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 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:25 PROFESSOR NELSON: All right, well last time we finished our 10 00:00:25 --> 00:00:29 sort of introduction to entropy which is a difficult topic, but 11 00:00:29 --> 00:00:32 I hope we got to the point where we have some working 12 00:00:32 --> 00:00:35 understanding of physically what it's representing for us, 13 00:00:35 --> 00:00:37 and also an ability to just calculate it. 14 00:00:37 --> 00:00:39 So we went through at the end of the last lecture, a few 15 00:00:39 --> 00:00:42 examples where we just calculated changes in entropy 16 00:00:42 --> 00:00:45 for simple processes like heating and cooling something 17 00:00:45 --> 00:00:49 or going through a phase transition where that process, 18 00:00:49 --> 00:00:51 that sort of process is relatively simple because 19 00:00:51 --> 00:00:53 there's no temperature change. 20 00:00:53 --> 00:00:56 While the ice is melting, for example, you're putting heat 21 00:00:56 --> 00:01:01 into it, but the temperature is staying at zero degree Celsius. 22 00:01:01 --> 00:01:04 And we saw, how to do these calculations, we need 23 00:01:04 --> 00:01:05 define reversible paths. 24 00:01:05 --> 00:01:09 So it was extremely straightforward to calculate 25 00:01:09 --> 00:01:12 the entropy of say ice melting at zero degrees Celsius, 26 00:01:12 --> 00:01:16 because there the process is reversible, because that's the 27 00:01:16 --> 00:01:18 melting temperature. 28 00:01:18 --> 00:01:22 But if we wanted to calculate the change in entropy of ice 29 00:01:22 --> 00:01:27 melting, you know at, once it had already been cooled to ten 30 00:01:27 --> 00:01:31 degrees above the melting point, to ten degrees Celsius, 31 00:01:31 --> 00:01:33 then in order to find a reversible path, we had to say, 32 00:01:33 --> 00:01:37 OK, let's first cool it down to zero degrees Celsius. 33 00:01:37 --> 00:01:40 Then let's have it melt, and then let's warm the liquid back 34 00:01:40 --> 00:01:43 up to ten degrees Celsius so we could construct the sequence of 35 00:01:43 --> 00:01:46 reversible steps that would get from the same starting point to 36 00:01:46 --> 00:01:49 the same end point, and we could calculate the change in 37 00:01:49 --> 00:01:52 entropy through that sort of sequence. 38 00:01:52 --> 00:01:56 So now that we've got at least some experience doing 39 00:01:56 --> 00:01:59 calculations of delta S and we're just thinking a little 40 00:01:59 --> 00:02:02 bit about entropy, what I'd like to do is to try to 41 00:02:02 --> 00:02:06 relate the state variables together in a useful way. 42 00:02:06 --> 00:02:10 And and the immediate problem that I'd like to address is 43 00:02:10 --> 00:02:14 the fact that right now we have kind of a cumbersome 44 00:02:14 --> 00:02:18 expression for energy. 45 00:02:18 --> 00:02:29 So you know we have u, we look at du, right it's dq plus 46 00:02:29 --> 00:02:34 dw, and you know, I don't like those. 47 00:02:34 --> 00:02:38 They're path specific, and it would be nice to be able to do 48 00:02:38 --> 00:02:43 a calculation of changes in energy that didn't 49 00:02:43 --> 00:02:44 depend on path. 50 00:02:44 --> 00:02:47 You know it's a state function. 51 00:02:47 --> 00:02:51 So in principle it seems like it sure ought to be possible, 52 00:02:51 --> 00:02:54 and yet so far when we've actually gone through 53 00:02:54 --> 00:02:57 calculations of du, we've had to go and consider the path 54 00:02:57 --> 00:03:01 and get the heat and get the work and so forth. 55 00:03:01 --> 00:03:04 And then we found special cases, you know an ideal gas 56 00:03:04 --> 00:03:07 where the temperature, where the change is only a function 57 00:03:07 --> 00:03:09 of temperature and so forth, where we could write this as a 58 00:03:09 --> 00:03:12 function of state variables, but nothing general that really 59 00:03:12 --> 00:03:17 allows us to do the calculation under all circumstances. 60 00:03:17 --> 00:03:21 So let's think about how to make this better. 61 00:03:21 --> 00:03:24 So, and what I mean by that is you've seen examples like, you 62 00:03:24 --> 00:03:32 know, some special examples you saw awhile back. 63 00:03:32 --> 00:03:39 The case where du was Cv dT minus Cv and this 64 00:03:39 --> 00:03:42 Joule coefficient d v. 65 00:03:42 --> 00:03:44 But you know you still need to find those coefficients 66 00:03:44 --> 00:03:46 for each system. 