1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,840 Your support will help MIT OpenCourseWare continue to 4 00:00:06,840 --> 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 --> 00:00:17,490 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,490 --> 00:00:21,060 ocw.mit.edu. 8 00:00:21,060 --> 00:00:21,660 PROFESSOR: Alright. 9 00:00:21,660 --> 00:00:24,690 Well, we've been looking in the last couple lectures at a 10 00:00:24,690 --> 00:00:27,690 really important topic in thermodynamics. 11 00:00:27,690 --> 00:00:31,500 Which is, how do you tell what's going to happen. 12 00:00:31,500 --> 00:00:33,800 Which way does a process want to go? 13 00:00:33,800 --> 00:00:36,040 Which way will it go spontaneously? 14 00:00:36,040 --> 00:00:38,470 And if it goes in one direction or another, where 15 00:00:38,470 --> 00:00:39,370 does is lead? 16 00:00:39,370 --> 00:00:42,660 In other words, what is the equilibrium state? 17 00:00:42,660 --> 00:00:44,660 And this is just an incredibly important area that 18 00:00:44,660 --> 00:00:48,330 thermodynamics allows us to speak to. 19 00:00:48,330 --> 00:00:52,180 So we started to see this. 20 00:00:52,180 --> 00:01:09,060 Sort of direction of spontaneous change. 21 00:01:09,060 --> 00:01:10,860 And where the equilibrium lies. 22 00:01:10,860 --> 00:01:18,520 So what we did is, remember we started with the second law. 23 00:01:18,520 --> 00:01:20,200 Right? 24 00:01:20,200 --> 00:01:27,490 That dS is greater than dq over T. And for the 25 00:01:27,490 --> 00:01:33,950 spontaneous change which happens irreversibly That 26 00:01:33,950 --> 00:01:37,180 means that'll be dq irreversible. 27 00:01:37,180 --> 00:01:39,690 It would be equal for the reversible case. 28 00:01:39,690 --> 00:01:49,790 And we combine this with first law, which for the case of 29 00:01:49,790 --> 00:01:57,610 pressure volume changes we write as this. 30 00:01:57,610 --> 00:02:01,590 And so what this gave us was a very, very useful general 31 00:02:01,590 --> 00:02:04,310 criterion for determining whether something happened 32 00:02:04,310 --> 00:02:06,220 spontaneously. 33 00:02:06,220 --> 00:02:18,620 Namely, du plus p external dV minus T for the surroundings 34 00:02:18,620 --> 00:02:26,780 dS, is greater than zero. 35 00:02:26,780 --> 00:02:31,950 Sorry. 36 00:02:31,950 --> 00:02:34,730 It's less than zero. 37 00:02:34,730 --> 00:02:43,360 And this is for any spontaneous change. 38 00:02:43,360 --> 00:02:50,390 If it equals zero, then we're at equilibrium. 39 00:02:50,390 --> 00:02:59,890 And if it's greater than zero, then the process 40 00:02:59,890 --> 00:03:01,830 goes the other way. 41 00:03:01,830 --> 00:03:04,565 We would write the process in the reverse to have it be less 42 00:03:04,565 --> 00:03:09,150 than 0 and it would go spontaneously. 43 00:03:09,150 --> 00:03:15,720 And based on this one result, we then looked under various 44 00:03:15,720 --> 00:03:19,720 constraints and said OK, what about looking at our 45 00:03:19,720 --> 00:03:23,540 variables, volume, pressure, temperature, other things, 46 00:03:23,540 --> 00:03:27,190 entropy, if we constrain those, what's the condition 47 00:03:27,190 --> 00:03:28,660 for equilibrium? 48 00:03:28,660 --> 00:03:32,240 And that's what led us to a number of results to determine 49 00:03:32,240 --> 00:03:35,980 what quantities we even need to be looking at. 50 00:03:35,980 --> 00:03:37,540 To figure out equilibrium. 51 00:03:37,540 --> 00:03:38,730 And what the conditions were. 52 00:03:38,730 --> 00:03:41,890 And so what we discovered were the following. 53 00:03:41,890 --> 00:03:49,670 This one, which we already had seen, which is dS, is 54 00:03:49,670 --> 00:03:50,760 greater than zero. 55 00:03:50,760 --> 00:03:53,770 Change in entropy is greater than zero, 56 00:03:53,770 --> 00:04:02,180 for an isolated system. 57 00:04:02,180 --> 00:04:08,130 We also saw that dS for constant H and p was greater 58 00:04:08,130 --> 00:04:20,110 than zero. du, regular energy, at constant entropy and volume 59 00:04:20,110 --> 00:04:41,370 is less then zero. 60 00:04:41,370 --> 00:04:43,300 And u is minimized at equilibrium. 61 00:04:43,300 --> 00:04:46,500 And this is the familiar result from ordinary 62 00:04:46,500 --> 00:04:48,560 mechanics, where you're not worrying about something like 63 00:04:48,560 --> 00:04:51,420 entropy for a whole collection of particles. 64 00:04:51,420 --> 00:04:54,850 That is, you minimize potential energy and you see 65 00:04:54,850 --> 00:04:57,930 things falling under the force of gravity and so forth, going 66 00:04:57,930 --> 00:05:05,560 to potential energy minima in conformance with this result. 67 00:05:05,560 --> 00:05:10,970 dH, S and p is less than zero. 68 00:05:10,970 --> 00:05:16,140 So our H is u plus pV, as you know. 69 00:05:16,140 --> 00:05:27,870 And H is minimized at equilibrium. 70 00:05:27,870 --> 00:05:33,290 And this is, of course, with constant S V. This is 71 00:05:33,290 --> 00:05:37,560 constant S and p. 72 00:05:37,560 --> 00:05:41,770 But of course, the need to have entropy constrained is 73 00:05:41,770 --> 00:05:44,250 never going to be the most convenient one experimental. 74 00:05:44,250 --> 00:05:47,040 There may be circumstances under which it's the case, but 75 00:05:47,040 --> 00:05:49,740 it's often difficult to control. 76 00:05:49,740 --> 00:05:53,100 On the other hand, temperature, volume and 77 00:05:53,100 --> 00:05:56,480 pressure are variables that are much easier in the lab to 78 00:05:56,480 --> 00:05:57,170 keep constant. 79 00:05:57,170 --> 00:05:58,530 To keep control over. 80 00:05:58,530 --> 00:06:02,160 And so that led us to the definitions of other energy 81 00:06:02,160 --> 00:06:05,860 quantities, the Helmholtz and Gibbs free energy. 82 00:06:05,860 --> 00:06:11,480 We discovered that the quantity dA, under conditions 83 00:06:11,480 --> 00:06:15,770 of constant volume and temperature, 84 00:06:15,770 --> 00:06:16,720 is less than zero. 85 00:06:16,720 --> 00:06:20,440 And A is u minus TS. 86 00:06:28,370 --> 00:06:34,830 And A is minimized at equilibrium, under conditions 87 00:06:34,830 --> 00:06:43,210 of constant T and V. And finally, and in many cases the 88 00:06:43,210 --> 00:06:45,670 most important of the results, because of the conditions it 89 00:06:45,670 --> 00:06:52,540 applies to, we saw that this Gibbs free energy is less than 90 00:06:52,540 --> 00:06:57,310 zero, that's our condition for spontaneous change. 