kilograms per meter second squared. 168 00:10:55,240 --> 00:11:11,370 In other words, it's force per unit area. 169 00:11:11,370 --> 00:11:13,090 And so that's what we have. 170 00:11:13,090 --> 00:11:15,430 And now, of course, we have the additional component. 171 00:11:15,430 --> 00:11:22,020 Due to the height here. 172 00:11:22,020 --> 00:11:24,740 Now, what do we know about the chemical potential 173 00:11:24,740 --> 00:11:25,770 in both parts? 174 00:11:25,770 --> 00:11:27,430 The chemical potential has to be equal 175 00:11:27,430 --> 00:11:31,590 outside and inside the container. 176 00:11:31,590 --> 00:11:45,880 So mu A at point alpha is just mu A 177 00:11:45,880 --> 00:11:54,070 of the liquid at pressure p plus pi and temperature T. 178 00:11:54,070 --> 00:12:00,170 And that has to be equal to mu A at point beta. 179 00:12:00,170 --> 00:12:03,400 But point beta is just in the pure liquid. 180 00:12:03,400 --> 00:12:08,155 So that's just mu A star of the liquid 181 00:12:08,155 --> 00:12:15,350 at pressure p and temperature T. So here's 182 00:12:15,350 --> 00:12:22,960 the chemical potential at point beta. 183 00:12:22,960 --> 00:12:24,470 Here it is at point alpha. 184 00:12:24,470 --> 00:12:26,570 Those two have to be equal to each other. 185 00:12:26,570 --> 00:12:32,310 And now we're just going to use Raoult's law. 186 00:12:32,310 --> 00:12:33,820 And what does that tell us? 187 00:12:33,820 --> 00:12:44,280 We have RT log of xA plus mu A star of the liquid at pressure 188 00:12:44,280 --> 00:12:49,460 p plus pi and T So there is our chemical potential 189 00:12:49,460 --> 00:12:51,780 inside the container. 190 00:12:51,780 --> 00:13:01,110 And that's equal to mu A star of the pure liquid at point beta. 191 00:13:01,110 --> 00:13:09,150 And so we can just rewrite this as RT log xA. 192 00:13:09,150 --> 00:13:16,080 Plus mu A star liquid at pressure p plus pi 193 00:13:16,080 --> 00:13:25,640 of T minus mu A star at pressure p is equal to zero. 194 00:13:25,640 --> 00:13:29,510 OK., now we need the pressure dependence 195 00:13:29,510 --> 00:13:31,610 of the chemical potential. 196 00:13:31,610 --> 00:13:37,320 But we certainly have an expression for that. 197 00:13:37,320 --> 00:13:47,440 So we know that dG is S dT plus V dp. 198 00:13:47,440 --> 00:13:49,450 The temperature is the same on both sides. 199 00:13:49,450 --> 00:13:53,880 But we need to worry about what happens at different pressures. 200 00:13:53,880 --> 00:14:04,930 So at constant T, dG is V dp. 201 00:14:04,930 --> 00:14:07,390 So now if we look at the chemical potential, which 202 00:14:07,390 --> 00:14:12,980 is just the Gibbs free energy per mole, 203 00:14:12,980 --> 00:14:28,050 then d mu A star is VA star molar dp. 204 00:14:28,050 --> 00:14:30,620 In other words, the difference in the chemical potential 205 00:14:30,620 --> 00:14:33,190 is, this changes as a function of pressure. 206 00:14:33,190 --> 00:14:36,900 Is going to be given by the molar volume of a 207 00:14:36,900 --> 00:14:39,630 under these conditions in the pure liquid dp. 208 00:14:45,720 --> 00:14:47,810 It's just the potential, the pressure dependence 209 00:14:47,810 --> 00:14:52,510 of G. It's not the kind of topic I would have thought 210 00:14:52,510 --> 00:14:54,110 would drive away a lot of people. 211 00:14:54,110 --> 00:14:56,560 But, you never know. 212 00:14:56,560 --> 00:14:59,264 OK. 213 00:14:59,264 --> 00:15:01,430 For the rest of you who are willing to bear with me, 214 00:15:01,430 --> 00:15:06,600 let's continue. 215 00:15:06,600 --> 00:15:08,620 So now let's just integrate. 216 00:15:08,620 --> 00:15:14,200 We need to know the change in mu A at a finite jump in pressure 217 00:15:14,200 --> 00:15:17,060 from inside to outside the container. 218 00:15:17,060 --> 00:15:19,540 So we're just going to integrate. 219 00:15:19,540 --> 00:15:30,380 So if I want mu A star at pressure p 220 00:15:30,380 --> 00:15:39,180 plus pi minus mu A star at pressure p, 221 00:15:39,180 --> 00:15:49,130 then I just have to integrate from p to p plus pi VA bar star 222 00:15:49,130 --> 00:15:50,207 dp. 223 00:15:50,207 --> 00:15:52,290 And now I'm going to assume, and this is certainly 224 00:15:52,290 --> 00:15:56,460 a safe assumption, that this quantity this, molar volume, 225 00:15:56,460 --> 00:15:57,690 isn't going to change. 226 00:15:57,690 --> 00:16:00,250 In other words, the molar volume of liquid A 227 00:16:00,250 --> 00:16:02,340 isn't going to change significantly, 228 00:16:02,340 --> 00:16:05,360 going from the pressure out here to the pressure in here. 229 00:16:05,360 --> 00:16:08,220 On the grand scale of things it's a small pressure change. 230 00:16:08,220 --> 00:16:10,930 We can assume that the liquid is incompressible 231 00:16:10,930 --> 00:16:13,877 over that small pressure change. 232 00:16:13,877 --> 00:16:15,710 Which means that this is constant, since all 233 00:16:15,710 --> 00:16:27,330 we have then is simply VA bar star times pi. 234 00:16:27,330 --> 00:16:30,930 So then, substituting back here, so now 235 00:16:30,930 --> 00:16:32,890 we've calculated this difference, 236 00:16:32,890 --> 00:16:35,340 it's a simple result. 