1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation or view additional materials from 7 00:00:13 --> 00:00:17 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:17 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:21 PROFESSOR: Alright. 10 00:00:21 --> 00:00:24 Well, we've been looking in the last couple lectures at a 11 00:00:24 --> 00:00:27 really important topic in thermodynamics. 12 00:00:27 --> 00:00:31 Which is, how do you tell what's going to happen. 13 00:00:31 --> 00:00:33 Which way does a process want to go? 14 00:00:33 --> 00:00:36 Which way will it go spontaneously? 15 00:00:36 --> 00:00:39 And if it goes in one direction or another, where does is lead? 16 00:00:39 --> 00:00:42 In other words, what is the equilibrium state? 17 00:00:42 --> 00:00:44 And this is just an incredibly important area that 18 00:00:44 --> 00:00:48 thermodynamics allows us to speak to. 19 00:00:48 --> 00:00:52 So we started to see this. 20 00:00:52 --> 00:01:09 Sort of direction of spontaneous change. 21 00:01:09 --> 00:01:10 And where the equilibrium lies. 22 00:01:10 --> 00:01:18 So what we did is, remember we started with the second law. 23 00:01:18 --> 00:01:20 Right? 24 00:01:20 --> 00:01:26 That dS is greater than dq over T. 25 00:01:26 --> 00:01:33 And for the spontaneous change which happens irreversibly That 26 00:01:33 --> 00:01:37 means that'll be dq irreversible. 27 00:01:37 --> 00:01:39 It would be equal for the reversible case. 28 00:01:39 --> 00:01:49 And we combine this with first law, which for the case of 29 00:01:49 --> 00:01:57 pressure volume changes we write as this. 30 00:01:57 --> 00:02:01 And so what this gave us was a very, very useful general 31 00:02:01 --> 00:02:03 criterion for determining whether something 32 00:02:03 --> 00:02:06 happened spontaneously. 33 00:02:06 --> 00:02:18 Namely, du plus p external dV minus T for the surroundings 34 00:02:18 --> 00:02:26 dS, is greater than zero. 35 00:02:26 --> 00:02:31 Sorry. 36 00:02:31 --> 00:02:34 It's less than zero. 37 00:02:34 --> 00:02:43 And this is for any spontaneous change. 38 00:02:43 --> 00:02:50 If it equals zero, then we're at equilibrium. 39 00:02:50 --> 00:02:59 And if it's greater than zero, then the process 40 00:02:59 --> 00:03:01 goes the other way. 41 00:03:01 --> 00:03:04 We would write the process in the reverse to have it be less 42 00:03:04 --> 00:03:09 than 0 and it would go spontaneously. 43 00:03:09 --> 00:03:15 And based on this one result, we then looked under various 44 00:03:15 --> 00:03:20 constraints and said OK, what about looking at our variables, 45 00:03:20 --> 00:03:24 volume, pressure, temperature, other things, entropy, if we 46 00:03:24 --> 00:03:28 constrain those, what's the condition for equilibrium? 47 00:03:28 --> 00:03:32 And that's what led us to a number of results to determine 48 00:03:32 --> 00:03:35 what quantities we even need to be looking at. 49 00:03:35 --> 00:03:37 To figure out equilibrium. 50 00:03:37 --> 00:03:38 And what the conditions were. 51 00:03:38 --> 00:03:41 And so what we discovered were the following. 52 00:03:41 --> 00:03:49 This one, which we already had seen, which is dS, 53 00:03:49 --> 00:03:50 is greater than zero. 54 00:03:50 --> 00:03:54 Change in entropy is greater than zero, for 55 00:03:54 --> 00:04:02 an isolated system. 56 00:04:02 --> 00:04:08 We also saw that dS for constant H and p was greater 57 00:04:08 --> 00:04:18 than zero. du, regular energy, at constant entropy and 58 00:04:18 --> 00:04:41 volume is less then zero. 59 00:04:41 --> 00:04:43 And u is minimized at equilibrium. 60 00:04:43 --> 00:04:47 And this is the familiar result from ordinary mechanics, where 61 00:04:47 --> 00:04:49 you're not worrying about something like entropy for a 62 00:04:49 --> 00:04:51 whole collection of particles. 63 00:04:51 --> 00:04:55 That is, you minimize potential energy and you see things 64 00:04:55 --> 00:04:58 falling under the force of gravity and so forth, going to 65 00:04:58 --> 00:05:05 potential energy minima in conformance with this result. 66 00:05:05 --> 00:05:10 dH, S and p is less than zero. 67 00:05:10 --> 00:05:16 So our H is u plus pV, as you know. 68 00:05:16 --> 00:05:27 And H is minimized at equilibrium. 69 00:05:27 --> 00:05:32 And this is, of course, with constant S V. 70 00:05:32 --> 00:05:37 This is constant S and p. 71 00:05:37 --> 00:05:41 But of course, the need to have entropy constrained is 72 00:05:41 --> 00:05:44 never going to be the most convenient one experimental. 73 00:05:44 --> 00:05:46 There may be circumstances under which it's the 74 00:05:46 --> 00:05:49 case, but it's often difficult to control. 75 00:05:49 --> 00:05:54 On the other hand, temperature, volume and pressure are 76 00:05:54 --> 00:05:57 variables that are much easier in the lab to keep constant. 77 00:05:57 --> 00:05:58 To keep control over. 78 00:05:58 --> 00:06:02 And so that led us to the definitions of other energy 79 00:06:02 --> 00:06:05 quantities, the Helmholtz and Gibbs free energy. 80 00:06:05 --> 00:06:11 We discovered that the quantity dA, under conditions of 81 00:06:11 --> 00:06:16 constant volume and temperature, is less than zero. 82 00:06:16 --> 00:06:20 And A is u minus TS. 83 00:06:20 --> 00:06:28 84 00:06:28 --> 00:06:34 And A is minimized at equilibrium, under conditions 85 00:06:34 --> 00:06:39 of constant T and V. 86 00:06:39 --> 00:06:44 And finally, and in many cases the most important of the 87 00:06:44 --> 00:06:46 results, because of the conditions it applies to, we 88 00:06:46 --> 00:06:53 saw that this Gibbs free energy is less than zero, 89 00:06:53 --> 00:06:57 that's our condition for spontaneous change. 