91 00:06:57,310 --> 00:07:01,850 Where the Gibbs free energy, u plus pV minus 92 00:07:01,850 --> 00:07:05,550 TS is H minus TS. 93 00:07:08,380 --> 00:07:22,330 Also A plus pV and G is minimized at equilibrium with 94 00:07:22,330 --> 00:07:26,090 constant temperature and pressure. 95 00:07:26,090 --> 00:07:27,760 And that's why the Gibbs free energy is just 96 00:07:27,760 --> 00:07:28,980 so enormously important. 97 00:07:28,980 --> 00:07:31,560 Because so much of what we do in chemistry does take place 98 00:07:31,560 --> 00:07:34,290 with constant temperature and pressure. 99 00:07:34,290 --> 00:07:37,880 So we have this condition that's established in a 100 00:07:37,880 --> 00:07:40,390 quantity that we know how to calculate. 101 00:07:40,390 --> 00:07:43,930 That tells us the direction of spontaneous change for 102 00:07:43,930 --> 00:07:47,140 ordinary processes, chemical processes, mixing and you name 103 00:07:47,140 --> 00:07:54,180 it, under conditions that are easy to achieve in the lab. 104 00:07:54,180 --> 00:08:01,030 OK, now what we'd like to do is be able to calculate any of 105 00:08:01,030 --> 00:08:04,940 these quantities in terms of temperature, pressure, volume 106 00:08:04,940 --> 00:08:05,370 properties. 107 00:08:05,370 --> 00:08:08,370 That is, in terms of equations of state. 108 00:08:08,370 --> 00:08:09,230 For any material. 109 00:08:09,230 --> 00:08:10,460 Then we would really be able to 110 00:08:10,460 --> 00:08:12,530 essentially calculate anything. 111 00:08:12,530 --> 00:08:14,250 Anything thermodynamic. 112 00:08:14,250 --> 00:08:15,990 About a material. 113 00:08:15,990 --> 00:08:18,760 Of course, that's assuming we know the equation of state. 114 00:08:18,760 --> 00:08:20,640 We may or may not. 115 00:08:20,640 --> 00:08:25,190 But because in many cases we can reasonably either model or 116 00:08:25,190 --> 00:08:28,470 measure equations of state, collect data for a material 117 00:08:28,470 --> 00:08:32,590 for its temperature, pressure, volume relations, then in fact 118 00:08:32,590 --> 00:08:35,290 if we can relate all these quantities to those, then in 119 00:08:35,290 --> 00:08:37,460 fact we really can calculate essentially all of the 120 00:08:37,460 --> 00:08:38,500 thermodynamics. 121 00:08:38,500 --> 00:08:42,830 For the material. 122 00:08:42,830 --> 00:08:58,450 So let's relate the thermodynamic quantities to 123 00:08:58,450 --> 00:09:07,620 equation of state p, V, T data. 124 00:09:07,620 --> 00:09:12,380 And we can do that by going through and deriving what 125 00:09:12,380 --> 00:09:15,800 we'll call the fundamental equations of thermodynamics. 126 00:09:15,800 --> 00:09:17,330 that'll provide these relations. 127 00:09:17,330 --> 00:09:19,250 And at this point we know enough to do this in a 128 00:09:19,250 --> 00:09:21,020 straightforward way. 129 00:09:21,020 --> 00:09:38,630 So if we start with a relation for energy, T dS minus p dV. 130 00:09:38,630 --> 00:09:40,270 Where u is written as a function 131 00:09:40,270 --> 00:09:43,870 of entropy and volume. 132 00:09:43,870 --> 00:09:47,790 And we've seen that that's generally the case. 133 00:09:47,790 --> 00:09:59,450 It comes from the fact that dq reversible is T dS, and dw 134 00:09:59,450 --> 00:10:04,760 reversible is minus p dV. 135 00:10:04,760 --> 00:10:08,310 And of course du is the some of those. 136 00:10:08,310 --> 00:10:14,180 So, this is generally true. 137 00:10:14,180 --> 00:10:15,930 Since these are all state functions. 138 00:10:15,930 --> 00:10:19,720 That is, this is derived in the case for reversible paths. 139 00:10:19,720 --> 00:10:22,590 But since these are all simply state functions and 140 00:10:22,590 --> 00:10:25,440 quantities, this is generally true. 141 00:10:25,440 --> 00:10:28,710 Now we can use it to derive differential relations for all 142 00:10:28,710 --> 00:10:29,960 of the thermodynamics quantities. 143 00:10:29,960 --> 00:10:32,010 So let's just go through and do that. 144 00:10:32,010 --> 00:10:38,590 So H is u plus pV. 145 00:10:41,180 --> 00:10:49,590 So dH is just du plus p dV plus V dp. 146 00:10:49,590 --> 00:10:53,940 And now we're just going to substitute du in here. 147 00:10:53,940 --> 00:10:58,380 And the p dV terms are going to cancel. 148 00:10:58,380 --> 00:11:08,970 So we have the result that dH is T dS plus V dp. 149 00:11:08,970 --> 00:11:11,570 Right? 150 00:11:11,570 --> 00:11:16,740 And that shows us that H is written naturally as a 151 00:11:16,740 --> 00:11:23,580 function of entropy and pressure. 152 00:11:23,580 --> 00:11:26,130 And now let's keep going. 153 00:11:26,130 --> 00:11:29,370 A is u minus TS. 154 00:11:31,880 --> 00:11:41,160 dA is du minus T dS minus S dT. 155 00:11:41,160 --> 00:11:42,360 We're going to do the same thing. 156 00:11:42,360 --> 00:11:45,660 Substitute this for du. 157 00:11:45,660 --> 00:11:51,570 This time, the T dS terms are going to cancel. 158 00:11:51,570 --> 00:11:58,900 So we have dA is minus S dT minus T dS. 159 00:12:02,570 --> 00:12:03,510 That can't be right. 160 00:12:03,510 --> 00:12:06,760 And it isn't. 161 00:12:06,760 --> 00:12:12,040 Minus S dT, that's the p dV term that's left, minus p dV. 162 00:12:12,040 --> 00:12:15,520 And it shows us that A is written naturally as a 163 00:12:15,520 --> 00:12:26,140 function of T and V. G, we can write in any 164 00:12:26,140 --> 00:12:27,100 of a number of ways. 165 00:12:27,100 --> 00:12:31,230 Let's write it as H minus TS. 166 00:12:31,230 --> 00:12:40,720 So dG is dH minus T dS minus S dT. 167 00:12:40,720 --> 00:12:43,000 Here's dH. 168 00:12:43,000 --> 00:12:46,420 We'll substitute that in, and the T dS terms 169 00:12:46,420 --> 00:12:48,990 are going to cancel. 170 00:12:48,990 --> 00:12:56,150 So dG is minus S dT plus V dp. 171 00:12:59,840 --> 00:13:02,650 And this shows that G is written naturally as a 172 00:13:02,650 --> 00:13:10,510 function of T and p. 173 00:13:10,510 --> 00:13:22,000 So these, which we will exalt and celebrate by our 174 00:13:22,000 --> 00:13:30,210 sparingly-used colored chalk, are our fundamental equations 175 00:13:30,210 --> 00:13:54,790 of thermodynamics. 