237 00:16:35,340 --> 00:16:42,980 So then we simply have that RT log xA plus VA bar 238 00:16:42,980 --> 00:16:50,810 star pi is equal to zero. 239 00:16:50,810 --> 00:16:55,020 Now, you saw last time, and I'll work through this 240 00:16:55,020 --> 00:17:01,950 quickly again, that the log of xA 241 00:17:01,950 --> 00:17:08,660 can be written approximately as minus nB over nA 242 00:17:08,660 --> 00:17:11,530 if you remember, this came from writing 243 00:17:11,530 --> 00:17:17,460 that this is equal to the log of one minus xB. 244 00:17:17,460 --> 00:17:21,650 But we're assuming that B is the minor constituent here. 245 00:17:21,650 --> 00:17:25,070 So xB is a small number. 246 00:17:25,070 --> 00:17:30,840 So that this is roughly equal to minus xB. 247 00:17:30,840 --> 00:17:38,430 And that's just minus nB over nA plus nB. 248 00:17:38,430 --> 00:17:40,760 And since nB is much smaller than nA, 249 00:17:40,760 --> 00:17:48,400 this is approximately equal to minus nB over nA. 250 00:17:48,400 --> 00:17:51,540 So we're going to make this substitution. 251 00:17:51,540 --> 00:17:55,300 Also, this quantity, this molar volume 252 00:17:55,300 --> 00:18:04,200 of A over the pure liquid, since the concentration of B is low, 253 00:18:04,200 --> 00:18:06,180 we can assume that this is just the molar 254 00:18:06,180 --> 00:18:09,610 volume of A in general. 255 00:18:09,610 --> 00:18:16,410 So, in other words, we can write VA bar star. 256 00:18:16,410 --> 00:18:21,600 We can just write it as VA bar. 257 00:18:21,600 --> 00:18:23,550 That is, we're not going to worry 258 00:18:23,550 --> 00:18:25,340 about changes in the molar volume, 259 00:18:25,340 --> 00:18:29,190 either as a function of pressure or a function of concentration 260 00:18:29,190 --> 00:18:32,350 at the low concentrations that we're working. 261 00:18:32,350 --> 00:18:39,730 And then, note that nA times VA per mole 262 00:18:39,730 --> 00:18:41,710 is just the total volume. 263 00:18:41,710 --> 00:18:43,450 Oops, not bar. 264 00:18:43,450 --> 00:18:44,550 Not the molar quantity. 265 00:18:44,550 --> 00:18:46,260 This is the molar quantity multiplied 266 00:18:46,260 --> 00:18:49,120 by the number of moles. 267 00:18:49,120 --> 00:18:54,360 So it just gives us VA, the total volume, occupied by A. 268 00:18:54,360 --> 00:19:05,040 And using these two results, then we have that RT times 269 00:19:05,040 --> 00:19:11,160 negative nB over nA, that's this result. 270 00:19:11,160 --> 00:19:21,970 Plus VA over nA times pi is equal to zero. nA will cancel. 271 00:19:21,970 --> 00:19:27,420 And finally, since almost all the volume is due to A, 272 00:19:27,420 --> 00:19:30,220 again because B is the minor constituent, 273 00:19:30,220 --> 00:19:39,060 we can approximate further that VA is approximately equal to V. 274 00:19:39,060 --> 00:19:43,900 So finally getting rid of the nA on the denominators, 275 00:19:43,900 --> 00:19:46,650 we're left with a simple expression. 276 00:19:46,650 --> 00:19:58,500 pi times V is RT nB. 277 00:19:58,500 --> 00:20:00,670 So we have a very simple expression 278 00:20:00,670 --> 00:20:04,190 which is called the van't Hoff expression, 279 00:20:04,190 --> 00:20:08,690 and look at how it resembles the ideal gas law. 280 00:20:08,690 --> 00:20:11,170 This is the pressure times the volume equals 281 00:20:11,170 --> 00:20:13,460 the number of moles times RT. 282 00:20:13,460 --> 00:20:17,090 Of course, really it's a change in pressure. 283 00:20:17,090 --> 00:20:18,950 And this number of moles is the number 284 00:20:18,950 --> 00:20:24,620 of moles of a guest constituent in a solution. 285 00:20:24,620 --> 00:20:29,860 But it has the same form that you're familiar with. 286 00:20:29,860 --> 00:20:33,840 We also, sometimes it's convenient to substitute, 287 00:20:33,840 --> 00:20:40,200 since nB over V, that's just the concentration. 288 00:20:40,200 --> 00:20:50,190 That's the number of moles over the volume. 289 00:20:50,190 --> 00:20:52,110 And in that case, the expression is 290 00:20:52,110 --> 00:20:58,100 rewritten as pi is RT times c, where it's understood that this 291 00:20:58,100 --> 00:21:06,510 is the concentration of the solute. 292 00:21:06,510 --> 00:21:14,320 OK, any questions? 293 00:21:14,320 --> 00:21:17,400 Let's work through a numerical problem. 294 00:21:17,400 --> 00:21:21,650 These things can seem a little bit, they can seem simple, 295 00:21:21,650 --> 00:21:24,650 but sometimes when confronted with a problem, 296 00:21:24,650 --> 00:21:27,580 sometimes it may seem less than obvious what to do. 297 00:21:27,580 --> 00:21:30,200 And also the results can seem a little bit surprising. 298 00:21:30,200 --> 00:21:33,190 So I just want to work through a simple numerical example 299 00:21:33,190 --> 00:21:34,850 to see how these play out. 300 00:21:34,850 --> 00:21:51,670 So let's try, let's imagine we have 10 grams of unknown solid. 301 00:21:51,670 --> 00:22:09,780 Dissolved in 1000 grams of water. 302 00:22:09,780 --> 00:22:22,550 The vapor pressure is 25.195 torr at 27 degrees C. 303 00:22:22,550 --> 00:22:35,190 And the pure water vapor pressure is 25.000. 304 00:22:35,190 --> 00:22:43,580 No, .200 torr at the same temperature. 305 00:22:43,580 --> 00:22:53,790 So first, let's just calculate the molecular weight 306 00:22:53,790 --> 00:22:54,770 of the solid. 