90 00:06:57 --> 00:07:05 Where the Gibbs free energy, u plus pV minus TS is H minus TS. 91 00:07:05 --> 00:07:08 92 00:07:08 --> 00:07:22 Also A plus pV and G is minimized at equilibrium 93 00:07:22 --> 00:07:26 with constant temperature and pressure. 94 00:07:26 --> 00:07:27 And that's why the Gibbs free energy is just so 95 00:07:27 --> 00:07:28 enormously important. 96 00:07:28 --> 00:07:31 Because so much of what we do in chemistry does take place 97 00:07:31 --> 00:07:34 with constant temperature and pressure. 98 00:07:34 --> 00:07:37 So we have this condition that's established in a 99 00:07:37 --> 00:07:40 quantity that we know how to calculate. 100 00:07:40 --> 00:07:44 That tells us the direction of spontaneous change for ordinary 101 00:07:44 --> 00:07:47 processes, chemical processes, mixing and you name it, under 102 00:07:47 --> 00:07:54 conditions that are easy to achieve in the lab. 103 00:07:54 --> 00:08:01 OK, now what we'd like to do is be able to calculate any of 104 00:08:01 --> 00:08:04 these quantities in terms of temperature, pressure, 105 00:08:04 --> 00:08:05 volume properties. 106 00:08:05 --> 00:08:08 That is, in terms of equations of state. 107 00:08:08 --> 00:08:09 For any material. 108 00:08:09 --> 00:08:12 Then we would really be able to essentially calculate anything. 109 00:08:12 --> 00:08:14 Anything thermodynamic. 110 00:08:14 --> 00:08:15 About a material. 111 00:08:15 --> 00:08:18 Of course, that's assuming we know the equation of state. 112 00:08:18 --> 00:08:20 We may or may not. 113 00:08:20 --> 00:08:25 But because in many cases we can reasonably either model or 114 00:08:25 --> 00:08:28 measure equations of state, collect data for a material for 115 00:08:28 --> 00:08:32 its temperature, pressure, volume relations, then in fact 116 00:08:32 --> 00:08:35 if we can relate all these quantities to those, then in 117 00:08:35 --> 00:08:37 fact we really can calculate essentially all of 118 00:08:37 --> 00:08:38 the thermodynamics. 119 00:08:38 --> 00:08:42 For the material. 120 00:08:42 --> 00:08:58 So let's relate the thermodynamic quantities to 121 00:08:58 --> 00:09:07 equation of state p, V, T data. 122 00:09:07 --> 00:09:12 And we can do that by going through and deriving what we'll 123 00:09:12 --> 00:09:16 call the fundamental equations of thermodynamics. that'll 124 00:09:16 --> 00:09:17 provide these relations. 125 00:09:17 --> 00:09:19 And at this point we know enough to do this in a 126 00:09:19 --> 00:09:21 straightforward way. 127 00:09:21 --> 00:09:38 So if we start with a relation for energy, T dS minus p dV. 128 00:09:38 --> 00:09:43 Where u is written as a function of entropy and volume. 129 00:09:43 --> 00:09:47 And we've seen that that's generally the case. 130 00:09:47 --> 00:09:59 It comes from the fact that dq reversible is T dS, and dw 131 00:09:59 --> 00:10:04 reversible is minus p dV. 132 00:10:04 --> 00:10:08 And of course du is the some of those. 133 00:10:08 --> 00:10:14 So, this is generally true. 134 00:10:14 --> 00:10:15 Since these are all state functions. 135 00:10:15 --> 00:10:19 That is, this is derived in the case for reversible paths. 136 00:10:19 --> 00:10:23 But since these are all simply state functions and quantities, 137 00:10:23 --> 00:10:25 this is generally true. 138 00:10:25 --> 00:10:28 Now we can use it to derive differential relations for 139 00:10:28 --> 00:10:29 all of the thermodynamics quantities. 140 00:10:29 --> 00:10:32 So let's just go through and do that. 141 00:10:32 --> 00:10:38 So H is u plus pV. 142 00:10:38 --> 00:10:41 143 00:10:41 --> 00:10:49 So dH is just du plus p dV plus V dp. 144 00:10:49 --> 00:10:53 And now we're just going to substitute du in here. 145 00:10:53 --> 00:10:58 And the p dV terms are going to cancel. 146 00:10:58 --> 00:11:08 So we have the result that dH is T dS plus V dp. 147 00:11:08 --> 00:11:11 Right? 148 00:11:11 --> 00:11:17 And that shows us that H is written naturally as a function 149 00:11:17 --> 00:11:23 of entropy and pressure. 150 00:11:23 --> 00:11:26 And now let's keep going. 151 00:11:26 --> 00:11:29 A is u minus TS. 152 00:11:29 --> 00:11:31 153 00:11:31 --> 00:11:41 dA is du minus T dS minus S dT. 154 00:11:41 --> 00:11:42 We're going to do the same thing. 155 00:11:42 --> 00:11:45 Substitute this for du. 156 00:11:45 --> 00:11:51 This time, the T dS terms are going to cancel. 157 00:11:51 --> 00:11:58 So we have dA is minus S dT minus T dS. 158 00:11:58 --> 00:12:02 159 00:12:02 --> 00:12:03 That can't be right. 160 00:12:03 --> 00:12:06 And it isn't. 161 00:12:06 --> 00:12:12 Minus S dT, that's the p dV term that's left, minus p dV. 162 00:12:12 --> 00:12:15 And it shows us that A is written naturally as a 163 00:12:15 --> 00:12:24 function of T and V. 164 00:12:24 --> 00:12:27 G, we can write in any of a number of ways. 165 00:12:27 --> 00:12:31 Let's write it as H minus TS. 166 00:12:31 --> 00:12:40 So dG is dH minus T dS minus S dT. 167 00:12:40 --> 00:12:43 Here's dH. 168 00:12:43 --> 00:12:46 We'll substitute that in, and the T dS terms 169 00:12:46 --> 00:12:48 are going to cancel. 170 00:12:48 --> 00:12:56 So dG is minus S dT plus V dp. 171 00:12:56 --> 00:12:59 172 00:12:59 --> 00:13:02 And this shows that G is written naturally as a 173 00:13:02 --> 00:13:10 function of T and p. 174 00:13:10 --> 00:13:22 So these, which we will exalt and celebrate by our 175 00:13:22 --> 00:13:30 sparingly-used colored chalk, are our fundamental equations 176 00:13:30 --> 00:13:54 of thermodynamics. 