67 00:03:46 --> 00:03:50 This isn't a general equation that tells us how energy 68 00:03:50 --> 00:03:54 changes in terms of only functions of state. 69 00:03:54 --> 00:03:58 Because of things like this and this -- what I'd really like 70 00:03:58 --> 00:04:11 is to be able to you know write du equals something. 71 00:04:11 --> 00:04:16 And that something, you know, it can have T and p and 72 00:04:16 --> 00:04:17 whatever else I need. 73 00:04:17 --> 00:04:23 It can have S, H, and of course differentials of any of those 74 00:04:23 --> 00:04:30 quantities. dT or dp or dS, dH, you name it, but all 75 00:04:30 --> 00:04:32 state variables. 76 00:04:32 --> 00:04:35 That's what I'd much rather have. 77 00:04:35 --> 00:04:39 And then, you know, if I want to, if I've got something like 78 00:04:39 --> 00:04:42 that and I want to find out how the energy changes as a 79 00:04:42 --> 00:04:46 function of volume, so I'll calculate du/dV With respect to 80 00:04:46 --> 00:04:51 some selected variable hold constant, I can do it. 81 00:04:51 --> 00:04:55 Right now, in a general sense, that's cumbersome. 82 00:04:55 --> 00:04:58 I've got to figure out how to do that for each particular 83 00:04:58 --> 00:05:01 case that I want to treat. 84 00:05:01 --> 00:05:13 So, let's see how we could construct such a thing. 85 00:05:13 --> 00:05:16 Let's consider just a reversible process, at 86 00:05:16 --> 00:05:32 constant pressure. 87 00:05:32 --> 00:05:43 So, OK, I've got one and I'm going to in some path wind up 88 00:05:43 --> 00:05:50 at state two and I'll write du is dq, it's reversible in 89 00:05:50 --> 00:05:54 this case, minus p dV. 90 00:05:54 --> 00:05:58 91 00:05:58 --> 00:06:06 And from the second law, we know that we can write 92 00:06:06 --> 00:06:12 dq reversible as T dS. 93 00:06:12 --> 00:06:18 dq over T is dS or entropy. 94 00:06:18 --> 00:06:28 So, we can write du is T dS minus p dV. 95 00:06:28 --> 00:06:35 96 00:06:35 --> 00:06:41 That's so important, we'll circle it with colored chalk. 97 00:06:41 --> 00:06:43 That's how important it is. 98 00:06:43 --> 00:06:48 You know it's a dramatic moment. 99 00:06:48 --> 00:06:49 So let's look at what we have here. 100 00:06:49 --> 00:06:55 Here's du and over on this side we have T, we have S, 101 00:06:55 --> 00:06:58 we have p and we have V. 102 00:06:58 --> 00:07:02 Suddenly and simply, it's only functions of state. 103 00:07:02 --> 00:07:05 Well that was pretty easy. 104 00:07:05 --> 00:07:12 So, what that's telling us is that we can write u this way, 105 00:07:12 --> 00:07:15 and you know, this is generally true. 106 00:07:15 --> 00:07:19 We got to this by considering a reversible constant 107 00:07:19 --> 00:07:21 pressure process. 108 00:07:21 --> 00:07:26 But we know u is a state variable right. 109 00:07:26 --> 00:07:29 So this result is going to be generally applicable. 110 00:07:29 --> 00:07:31 And it tells us a couple of things too. 111 00:07:31 --> 00:07:38 It tells us that in some sense, the natural variables for u are 112 00:07:38 --> 00:07:50 these, right, it's a function of S and V. 113 00:07:50 --> 00:07:53 Those are natural variables in the sense that then it written 114 00:07:53 --> 00:08:01 as functions of those variables, we only have state 115 00:08:01 --> 00:08:06 quantities on the right-hand side. 116 00:08:06 --> 00:08:09 Very, very valuable expression. 117 00:08:09 --> 00:08:12 And of course coming out of that then, we can take 118 00:08:12 --> 00:08:17 derivatives and at least for those particular variables, 119 00:08:17 --> 00:08:26 we can see that du/dS at constant V is minus p. 120 00:08:26 --> 00:08:41 And du/dV at constant S is T. 121 00:08:41 --> 00:08:43 All right, those fall right out. 122 00:08:43 --> 00:08:49 Now we can have a similar set of steps for H, for 123 00:08:49 --> 00:09:10 the enthalpy, so let's just look at that. 124 00:09:10 --> 00:09:21 So H, of course, it's u plus pV, so dH is just du plus 125 00:09:21 --> 00:09:26 d(pV), and now there's our expression for du. 126 00:09:26 --> 00:09:29 We're going to use it this way. 127 00:09:29 --> 00:09:39 So it's T dS minus p dV plus p dV plus V dp. 128 00:09:39 --> 00:09:43 And of course these are going to cancel. 129 00:09:43 --> 00:09:51 So we can write dH is T dS plus V dp. 130 00:09:51 --> 00:09:55 131 00:09:55 --> 00:09:58 Also important enough that we'll highlight 132 00:09:58 --> 00:10:00 it a little bit. 133 00:10:00 --> 00:10:03 So again let's look at what we've got on the right-hand 134 00:10:03 --> 00:10:08 side here, T, S, V, p, only quantities that are 135 00:10:08 --> 00:10:13 functions of state. 