307 00:22:54,770 --> 00:22:58,346 Now, this you I think, saw from last time. 308 00:22:58,346 --> 00:22:59,470 So this is straightforward. 309 00:22:59,470 --> 00:23:02,220 It's going to come from the vapor pressure lowering 310 00:23:02,220 --> 00:23:05,870 not from the osmotic pressure, which we'll do next. 311 00:23:05,870 --> 00:23:19,210 So the delta p of H2O is minus xB times p star of H2O. 312 00:23:19,210 --> 00:23:21,490 Alright, this is from last time. 313 00:23:21,490 --> 00:23:24,600 The expression for the vapor pressure reduction. 314 00:23:24,600 --> 00:23:31,420 Due to the concentration of the mole fraction of B. 315 00:23:31,420 --> 00:23:42,590 And then xB is 0.005 torr over 25.2 torr. 316 00:23:42,590 --> 00:23:45,410 Right? 317 00:23:45,410 --> 00:23:49,470 And that's 1.98 times 10 to the minus 4. 318 00:23:49,470 --> 00:23:57,860 So that's our mole fraction of B. 319 00:23:57,860 --> 00:24:06,750 And then xB is nB over nA plus nB. 320 00:24:06,750 --> 00:24:12,330 So, now we have this, it's 10 grams 321 00:24:12,330 --> 00:24:18,210 divided by the molecular weight, which is in grams per mole, 322 00:24:18,210 --> 00:24:23,480 over 1000 grams of water. 323 00:24:23,480 --> 00:24:34,560 Divided by 18 grams per mole plus 10 grams 324 00:24:34,560 --> 00:24:40,900 divided by the molecular weight in grams per mole. 325 00:24:40,900 --> 00:24:44,080 But since the solution is overwhelmingly water, 326 00:24:44,080 --> 00:24:47,712 we can get rid of this term and make the math simple. 327 00:24:47,712 --> 00:24:48,420 Can you see that? 328 00:24:48,420 --> 00:24:52,510 I guess barely. 329 00:24:52,510 --> 00:25:00,220 So this turns out to be about 1.98 time 10 to the minus 4. 330 00:25:00,220 --> 00:25:02,770 And if we solve for molecular weight, 331 00:25:02,770 --> 00:25:13,030 then we wind up with 907 grams per mole. 332 00:25:13,030 --> 00:25:16,520 So that's using one of the colligative properties 333 00:25:16,520 --> 00:25:17,960 that you saw last time. 334 00:25:17,960 --> 00:25:27,030 Now let's, though, finish up by calculating 335 00:25:27,030 --> 00:25:30,250 the additional pressure that we'll find. 336 00:25:30,250 --> 00:25:42,730 So what's the osmotic pressure of the solution. 337 00:25:42,730 --> 00:25:45,840 So now we're going to assume that we have that solution 338 00:25:45,840 --> 00:25:48,370 inside, in a situation like this where there's 339 00:25:48,370 --> 00:25:50,650 pure water outside that can go in. 340 00:25:50,650 --> 00:25:53,050 And there's going to be excess pressure because 341 00:25:53,050 --> 00:25:55,600 of the water rushing into the solution. 342 00:25:55,600 --> 00:26:01,940 So what happens? 343 00:26:01,940 --> 00:26:05,570 And let's use a value of the density. 344 00:26:05,570 --> 00:26:15,600 0.995 grams per centimeters cubed. 345 00:26:15,600 --> 00:26:24,000 So, pi is RT times c. 346 00:26:24,000 --> 00:26:35,270 That's 0.08314 bar per Kelvin mole. 347 00:26:35,270 --> 00:26:48,400 Times 300 K times c. c is nB over V. 348 00:26:48,400 --> 00:26:57,500 So it's 10 grams divided by 907 grams per mole, 349 00:26:57,500 --> 00:27:08,100 over 1010 grams over 0.995 grams per milliliter, 350 00:27:08,100 --> 00:27:12,080 I'm equating with centimeter cubed. 351 00:27:12,080 --> 00:27:17,160 And that turns out to be 1.09 times 10 352 00:27:17,160 --> 00:27:22,890 to the minus 5th mole per millileter. 353 00:27:22,890 --> 00:27:25,330 Let's get that into liters. 354 00:27:25,330 --> 00:27:31,340 So we'll multiply by 1000 milliliters per liter. 355 00:27:31,340 --> 00:27:39,280 So it's 1.09 times 10 to the 5th moles per liter. 356 00:27:39,280 --> 00:27:40,230 OK, yeah. 357 00:27:40,230 --> 00:27:44,780 STUDENT: [INAUDIBLE] 358 00:27:44,780 --> 00:27:49,870 PROFESSOR: Let's see. 359 00:27:49,870 --> 00:27:51,890 Oh, don't forget that, right. 360 00:27:51,890 --> 00:27:56,160 STUDENT: [INAUDIBLE] 361 00:27:56,160 --> 00:27:58,750 PROFESSOR: Ah, yes, yes, yes. 362 00:27:58,750 --> 00:28:05,830 And I think that's the case in the notes also. 363 00:28:05,830 --> 00:28:06,840 Yep. 364 00:28:06,840 --> 00:28:09,450 Thank you. 365 00:28:09,450 --> 00:28:13,925 OK, so the concentration, just to calibrate us here, 366 00:28:13,925 --> 00:28:15,550 it's a pretty low concentration, right? 367 00:28:15,550 --> 00:28:20,770 It's about 10 to the minus 5 moles per liter. 368 00:28:20,770 --> 00:28:27,169 The excess pressure is 0.27 bar. 369 00:28:27,169 --> 00:28:29,210 In other words, about a quarter of an atmosphere. 370 00:28:29,210 --> 00:28:30,210 Uh-oh, what's happening? 371 00:28:30,210 --> 00:28:34,240 STUDENT: [INAUDIBLE] 372 00:28:34,240 --> 00:28:35,240 PROFESSOR: Where are we? 373 00:28:35,240 --> 00:28:38,110 STUDENT: [INAUDIBLE] 374 00:28:38,110 --> 00:28:40,010 PROFESSOR: Sorry, where exactly? 375 00:28:40,010 --> 00:28:48,490 STUDENT: [INAUDIBLE] 376 00:28:48,490 --> 00:28:50,230 PROFESSOR: Oh, yes. 377 00:28:50,230 --> 00:28:53,720 Yes, sorry. 378 00:28:53,720 --> 00:28:54,890 Yeah. 379 00:28:54,890 --> 00:28:57,650 10 to the minus 2 moles per liter. 380 00:28:57,650 --> 00:28:58,490 All right. 381 00:28:58,490 --> 00:29:01,650 And so the pressure that results is a modest pressure, right? 382 00:29:01,650 --> 00:29:04,800 It's about a quarter of an atmosphere of excess pressure. 