177 00:13:54 --> 00:13:58 So what they do is, they're describing how these 178 00:13:58 --> 00:14:03 thermodynamic properties change, in terms of only state 179 00:14:03 --> 00:14:07 functions and state variables. 180 00:14:07 --> 00:14:08 Very, very useful. 181 00:14:08 --> 00:14:11 And that's what it means, when we say well, it's natural then, 182 00:14:11 --> 00:14:14 to express say, G as a function of T and p, that's 183 00:14:14 --> 00:14:14 what we're saying. 184 00:14:14 --> 00:14:17 Is that we can express its changes in terms 185 00:14:17 --> 00:14:19 of these variables. 186 00:14:19 --> 00:14:25 Related only through quantities that are functions of state. 187 00:14:25 --> 00:14:29 I don't need to know about a specific path here. 188 00:14:29 --> 00:14:33 If I know about the states involved, I just need to 189 00:14:33 --> 00:14:40 know what the volume was in each of them. 190 00:14:40 --> 00:14:45 Now, before, of course, in the first part of the class we 191 00:14:45 --> 00:14:49 started out looking at u and then looking at H not as 192 00:14:49 --> 00:14:53 functions of S and V or S and p, but as functions of 193 00:14:53 --> 00:14:55 temperature, mostly. 194 00:14:55 --> 00:14:58 In general, temperature and volume or pressure. 195 00:14:58 --> 00:15:01 And it doesn't mean that something was somehow 196 00:15:01 --> 00:15:04 wrong with that. 197 00:15:04 --> 00:15:07 It certainly is, it still is going to be useful 198 00:15:07 --> 00:15:08 to do thermochemistry. 199 00:15:08 --> 00:15:11 To ask questions like how much heat is released in a chemical 200 00:15:11 --> 00:15:15 reaction that takes place at constant temperature. 201 00:15:15 --> 00:15:17 Not one of these variables. 202 00:15:17 --> 00:15:19 And we can calculate that. 203 00:15:19 --> 00:15:22 So it's not that we're somehow throwing away 204 00:15:22 --> 00:15:23 our ability to do that. 205 00:15:23 --> 00:15:28 However, the thing to remember is, when you look at heats of 206 00:15:28 --> 00:15:32 reaction under those conditions it's all well and good. 207 00:15:32 --> 00:15:35 But it doesn't tell you, this is the direction that the 208 00:15:35 --> 00:15:37 reaction is going to go. 209 00:15:37 --> 00:15:41 It doesn't tell you, this is the equilibrium concentration 210 00:15:41 --> 00:15:43 that you'll end up with. 211 00:15:43 --> 00:15:46 That doesn't come out of what we calculated before 212 00:15:46 --> 00:15:47 in thermochemistry. 213 00:15:47 --> 00:15:51 What does come out, which is very useful is, if you do run 214 00:15:51 --> 00:15:54 the reaction, here's how much heat comes out. 215 00:15:54 --> 00:15:57 And if you want to run a furnace and provide energy, 216 00:15:57 --> 00:16:01 that's an extremely important thing to be able to calculate. 217 00:16:01 --> 00:16:03 Because you're going to run it and you'll probably find 218 00:16:03 --> 00:16:05 conditions under which you can run it more or less 219 00:16:05 --> 00:16:07 to completion. 220 00:16:07 --> 00:16:10 But it doesn't tell you, by itself, which direction 221 00:16:10 --> 00:16:14 things run in. 222 00:16:14 --> 00:16:17 Whereas under these conditions, these quantities, if you look 223 00:16:17 --> 00:16:20 at free energy change, for example, at constant 224 00:16:20 --> 00:16:24 temperature and pressure, you can still calculate H. 225 00:16:24 --> 00:16:26 You can still calculate the heat that's released. 226 00:16:26 --> 00:16:28 This is what will tell you under some particular 227 00:16:28 --> 00:16:31 conditions what will actually happen. 228 00:16:31 --> 00:16:33 Where will you end up. 229 00:16:33 --> 00:16:34 Very, very important, of course, to be able 230 00:16:34 --> 00:16:39 to understand that. 231 00:16:39 --> 00:16:55 Now, it's also very useful to look at some of the relations 232 00:16:55 --> 00:16:57 that come out of these fundamental equations. 233 00:16:57 --> 00:16:59 And they're straightforward to derive. 234 00:16:59 --> 00:17:04 So, all I want to do now is look at the derivatives of the 235 00:17:04 --> 00:17:06 free energies with respect to temperature and 236 00:17:06 --> 00:17:10 volume and pressure. 237 00:17:10 --> 00:17:15 So for example, if I look at A, which we now have written as 238 00:17:15 --> 00:17:19 the function of T and V, of course, in general I can always 239 00:17:19 --> 00:17:25 write dA as partial of A, with respect to T at constant volume 240 00:17:25 --> 00:17:29 dT, plus partial of A with respect to V, at constant 241 00:17:29 --> 00:17:32 temperature dV. 242 00:17:32 --> 00:17:35 And I know what those turn out to be. 243 00:17:35 --> 00:17:41 It's minus S dT minus p dV. 244 00:17:41 --> 00:17:42 So what does that tell me? 245 00:17:42 --> 00:17:47 It tells me that the partial of A with respect to T at 246 00:17:47 --> 00:17:51 constant V is minus S. 247 00:17:51 --> 00:17:52 Right? 248 00:17:52 --> 00:17:55 In other words, now I know how to tell how the Helmholtz 249 00:17:55 --> 00:17:59 free energy changes as a function of temperature. 250 00:17:59 --> 00:18:04 Or as a function of volume. dA/dV, at constant T, 251 00:18:04 --> 00:18:08 must be negative p. 252 00:18:08 --> 00:18:12 Things that I can measure. 253 00:18:12 --> 00:18:14 So I can in a very straightforward way say, 254 00:18:14 --> 00:18:19 OK, well, here is my Helmholtz free energy. 255 00:18:19 --> 00:18:22 If I'm working under conditions of constant temperature and 256 00:18:22 --> 00:18:23 volume, that's very useful. 257 00:18:23 --> 00:18:26 Now, if I want to change those quantities; change the 258 00:18:26 --> 00:18:28 temperature, change the volume, how will it change? 