136 00:10:13 --> 00:10:17 And of course, we can take the corresponding derivative, so 137 00:10:17 --> 00:10:21 let's also be explicit here, that means that were writing H 138 00:10:21 --> 00:10:28 as a function of the variables S and p. 139 00:10:28 --> 00:10:53 And dH/dS at constant p is T, and dH/dp at constant S is V. 140 00:10:53 --> 00:10:58 So now we've got a couple of really surprisingly simple 141 00:10:58 --> 00:11:03 expressions that we can use to describe u and H in terms 142 00:11:03 --> 00:11:05 of only state variables. 143 00:11:05 --> 00:11:10 All right? 144 00:11:10 --> 00:11:14 We also can go from these expressions, using the chain 145 00:11:14 --> 00:11:18 rule, to expressions for particularly useful expressions 146 00:11:18 --> 00:11:23 for the entropy as a function of temperature. 147 00:11:23 --> 00:11:32 So, you know, from du is T dS minus p dV. 148 00:11:32 --> 00:11:44 We can rewrite this as dS is one over T du plus p over T dV. 149 00:11:44 --> 00:11:48 150 00:11:48 --> 00:11:52 And now we can go back, you know, if we can go back to our 151 00:11:52 --> 00:11:58 writing of u in terms of, as a function of T and V, right. 152 00:11:58 --> 00:12:09 So we can write here du as a function of T and V, Cv dT 153 00:12:09 --> 00:12:17 plus du/dV at constant T dV. 154 00:12:17 --> 00:12:22 The reason we're doing that is now we can rewrite dS is one 155 00:12:22 --> 00:12:28 over T times Cv dT, and that's the only temperature dependence 156 00:12:28 --> 00:12:29 we're going to have. 157 00:12:29 --> 00:12:33 The other part is going to be a function of volume. 158 00:12:33 --> 00:12:45 So it's, we've got p over T plus du/dV at constant T dV, 159 00:12:45 --> 00:12:53 and what this says then is that dS/dT at constant 160 00:12:53 --> 00:13:01 V is just Cv over T. 161 00:13:01 --> 00:13:04 Very useful, not surprising because of the relation 162 00:13:04 --> 00:13:08 between heat at constant volume and Cv, right. 163 00:13:08 --> 00:13:15 And of course dS is just dq reversible over T, but this is 164 00:13:15 --> 00:13:19 telling us, in general, how the entropy changes with 165 00:13:19 --> 00:13:23 temperature at constant volume. 166 00:13:23 --> 00:13:28 We can go through the exact same procedure with the H to 167 00:13:28 --> 00:13:32 look at how entropy varies with temperature at constant 168 00:13:32 --> 00:13:36 pressure, and we'll get exactly an analogous set of steps that 169 00:13:36 --> 00:13:39 will be Cp over T, right. 170 00:13:39 --> 00:13:51 So also dS/dT at constant pressure is Cp over T. 171 00:13:51 --> 00:14:00 OK? 172 00:14:00 --> 00:14:05 Now I want to carry our discussion a little bit further 173 00:14:05 --> 00:14:10 and look at entropy a little more carefully and in 174 00:14:10 --> 00:14:15 particular, how it varies with temperature. 175 00:14:15 --> 00:14:18 And here's what I really want to look at. 176 00:14:18 --> 00:14:21 You know, we've talked about when we look at changes in u 177 00:14:21 --> 00:14:24 and changes in H, and we've done this under lots of 178 00:14:24 --> 00:14:26 circumstances at this point. 179 00:14:26 --> 00:14:29 And at various times I've emphasized, and I'm sure 180 00:14:29 --> 00:14:33 Professor Bawendi did too, that when we look at these 181 00:14:33 --> 00:14:36 quantities we can only define changes in them. 182 00:14:36 --> 00:14:41 There's not an absolute scales for energy or for enthalpy. 183 00:14:41 --> 00:14:46 We can set the zero in a particular problem, 184 00:14:46 --> 00:14:49 arbitrarily. 185 00:14:49 --> 00:14:55 And so, for example, when we talked about thermochemistry, 186 00:14:55 --> 00:15:00 we defined heats of formation, and then we said, well, the 187 00:15:00 --> 00:15:05 heat of formation of an element in its natural state at room 188 00:15:05 --> 00:15:08 temperature and pressure we'll call zero. 189 00:15:08 --> 00:15:09 We called it zero. 190 00:15:09 --> 00:15:14 If we wanted to put some number on it, and put energy in it, 191 00:15:14 --> 00:15:15 we could have done that. 192 00:15:15 --> 00:15:18 We defined the zero. 193 00:15:18 --> 00:15:22 So far, that's, well not just so far, that's always the way 194 00:15:22 --> 00:15:25 it will be for quantities like energy and enthalpy. 195 00:15:25 --> 00:15:30 Entropy is different. 196 00:15:30 --> 00:15:33 So let's just see how that works. 197 00:15:33 --> 00:15:52 So, let's consider the entropy, we'll consider as a function 198 00:15:52 --> 00:16:00 of temperature and pressure. 