383 00:29:04,800 --> 00:29:13,340 2.7 time 10 to the 4 pascals. 384 00:29:13,340 --> 00:29:17,900 So let's just calculate, given this small additional pressure, 385 00:29:17,900 --> 00:29:20,860 how high is the liquid going to go, 386 00:29:20,860 --> 00:29:26,380 above the level of the solution. 387 00:29:26,380 --> 00:29:28,080 Anybody want to guess? 388 00:29:28,080 --> 00:29:29,460 You know, a centimeter. 389 00:29:29,460 --> 00:29:30,430 10 centimeters. 390 00:29:30,430 --> 00:29:32,040 100 centimeters? 391 00:29:32,040 --> 00:29:33,520 Give me an order of magnitude. 392 00:29:33,520 --> 00:29:35,700 Who guesses one centimeter? 393 00:29:35,700 --> 00:29:36,690 You got three choices. 394 00:29:36,690 --> 00:29:39,040 One, 10 and 100, and you have to make one. 395 00:29:39,040 --> 00:29:41,250 Who guesses one centimeter? 396 00:29:41,250 --> 00:29:43,040 Who guesses 10 centimeters? 397 00:29:43,040 --> 00:29:46,020 Who guesses 100 centimeters? 398 00:29:46,020 --> 00:29:49,260 OK. 399 00:29:49,260 --> 00:29:53,520 Now, fortunately we don't do problem sets. 400 00:29:53,520 --> 00:29:58,320 Or other scientific problems by popular vote. 401 00:29:58,320 --> 00:29:59,830 Rather, we just solve them. 402 00:29:59,830 --> 00:30:01,500 Whenever possible. 403 00:30:01,500 --> 00:30:15,030 So, what's the height of a column of solution? 404 00:30:15,030 --> 00:30:22,050 Well, that pi, remember, that's equal to rho times g times h, 405 00:30:22,050 --> 00:30:22,830 right? 406 00:30:22,830 --> 00:30:24,520 And we know pi now. 407 00:30:24,520 --> 00:30:30,240 So h is pi over rho times g. 408 00:30:30,240 --> 00:30:36,410 So it's 2.7 times 10 to the 4 pascal 409 00:30:36,410 --> 00:30:43,870 over rho, that's 0.995 grams per centimeters cubed. 410 00:30:43,870 --> 00:30:56,010 Times 1 kilogram over 1000 grams times 100 centimeters per meter 411 00:30:56,010 --> 00:30:56,510 cubed. 412 00:30:56,510 --> 00:30:58,670 I'm just getting this into units that 413 00:30:58,670 --> 00:31:01,350 will give me, that will be in meters, 414 00:31:01,350 --> 00:31:06,280 times 9.8 meter per a second squared. 415 00:31:06,280 --> 00:31:13,230 And when I multiply all that out, it's 2.8 meters. 416 00:31:13,230 --> 00:31:18,360 So that puny little pressure ends up being meters high. 417 00:31:18,360 --> 00:31:21,550 Taller than any of us, by far. 418 00:31:21,550 --> 00:31:25,270 Sorry, it's up to there. 419 00:31:25,270 --> 00:31:26,730 And that's actually pretty typical. 420 00:31:26,730 --> 00:31:31,450 And what that means is, it actually makes osmotic pressure 421 00:31:31,450 --> 00:31:35,240 a very, very sensitive measurement method. 422 00:31:35,240 --> 00:31:39,430 If you want to determine a molecular weight, for example. 423 00:31:39,430 --> 00:31:41,220 Because it's so high, that means, 424 00:31:41,220 --> 00:31:43,720 and of course you can measure that height pretty accurately. 425 00:31:43,720 --> 00:31:46,420 Certainly within about a millimeter. 426 00:31:46,420 --> 00:31:50,106 And so even relatively modest concentrations, 427 00:31:50,106 --> 00:31:52,480 it's still easy to see the effect and measure the impact. 428 00:31:52,480 --> 00:31:58,860 And use that to calculate the properties of the solute. 429 00:31:58,860 --> 00:32:02,990 OK, any questions? 430 00:32:02,990 --> 00:32:07,840 Alright. 431 00:32:07,840 --> 00:32:10,770 Now we're going to start something new. 432 00:32:10,770 --> 00:32:17,550 So, so far what you've seen, and I hope come to appreciate, 433 00:32:17,550 --> 00:32:20,370 is the whole structure of thermodynamics 434 00:32:20,370 --> 00:32:25,560 is built on empirical macroscopic observation 435 00:32:25,560 --> 00:32:27,750 and deduction. 436 00:32:27,750 --> 00:32:29,030 We observe things. 437 00:32:29,030 --> 00:32:33,380 We formulate these broad thermodynamic laws. 438 00:32:33,380 --> 00:32:34,370 All macroscopic. 439 00:32:34,370 --> 00:32:37,790 It doesn't depend on any microscopic model at all. 440 00:32:37,790 --> 00:32:39,600 And in fact much of it was formulated 441 00:32:39,600 --> 00:32:42,930 before there was a good microscopic model of matter. 442 00:32:42,930 --> 00:32:45,610 Before the atomic theory of matter, and molecules, 443 00:32:45,610 --> 00:32:50,230 and so forth was well worked out. 444 00:32:50,230 --> 00:32:56,570 And it's incredible how powerful that whole formalism is, right? 445 00:32:56,570 --> 00:32:58,830 In some sense, although it may seem 446 00:32:58,830 --> 00:33:03,460 like neglecting what we now know to be 447 00:33:03,460 --> 00:33:07,190 an important part of nature, actually part of its power 448 00:33:07,190 --> 00:33:09,090 is its empiricism. 449 00:33:09,090 --> 00:33:09,680 Right? 450 00:33:09,680 --> 00:33:12,010 There are these very small number 451 00:33:12,010 --> 00:33:16,579 of fundamental laws from which everything else follows. 452 00:33:16,579 --> 00:33:18,120 And at this point, of course, there's 453 00:33:18,120 --> 00:33:21,240 just enormous confidence in that small number of laws. 454 00:33:21,240 --> 00:33:24,690 Because of their being verified in so many context. 455 00:33:24,690 --> 00:33:29,180 But, at the same time, we do know 456 00:33:29,180 --> 00:33:31,340 about the atomic theory of matter. 