259 00:18:28 --> 00:18:32 Well, I can, for any given case, measure the pressure, 260 00:18:32 --> 00:18:35 determine the entropy and I'll know what the slope 261 00:18:35 --> 00:18:38 of change will be. 262 00:18:38 --> 00:18:45 Similarly for G as a function of temperature and pressure, I 263 00:18:45 --> 00:18:47 can go through the same procedure. 264 00:18:47 --> 00:18:54 That is, it's easy to write down straight away that dG, 265 00:18:54 --> 00:18:59 with respect to temperature at constant pressure is minus S. 266 00:18:59 --> 00:19:03 That is, this is, dG/dT at constant pressure. 267 00:19:03 --> 00:19:18 And this is dG/dp at constant temperature. 268 00:19:18 --> 00:19:22 So again with the Gibbs free energy, now I see how to 269 00:19:22 --> 00:19:25 determine, if I change the pressure, if I change the 270 00:19:25 --> 00:19:29 temperature by some modest amount, how much is the Gibbs 271 00:19:29 --> 00:19:30 free energy going to change? 272 00:19:30 --> 00:19:39 Well, it's easy to see. 273 00:19:39 --> 00:19:43 These two relations involving entropy are also useful because 274 00:19:43 --> 00:19:48 they'll let us see how entropy depends on volume and pressure. 275 00:19:48 --> 00:20:09 And let me show you how that goes. 276 00:20:09 --> 00:20:22 Now, you've already seen how entropy depends on temperature. 277 00:20:22 --> 00:20:26 We've already seen that, going to write dS as 278 00:20:26 --> 00:20:32 dq reversible over T. 279 00:20:32 --> 00:20:42 And it's Cv dT over T at constant volume. 280 00:20:42 --> 00:20:51 It's Cp dT over T at constant pressure. 281 00:20:51 --> 00:20:56 So we know that dS/dT at constant volume is Cv over 282 00:20:56 --> 00:21:03 T, and dS/dT at constant pressure is Cp, over T. 283 00:21:03 --> 00:21:08 And we've seen that on a number of occasions. 284 00:21:08 --> 00:21:11 So that tells us what to do to know the entropy as 285 00:21:11 --> 00:21:13 the temperature changes. 286 00:21:13 --> 00:21:19 But now, what happens if, instead we look at what happens 287 00:21:19 --> 00:21:21 when we go to some state one to some other state two 288 00:21:21 --> 00:21:22 and it's the pressure. 289 00:21:22 --> 00:21:24 Or the volume, that changes. 290 00:21:24 --> 00:21:28 And by the way, just to be explicit about this, let's take 291 00:21:28 --> 00:21:32 this example, it means that delta S, if we undergo a 292 00:21:32 --> 00:21:35 change from, say, T1 to T2. 293 00:21:35 --> 00:21:42 There's Cp over T dT. 294 00:21:42 --> 00:21:51 So it's Cp log of T2 over T1, and we saw this before. 295 00:21:51 --> 00:21:56 So now, instead, let's look at some process. 296 00:21:56 --> 00:21:59 State one goes to state two. 297 00:21:59 --> 00:22:04 Let's have constant T. 298 00:22:04 --> 00:22:09 And look at what happens if pressure goes from p1 to p2. 299 00:22:09 --> 00:22:14 Or volume goes from V1 to V2. 300 00:22:14 --> 00:22:17 And see what happens there. 301 00:22:17 --> 00:22:20 We looked at pressure change before, actually, in discussing 302 00:22:20 --> 00:22:24 the third law, the fact that the entropy goes to zero as the 303 00:22:24 --> 00:22:27 absolute temperature goes to zero for a pure, 304 00:22:27 --> 00:22:28 perfect crystal. 305 00:22:28 --> 00:22:31 But, actually, we didn't do that in a general way. 306 00:22:31 --> 00:22:34 We just treated the one case of an ideal gas as the 307 00:22:34 --> 00:22:36 temperature is reduced. 308 00:22:36 --> 00:22:39 But we can do this, generally, by using what are called 309 00:22:39 --> 00:22:41 Maxwell relations. 310 00:22:41 --> 00:22:46 And all this is, is saying that when you take a mixed second 311 00:22:46 --> 00:22:50 derivative, it doesn't matter in which order you take 312 00:22:50 --> 00:22:54 the two derivatives. 313 00:22:54 --> 00:23:03 So, let's, we're going to use this relationship. 314 00:23:03 --> 00:23:13 And we're going to use these two. 315 00:23:13 --> 00:23:18 So, using those, now, what happens if we take the second 316 00:23:18 --> 00:23:22 derivative of A, the mixed derivative, partial with 317 00:23:22 --> 00:23:27 respect to T and the partial with respect to V. 318 00:23:27 --> 00:23:35 So let's leave these off for a moment, and now let's try that. 319 00:23:35 --> 00:23:41 And the point is that the second derivative of A, with 320 00:23:41 --> 00:23:48 respect to V and T in this order is the same as the second 321 00:23:48 --> 00:23:52 derivative of a with respect to T and V in this order. 322 00:23:52 --> 00:23:56 It doesn't matter which order. 323 00:23:56 --> 00:23:58 But that turns out to be useful. 324 00:23:58 --> 00:24:01 So let's do this explicitly. 325 00:24:01 --> 00:24:04 Which means we're going to take the derivative with respect 326 00:24:04 --> 00:24:06 to volume of dA/dT. 327 00:24:06 --> 00:24:09 328 00:24:09 --> 00:24:15 Now, the dA/dT isn't constant volume. 329 00:24:15 --> 00:24:18 The derivative we're taking with respect to volume, 330 00:24:18 --> 00:24:23 when we take that it's at constant temperature. 331 00:24:23 --> 00:24:24 But what is it? 332 00:24:24 --> 00:24:30 Well, we already know what dA/dT at constant V is. 333 00:24:30 --> 00:24:32 It's negative S. 334 00:24:32 --> 00:24:35 So this is negative dS/dV. 335 00:24:35 --> 00:24:38 336 00:24:38 --> 00:24:43 At constant temperature. 337 00:24:43 --> 00:24:46 Now let's take it in the other order. 338 00:24:46 --> 00:24:57 So d/dT of dA/dV, just like this. 339 00:24:57 --> 00:25:01 The dA/dV is calculated at constant temperature. 340 00:25:01 --> 00:25:02 We know it. 341 00:25:02 --> 00:25:05 Then we can take the derivative of that quantity, when we vary 342 00:25:05 --> 00:25:07 the temperature, holding the volume constant. 