176 00:13:54,790 --> 00:13:58,300 So what they do is, they're describing how these 177 00:13:58,300 --> 00:14:03,500 thermodynamic properties change, in terms of only state 178 00:14:03,500 --> 00:14:07,180 functions and state variables. 179 00:14:07,180 --> 00:14:08,520 Very, very useful. 180 00:14:08,520 --> 00:14:11,010 And that's what it means, when we say well, it's natural 181 00:14:11,010 --> 00:14:14,180 then, to express say, G as a function of T and p, that's 182 00:14:14,180 --> 00:14:14,830 what we're saying. 183 00:14:14,830 --> 00:14:16,870 Is that we can express its changes in 184 00:14:16,870 --> 00:14:19,530 terms of these variables. 185 00:14:19,530 --> 00:14:23,300 Related only through quantities that are 186 00:14:23,300 --> 00:14:25,740 functions of state. 187 00:14:25,740 --> 00:14:29,360 I don't need to know about a specific path here. 188 00:14:29,360 --> 00:14:34,030 If I know about the states involved, I just need to know 189 00:14:34,030 --> 00:14:40,940 what the volume was in each of them. 190 00:14:40,940 --> 00:14:45,130 Now, before, of course, in the first part of the class we 191 00:14:45,130 --> 00:14:49,470 started out looking at u and then looking at H not as 192 00:14:49,470 --> 00:14:53,810 functions of S and V or S and p, but as functions of 193 00:14:53,810 --> 00:14:55,640 temperature, mostly. 194 00:14:55,640 --> 00:14:58,900 In general, temperature and volume or pressure. 195 00:14:58,900 --> 00:15:00,740 And it doesn't mean that something was 196 00:15:00,740 --> 00:15:04,100 somehow wrong with that. 197 00:15:04,100 --> 00:15:07,960 It certainly is, it still is going to be useful to do 198 00:15:07,960 --> 00:15:08,470 thermochemistry. 199 00:15:08,470 --> 00:15:11,890 To ask questions like how much heat is released in a chemical 200 00:15:11,890 --> 00:15:15,640 reaction that takes place at constant temperature. 201 00:15:15,640 --> 00:15:17,130 Not one of these variables. 202 00:15:17,130 --> 00:15:19,740 And we can calculate that. 203 00:15:19,740 --> 00:15:22,870 So it's not that we're somehow throwing away our 204 00:15:22,870 --> 00:15:23,940 ability to do that. 205 00:15:23,940 --> 00:15:28,710 However, the thing to remember is, when you look at heats of 206 00:15:28,710 --> 00:15:30,510 reaction under those conditions it's 207 00:15:30,510 --> 00:15:32,440 all well and good. 208 00:15:32,440 --> 00:15:35,940 But it doesn't tell you, this is the direction that the 209 00:15:35,940 --> 00:15:37,930 reaction is going to go. 210 00:15:37,930 --> 00:15:41,380 It doesn't tell you, this is the equilibrium concentration 211 00:15:41,380 --> 00:15:43,950 that you'll end up with. 212 00:15:43,950 --> 00:15:46,950 That doesn't come out of what we calculated before in 213 00:15:46,950 --> 00:15:47,650 thermochemistry. 214 00:15:47,650 --> 00:15:51,470 What does come out, which is very useful is, if you do run 215 00:15:51,470 --> 00:15:54,360 the reaction, here's how much heat comes out. 216 00:15:54,360 --> 00:15:57,950 And if you want to run a furnace and provide energy, 217 00:15:57,950 --> 00:16:01,840 that's an extremely important thing to be able to calculate. 218 00:16:01,840 --> 00:16:03,940 Because you're going to run it and you'll probably find 219 00:16:03,940 --> 00:16:05,940 conditions under which you can run it more or less to 220 00:16:05,940 --> 00:16:07,170 completion. 221 00:16:07,170 --> 00:16:10,100 But it doesn't tell you, by itself, which 222 00:16:10,100 --> 00:16:14,060 direction things run in. 223 00:16:14,060 --> 00:16:16,710 Whereas under these conditions, these quantities, 224 00:16:16,710 --> 00:16:19,610 if you look at free energy change, for example, at 225 00:16:19,610 --> 00:16:23,180 constant temperature and pressure, you can still 226 00:16:23,180 --> 00:16:25,180 calculate H. You can still calculate 227 00:16:25,180 --> 00:16:26,320 the heat that's released. 228 00:16:26,320 --> 00:16:28,410 This is what will tell you under some particular 229 00:16:28,410 --> 00:16:31,110 conditions what will actually happen. 230 00:16:31,110 --> 00:16:33,040 Where will you end up. 231 00:16:33,040 --> 00:16:34,500 Very, very important, of course, to be able to 232 00:16:34,500 --> 00:16:39,220 understand that. 233 00:16:39,220 --> 00:16:55,520 Now, it's also very useful to look at some of the relations 234 00:16:55,520 --> 00:16:57,970 that come out of these fundamental equations. 235 00:16:57,970 --> 00:16:59,990 And they're straightforward to derive. 236 00:16:59,990 --> 00:17:04,200 So, all I want to do now is look at the derivatives of the 237 00:17:04,200 --> 00:17:06,680 free energies with respect to temperature 238 00:17:06,680 --> 00:17:10,070 and volume and pressure. 239 00:17:10,070 --> 00:17:14,830 So for example, if I look at A, which we now have written 240 00:17:14,830 --> 00:17:18,800 as the function of T and V, of course, in general I can 241 00:17:18,800 --> 00:17:23,550 always write dA as partial of A, with respect to T at 242 00:17:23,550 --> 00:17:28,750 constant volume dT, plus partial of A with respect to 243 00:17:28,750 --> 00:17:32,210 V, at constant temperature dV. 244 00:17:32,210 --> 00:17:35,330 And I know what those turn out to be. 245 00:17:35,330 --> 00:17:41,320 It's minus S dT minus p dV. 246 00:17:41,320 --> 00:17:42,840 So what does that tell me? 247 00:17:42,840 --> 00:17:47,690 It tells me that the partial of A with respect to T at 248 00:17:47,690 --> 00:17:52,260 constant V is minus S. Right? 249 00:17:52,260 --> 00:17:55,760 In other words, now I know how to tell how the Helmholtz free 250 00:17:55,760 --> 00:17:59,300 energy changes as a function of temperature. 251 00:17:59,300 --> 00:18:05,040 Or as a function of volume. dA/dV, at constant T, must be 252 00:18:05,040 --> 00:18:08,230 negative p. 253 00:18:08,230 --> 00:18:12,770 Things that I can measure. 254 00:18:12,770 --> 00:18:15,150 So I can in a very straightforward way say, OK, 255 00:18:15,150 --> 00:18:19,560 well, here is my Helmholtz free energy. 256 00:18:19,560 --> 00:18:21,870 If I'm working under conditions of constant 257 00:18:21,870 --> 00:18:23,370 temperature and volume, that's very useful. 258 00:18:23,370 --> 00:18:26,130 Now, if I want to change those quantities; change the 259 00:18:26,130 --> 00:18:28,580 temperature, change the volume, how will it change? 