457 00:33:31,340 --> 00:33:33,920 We know there are atoms and molecules. 458 00:33:33,920 --> 00:33:37,650 So it ought to be possible, at least in principle, 459 00:33:37,650 --> 00:33:42,350 to start from a purely microscopic approach to nature. 460 00:33:42,350 --> 00:33:48,650 And just based on figuring out the microscopic properties. 461 00:33:48,650 --> 00:33:50,970 And then saying, well, OK, my macroscopic stuff 462 00:33:50,970 --> 00:33:55,240 is just a collection of those microscopic entities. 463 00:33:55,240 --> 00:33:59,240 I should also be able to figure out macroscopic thermodynamics. 464 00:33:59,240 --> 00:34:02,380 I should be able to start from my microscopic picture 465 00:34:02,380 --> 00:34:06,890 and get to macroscopic thermodynamic results. 466 00:34:06,890 --> 00:34:08,760 And in fact, that is possible. 467 00:34:08,760 --> 00:34:11,210 And the theoretical formulation for it 468 00:34:11,210 --> 00:34:13,180 is what's called statistical mechanics. 469 00:34:13,180 --> 00:34:15,710 Called that because, of course, you 470 00:34:15,710 --> 00:34:18,650 won't be surprised to learn that it's 471 00:34:18,650 --> 00:34:21,090 going to require a statistical treatment. 472 00:34:21,090 --> 00:34:23,660 We're going to be dealing with moles of material. 473 00:34:23,660 --> 00:34:26,030 But now we're going to be trying to think 474 00:34:26,030 --> 00:34:28,920 about their microscopic properties. 475 00:34:28,920 --> 00:34:33,500 And so we'll be dealing not with n number of moles, but with 10 476 00:34:33,500 --> 00:34:37,650 to the 24th number of molecules or atoms. 477 00:34:37,650 --> 00:34:40,180 And you know we won't be able to keep track 478 00:34:40,180 --> 00:34:44,890 of every one of their individual states, all the time. 479 00:34:44,890 --> 00:34:48,330 We may be able to do a terrific calculation of the quantum 480 00:34:48,330 --> 00:34:50,660 mechanics or the classical mechanics that 481 00:34:50,660 --> 00:34:54,070 describe the states that they could be in. 482 00:34:54,070 --> 00:34:56,070 But there's no way in practice that we're either 483 00:34:56,070 --> 00:34:59,070 going to experimentally or theoretically keep 484 00:34:59,070 --> 00:35:01,580 track of all of that. 485 00:35:01,580 --> 00:35:03,280 So we're going to, at some point, 486 00:35:03,280 --> 00:35:05,700 have to introduce statistics. 487 00:35:05,700 --> 00:35:08,750 To take what we know about the microscopic properties 488 00:35:08,750 --> 00:35:11,920 and try to go from there to the macroscopic results. 489 00:35:11,920 --> 00:35:14,100 The thermodynamics that we've seen so far. 490 00:35:14,100 --> 00:35:34,050 And that's what statistical mechanics is all about. 491 00:35:34,050 --> 00:35:42,560 So let's try to introduce a little bit 492 00:35:42,560 --> 00:35:44,070 of statistical mechanics. 493 00:35:44,070 --> 00:35:46,400 Where we're going to go from what 494 00:35:46,400 --> 00:35:49,010 we know about microscopic properties 495 00:35:49,010 --> 00:35:51,870 all the way to macroscopic thermodynamics. 496 00:35:51,870 --> 00:35:58,510 That's our objective. 497 00:35:58,510 --> 00:36:04,100 So let's start by just trying to calculate 498 00:36:04,100 --> 00:36:06,080 energies of individual molecules, 499 00:36:06,080 --> 00:36:08,500 or individual particles. 500 00:36:08,500 --> 00:36:13,790 And what's the probability that some molecule, one 501 00:36:13,790 --> 00:36:16,870 of the oxygen molecules somewhere in this room, 502 00:36:16,870 --> 00:36:18,870 is in a certain energy state. 503 00:36:18,870 --> 00:36:20,380 Right? 504 00:36:20,380 --> 00:36:22,280 And what our strategy is going to be, 505 00:36:22,280 --> 00:36:26,420 is to determine what are the probabilities that molecules 506 00:36:26,420 --> 00:36:29,880 are in certain states with certain energies. 507 00:36:29,880 --> 00:36:33,820 And if we can determine that for all the possible states, 508 00:36:33,820 --> 00:36:36,250 then we can average over those. 509 00:36:36,250 --> 00:36:40,450 So without keeping track of every individual molecule, 510 00:36:40,450 --> 00:36:42,879 we could then calculate, an average energy, 511 00:36:42,879 --> 00:36:44,920 which is what you would measure thermodynamically 512 00:36:44,920 --> 00:36:47,169 when you look at the whole collection and measure what 513 00:36:47,169 --> 00:36:48,790 we call u, right? 514 00:36:48,790 --> 00:36:50,190 Of course, we're really averaging 515 00:36:50,190 --> 00:36:53,750 over disparate energies of lots of different atoms 516 00:36:53,750 --> 00:36:54,820 or molecules. 517 00:36:54,820 --> 00:36:58,210 They don't all have the same molecular energy. 518 00:36:58,210 --> 00:37:01,930 And we don't try to measure their individual energies. 519 00:37:01,930 --> 00:37:03,690 So that's what we'd like to calculate. 520 00:37:03,690 --> 00:37:06,620 And so we'd like to be able to know 521 00:37:06,620 --> 00:37:10,790 what are all these probabilities of different energy states. 522 00:37:10,790 --> 00:37:13,440 And then from that, statistically 523 00:37:13,440 --> 00:37:17,210 averaging, what are the macroscopic average energies. 524 00:37:17,210 --> 00:37:19,710 And other macroscopic quantities. 525 00:37:19,710 --> 00:37:22,550 So, let's start there. 