343 00:25:07 --> 00:25:11 But again, dA/dV dT, there it is. 344 00:25:11 --> 00:25:14 It's negative p. 345 00:25:14 --> 00:25:24 So this is just negative dp/dT at constant volume. 346 00:25:24 --> 00:25:31 These things have to be equal to each other. 347 00:25:31 --> 00:25:36 Because these mixed second derivatives are the same thing. 348 00:25:36 --> 00:25:37 But that's very useful. 349 00:25:37 --> 00:25:41 Because this is what comes directly out of an 350 00:25:41 --> 00:25:43 equation of state, right? 351 00:25:43 --> 00:25:46 You know how pressure changes with temperature at constant 352 00:25:46 --> 00:25:48 volume if you know the equation of state. 353 00:25:48 --> 00:25:50 It relates the pressure, volume, and 354 00:25:50 --> 00:26:03 temperature together. 355 00:26:03 --> 00:26:06 So from measured equation of state data, or from a model 356 00:26:06 --> 00:26:09 like the ideal gas or the van der Waal's gas or another 357 00:26:09 --> 00:26:11 equation of state, you know this. 358 00:26:11 --> 00:26:14 Can determine how entropy is going to behave as 359 00:26:14 --> 00:26:31 the volume changes. 360 00:26:31 --> 00:26:42 If we try that for an ideal gas, pV is nRT. 361 00:26:42 --> 00:26:54 So dp/dT at constant volume, it's just nR over V. 362 00:26:54 --> 00:27:03 And that, now, we know must equal dS/dV, 363 00:27:03 --> 00:27:04 with a positive sign. 364 00:27:04 --> 00:27:08 At constant temperature. 365 00:27:08 --> 00:27:11 So now let's try looking at something where 366 00:27:11 --> 00:27:13 are V1 goes to V2. 367 00:27:13 --> 00:27:17 The volume is going to change, and we can see 368 00:27:17 --> 00:27:20 how the entropy changes. 369 00:27:20 --> 00:27:29 So, if one goes to two and V1 goes to V2, and it's constant 370 00:27:29 --> 00:27:33 temperature, just what we've specified there. 371 00:27:33 --> 00:27:48 Delta S is S(T, V2) minus S(T, V1), T's staying the same. 372 00:27:48 --> 00:27:57 So it's just the integral from V1 to V2 of dS/dV At 373 00:27:57 --> 00:27:59 constant temperature dV. 374 00:27:59 --> 00:28:00 And now we know what that is. 375 00:28:00 --> 00:28:08 So it's nR integral from V1 to V2 dV over V. 376 00:28:08 --> 00:28:16 So it's nR log V2 over V1. 377 00:28:16 --> 00:28:18 There's our delta S. 378 00:28:18 --> 00:28:22 So we know how to calculate it. 379 00:28:22 --> 00:28:30 Make sense? 380 00:28:30 --> 00:28:35 Now, we can do the same procedure for the 381 00:28:35 --> 00:28:38 pressure change. 382 00:28:38 --> 00:28:41 And all we do is, I'll just outline this, I think. 383 00:28:41 --> 00:28:45 I won't write it all on the board. 384 00:28:45 --> 00:28:48 But, of course, it's going to come from the fact that these 385 00:28:48 --> 00:28:52 second derivatives are also equal. 386 00:28:52 --> 00:29:05 So d squared G dT dp is equal to d squared G dp dT. 387 00:29:05 --> 00:29:08 In other words, the order of taking the derivatives with 388 00:29:08 --> 00:29:13 respect to pressure and temperature doesn't matter. 389 00:29:13 --> 00:29:20 And what this will show is that dS/dp at constant temperature, 390 00:29:20 --> 00:29:24 here we saw how entropy varies with volume, this is going to 391 00:29:24 --> 00:29:27 show us how it varies with pressure. 392 00:29:27 --> 00:29:34 Is equal to minus dV/dT at constant pressure. 393 00:29:34 --> 00:29:38 And again, this is something that comes from an 394 00:29:38 --> 00:29:40 equation of state. 395 00:29:40 --> 00:29:44 We know how the volume and temperature vary with 396 00:29:44 --> 00:29:46 respect to each other at constant pressure. 397 00:29:46 --> 00:29:53 That's what the equation of state tells us. 398 00:29:53 --> 00:29:57 So, again, I can just use that result. 399 00:29:57 --> 00:30:03 So, if we do a process where one goes to two at constant 400 00:30:03 --> 00:30:11 temperature, and now the pressure, p1, goes to p2, well 401 00:30:11 --> 00:30:21 then delta S is just the integral from p1 to p2 of dS/dp 402 00:30:21 --> 00:30:26 times dS, so it's just this. 403 00:30:26 --> 00:30:30 And so of course it's still pV equals nRT. 404 00:30:30 --> 00:30:38 So now we just have nR over p dp. 405 00:30:38 --> 00:30:39 Right? 406 00:30:39 --> 00:30:41 So we're going to see the same story. 407 00:30:41 --> 00:30:49 It's nR log of p2 over p1 for the process where there's 408 00:30:49 --> 00:31:01 a pressure change. 409 00:31:01 --> 00:31:04 Any questions about this part? 410 00:31:04 --> 00:31:07 So what we've done is take one step further. 411 00:31:07 --> 00:31:15 We've used the fundamental equations that are hiding 412 00:31:15 --> 00:31:19 down here, out of sight but never out of mind. 413 00:31:19 --> 00:31:23 And what we've done is look at the derivatives of the new free 414 00:31:23 --> 00:31:28 energies that we've just recently introduced, A and G. 415 00:31:28 --> 00:31:33 And then, the only thing we've done beyond that is say, OK, 416 00:31:33 --> 00:31:35 well now let's just take the mixed second derivatives, they 417 00:31:35 --> 00:31:37 have to be equal to each other. 418 00:31:37 --> 00:31:42 And what's fallen out when we do that, because in each case, 419 00:31:42 --> 00:31:46 one of the first derivatives gives us the entropy. 420 00:31:46 --> 00:31:48 Then the second derivative gives the change in entropy 421 00:31:48 --> 00:31:51 with respect to the variable that we're differentiating, 422 00:31:51 --> 00:31:54 with respect to which is either pressure or volume. 