199 00:16:00 --> 00:16:03 First let's just see how it varies with pressure. 200 00:16:03 --> 00:16:05 We're going to see -- what we'll do is consider its 201 00:16:05 --> 00:16:08 variation to both pressure and temperature, and the objective 202 00:16:08 --> 00:16:10 is to say all right, if I've got some substance at any 203 00:16:10 --> 00:16:15 arbitrary temperature and pressure, can I define and 204 00:16:15 --> 00:16:19 calculate an absolute number for the entropy? 205 00:16:19 --> 00:16:22 Not just a change in entropy, unlike the cases with delta u 206 00:16:22 --> 00:16:26 and delta H, but an absolute number that says in absolute 207 00:16:26 --> 00:16:29 terms the entropy of this substance at room temperature 208 00:16:29 --> 00:16:31 and pressure or whatever temperature and pressure 209 00:16:31 --> 00:16:33 is this amount. 210 00:16:33 --> 00:16:39 Something that I can do by choice of a zero for energy or 211 00:16:39 --> 00:16:43 for u or H, but here, I want to look for an absolute answer. 212 00:16:43 --> 00:16:46 All right, so let's start by looking at the 213 00:16:46 --> 00:16:48 pressure dependence. 214 00:16:48 --> 00:16:58 So we're going to start with du is T dS minus p dV, so 215 00:16:58 --> 00:17:07 dS is du plus p dV over T. 216 00:17:07 --> 00:17:14 Now let's look, T being constant. 217 00:17:14 --> 00:17:16 OK, and now let's specify a little bit. 218 00:17:16 --> 00:17:20 I want to make it something as tractable as possible. 219 00:17:20 --> 00:17:27 Let's go to an ideal gas. 220 00:17:27 --> 00:17:31 So then at constant temperature, that says 221 00:17:31 --> 00:17:34 du is equal to zero. 222 00:17:34 --> 00:17:41 So dS at constant temperature is just p over T dV. 223 00:17:41 --> 00:17:45 224 00:17:45 --> 00:17:57 And in the case of an ideal gas, that's nR dV over V. 225 00:17:57 --> 00:18:08 And at constant temperature, that means that d(nRT), which 226 00:18:08 --> 00:18:19 is the same as d(pV) is equal to zero, but this 227 00:18:19 --> 00:18:23 is p dV plus V dp. 228 00:18:23 --> 00:18:27 229 00:18:27 --> 00:18:33 So this says that dV over V, that I've got there, 230 00:18:33 --> 00:18:39 is the same thing as negative dp over p. 231 00:18:39 --> 00:18:58 Right, so I can write that dS as constant temperature 232 00:18:58 --> 00:19:06 is minus nR dp over p. 233 00:19:06 --> 00:19:07 So that's great. 234 00:19:07 --> 00:19:10 That says now if I know the entropy at some particular 235 00:19:10 --> 00:19:13 pressure, I can calculate how it changes as a function 236 00:19:13 --> 00:19:20 of pressure, right. 237 00:19:20 --> 00:19:33 If I know S at some standard pressure that we can define, 238 00:19:33 --> 00:19:43 then S at some arbitrary pressure, is just S of p naught 239 00:19:43 --> 00:19:53 and T minus the integral from p naught to p of nR dp over p. 240 00:19:53 --> 00:20:00 All right, which is to say it's S of p naught T minus 241 00:20:00 --> 00:20:06 nR log of p over p naught. 242 00:20:06 --> 00:20:18 Right, now normally we'll define p naught as 243 00:20:18 --> 00:20:23 equal to one bar. 244 00:20:23 --> 00:20:28 And often you'll see this simply written as nR log p. 245 00:20:28 --> 00:20:31 I don't particularly like to do that because of course, then, 246 00:20:31 --> 00:20:34 formally speaking were looking at something that's written 247 00:20:34 --> 00:20:36 that has units inside as the argument of a log. 248 00:20:36 --> 00:20:40 Of course it's understood when you see that, and you're likely 249 00:20:40 --> 00:20:43 to see it in various places, it's understood when you see 250 00:20:43 --> 00:20:47 that the quantity p is always supposed to be divided by one 251 00:20:47 --> 00:20:53 bar, and the units then are taken care of. 252 00:20:53 --> 00:21:01 For one mole, we can write the molar quantities S of p and T, 253 00:21:01 --> 00:21:16 is S, S naught of T minus R log p over p naught. 254 00:21:16 --> 00:21:24 All right, so that's our pressure dependence. 255 00:21:24 --> 00:21:25 What about that? 256 00:21:25 --> 00:21:29 We still don't really have a formulation for calculating 257 00:21:29 --> 00:21:32 this, or you know, defining it or whatever we're going to 258 00:21:32 --> 00:21:57 do to allow us to know it. 259 00:21:57 --> 00:22:02 Well, let's just consider the entropy as a function of 260 00:22:02 --> 00:22:06 temperature, starting all the way down at zero degrees 261 00:22:06 --> 00:22:09 Kelvin, and going up to whatever temperature 262 00:22:09 --> 00:22:23 we want to consider. 