526 00:37:22,550 --> 00:37:40,030 Probability that a molecule, must be specific 527 00:37:40,030 --> 00:37:50,910 as possible is in state i with energy Ei. 528 00:37:50,910 --> 00:37:52,825 And right now we're not even going 529 00:37:52,825 --> 00:37:55,580 to specify the nature of the state. 530 00:37:55,580 --> 00:37:59,190 We could be worrying about translational energy. 531 00:37:59,190 --> 00:38:02,800 What state it is, how fast it's whizzing around the room. 532 00:38:02,800 --> 00:38:05,680 We could worry about its vibrational or rotational 533 00:38:05,680 --> 00:38:07,500 energy, or electronic state. 534 00:38:07,500 --> 00:38:09,870 For now, let's not even specify it. 535 00:38:09,870 --> 00:38:11,830 Let's just say the only thing we're really 536 00:38:11,830 --> 00:38:14,360 specifying about the state is it has 537 00:38:14,360 --> 00:38:19,200 some energy that we presumably know. 538 00:38:19,200 --> 00:38:27,460 Now, what we'd like is to know a functional 539 00:38:27,460 --> 00:38:29,930 form for the probability. 540 00:38:29,930 --> 00:38:32,530 And we don't know one a priori. 541 00:38:32,530 --> 00:38:38,570 But let's just think about two molecules. 542 00:38:38,570 --> 00:38:40,990 And the probability that one of them 543 00:38:40,990 --> 00:38:45,450 has energy i, and another one has energy Ej. 544 00:38:45,450 --> 00:38:48,370 It also, let me say, would be sufficient to think about even 545 00:38:48,370 --> 00:38:50,040 one molecule and say, we'll let's 546 00:38:50,040 --> 00:38:52,417 think about independent parts of its energy. 547 00:38:52,417 --> 00:38:54,500 Let's say, translational energy in this direction. 548 00:38:54,500 --> 00:38:56,680 And in an orthogonal direction. 549 00:38:56,680 --> 00:39:02,190 What matters is, we're thinking about two independent energies. 550 00:39:02,190 --> 00:39:05,160 Could be separate molecules that aren't interacting. 551 00:39:05,160 --> 00:39:08,930 Or independent degrees of freedom on the same molecule. 552 00:39:08,930 --> 00:39:12,500 But let's just consider it. 553 00:39:12,500 --> 00:39:25,360 So a molecule is in state i, and that another molecule 554 00:39:25,360 --> 00:39:35,870 is in state j, with energy Ej. 555 00:39:35,870 --> 00:39:39,110 So what we're going to calculate is the joint probability 556 00:39:39,110 --> 00:39:42,830 that the two molecules are in those states. 557 00:39:42,830 --> 00:39:46,040 So we want a probability. 558 00:39:46,040 --> 00:39:52,090 This one, we'll call Pi(Ei). 559 00:39:52,090 --> 00:39:58,100 This would be Pj of Ej. 560 00:39:58,100 --> 00:40:00,370 And this probability together will be Pij(Ei + Ej). 561 00:40:05,680 --> 00:40:09,400 I could just write Ei comma Ej, but the energy's add. 562 00:40:09,400 --> 00:40:14,120 And that's crucial. 563 00:40:14,120 --> 00:40:17,600 Now, since the molecules are completely independent, 564 00:40:17,600 --> 00:40:29,850 this should just be the product of the separate probabilities. 565 00:40:29,850 --> 00:40:32,420 Two completely independent events, 566 00:40:32,420 --> 00:40:37,210 or things whose probability we want to determine together. 567 00:40:37,210 --> 00:40:41,320 So, that suggests a really simple functional form. 568 00:40:41,320 --> 00:40:48,070 Because what we're saying here is we want a function of a plus 569 00:40:48,070 --> 00:40:54,000 b, which is equal to the same sort of function of a times 570 00:40:54,000 --> 00:40:55,910 the same sort of function of b. 571 00:40:55,910 --> 00:40:59,100 In other words, when we multiply these functions, 572 00:40:59,100 --> 00:41:05,460 we get a function of the sum of those arguments. 573 00:41:05,460 --> 00:41:11,800 What functional form does that? 574 00:41:11,800 --> 00:41:12,770 Exponential. 575 00:41:12,770 --> 00:41:17,150 Sure. 576 00:41:17,150 --> 00:41:21,000 So in other words, e to the a plus 577 00:41:21,000 --> 00:41:27,240 b power equals e to the a times e to the b. 578 00:41:27,240 --> 00:41:35,000 So it suggests that there's an exponential probability. 579 00:41:35,000 --> 00:41:37,740 Suggests it. 580 00:41:37,740 --> 00:41:49,160 Now, there's also some physical insight 581 00:41:49,160 --> 00:41:51,940 that we can apply to the problem. 582 00:41:51,940 --> 00:41:56,040 First of all, we expect that states 583 00:41:56,040 --> 00:41:57,550 with higher energy in general are 584 00:41:57,550 --> 00:42:02,700 going to be less probable than states with lower energy. 585 00:42:02,700 --> 00:42:06,205 If it's cold, not much thermal energy around, 586 00:42:06,205 --> 00:42:07,580 and you say, how much energy does 587 00:42:07,580 --> 00:42:11,570 a particle in an equilibrium collection of particles have, 588 00:42:11,570 --> 00:42:15,370 at thermal equilibrium, if there's not much thermal energy 589 00:42:15,370 --> 00:42:17,830 around, you don't expect it to be very likely 590 00:42:17,830 --> 00:42:19,760 that a particle has a huge amount of energy. 591 00:42:19,760 --> 00:42:21,390 That one individual particle has it. 592 00:42:21,390 --> 00:42:22,970 Could happen, right? 593 00:42:22,970 --> 00:42:24,660 From some random collisions that just 594 00:42:24,660 --> 00:42:28,780 happened to bulk up the energy of that one particle. 595 00:42:28,780 --> 00:42:31,330 But on a balance it's not going to be very probable. 