260 00:18:28,580 --> 00:18:32,360 Well, I can, for any given case, measure the pressure, 261 00:18:32,360 --> 00:18:35,010 determine the entropy and I'll know what the slope 262 00:18:35,010 --> 00:18:38,680 of change will be. 263 00:18:38,680 --> 00:18:45,040 Similarly for G as a function of temperature and pressure, I 264 00:18:45,040 --> 00:18:47,560 can go through the same procedure. 265 00:18:47,560 --> 00:18:54,210 That is, it's easy to write down straight away that dG, 266 00:18:54,210 --> 00:18:59,060 with respect to temperature at constant pressure is minus S. 267 00:18:59,060 --> 00:19:03,090 That is, this is, dG/dT at constant pressure. 268 00:19:03,090 --> 00:19:18,020 And this is dG/dp at constant temperature. 269 00:19:18,020 --> 00:19:22,450 So again with the Gibbs free energy, now I see how to 270 00:19:22,450 --> 00:19:25,680 determine, if I change the pressure, if I change the 271 00:19:25,680 --> 00:19:29,200 temperature by some modest amount, how much is the Gibbs 272 00:19:29,200 --> 00:19:30,300 free energy going to change? 273 00:19:30,300 --> 00:19:39,240 Well, it's easy to see. 274 00:19:39,240 --> 00:19:42,990 These two relations involving entropy are also useful 275 00:19:42,990 --> 00:19:46,890 because they'll let us see how entropy depends 276 00:19:46,890 --> 00:19:48,680 on volume and pressure. 277 00:19:48,680 --> 00:20:09,410 And let me show you how that goes. 278 00:20:09,410 --> 00:20:11,930 Now, you've already seen how entropy depends on 279 00:20:11,930 --> 00:20:22,890 temperature. 280 00:20:22,890 --> 00:20:30,630 We've already seen that, going to write dS as dq reversible 281 00:20:30,630 --> 00:20:42,050 over T. And it's Cv dT over T at constant volume. 282 00:20:42,050 --> 00:20:51,040 It's Cp dT over T at constant pressure. 283 00:20:51,040 --> 00:20:56,500 So we know that dS/dT at constant volume is Cv over T, 284 00:20:56,500 --> 00:21:04,430 and dS/dT at constant pressure is Cp, over T. And we've seen 285 00:21:04,430 --> 00:21:08,180 that on a number of occasions. 286 00:21:08,180 --> 00:21:11,420 So that tells us what to do to know the entropy as the 287 00:21:11,420 --> 00:21:13,280 temperature changes. 288 00:21:13,280 --> 00:21:19,360 But now, what happens if, instead we look at what 289 00:21:19,360 --> 00:21:21,780 happens when we go to some state one to some other state 290 00:21:21,780 --> 00:21:22,840 two and it's the pressure. 291 00:21:22,840 --> 00:21:24,420 Or the volume, that changes. 292 00:21:24,420 --> 00:21:27,930 And by the way, just to be explicit about this, let's 293 00:21:27,930 --> 00:21:32,910 take this example, it means that delta S, if we undergo a 294 00:21:32,910 --> 00:21:35,110 change from, say, T1 to T2. 295 00:21:35,110 --> 00:21:42,140 There's Cp over T dT. 296 00:21:42,140 --> 00:21:51,370 So it's Cp log of T2 over T1, and we saw this before. 297 00:21:51,370 --> 00:21:56,260 So now, instead, let's look at some process. 298 00:21:56,260 --> 00:21:59,890 State one goes to state two. 299 00:21:59,890 --> 00:22:06,190 Let's have constant T. And look at what happens if 300 00:22:06,190 --> 00:22:09,730 pressure goes from p1 to p2. 301 00:22:09,730 --> 00:22:14,760 Or volume goes from V1 to V2. 302 00:22:14,760 --> 00:22:17,650 And see what happens there. 303 00:22:17,650 --> 00:22:20,030 We looked at pressure change before, actually, in 304 00:22:20,030 --> 00:22:23,740 discussing the third law, the fact that the entropy goes to 305 00:22:23,740 --> 00:22:27,105 zero as the absolute temperature goes to zero for a 306 00:22:27,105 --> 00:22:28,350 pure, perfect crystal. 307 00:22:28,350 --> 00:22:31,320 But, actually, we didn't do that in a general way. 308 00:22:31,320 --> 00:22:34,830 We just treated the one case of an ideal gas as the 309 00:22:34,830 --> 00:22:36,850 temperature is reduced. 310 00:22:36,850 --> 00:22:39,890 But we can do this, generally, by using what are called 311 00:22:39,890 --> 00:22:41,450 Maxwell relations. 312 00:22:41,450 --> 00:22:45,990 And all this is, is saying that when you take a mixed 313 00:22:45,990 --> 00:22:50,470 second derivative, it doesn't matter in which order you take 314 00:22:50,470 --> 00:22:54,980 the two derivatives. 315 00:22:54,980 --> 00:23:03,300 So, let's, we're going to use this relationship. 316 00:23:03,300 --> 00:23:13,830 And we're going to use these two. 317 00:23:13,830 --> 00:23:18,770 So, using those, now, what happens if we take the second 318 00:23:18,770 --> 00:23:22,170 derivative of A, the mixed derivative, partial with 319 00:23:22,170 --> 00:23:29,900 respect to T and the partial with respect to V. So let's 320 00:23:29,900 --> 00:23:35,910 leave these off for a moment, and now let's try that. 321 00:23:35,910 --> 00:23:41,110 And the point is that the second derivative of A, with 322 00:23:41,110 --> 00:23:47,630 respect to V and T in this order is the same as the 323 00:23:47,630 --> 00:23:50,760 second derivative of a with respect to T 324 00:23:50,760 --> 00:23:52,160 and V in this order. 325 00:23:52,160 --> 00:23:56,340 It doesn't matter which order. 326 00:23:56,340 --> 00:23:58,840 But that turns out to be useful. 327 00:23:58,840 --> 00:24:01,230 So let's do this explicitly. 328 00:24:01,230 --> 00:24:03,900 Which means we're going to take the derivative with 329 00:24:03,900 --> 00:24:06,620 respect to volume of dA/dT. 330 00:24:09,680 --> 00:24:15,590 Now, the dA/dT isn't constant volume. 331 00:24:15,590 --> 00:24:18,700 The derivative we're taking with respect to volume, when 332 00:24:18,700 --> 00:24:23,570 we take that it's at constant temperature. 333 00:24:23,570 --> 00:24:24,770 But what is it? 334 00:24:24,770 --> 00:24:30,360 Well, we already know what dA/dT at constant V is. 335 00:24:30,360 --> 00:24:35,520 It's negative S. So this is negative dS/dV. 336 00:24:38,440 --> 00:24:43,110 At constant temperature. 337 00:24:43,110 --> 00:24:46,150 Now let's take it in the other order. 338 00:24:46,150 --> 00:24:57,410 So d/dT of dA/dV, just like this. 339 00:24:57,410 --> 00:25:01,410 The dA/dV is calculated at constant temperature. 340 00:25:01,410 --> 00:25:02,290 We know it. 341 00:25:02,290 --> 00:25:04,460 Then we can take the derivative of that quantity, 342 00:25:04,460 --> 00:25:07,820 when we vary the temperature, holding the volume constant. 