263 00:22:23 --> 00:22:28 Now, we certainly do know how to calculate delta S for all 264 00:22:28 --> 00:22:30 that because we've seen how to calculate delta S if you just 265 00:22:30 --> 00:22:34 heat something up, and we've seen how to calculate delta S 266 00:22:34 --> 00:22:37 when something under goes a phase transition, right. 267 00:22:37 --> 00:22:39 Presumably, if we're starting at zero Kelvin, we're 268 00:22:39 --> 00:22:42 starting in a solid state. 269 00:22:42 --> 00:22:45 As we heat it up, depending on the material, it may 270 00:22:45 --> 00:22:47 melt at some temperature. 271 00:22:47 --> 00:22:50 If we keep keep heating it up, it'll boil at some temperature, 272 00:22:50 --> 00:22:54 but we know how to treat all of that, right. 273 00:22:54 --> 00:22:56 So let's just consider something that undergoes 274 00:22:56 --> 00:22:58 that set of changes. 275 00:22:58 --> 00:23:05 So, we've got some substance A, solid, zero degrees 276 00:23:05 --> 00:23:13 Kelvin, one bar. 277 00:23:13 --> 00:23:16 Here's process one. 278 00:23:16 --> 00:23:20 It goes to A, it's a solid at the melting 279 00:23:20 --> 00:23:29 temperature and one bar. 280 00:23:29 --> 00:23:34 Process two is it turns into a liquid at the melting 281 00:23:34 --> 00:23:41 temperature and one bar. 282 00:23:41 --> 00:23:50 Process three is we heat it up some more, up to the boiling 283 00:23:50 --> 00:23:56 temperature at one bar. 284 00:23:56 --> 00:24:03 Process four is it evaporates, so now it's a gas at the 285 00:24:03 --> 00:24:07 boiling temperature and one bar. 286 00:24:07 --> 00:24:13 Finally, we heat it up some more, so now it's a gas at 287 00:24:13 --> 00:24:19 temperature T and one bar. 288 00:24:19 --> 00:24:21 And if we wanted to, we can go further. 289 00:24:21 --> 00:24:26 We can make it a gas at temperature and whatever 290 00:24:26 --> 00:24:26 pressure we want. 291 00:24:26 --> 00:24:36 That part we already know how to take care of, right. 292 00:24:36 --> 00:24:41 Well, let's look at what happens to S, all right. 293 00:24:41 --> 00:24:46 S, a molar enthalpy at T and p, where we're going to finally 294 00:24:46 --> 00:24:56 end up, is, well it's s zero at zero Kelvin and one bar or one 295 00:24:56 --> 00:25:02 bar is implied by the superscripts here. 296 00:25:02 --> 00:25:06 And then we have delta S for step one, and delta S 297 00:25:06 --> 00:25:09 for step two and so on. 298 00:25:09 --> 00:25:13 So all right, let's, we can label this six so we to all the 299 00:25:13 --> 00:25:37 way to delta S for step six. 300 00:25:37 --> 00:25:40 Well, so we can do that. 301 00:25:40 --> 00:25:52 It's S of the material at T and p is S naught at zero Kelvin, 302 00:25:52 --> 00:25:55 plus, here's for process one. 303 00:25:55 --> 00:26:01 We heat it up from zero Kelvin up to the melting point. 304 00:26:01 --> 00:26:08 Cp of the solid more heat capacity, divided by T dT. 305 00:26:08 --> 00:26:15 We can calculate delta S for heating something up, right. 306 00:26:15 --> 00:26:21 Plus, now we've got the heat of fusion to melt the stuff, so 307 00:26:21 --> 00:26:28 it's just delta H naught of fusion, divided by Tm right. 308 00:26:28 --> 00:26:29 We saw that last time. 309 00:26:29 --> 00:26:33 In other words, remember, we're just looking at q reversible 310 00:26:33 --> 00:26:38 over T to get delta S, and it's just given by the 311 00:26:38 --> 00:26:40 heat of fusion. 312 00:26:40 --> 00:26:44 All right, then let's go from the melting point 313 00:26:44 --> 00:26:45 to the boiling point. 314 00:26:45 --> 00:26:50 So it's Cp now it's the heat capacity, the molar heat 315 00:26:50 --> 00:26:54 capacity of the liquid, divided by T dT. 316 00:26:54 --> 00:26:59 We're heating up the liquid. 317 00:26:59 --> 00:27:01 And then there's vaporization. 318 00:27:01 --> 00:27:11 Delta H of vaporization over T at the boiling point. 319 00:27:11 --> 00:27:13 Then we can go from the boiling point to our 320 00:27:13 --> 00:27:16 final temperature T. 321 00:27:16 --> 00:27:23 Now it's the molar heat capacity of the gas over T dT 322 00:27:23 --> 00:27:29 minus R log p over p naught. 323 00:27:29 --> 00:27:33 OK, so that's everything, and these are all things 324 00:27:33 --> 00:27:37 that we know how to do. 325 00:27:37 --> 00:27:38 Just about. 326 00:27:38 --> 00:27:42 OK, this one we're going to have to think about, but all 327 00:27:42 --> 00:27:44 the changes we know how to calculate, right. 