596 00:42:31,330 --> 00:42:34,200 Lower energies will be more probable. 597 00:42:34,200 --> 00:42:37,030 And the other thing is, even in thinking about something 598 00:42:37,030 --> 00:42:39,880 like that, temperature comes in. 599 00:42:39,880 --> 00:42:40,950 Immediately. 600 00:42:40,950 --> 00:42:42,262 Right? 601 00:42:42,262 --> 00:42:43,720 If you raise the temperature a lot, 602 00:42:43,720 --> 00:42:46,800 surely we have to expect many more molecules will 603 00:42:46,800 --> 00:42:48,410 start to become energized. 604 00:42:48,410 --> 00:42:50,890 Will have more energy. 605 00:42:50,890 --> 00:42:55,230 So, lower probability for higher energy. 606 00:42:55,230 --> 00:42:56,730 But the probability of higher energy 607 00:42:56,730 --> 00:42:59,780 should go up if the temperature goes up. 608 00:42:59,780 --> 00:43:03,930 In other words, the ratio of energy to temperature 609 00:43:03,930 --> 00:43:06,330 should be involved here. 610 00:43:06,330 --> 00:43:20,910 So, P of Ei should go down as Ei goes up. 611 00:43:20,910 --> 00:43:30,930 And should depend on Ei over T. Right? 612 00:43:30,930 --> 00:43:35,800 So, we can start with this functional form. 613 00:43:35,800 --> 00:43:40,070 And by the way, there's no reason we can't have, 614 00:43:40,070 --> 00:43:42,740 in general we would have some constants here, right? 615 00:43:42,740 --> 00:43:45,870 We haven't changed anything to do that. 616 00:43:45,870 --> 00:43:48,900 And now let's just use the little bit 617 00:43:48,900 --> 00:43:53,220 of physical intuition that we're thinking about here, 618 00:43:53,220 --> 00:43:55,640 to refine this just a little bit. 619 00:43:55,640 --> 00:44:00,570 Let's write, we expect that Pi of Ei 620 00:44:00,570 --> 00:44:10,210 is some exponential to the minus C Ei over T, 621 00:44:10,210 --> 00:44:13,130 where C is some constant greater than zero. 622 00:44:13,130 --> 00:44:17,620 So this will have the property, then, that as energy goes up, 623 00:44:17,620 --> 00:44:18,940 the probability goes down. 624 00:44:18,940 --> 00:44:20,560 But it's scaled by temperature. 625 00:44:20,560 --> 00:44:24,570 If I raise the temperature, that makes the higher energy states 626 00:44:24,570 --> 00:44:30,500 get more and more likely. 627 00:44:30,500 --> 00:44:39,270 Well, this is our functional form 628 00:44:39,270 --> 00:44:43,320 for probability of a molecule being in a state 629 00:44:43,320 --> 00:44:44,980 with energy Ei. 630 00:44:44,980 --> 00:44:47,170 And the only difference between this and what's 631 00:44:47,170 --> 00:44:51,930 written conventionally is the way the constant is labeled. 632 00:44:51,930 --> 00:44:58,070 So really what we have is Pi of Ei 633 00:44:58,070 --> 00:45:06,640 is proportional to e to the minus Ei over kB T, or just k 634 00:45:06,640 --> 00:45:17,050 T, where kB is called the Boltzmann constant. 635 00:45:17,050 --> 00:45:21,680 And it's just equal to R over Avogadro's number. 636 00:45:21,680 --> 00:45:30,700 It's the gas constant per molecule, rather than per mole. 637 00:45:30,700 --> 00:45:37,080 One way to try to rationalize this is, you've probably seen, 638 00:45:37,080 --> 00:45:40,580 you've all seen Arrhenius kinetics, right? 639 00:45:40,580 --> 00:45:44,540 Arrhenius rate laws? 640 00:45:44,540 --> 00:45:53,140 If you remember what that looks like, 641 00:45:53,140 --> 00:45:56,090 you get this rate constant. 642 00:45:56,090 --> 00:46:00,663 Arrhenius rate constant is some constant A times e 643 00:46:00,663 --> 00:46:05,460 to the minus Ea over RT. 644 00:46:05,460 --> 00:46:07,510 Remember that? 645 00:46:07,510 --> 00:46:10,720 You've all seen that before? 646 00:46:10,720 --> 00:46:14,720 And so what's happening here, Ea is 647 00:46:14,720 --> 00:46:18,300 what is an activation energy. 648 00:46:18,300 --> 00:46:20,390 Remember, the idea is that you've 649 00:46:20,390 --> 00:46:24,180 got reactants and products, and there's 650 00:46:24,180 --> 00:46:26,140 some barrier you've got to get over. 651 00:46:26,140 --> 00:46:29,480 The activation energy, before you can have reaction. 652 00:46:29,480 --> 00:46:33,640 So the rate depends on surmounting that activation 653 00:46:33,640 --> 00:46:36,390 barrier. 654 00:46:36,390 --> 00:46:39,960 This is really coming from the same idea 655 00:46:39,960 --> 00:46:44,340 as this, which is the probability of one 656 00:46:44,340 --> 00:46:47,920 of the molecules having this much energy. 657 00:46:47,920 --> 00:46:51,230 Depends on e to the minus energy over RT. 658 00:46:51,230 --> 00:46:53,320 This is per mole, so this is per mole. 659 00:46:53,320 --> 00:46:55,980 The exact same relation. 660 00:46:55,980 --> 00:46:59,100 So the idea in the context of kinetics 661 00:46:59,100 --> 00:47:04,540 is that the rate depends on how many molecules can get up here. 662 00:47:04,540 --> 00:47:07,690 And what the probability is of their getting enough energy 663 00:47:07,690 --> 00:47:10,040 to go over the barrier. 664 00:47:10,040 --> 00:47:11,700 And that energy, that probability, 665 00:47:11,700 --> 00:47:13,520 is given by this expression. 666 00:47:13,520 --> 00:47:16,320 So in this form you've seen this kind of dependence 667 00:47:16,320 --> 00:47:29,080 before in a very explicit way. 668 00:47:29,080 --> 00:47:36,740 Let me just take it one small but important step farther. 