423 00:31:54 --> 00:31:59 And the useful outcome of all that is that we get to see how 424 00:31:59 --> 00:32:03 entropy changes with one of those variables in terms of 425 00:32:03 --> 00:32:09 only V, T, and p, which come out of some equation of state. 426 00:32:09 --> 00:32:11 And all we did, further, is take that second derivative. 427 00:32:11 --> 00:32:13 That mixed second derivative. 428 00:32:13 --> 00:32:15 And, of course, see that either way we do that 429 00:32:15 --> 00:32:24 we'll have an equality. 430 00:32:24 --> 00:32:34 Now, let's go back to our older friends u and H. 431 00:32:34 --> 00:32:46 Which we've expressed now in terms of S and V, S and p. 432 00:32:46 --> 00:32:52 So, so far we don't have a way to just write off, relate them 433 00:32:52 --> 00:32:53 to equation of state data. 434 00:32:53 --> 00:32:57 Which also would be very useful. 435 00:32:57 --> 00:33:02 Here, A and G, we've already got as functions of these 436 00:33:02 --> 00:33:05 easily controlled, conveniently controlled state variables. 437 00:33:05 --> 00:33:13 Let's look at those quantities. u and H. 438 00:33:13 --> 00:33:17 And look at, for example, the V dependence of u. 439 00:33:17 --> 00:33:35 The volume dependence. 440 00:33:35 --> 00:33:44 And in particular let's look at, for example, du/dV 441 00:33:44 --> 00:33:46 at constant temperature. 442 00:33:46 --> 00:33:53 Now, we can immediately see what du/dV at 443 00:33:53 --> 00:33:57 constant entropy is. 444 00:33:57 --> 00:33:59 Experimentally, though, that's not such an easy 445 00:33:59 --> 00:34:02 situation to arrange. 446 00:34:02 --> 00:34:07 Of course, this is a much more practical one. 447 00:34:07 --> 00:34:10 But it doesn't just fall out immediately from the one 448 00:34:10 --> 00:34:14 fundamental equation for du. 449 00:34:14 --> 00:34:16 But we can start there. 450 00:34:16 --> 00:34:24 So, du is T dS minus p dV. 451 00:34:24 --> 00:34:34 And I can take this derivative. du/dV at constant T. 452 00:34:34 --> 00:34:35 And so, what is it? 453 00:34:35 --> 00:34:39 Well, it's not just p because there's some 454 00:34:39 --> 00:34:43 dS/dV at constant T. 455 00:34:43 --> 00:34:44 This isn't zero. 456 00:34:44 --> 00:34:50 There's some variation, dS/dV, at constant temperature. 457 00:34:50 --> 00:34:53 That's going to matter. 458 00:34:53 --> 00:35:00 This part, of course, is just minus p. 459 00:35:00 --> 00:35:15 But we just figured out what dS/dV at constant T is. 460 00:35:15 --> 00:35:19 This is dp/dT at constant V. 461 00:35:19 --> 00:35:20 So that's neat. 462 00:35:20 --> 00:35:24 So in other words, we can write this as T, dp/dT 463 00:35:24 --> 00:35:29 at constant V, minus p. 464 00:35:29 --> 00:35:35 Let's just check T, p, T, V, p. 465 00:35:35 --> 00:35:37 Right? 466 00:35:37 --> 00:35:38 In other words, we just have pressure, 467 00:35:38 --> 00:35:40 temperature and volume. 468 00:35:40 --> 00:35:42 Again, if we know the equation of state, we 469 00:35:42 --> 00:35:45 know all this stuff. 470 00:35:45 --> 00:35:47 So again, we can measure equation of state data. 471 00:35:47 --> 00:35:50 Or, if we know the equation of state from a model, ideal gas, 472 00:35:50 --> 00:35:56 van der Waal's gas, whatever, now we can determine u. 473 00:35:56 --> 00:35:58 From equation of state data. 474 00:35:58 --> 00:36:17 Terrific, right? 475 00:36:17 --> 00:36:22 So let's take our one model that we keep going back to. 476 00:36:22 --> 00:36:25 Equation of state, and just see how it works. 477 00:36:25 --> 00:36:28 That is, ideal gas. 478 00:36:28 --> 00:36:30 And see how it works with that. 479 00:36:30 --> 00:36:37 Now, we saw before, or really I should say we accepted before, 480 00:36:37 --> 00:36:40 that for an ideal gas, u was a function of temperature only. 481 00:36:40 --> 00:36:43 Well, now let's try it. 482 00:36:43 --> 00:36:50 So, dp/dT, for our ideal gas, at constant volume, 483 00:36:50 --> 00:36:54 remember pV is nRT. 484 00:36:54 --> 00:36:59 So this nR over V. 485 00:36:59 --> 00:37:01 And then, using the relation again, we can just 486 00:37:01 --> 00:37:05 write this as p over T. 487 00:37:05 --> 00:37:07 In other words, we're taking advantage of the fact that 488 00:37:07 --> 00:37:10 we now know that quantity. 489 00:37:10 --> 00:37:12 In the case of the ideal gas, we just have a 490 00:37:12 --> 00:37:13 simple model for it. 491 00:37:13 --> 00:37:14 More generally, we could measure it. 492 00:37:14 --> 00:37:16 We could just collect a bunch of data. 493 00:37:16 --> 00:37:17 For a material. 494 00:37:17 --> 00:37:21 What's the volume it occupies at some pressure 495 00:37:21 --> 00:37:23 and temperature? 496 00:37:23 --> 00:37:25 Now let's change the pressure and temperature and sweep 497 00:37:25 --> 00:37:27 through a whole range of pressures and temperatures and 498 00:37:27 --> 00:37:28 measure the volume in every one of them. 499 00:37:28 --> 00:37:31 Well, then, we could just use that for our equation of state. 500 00:37:31 --> 00:37:33 One way or another, we can determine this quantity. 501 00:37:33 --> 00:37:35 For the ideal gas it's this. 502 00:37:35 --> 00:37:45 So now our du/dV, at constant T is just T times dp/dT, which is 503 00:37:45 --> 00:37:52 just p over T minus p, it's zero. 504 00:37:52 --> 00:37:55 Remember the Joule expansion. 505 00:37:55 --> 00:37:58 And we saw that, you saw that the Joule coefficient for 506 00:37:58 --> 00:38:01 an ideal gas was zero. 507 00:38:01 --> 00:38:06 So that you could see that for the ideal gas, u would not be a 508 00:38:06 --> 00:38:08 function of volume, but only of temperature. 