328 00:27:44 --> 00:27:56 So if we plot this, S, and let's just do this as a 329 00:27:56 --> 00:27:58 function of temperature. 330 00:27:58 --> 00:28:02 I don't have pressure in here explicitly. 331 00:28:02 --> 00:28:08 Well, it's going to change as I warm up the solid, soon we're 332 00:28:08 --> 00:28:14 really starting at zero Kelvin. 333 00:28:14 --> 00:28:18 This stuff is all positive, right, so the change in entropy 334 00:28:18 --> 00:28:19 is going to be positive. 335 00:28:19 --> 00:28:23 Entropy is going to increase as this happens, and then there's 336 00:28:23 --> 00:28:26 a change right at some fixed temperature as the 337 00:28:26 --> 00:28:30 material melts. 338 00:28:30 --> 00:28:33 So here is step one. 339 00:28:33 --> 00:28:36 Here is step two, right, this must be the 340 00:28:36 --> 00:28:40 melting temperature. 341 00:28:40 --> 00:28:45 And then there's another heating step. 342 00:28:45 --> 00:28:48 Well, strictly speaking, I'm going to run out of space here 343 00:28:48 --> 00:28:52 if I'm not careful, so I'm going to be a little 344 00:28:52 --> 00:28:55 more careful here. 345 00:28:55 --> 00:28:59 One, two, three. 346 00:28:59 --> 00:29:01 I'm heating it up a little more. 347 00:29:01 --> 00:29:03 Entropy is still increasing, right. 348 00:29:03 --> 00:29:05 So I've done this. 349 00:29:05 --> 00:29:06 I've done this. 350 00:29:06 --> 00:29:09 Now I've heated up the liquid. 351 00:29:09 --> 00:29:13 Now, I'm going to boil the liquid, so it's going to have 352 00:29:13 --> 00:29:17 some change in entropy. 353 00:29:17 --> 00:29:21 This must be my boiling point, and now there's some further 354 00:29:21 --> 00:29:25 change in the gas, and that gets me to whatever my final 355 00:29:25 --> 00:29:32 temperature is, right, that I'm going to reach. 356 00:29:32 --> 00:29:36 Four and five, great. 357 00:29:36 --> 00:29:42 So there is monotonic increase in the entropy. 358 00:29:42 --> 00:29:52 OK, so we're there, except for this value. 359 00:29:52 --> 00:30:03 That one stinking little number -- S naught at zero Kelvin. 360 00:30:03 --> 00:30:07 That's the only thing we don't know so far. 361 00:30:07 --> 00:30:13 So, for this we need some additional input. 362 00:30:13 --> 00:30:17 We got some input of the sort that we need in 363 00:30:17 --> 00:30:18 1905 from Nernst. 364 00:30:18 --> 00:30:26 Nernst deduced that as you go down from zero Kelvin for any 365 00:30:26 --> 00:30:31 process, the change in entropy gets smaller and smaller. 366 00:30:31 --> 00:30:35 It approaches zero. 367 00:30:35 --> 00:30:38 Now, that actually was certainly an important advance, 368 00:30:38 --> 00:30:43 but it was superseded by such an important advance that I'm 369 00:30:43 --> 00:30:49 not even going to reward it by placing it on the blackboard. 370 00:30:49 --> 00:30:52 Forget highlight, color, forget it. 371 00:30:52 --> 00:30:59 Because Planck, six years later, in 1911, deduced a 372 00:30:59 --> 00:31:04 stronger statement which is extremely useful, 373 00:31:04 --> 00:31:06 and it's the following. 374 00:31:06 --> 00:31:11 What he showed is that as temperature approaches zero 375 00:31:11 --> 00:31:16 Kelvin, for a pure substance in it's crystalline 376 00:31:16 --> 00:31:20 state, S is zero. 377 00:31:20 --> 00:31:22 A much stronger statement, right. 378 00:31:22 --> 00:31:26 A stronger statement than the idea that changes in 379 00:31:26 --> 00:31:30 S get very small as you approach zero Kelvin. 380 00:31:30 --> 00:31:34 No, he's saying we can make a statement about that absolute 381 00:31:34 --> 00:31:39 number S goes to zero as temperature goes to zero. 382 00:31:39 --> 00:31:43 For a pure substance in it's crystalline state. 383 00:31:43 --> 00:31:48 So that is monumentally important. 384 00:31:48 --> 00:32:14 So as T goes to zero Kelvin, S goes to zero, for every pure 385 00:32:14 --> 00:32:22 substance in its, and I'll sort of interject here, perfect 386 00:32:22 --> 00:32:30 crystalline state. 387 00:32:30 --> 00:32:32 That's really an amazing result. 388 00:32:32 --> 00:32:36 So what it's saying is I'm down at zero Kelvin. 389 00:32:36 --> 00:32:41 Minimally, I've somehow cooled it as much as I possibly could, 390 00:32:41 --> 00:32:44 and I've got my perfect crystal lattice. 391 00:32:44 --> 00:32:47 It could be an atomic crystal like this, or, you know, 392 00:32:47 --> 00:32:50 it could be molecules. 393 00:32:50 --> 00:32:54 But they're all exactly where they belong in their locations 394 00:32:54 --> 00:32:57 in the crystal, and the absolute entropy is something 395 00:32:57 --> 00:33:02 I can define and it's zero. 396 00:33:02 --> 00:33:07 So S equals zero. 397 00:33:07 --> 00:33:19 Perfect, pure crystal, all right. 