669 00:47:36,740 --> 00:47:42,790 Which is that proportionality constant. 670 00:47:42,790 --> 00:47:45,450 Couldn't we do better? 671 00:47:45,450 --> 00:47:49,560 Couldn't we say, couldn't we figure out what exactly it is, 672 00:47:49,560 --> 00:47:52,520 not just what it's proportional to? 673 00:47:52,520 --> 00:47:53,260 Well, let's try. 674 00:47:53,260 --> 00:47:58,850 So as we've written it, we've got Pi of Ei 675 00:47:58,850 --> 00:48:03,660 is proportional to e to the minus Ei over kT. 676 00:48:03,660 --> 00:48:08,150 We could write that as equals A e to the minus Ei over kT. 677 00:48:08,150 --> 00:48:12,050 This is just some proportionality constant. 678 00:48:12,050 --> 00:48:14,710 But we can determine what that constant is, 679 00:48:14,710 --> 00:48:22,480 because we know that if we sum over all the possible states, 680 00:48:22,480 --> 00:48:25,550 the molecule has to be in some state, right? 681 00:48:25,550 --> 00:48:30,930 So that sum of all those probabilities has to equal one. 682 00:48:30,930 --> 00:48:37,560 So sum over i of Pi of Ei, is equal to one. 683 00:48:37,560 --> 00:48:44,270 Because the molecule must be in one of the states. 684 00:48:44,270 --> 00:48:49,030 OK, now this is our expression for all those p i's. 685 00:48:49,030 --> 00:48:52,790 So let's just write that way. a is just a constant times 686 00:48:52,790 --> 00:49:03,130 the sum over all the i's of e to the minus Ei over kT. 687 00:49:03,130 --> 00:49:09,480 And so there it is. a is equal to one over the sum over i, e 688 00:49:09,480 --> 00:49:17,070 to the minus Ei over kT. 689 00:49:17,070 --> 00:49:25,160 So now we can rewrite Pi of Ei is 690 00:49:25,160 --> 00:49:32,780 equal to e to the minus Ei over kT over, the sum over i, e 691 00:49:32,780 --> 00:49:41,600 to the minus Ei over kT. 692 00:49:41,600 --> 00:49:47,190 Now, just so you're not confused, this i matters. 693 00:49:47,190 --> 00:49:48,772 That's the i we're talking about. 694 00:49:48,772 --> 00:49:51,230 What's the probability of it being in some particular state 695 00:49:51,230 --> 00:49:53,540 i with energy e i. 696 00:49:53,540 --> 00:49:55,390 This i is just a dummy variable. 697 00:49:55,390 --> 00:49:57,480 So just to be explicit, we could write this just 698 00:49:57,480 --> 00:50:01,220 as well as e to the minus Ei over kT 699 00:50:01,220 --> 00:50:09,760 times the sum over j, e to the minus Ej over kT. 700 00:50:09,760 --> 00:50:11,890 It could be confusing, but this is usually the way 701 00:50:11,890 --> 00:50:13,330 that it's written. 702 00:50:13,330 --> 00:50:15,760 Even though this dummy variable here 703 00:50:15,760 --> 00:50:17,710 has nothing to do with the particular choice 704 00:50:17,710 --> 00:50:23,550 that's made here. 705 00:50:23,550 --> 00:50:24,780 So now we know. 706 00:50:24,780 --> 00:50:29,370 And we can calculate what the probability is for a molecule 707 00:50:29,370 --> 00:50:31,030 to be in any particular state if we 708 00:50:31,030 --> 00:50:32,765 know the energy of that state. 709 00:50:32,765 --> 00:50:34,890 Now, notice we need to know all the energies of all 710 00:50:34,890 --> 00:50:36,160 the other states, too. 711 00:50:36,160 --> 00:50:40,590 Because this thing, in other words, 712 00:50:40,590 --> 00:50:45,440 this a, right, depends on summing over all of those. 713 00:50:45,440 --> 00:50:48,020 But if we know those, then we can do it. 714 00:50:48,020 --> 00:50:49,390 We can do the whole calculation. 715 00:50:49,390 --> 00:50:52,610 And for a good number of cases, we realistically 716 00:50:52,610 --> 00:50:55,600 do know it all. 717 00:50:55,600 --> 00:51:00,120 So from the proportionality relationship alone, 718 00:51:00,120 --> 00:51:03,610 of course, we can tell the ratio of chances 719 00:51:03,610 --> 00:51:06,760 that you're in one state or another. 720 00:51:06,760 --> 00:51:10,430 Even without this a constant, we already 721 00:51:10,430 --> 00:51:13,640 could've said well, if what's the ratio of, 722 00:51:13,640 --> 00:51:18,890 the probability of being in state i with energy Ei to state 723 00:51:18,890 --> 00:51:21,460 j with energy Ej. 724 00:51:21,460 --> 00:51:26,180 Well, it's just e to the minus Ei over kT over e 725 00:51:26,180 --> 00:51:29,730 to the minus Ej over kT. 726 00:51:29,730 --> 00:51:37,110 It's e to the minus Ei minus Ej over kT. 727 00:51:37,110 --> 00:51:39,440 But apart from just being able to get the ratio, 728 00:51:39,440 --> 00:51:42,020 we can get the absolute number. 729 00:51:42,020 --> 00:51:44,225 What's the absolute probability that a molecule 730 00:51:44,225 --> 00:51:50,260 is in any particular state of a given energy. 731 00:51:50,260 --> 00:51:53,290 Next time what you'll see is what 732 00:51:53,290 --> 00:51:56,390 can be calculated based on the results that you've seen, 733 00:51:56,390 --> 00:52:00,570 just the results you've seen so far. 734 00:52:00,570 --> 00:52:04,630 From these quantities alone, it turns out 735 00:52:04,630 --> 00:52:07,930 you'll be able to calculate every single macroscopic 736 00:52:07,930 --> 00:52:12,560 thermodynamic quantity. 737 00:52:12,560 --> 00:52:14,569 So you'll see some of that next time.