343 00:25:07,820 --> 00:25:11,380 But again, dA/dV dT, there it is. 344 00:25:11,380 --> 00:25:14,300 It's negative p. 345 00:25:14,300 --> 00:25:24,100 So this is just negative dp/dT at constant volume. 346 00:25:24,100 --> 00:25:31,850 These things have to be equal to each other. 347 00:25:31,850 --> 00:25:33,610 Because these mixed second derivatives 348 00:25:33,610 --> 00:25:36,530 are the same thing. 349 00:25:36,530 --> 00:25:37,640 But that's very useful. 350 00:25:37,640 --> 00:25:42,570 Because this is what comes directly out of an equation of 351 00:25:42,570 --> 00:25:43,930 state, right? 352 00:25:43,930 --> 00:25:46,660 You know how pressure changes with temperature at constant 353 00:25:46,660 --> 00:25:48,490 volume if you know the equation of state. 354 00:25:48,490 --> 00:25:50,280 It relates the pressure, volume, 355 00:25:50,280 --> 00:26:03,690 and temperature together. 356 00:26:03,690 --> 00:26:06,600 So from measured equation of state data, or from a model 357 00:26:06,600 --> 00:26:09,380 like the ideal gas or the van der Waal's gas or another 358 00:26:09,380 --> 00:26:11,910 equation of state, you know this. 359 00:26:11,910 --> 00:26:14,620 Can determine how entropy is going to behave 360 00:26:14,620 --> 00:26:31,510 as the volume changes. 361 00:26:31,510 --> 00:26:42,070 If we try that for an ideal gas, pV is nRT. 362 00:26:42,070 --> 00:26:54,690 So dp/dT at constant volume, it's just nR over V. And that, 363 00:26:54,690 --> 00:27:04,725 now, we know must equal dS/dV, with a positive sign. 364 00:27:04,725 --> 00:27:08,280 At constant temperature. 365 00:27:08,280 --> 00:27:11,710 So now let's try looking at something where 366 00:27:11,710 --> 00:27:13,720 are V1 goes to V2. 367 00:27:13,720 --> 00:27:17,790 The volume is going to change, and we can see 368 00:27:17,790 --> 00:27:20,310 how the entropy changes. 369 00:27:20,310 --> 00:27:29,500 So, if one goes to two and V1 goes to V2, and it's constant 370 00:27:29,500 --> 00:27:33,770 temperature, just what we've specified there. 371 00:27:33,770 --> 00:27:48,930 Delta S is S(T, V2) minus S(T, V1), T's staying the same. 372 00:27:48,930 --> 00:27:57,900 So it's just the integral from V1 to V2 of dS/dV At constant 373 00:27:57,900 --> 00:27:59,810 temperature dV. 374 00:27:59,810 --> 00:28:00,900 And now we know what that is. 375 00:28:00,900 --> 00:28:11,380 So it's nR integral from V1 to V2 dV over V. So it's nR log 376 00:28:11,380 --> 00:28:16,690 V2 over V1. 377 00:28:16,690 --> 00:28:22,990 There's our delta S. So we know how to calculate it. 378 00:28:22,990 --> 00:28:30,440 Make sense? 379 00:28:30,440 --> 00:28:33,890 Now, we can do the same procedure for 380 00:28:33,890 --> 00:28:38,300 the pressure change. 381 00:28:38,300 --> 00:28:41,840 And all we do is, I'll just outline this, I think. 382 00:28:41,840 --> 00:28:45,780 I won't write it all on the board. 383 00:28:45,780 --> 00:28:48,970 But, of course, it's going to come from the fact that these 384 00:28:48,970 --> 00:28:52,590 second derivatives are also equal. 385 00:28:52,590 --> 00:29:05,530 So d squared G dT dp is equal to d squared G dp dT. 386 00:29:05,530 --> 00:29:08,560 In other words, the order of taking the derivatives with 387 00:29:08,560 --> 00:29:13,150 respect to pressure and temperature doesn't matter. 388 00:29:13,150 --> 00:29:20,590 And what this will show is that dS/dp at constant 389 00:29:20,590 --> 00:29:23,590 temperature, here we saw how entropy varies with volume, 390 00:29:23,590 --> 00:29:27,210 this is going to show us how it varies with pressure. 391 00:29:27,210 --> 00:29:34,470 Is equal to minus dV/dT at constant pressure. 392 00:29:34,470 --> 00:29:37,905 And again, this is something that comes from 393 00:29:37,905 --> 00:29:40,750 an equation of state. 394 00:29:40,750 --> 00:29:44,940 We know how the volume and temperature vary with respect 395 00:29:44,940 --> 00:29:46,400 to each other at constant pressure. 396 00:29:46,400 --> 00:29:53,030 That's what the equation of state tells us. 397 00:29:53,030 --> 00:29:57,050 So, again, I can just use that result. 398 00:29:57,050 --> 00:30:03,260 So, if we do a process where one goes to two at constant 399 00:30:03,260 --> 00:30:11,590 temperature, and now the pressure, p1, goes to p2, well 400 00:30:11,590 --> 00:30:19,750 then delta S is just the integral from p1 to p2 of 401 00:30:19,750 --> 00:30:26,910 dS/dp times dS, so it's just this. 402 00:30:26,910 --> 00:30:30,680 And so of course it's still pV equals nRT. 403 00:30:30,680 --> 00:30:38,680 So now we just have nR over p dp. 404 00:30:38,680 --> 00:30:39,890 Right? 405 00:30:39,890 --> 00:30:41,930 So we're going to see the same story. 406 00:30:41,930 --> 00:30:49,590 It's nR log of p2 over p1 for the process where there's a 407 00:30:49,590 --> 00:31:01,180 pressure change. 408 00:31:01,180 --> 00:31:04,030 Any questions about this part? 409 00:31:04,030 --> 00:31:07,140 So what we've done is take one step further. 410 00:31:07,140 --> 00:31:16,020 We've used the fundamental equations that are hiding down 411 00:31:16,020 --> 00:31:19,670 here, out of sight but never out of mind. 412 00:31:19,670 --> 00:31:23,390 And what we've done is look at the derivatives of the new 413 00:31:23,390 --> 00:31:28,740 free energies that we've just recently introduced, A and G. 414 00:31:28,740 --> 00:31:33,330 And then, the only thing we've done beyond that is say, OK, 415 00:31:33,330 --> 00:31:35,940 well now let's just take the mixed second derivatives, they 416 00:31:35,940 --> 00:31:37,990 have to be equal to each other. 417 00:31:37,990 --> 00:31:42,660 And what's fallen out when we do that, because in each case, 418 00:31:42,660 --> 00:31:46,370 one of the first derivatives gives us the entropy. 419 00:31:46,370 --> 00:31:48,630 Then the second derivative gives the change in entropy 420 00:31:48,630 --> 00:31:51,500 with respect to the variable that we're differentiating, 421 00:31:51,500 --> 00:31:54,790 with respect to which is either pressure or volume. 422 00:31:54,790 --> 00:31:59,520 And the useful outcome of all that is that we get to see how 423 00:31:59,520 --> 00:32:03,910 entropy changes with one of those variables in terms of 424 00:32:03,910 --> 00:32:09,200 only V, T, and p, which come out of some equation of state. 