398 00:33:19 --> 00:33:26 OK, well this came out of a microscopic description of 399 00:33:26 --> 00:33:32 entropy that I briefly alluded to last lecture, and again 400 00:33:32 --> 00:33:36 we'll go into in more detail in a few lectures hence. 401 00:33:36 --> 00:33:42 But the result that I mentioned, the general result, 402 00:33:42 --> 00:33:48 was that S was R over Na Avogadro's number, times the 403 00:33:48 --> 00:34:07 log of this omega number of microscopic states available 404 00:34:07 --> 00:34:10 to the system that I'm considering. 405 00:34:10 --> 00:34:14 Now normally for a macroscopic system, I've got just an 406 00:34:14 --> 00:34:19 astronomical number of microscopic states. 407 00:34:19 --> 00:34:22 You know, that could mean in a liquid, different little 408 00:34:22 --> 00:34:25 configurations of the molecules around each other. 409 00:34:25 --> 00:34:27 They're all different states. 410 00:34:27 --> 00:34:33 Huge amounts of possible states, and the gas even more. 411 00:34:33 --> 00:34:38 But if I go to zero Kelvin, and I've got a perfect crystal, 412 00:34:38 --> 00:34:42 every atom, every molecule is exactly in its place. 413 00:34:42 --> 00:34:46 How many possible different states is that? 414 00:34:46 --> 00:34:48 It's one. 415 00:34:48 --> 00:34:51 There aren't any more possible states. 416 00:34:51 --> 00:34:55 I've localized every identical lateral molecule in its 417 00:34:55 --> 00:34:59 particular place, and it's done. 418 00:34:59 --> 00:35:03 And you know, if I start worrying about the various 419 00:35:03 --> 00:35:06 things that would matter under ordinary conditions, right, you 420 00:35:06 --> 00:35:09 know, maybe at higher temperature, I'd have some 421 00:35:09 --> 00:35:13 molecules in excited vibration levels or 422 00:35:13 --> 00:35:14 maybe electronic levels. 423 00:35:14 --> 00:35:17 Maybe if it's hydrogen, maybe everything isn't 424 00:35:17 --> 00:35:18 in the ground state. 425 00:35:18 --> 00:35:21 It's not all in the 1s orbital but in higher levels. 426 00:35:21 --> 00:35:24 Then there's be lots of states available, right, even of only 427 00:35:24 --> 00:35:28 one atom in the whole crystal is excited, well there's one 428 00:35:28 --> 00:35:30 state that would have it be this atom. 429 00:35:30 --> 00:35:33 A different one would have it be this atom, and so forth. 430 00:35:33 --> 00:35:37 Already, there'd be an enormous number of states, but at zero 431 00:35:37 --> 00:35:41 Kelvin, there's no thermal lexitation of any 432 00:35:41 --> 00:35:42 of that stuff. 433 00:35:42 --> 00:35:46 Things are in the lowest states, and they're in 434 00:35:46 --> 00:35:48 their proper positions. 435 00:35:48 --> 00:35:52 There's only one state for the whole system, so that's why the 436 00:35:52 --> 00:35:56 entropy is zero in a perfect crystal. 437 00:35:56 --> 00:36:01 At zero degrees Kelvin. 438 00:36:01 --> 00:36:05 Now, there are things that may appear to violate that. 439 00:36:05 --> 00:36:07 Now of course you can make measurements of entropy, 440 00:36:07 --> 00:36:14 right, so it can be verified that this is the case. 441 00:36:14 --> 00:36:18 There are some things that would appear to 442 00:36:18 --> 00:36:21 violate that result. 443 00:36:21 --> 00:36:25 You know, you can make measurements of entropies, 444 00:36:25 --> 00:36:29 and for example let's take carbon monoxide crystal, CO. 445 00:36:29 --> 00:36:32 So let's say this is a crystal lattice, it's 446 00:36:32 --> 00:36:33 diatomic molecules. 447 00:36:33 --> 00:36:38 It's carbon oxygen carbon oxygen carbon oxygen. 448 00:36:38 --> 00:36:42 All there in perfect place in the crystal lattice. 449 00:36:42 --> 00:36:46 Well it turns out when you form the crystal every now and then 450 00:36:46 --> 00:36:51 -- you know, let's put it in color. 451 00:36:51 --> 00:36:56 Let's put it in the color that we'll use to signify 452 00:36:56 --> 00:37:01 something in some way evil. 453 00:37:01 --> 00:37:03 No insult to people who like that color. 454 00:37:03 --> 00:37:05 I kind of like it in fact. 455 00:37:05 --> 00:37:12 OK, so you know, you're making the crystal, cooling it, 456 00:37:12 --> 00:37:14 started out maybe in the gas phase, start cooling it, 457 00:37:14 --> 00:37:15 starts crystallizing. 458 00:37:15 --> 00:37:23 Gets colder and colder, but you know, carbon monoxide is pretty 459 00:37:23 --> 00:37:25 easy for those things to flip sides. 