509 00:38:08 --> 00:38:13 But actually, when you saw that before, you weren't 510 00:38:13 --> 00:38:15 given any proof of that. 511 00:38:15 --> 00:38:19 It was just that when the good Mr. Joule made the 512 00:38:19 --> 00:38:23 measurements, to the precision that he could measure, he 513 00:38:23 --> 00:38:26 discovered that for some gases it was extremely small. 514 00:38:26 --> 00:38:29 At least, smaller than anything he could detect. 515 00:38:29 --> 00:38:32 So it sure seemed like it was going to zero, under 516 00:38:32 --> 00:38:33 ideal gas conditions. 517 00:38:33 --> 00:38:36 And that was the result that we came to accept. 518 00:38:36 --> 00:38:38 Here, though, you can just derive straight away. 519 00:38:38 --> 00:38:41 That for an ideal gas it has to be the case that there's no 520 00:38:41 --> 00:38:43 volume dependence of the energy. 521 00:38:43 --> 00:38:53 Only a temperature dependence. 522 00:38:53 --> 00:39:21 It's the same for H. 523 00:39:21 --> 00:39:25 Just like u, we'd like to be able to express it in a way 524 00:39:25 --> 00:39:28 that allows us to calculate what happens only from 525 00:39:28 --> 00:39:30 equation of state data. 526 00:39:30 --> 00:39:36 But, again, our fundamental equations show us how it 527 00:39:36 --> 00:39:45 changes as a function of entropy and pressure. 528 00:39:45 --> 00:39:53 So, dH is T dS plus V dp. 529 00:39:53 --> 00:39:55 So let's look at dH/dp. 530 00:39:55 --> 00:39:58 531 00:39:58 --> 00:40:00 We know how to get it immediately if we keep 532 00:40:00 --> 00:40:01 entropy constant. 533 00:40:01 --> 00:40:04 But we'd like to relate it to what happens if we keep 534 00:40:04 --> 00:40:06 the temperature constant. 535 00:40:06 --> 00:40:09 So then, just like we saw, analogous to what saw just 536 00:40:09 --> 00:40:16 before, it's T dS/dp at constant T. 537 00:40:16 --> 00:40:18 Plus V. 538 00:40:18 --> 00:40:24 But now we've seen from the Maxwell relations that dS/dp is 539 00:40:24 --> 00:40:26 minus dV/dT, for constant p. 540 00:40:26 --> 00:40:29 Again, this is this quantity, one of these quantities that 541 00:40:29 --> 00:40:31 again we can determine from equation of state data. 542 00:40:31 --> 00:40:35 Only V, p and T appear. 543 00:40:35 --> 00:40:44 So it's minus T dV/dT at constant p, plus V. 544 00:40:44 --> 00:41:03 And so, again, this can come from equation of state data. 545 00:41:03 --> 00:41:09 And if you do this again for an ideal gas, let me see. 546 00:41:09 --> 00:41:11 So we have pV is nRT. 547 00:41:11 --> 00:41:24 So dV/dT at constant pressure is just nR over p. 548 00:41:24 --> 00:41:26 But we can plug that in again just like we did before. 549 00:41:26 --> 00:41:31 It's just equal to V over T. 550 00:41:31 --> 00:41:37 And so dH/dp under our condition of constant 551 00:41:37 --> 00:41:46 temperature is just minus T times V over T plus V, 552 00:41:46 --> 00:41:50 everything cancels, and that's zero. 553 00:41:50 --> 00:41:54 That's our Joule - Thompson expansion. 554 00:41:54 --> 00:41:56 That was a constant enthalpy change. 555 00:41:56 --> 00:42:01 And again there, too, you saw an experimental result you were 556 00:42:01 --> 00:42:04 presented with that says, well at least to the extent that it 557 00:42:04 --> 00:42:06 could be measured, it was obviously getting very small. 558 00:42:06 --> 00:42:11 For gases that approach ideal gas conditions. 559 00:42:11 --> 00:42:13 Well, there you can see it. 560 00:42:13 --> 00:42:16 Sure better have gotten small because in fact 561 00:42:16 --> 00:42:20 it has to be zero. 562 00:42:20 --> 00:42:30 Now let's take just one somewhat more complicated case. 563 00:42:30 --> 00:42:37 Let's look at a van der Waal's gas. 564 00:42:37 --> 00:42:40 Let's try it with a different equation of state, that 565 00:42:40 --> 00:42:47 isn't quite as simple as the ideal gas case. 566 00:42:47 --> 00:42:58 So, then p plus a over molar volume squared times V minus 567 00:42:58 --> 00:43:07 b molar volume V minus b is equal to RT, remember? 568 00:43:07 --> 00:43:14 This was back from the first or second lecture in the course. 569 00:43:14 --> 00:43:18 So, we can separate out p. 570 00:43:18 --> 00:43:25 It's RT over molar volume minus b minus a over 571 00:43:25 --> 00:43:31 molar volume V squared. 572 00:43:31 --> 00:43:33 And then we can take the derivative with respect to 573 00:43:33 --> 00:43:38 temperature, it's just R over molar volume minus b. 574 00:43:38 --> 00:43:53 So it's dp/dT at constant V is just R over V bar minus b. 575 00:43:53 --> 00:44:04 Well, let's now look, given this, let's now look in that 576 00:44:04 --> 00:44:08 case, at what happens to u as a function of V. 577 00:44:08 --> 00:44:12 For the ideal gas, we know that u is volume independent. 578 00:44:12 --> 00:44:16 It only depends on the temperature. 579 00:44:16 --> 00:44:22 But for the van der Waal's gas, now it's going to be different. 580 00:44:22 --> 00:44:27 And that's because this is different from what it is 581 00:44:27 --> 00:44:28 in the ideal gas case. 582 00:44:28 --> 00:44:49 Namely, now du/dV at constant T, for the van der Waal's gas. 583 00:44:49 --> 00:44:57 So it's this. 584 00:44:57 --> 00:45:04 So it's RT over molar volume minus b. 585 00:45:04 --> 00:45:09 Minus p, right? 586 00:45:09 --> 00:45:13 But in fact, if you go back to the van der Waal's equation of 587 00:45:13 --> 00:45:17 state, here's RT over v minus b. 588 00:45:17 --> 00:45:22 If we put it as minus b, that's just equal to a over V squared. 589 00:45:22 --> 00:45:28 Equals a over molar volume squared. 590 00:45:28 --> 00:45:33 But the point is, the main point is, it's not zero. 591 00:45:33 --> 00:45:37 It's some number. a over the molar volume squared. a is a 592 00:45:37 --> 00:45:40 positive number in the van der Waal's equation of state. 