425 00:32:09,200 --> 00:32:11,660 And all we did, further, is take that second derivative. 426 00:32:11,660 --> 00:32:13,060 That mixed second derivative. 427 00:32:13,060 --> 00:32:16,340 And, of course, see that either way we do that we'll 428 00:32:16,340 --> 00:32:24,600 have an equality. 429 00:32:24,600 --> 00:32:34,840 Now, let's go back to our older friends u and H. Which 430 00:32:34,840 --> 00:32:46,590 we've expressed now in terms of S and V, S and p. 431 00:32:46,590 --> 00:32:52,500 So, so far we don't have a way to just write off, relate them 432 00:32:52,500 --> 00:32:53,940 to equation of state data. 433 00:32:53,940 --> 00:32:57,860 Which also would be very useful. 434 00:32:57,860 --> 00:33:02,110 Here, A and G, we've already got as functions of these 435 00:33:02,110 --> 00:33:03,710 easily controlled, conveniently 436 00:33:03,710 --> 00:33:05,560 controlled state variables. 437 00:33:05,560 --> 00:33:13,450 Let's look at those quantities. u and H. And look 438 00:33:13,450 --> 00:33:17,270 at, for example, the V dependence of u. 439 00:33:17,270 --> 00:33:35,780 The volume dependence. 440 00:33:35,780 --> 00:33:44,810 And in particular let's look at, for example, du/dV at 441 00:33:44,810 --> 00:33:46,690 constant temperature. 442 00:33:46,690 --> 00:33:53,540 Now, we can immediately see what du/dV at 443 00:33:53,540 --> 00:33:57,210 constant entropy is. 444 00:33:57,210 --> 00:33:59,270 Experimentally, though, that's not such an easy 445 00:33:59,270 --> 00:34:02,330 situation to arrange. 446 00:34:02,330 --> 00:34:07,440 Of course, this is a much more practical one. 447 00:34:07,440 --> 00:34:10,570 But it doesn't just fall out immediately from the one 448 00:34:10,570 --> 00:34:14,980 fundamental equation for du. 449 00:34:14,980 --> 00:34:16,810 But we can start there. 450 00:34:16,810 --> 00:34:24,190 So, du is T dS minus p dV. 451 00:34:24,190 --> 00:34:33,560 And I can take this derivative. du/dV at constant 452 00:34:33,560 --> 00:34:35,380 T. And so, what is it? 453 00:34:35,380 --> 00:34:42,650 Well, it's not just p because there's some dS/dV at constant 454 00:34:42,650 --> 00:34:44,270 T. This isn't zero. 455 00:34:44,270 --> 00:34:50,370 There's some variation, dS/dV, at constant temperature. 456 00:34:50,370 --> 00:34:53,430 That's going to matter. 457 00:34:53,430 --> 00:35:00,050 This part, of course, is just minus p. 458 00:35:00,050 --> 00:35:15,160 But we just figured out what dS/dV at constant T is. 459 00:35:15,160 --> 00:35:20,310 This is dp/dT at constant V. So that's neat. 460 00:35:20,310 --> 00:35:25,230 So in other words, we can write this as T, dp/dT at 461 00:35:25,230 --> 00:35:29,420 constant V, minus p. 462 00:35:29,420 --> 00:35:35,970 Let's just check T, p, T, V, p. 463 00:35:35,970 --> 00:35:37,070 Right? 464 00:35:37,070 --> 00:35:38,920 In other words, we just have pressure, 465 00:35:38,920 --> 00:35:40,420 temperature and volume. 466 00:35:40,420 --> 00:35:42,820 Again, if we know the equation of state, we 467 00:35:42,820 --> 00:35:45,390 know all this stuff. 468 00:35:45,390 --> 00:35:47,670 So again, we can measure equation of state data. 469 00:35:47,670 --> 00:35:50,940 Or, if we know the equation of state from a model, ideal gas, 470 00:35:50,940 --> 00:35:56,260 van der Waal's gas, whatever, now we can determine u. 471 00:35:56,260 --> 00:35:58,810 From equation of state data. 472 00:35:58,810 --> 00:36:17,960 Terrific, right? 473 00:36:17,960 --> 00:36:22,000 So let's take our one model that we keep going back to. 474 00:36:22,000 --> 00:36:25,520 Equation of state, and just see how it works. 475 00:36:25,520 --> 00:36:28,110 That is, ideal gas. 476 00:36:28,110 --> 00:36:30,790 And see how it works with that. 477 00:36:30,790 --> 00:36:36,070 Now, we saw before, or really I should say we accepted 478 00:36:36,070 --> 00:36:39,770 before, that for an ideal gas, u was a function of 479 00:36:39,770 --> 00:36:40,950 temperature only. 480 00:36:40,950 --> 00:36:43,135 Well, now let's try it. 481 00:36:43,135 --> 00:36:50,640 So, dp/dT, for our ideal gas, at constant volume, 482 00:36:50,640 --> 00:36:54,680 remember pV is nRT. 483 00:36:54,680 --> 00:37:01,460 So this nR over V. And then, using the relation again, we 484 00:37:01,460 --> 00:37:06,380 can just write this as p over T. In other words, we're 485 00:37:06,380 --> 00:37:07,810 taking advantage of the fact that we 486 00:37:07,810 --> 00:37:10,420 now know that quantity. 487 00:37:10,420 --> 00:37:12,110 In the case of the ideal gas, we just have a 488 00:37:12,110 --> 00:37:13,180 simple model for it. 489 00:37:13,180 --> 00:37:14,610 More generally, we could measure it. 490 00:37:14,610 --> 00:37:16,940 We could just collect a bunch of data. 491 00:37:16,940 --> 00:37:17,640 For a material. 492 00:37:17,640 --> 00:37:22,040 What's the volume it occupies at some pressure and 493 00:37:22,040 --> 00:37:23,080 temperature? 494 00:37:23,080 --> 00:37:25,190 Now let's change the pressure and temperature and sweep 495 00:37:25,190 --> 00:37:27,030 through a whole range of pressures and temperatures and 496 00:37:27,030 --> 00:37:28,590 measure the volume in every one of them. 497 00:37:28,590 --> 00:37:30,680 Well, then, we could just use that for 498 00:37:30,680 --> 00:37:31,210 our equation of state. 499 00:37:31,210 --> 00:37:33,710 One way or another, we can determine this quantity. 500 00:37:33,710 --> 00:37:35,780 For the ideal gas it's this. 501 00:37:35,780 --> 00:37:44,970 So now our du/dV, at constant T is just T times dp/dT, which 502 00:37:44,970 --> 00:37:52,510 is just p over T minus p, it's zero. 503 00:37:52,510 --> 00:37:55,560 Remember the Joule expansion. 504 00:37:55,560 --> 00:37:58,880 And we saw that, you saw that the Joule coefficient for an 505 00:37:58,880 --> 00:38:01,390 ideal gas was zero. 506 00:38:01,390 --> 00:38:06,160 So that you could see that for the ideal gas, u would not be 507 00:38:06,160 --> 00:38:08,860 a function of volume, but only of temperature. 508 00:38:08,860 --> 00:38:13,350 But actually, when you saw that before, you weren't given 509 00:38:13,350 --> 00:38:15,220 any proof of that. 510 00:38:15,220 --> 00:38:19,160 It was just that when the good Mr. Joule made the 511 00:38:19,160 --> 00:38:23,450 measurements, to the precision that he could measure, he 512 00:38:23,450 --> 00:38:26,680 discovered that for some gases it was extremely small. 513 00:38:26,680 --> 00:38:29,920 At least, smaller than anything he could detect. 514 00:38:29,920 --> 00:38:33,280 So it sure seemed like it was going to zero, under ideal gas 515 00:38:33,280 --> 00:38:33,800 conditions. 516 00:38:33,800 --> 00:38:36,260 And that was the result that we came to accept. 517 00:38:36,260 --> 00:38:38,340 Here, though, you can just derive straight away. 518 00:38:38,340 --> 00:38:41,160 That for an ideal gas it has to be the case that there's no 519 00:38:41,160 --> 00:38:43,060 volume dependence of the energy. 520 00:38:43,060 --> 00:38:53,460 Only a temperature dependence. 521 00:38:53,460 --> 00:39:23,690 It's the same for H. Just like u, we'd like to be able to 522 00:39:23,690 --> 00:39:27,190 express it in a way that allows us to calculate what 523 00:39:27,190 --> 00:39:30,820 happens only from equation of state data. 524 00:39:30,820 --> 00:39:36,880 But, again, our fundamental equations show us how it 525 00:39:36,880 --> 00:39:45,690 changes as a function of entropy and pressure. 526 00:39:45,690 --> 00:39:53,920 So, dH is T dS plus V dp. 527 00:39:53,920 --> 00:39:55,900 So let's look at dH/dp. 528 00:39:58,930 --> 00:40:00,470 We know how to get it immediately if we 529 00:40:00,470 --> 00:40:01,640 keep entropy constant. 530 00:40:01,640 --> 00:40:04,350 But we'd like to relate it to what happens if we keep the 531 00:40:04,350 --> 00:40:06,950 temperature constant. 532 00:40:06,950 --> 00:40:09,890 So then, just like we saw, analogous to what saw just 533 00:40:09,890 --> 00:40:19,380 before, it's T dS/dp at constant T. Plus V. But now 534 00:40:19,380 --> 00:40:24,830 we've seen from the Maxwell relations that dS/dp is minus 535 00:40:24,830 --> 00:40:26,860 dV/dT, for constant p. 536 00:40:26,860 --> 00:40:29,530 Again, this is this quantity, one of these quantities that 537 00:40:29,530 --> 00:40:31,850 again we can determine from equation of state data. 538 00:40:31,850 --> 00:40:35,180 Only V, p and T appear. 539 00:40:35,180 --> 00:40:45,170 So it's minus T dV/dT at constant p, plus V. And so, 540 00:40:45,170 --> 00:41:03,120 again, this can come from equation of state data. 541 00:41:03,120 --> 00:41:09,550 And if you do this again for an ideal gas, let me see. 542 00:41:09,550 --> 00:41:11,810 So we have pV is nRT. 543 00:41:11,810 --> 00:41:24,100 So dV/dT at constant pressure is just nR over p. 544 00:41:24,100 --> 00:41:26,810 But we can plug that in again just like we did before. 545 00:41:26,810 --> 00:41:36,480 It's just equal to V over T. And so dH/dp under our 546 00:41:36,480 --> 00:41:43,170 condition of constant temperature is just minus T 547 00:41:43,170 --> 00:41:47,220 times V over T plus V, everything 548 00:41:47,220 --> 00:41:50,820 cancels, and that's zero. 549 00:41:50,820 --> 00:41:54,430 That's our Joule - Thompson expansion. 550 00:41:54,430 --> 00:41:56,790 That was a constant enthalpy change. 551 00:41:56,790 --> 00:42:01,700 And again there, too, you saw an experimental result you 552 00:42:01,700 --> 00:42:04,060 were presented with that says, well at least to the extent 553 00:42:04,060 --> 00:42:05,410 that it could be measured, it was obviously 554 00:42:05,410 --> 00:42:06,950 getting very small. 555 00:42:06,950 --> 00:42:11,530 For gases that approach ideal gas conditions. 556 00:42:11,530 --> 00:42:13,910 Well, there you can see it. 557 00:42:13,910 --> 00:42:16,020 Sure better have gotten small because in fact 558 00:42:16,020 --> 00:42:20,790 it has to be zero. 559 00:42:20,790 --> 00:42:27,550 Now let's take just one somewhat 560 00:42:27,550 --> 00:42:30,490 more complicated case. 561 00:42:30,490 --> 00:42:37,290 Let's look at a van der Waal's gas. 562 00:42:37,290 --> 00:42:40,710 Let's try it with a different equation of state, that isn't 563 00:42:40,710 --> 00:42:47,230 quite as simple as the ideal gas case. 564 00:42:47,230 --> 00:42:59,280 So, then p plus a over molar volume squared times V minus b 565 00:42:59,280 --> 00:43:07,460 molar volume V minus b is equal to RT, remember? 566 00:43:07,460 --> 00:43:10,340 This was back from the first or second 567 00:43:10,340 --> 00:43:14,350 lecture in the course. 568 00:43:14,350 --> 00:43:18,970 So, we can separate out p. 569 00:43:18,970 --> 00:43:25,730 It's RT over molar volume minus b minus a over molar 570 00:43:25,730 --> 00:43:31,670 volume V squared. 571 00:43:31,670 --> 00:43:33,440 And then we can take the derivative with respect to 572 00:43:33,440 --> 00:43:38,190 temperature, it's just R over molar volume minus b. 573 00:43:38,190 --> 00:43:53,870 So it's dp/dT at constant V is just R over V bar minus b. 574 00:43:53,870 --> 00:44:04,190 Well, let's now look, given this, let's now look in that 575 00:44:04,190 --> 00:44:09,100 case, at what happens to u as a function of V. For the ideal 576 00:44:09,100 --> 00:44:12,860 gas, we know that u is volume independent. 577 00:44:12,860 --> 00:44:16,270 It only depends on the temperature. 578 00:44:16,270 --> 00:44:20,650 But for the van der Waal's gas, now 579 00:44:20,650 --> 00:44:22,080 it's going to be different. 580 00:44:22,080 --> 00:44:27,120 And that's because this is different from what it is in 581 00:44:27,120 --> 00:44:28,760 the ideal gas case. 582 00:44:28,760 --> 00:44:49,770 Namely, now du/dV at constant T, for the van der Waal's gas. 583 00:44:49,770 --> 00:44:57,010 So it's this. 584 00:44:57,010 --> 00:45:04,550 So it's RT over molar volume minus b. 585 00:45:04,550 --> 00:45:09,750 Minus p, right? 586 00:45:09,750 --> 00:45:13,130 But in fact, if you go back to the van der Waal's equation of 587 00:45:13,130 --> 00:45:17,690 state, here's RT over v minus b. 588 00:45:17,690 --> 00:45:19,220 If we put it as minus b, that's just 589 00:45:19,220 --> 00:45:22,100 equal to a over V squared. 590 00:45:22,100 --> 00:45:28,610 Equals a over molar volume squared. 591 00:45:28,610 --> 00:45:33,220 But the point is, the main point is, it's not zero. 592 00:45:33,220 --> 00:45:37,110 It's some number. a over the molar volume squared. a is a 593 00:45:37,110 --> 00:45:40,220 positive number in the van der Waal's equation of state. 