593 00:45:40 --> 00:45:44 So this is greater than zero. 594 00:45:44 --> 00:45:52 In other words, u is a function of T and V. 595 00:45:52 --> 00:45:59 If we don't have an ideal gas. 596 00:45:59 --> 00:46:01 By the way, just to think about it a little bit, 597 00:46:01 --> 00:46:02 it's a positive number. 598 00:46:02 --> 00:46:08 What that means is, I've got my ideal gas in some container. 599 00:46:08 --> 00:46:12 There's some energy, some internal energy. 600 00:46:12 --> 00:46:13 Now I make the volume bigger. 601 00:46:13 --> 00:46:17 I allow it to expand. 602 00:46:17 --> 00:46:22 And the energy changes, it goes up. 603 00:46:22 --> 00:46:25 In some sense it's less favorable energetically. 604 00:46:25 --> 00:46:31 What's happening there, that a term in the van der Waal's 605 00:46:31 --> 00:46:35 equation of state, is describing interactions, 606 00:46:35 --> 00:46:38 favorable attractions, between gas molecules. 607 00:46:38 --> 00:46:42 The b is describing repulsions. 608 00:46:42 --> 00:46:44 Effectively, the volume changes. 609 00:46:44 --> 00:46:47 The molar volume is being changed a little bit by the 610 00:46:47 --> 00:46:50 fact that if you're really trying to make things collide 611 00:46:50 --> 00:46:54 with each other, they can't occupy the same volume. 612 00:46:54 --> 00:46:56 And that is being expressed here. 613 00:46:56 --> 00:47:02 The a, though is expressing attraction between molecules 614 00:47:02 --> 00:47:04 at somewhat longer range. 615 00:47:04 --> 00:47:07 So now, if you make the volume bigger, those 616 00:47:07 --> 00:47:08 attractions die out. 617 00:47:08 --> 00:47:11 Because the molecules are farther apart from each other. 618 00:47:11 --> 00:47:13 So the energy goes up. 619 00:47:13 --> 00:47:17 Now, you might ask, well why does it do that, right? 620 00:47:17 --> 00:47:22 I mean, if the energy is lower to occupy a smaller volume, 621 00:47:22 --> 00:47:26 then if I have this room and a bunch of molecules of oxygen, 622 00:47:26 --> 00:47:29 and nitrogen and what have you in the air, and there are weak 623 00:47:29 --> 00:47:31 attractions between them, why don't they all just sort 624 00:47:31 --> 00:47:34 of glum together and find whatever volume they like. 625 00:47:34 --> 00:47:37 So that the attractive forces can exert 626 00:47:37 --> 00:47:38 themselves a little bit. 627 00:47:38 --> 00:47:39 Not too close, right? 628 00:47:39 --> 00:47:42 Not so close that the repulsions dominate. 629 00:47:42 --> 00:47:44 Why don't they do that? 630 00:47:44 --> 00:47:54 What else matters besides any of those considerations? 631 00:47:54 --> 00:47:57 What else matters that I haven't considered in 632 00:47:57 --> 00:48:02 this little discussion? 633 00:48:02 --> 00:48:04 Yeah. 634 00:48:04 --> 00:48:06 What about entropy, right? 635 00:48:06 --> 00:48:08 If I only worry about minimizing the 636 00:48:08 --> 00:48:10 energy, it's true. 637 00:48:10 --> 00:48:13 They'll stick together a little bit. 638 00:48:13 --> 00:48:15 But entropy also matters. 639 00:48:15 --> 00:48:18 And there's disorder achieved by occupying the full 640 00:48:18 --> 00:48:19 available volume. 641 00:48:19 --> 00:48:21 Many more states possible. 642 00:48:21 --> 00:48:24 And that will end up winning out at basically any realistic 643 00:48:24 --> 00:48:28 temperature where the stuff really is a gas. 644 00:48:28 --> 00:48:29 OK. 645 00:48:29 --> 00:48:32 Next time, what you're going to see is the following. 646 00:48:32 --> 00:48:37 It turns out we can express all these functions in terms of G, 647 00:48:37 --> 00:48:40 we wouldn't need to choose G, but it's a very useful 648 00:48:40 --> 00:48:40 function to choose. 649 00:48:40 --> 00:48:46 Because of its natural expression in terms of T and p. 650 00:48:46 --> 00:48:49 So we can write any of these functions in terms of G. 651 00:48:49 --> 00:48:51 Which means that we can really calculate all 652 00:48:51 --> 00:48:55 the thermodynamics in terms of only g. 653 00:48:55 --> 00:48:59 It's not necessary to do that, but it can be quite convenient. 654 00:48:59 --> 00:49:03 And then we can say, OK, if we have many constituents, what if 655 00:49:03 --> 00:49:04 we have a mixture of stuff? 656 00:49:04 --> 00:49:08 We can take the derivative of G with respect to how 657 00:49:08 --> 00:49:10 much material there is. 658 00:49:10 --> 00:49:13 With respect to n, the number of moles. 659 00:49:13 --> 00:49:15 And if there are one, and two, and three constituents with 660 00:49:15 --> 00:49:18 respect to n1, and n2, and n3. 661 00:49:18 --> 00:49:21 Each individual amount of stuff. 662 00:49:21 --> 00:49:23 What that's going to allow us to do is, if we say, OK I have 663 00:49:23 --> 00:49:26 a mixture of stuff, how does the free energy change? 664 00:49:26 --> 00:49:29 If I change the composition of the mixture? 665 00:49:29 --> 00:49:32 If I take something away, or put something else in? 666 00:49:32 --> 00:49:33 And we'll be able to determine equilibrium 667 00:49:33 --> 00:49:35 under those conditions. 668 00:49:35 --> 00:49:38 That's very useful for things like chemical reactions, 669 00:49:38 --> 00:49:41 where this constituent changes to this one. 670 00:49:41 --> 00:49:43 And I can calculate what happens to G under 671 00:49:43 --> 00:49:44 those conditions. 672 00:49:44 --> 00:49:46 And that's what you'll see starting next time. 673 00:49:46 --> 00:49:48 And professor Blendi will be taking over for 674 00:49:48 --> 00:49:49 that set of lectures. 675 00:49:49 --> 00:49:50