594 00:45:40,220 --> 00:45:44,050 So this is greater than zero. 595 00:45:44,050 --> 00:45:52,990 In other words, u is a function of T and V. If we 596 00:45:52,990 --> 00:45:59,150 don't have an ideal gas. 597 00:45:59,150 --> 00:46:01,260 By the way, just to think about it a little bit, it's a 598 00:46:01,260 --> 00:46:02,620 positive number. 599 00:46:02,620 --> 00:46:04,490 What that means is, I've got my 600 00:46:04,490 --> 00:46:08,720 ideal gas in some container. 601 00:46:08,720 --> 00:46:12,490 There's some energy, some internal energy. 602 00:46:12,490 --> 00:46:13,760 Now I make the volume bigger. 603 00:46:13,760 --> 00:46:17,630 I allow it to expand. 604 00:46:17,630 --> 00:46:22,790 And the energy changes, it goes up. 605 00:46:22,790 --> 00:46:25,840 In some sense it's less favorable energetically. 606 00:46:25,840 --> 00:46:31,310 What's happening there, that a term in the van der Waal's 607 00:46:31,310 --> 00:46:35,730 equation of state, is describing interactions, 608 00:46:35,730 --> 00:46:38,660 favorable attractions, between gas molecules. 609 00:46:38,660 --> 00:46:42,290 The b is describing repulsions. 610 00:46:42,290 --> 00:46:44,290 Effectively, the volume changes. 611 00:46:44,290 --> 00:46:47,300 The molar volume is being changed a little bit by the 612 00:46:47,300 --> 00:46:50,160 fact that if you're really trying to make things collide 613 00:46:50,160 --> 00:46:54,110 with each other, they can't occupy the same volume. 614 00:46:54,110 --> 00:46:56,270 And that is being expressed here. 615 00:46:56,270 --> 00:47:02,800 The a, though is expressing attraction between molecules 616 00:47:02,800 --> 00:47:04,740 at somewhat longer range. 617 00:47:04,740 --> 00:47:07,380 So now, if you make the volume bigger, those 618 00:47:07,380 --> 00:47:08,670 attractions die out. 619 00:47:08,670 --> 00:47:11,500 Because the molecules are farther apart from each other. 620 00:47:11,500 --> 00:47:13,520 So the energy goes up. 621 00:47:13,520 --> 00:47:17,970 Now, you might ask, well why does it do that, right? 622 00:47:17,970 --> 00:47:22,160 I mean, if the energy is lower to occupy a smaller volume, 623 00:47:22,160 --> 00:47:26,290 then if I have this room and a bunch of molecules of oxygen, 624 00:47:26,290 --> 00:47:29,120 and nitrogen and what have you in the air, and there are weak 625 00:47:29,120 --> 00:47:31,490 attractions between them, why don't they all just sort of 626 00:47:31,490 --> 00:47:34,890 glum together and find whatever volume they like. 627 00:47:34,890 --> 00:47:37,720 So that the attractive forces can exert 628 00:47:37,720 --> 00:47:38,580 themselves a little bit. 629 00:47:38,580 --> 00:47:39,980 Not too close, right? 630 00:47:39,980 --> 00:47:42,530 Not so close that the repulsions dominate. 631 00:47:42,530 --> 00:47:44,630 Why don't they do that? 632 00:47:44,630 --> 00:47:54,790 What else matters besides any of those considerations? 633 00:47:54,790 --> 00:47:58,720 What else matters that I haven't considered in this 634 00:47:58,720 --> 00:48:02,370 little discussion? 635 00:48:02,370 --> 00:48:04,350 Yeah. 636 00:48:04,350 --> 00:48:06,750 What about entropy, right? 637 00:48:06,750 --> 00:48:08,710 If I only worry about minimizing the 638 00:48:08,710 --> 00:48:10,900 energy, it's true. 639 00:48:10,900 --> 00:48:13,660 They'll stick together a little bit. 640 00:48:13,660 --> 00:48:15,540 But entropy also matters. 641 00:48:15,540 --> 00:48:18,680 And there's disorder achieved by occupying the full 642 00:48:18,680 --> 00:48:19,930 available volume. 643 00:48:19,930 --> 00:48:21,990 Many more states possible. 644 00:48:21,990 --> 00:48:24,950 And that will end up winning out at basically any realistic 645 00:48:24,950 --> 00:48:28,040 temperature where the stuff really is a gas. 646 00:48:28,040 --> 00:48:29,000 OK. 647 00:48:29,000 --> 00:48:32,540 Next time, what you're going to see is the following. 648 00:48:32,540 --> 00:48:36,120 It turns out we can express all these functions in terms 649 00:48:36,120 --> 00:48:39,730 of G, we wouldn't need to choose G, but it's a very 650 00:48:39,730 --> 00:48:40,820 useful function to choose. 651 00:48:40,820 --> 00:48:44,280 Because of its natural expression in 652 00:48:44,280 --> 00:48:46,630 terms of T and p. 653 00:48:46,630 --> 00:48:49,690 So we can write any of these functions in terms of G. Which 654 00:48:49,690 --> 00:48:51,970 means that we can really calculate all the 655 00:48:51,970 --> 00:48:55,350 thermodynamics in terms of only g. 656 00:48:55,350 --> 00:48:58,080 It's not necessary to do that, but it can be quite 657 00:48:58,080 --> 00:48:59,450 convenient. 658 00:48:59,450 --> 00:49:03,190 And then we can say, OK, if we have many constituents, what 659 00:49:03,190 --> 00:49:04,800 if we have a mixture of stuff? 660 00:49:04,800 --> 00:49:08,590 We can take the derivative of G with respect to how much 661 00:49:08,590 --> 00:49:10,510 material there is. 662 00:49:10,510 --> 00:49:13,010 With respect to n, the number of moles. 663 00:49:13,010 --> 00:49:15,420 And if there are one, and two, and three constituents with 664 00:49:15,420 --> 00:49:18,280 respect to n1, and n2, and n3. 665 00:49:18,280 --> 00:49:21,160 Each individual amount of stuff. 666 00:49:21,160 --> 00:49:23,750 What that's going to allow us to do is, if we say, OK I have 667 00:49:23,750 --> 00:49:26,560 a mixture of stuff, how does the free energy change? 668 00:49:26,560 --> 00:49:29,020 If I change the composition of the mixture? 669 00:49:29,020 --> 00:49:32,320 If I take something away, or put something else in? 670 00:49:32,320 --> 00:49:34,440 And we'll be able to determine equilibrium under those 671 00:49:34,440 --> 00:49:35,140 conditions. 672 00:49:35,140 --> 00:49:38,730 That's very useful for things like chemical reactions, where 673 00:49:38,730 --> 00:49:41,380 this constituent changes to this one. 674 00:49:41,380 --> 00:49:43,530 And I can calculate what happens to G under those 675 00:49:43,530 --> 00:49:44,360 conditions. 676 00:49:44,360 --> 00:49:46,390 And that's what you'll see starting next time. 677 00:49:46,390 --> 00:49:47,970 And professor Blendi will be taking over 678 00:49:47,970 --> 00:49:49,860 for that set of lectures.