460 00:37:25 --> 00:37:28 And even in the crystalline state, even though it's a 461 00:37:28 --> 00:37:30 crystal, so the molecules center of mass are all where 462 00:37:30 --> 00:37:34 they belong, still at ordinary temperatures they will 463 00:37:34 --> 00:37:36 be able to rotate a bit. 464 00:37:36 --> 00:37:40 So even in the crystalline state, when it's originally 465 00:37:40 --> 00:37:43 formed, not at zero Kelvin, there's thermal energy around. 466 00:37:43 --> 00:37:45 These things can to get knocked around, and the 467 00:37:45 --> 00:37:47 orientations can change. 468 00:37:47 --> 00:37:50 Now you start cooling it, and you know, by and large 469 00:37:50 --> 00:37:54 they'll all go into the proper orientation. 470 00:37:54 --> 00:37:57 Right, that's the lowest energy state, but you know, there are 471 00:37:57 --> 00:37:59 all sorts of kinetic things involved. 472 00:37:59 --> 00:38:01 There it takes time for the flipping, depending on how 473 00:38:01 --> 00:38:05 long, how slowly it was cooled, and so forth. 474 00:38:05 --> 00:38:08 May never happen, and then it's cooled, and then anything 475 00:38:08 --> 00:38:12 that's left in the other orientation is frozen in there. 476 00:38:12 --> 00:38:13 There's no thermal energy anymore. 477 00:38:13 --> 00:38:16 It can't find a way to reorient. 478 00:38:16 --> 00:38:18 That's it. 479 00:38:18 --> 00:38:21 All right, let's say we're down to zero Kelvin, and out of the 480 00:38:21 --> 00:38:24 whole crystal, we've got a mole of molecules. 481 00:38:24 --> 00:38:27 One of them is in the wrong orientation. 482 00:38:27 --> 00:38:30 Now how many states do we have like that that 483 00:38:30 --> 00:38:32 would be possible? 484 00:38:32 --> 00:38:34 We'd have a mole of states, right. 485 00:38:34 --> 00:38:34 It could be this one. 486 00:38:34 --> 00:38:36 It could be that one, right. 487 00:38:36 --> 00:38:39 Or in general, of course really there's a whole distribution of 488 00:38:39 --> 00:38:41 them, and they could be anywhere, and pairs of them 489 00:38:41 --> 00:38:44 could be, and it doesn't take long to get to really 490 00:38:44 --> 00:38:47 large numbers. 491 00:38:47 --> 00:38:49 So the entropy won't be zero. 492 00:38:49 --> 00:38:51 Entropy of a perfect crystal would mean it's 493 00:38:51 --> 00:38:54 perfectly ordered. 494 00:38:54 --> 00:38:56 So that's zero. 495 00:38:56 --> 00:38:58 But things like that not withstanding, and of course 496 00:38:58 --> 00:39:00 it's the same if you have a mixed crystal. 497 00:39:00 --> 00:39:04 In a sense I've described a mixed crystal where the mixture 498 00:39:04 --> 00:39:07 is a mixture of carbon monoxide pointing this way, and carbon 499 00:39:07 --> 00:39:08 monoxide pointing this way. 500 00:39:08 --> 00:39:12 But a real mixed crystal with two different constituents, 501 00:39:12 --> 00:39:15 well of course, then you have all the possible configurations 502 00:39:15 --> 00:39:18 where, you know, they could be here and here and here, and 503 00:39:18 --> 00:39:20 then you could move one of them around and so forth, 504 00:39:20 --> 00:39:25 there are zillions and zillions of states. 505 00:39:25 --> 00:39:30 But for a pure crystal, in perfect form, you really have 506 00:39:30 --> 00:39:34 only one configuration, and your entropy is therefore zero. 507 00:39:34 --> 00:39:44 OK, so now, we can go back and we can do this. 508 00:39:44 --> 00:39:47 And at least in principle, even for things that don't form 509 00:39:47 --> 00:39:50 perfect crystals, we could calculate the change in entropy 510 00:39:50 --> 00:39:54 going from the perfect ordered crystal to something else with 511 00:39:54 --> 00:39:57 some degree of disorder and keep going and change the 512 00:39:57 --> 00:40:00 temperature and do all these things. 513 00:40:00 --> 00:40:03 So the real point is that this is extremely powerful because 514 00:40:03 --> 00:40:07 given this, we really can calculate absolute numbers for 515 00:40:07 --> 00:40:10 the entropies of substances, at ordinary, not just at zero 516 00:40:10 --> 00:40:13 Kelvin, but using this which, you know, this is a really very 517 00:40:13 --> 00:40:15 straightforward procedure. 518 00:40:15 --> 00:40:18 And in fact these things are really quite easy to measure. 519 00:40:18 --> 00:40:21 You know, you do calorimetry, you can measure those delta H 520 00:40:21 --> 00:40:25 of fusions, right, delta H of vaporization. 521 00:40:25 --> 00:40:27 You can measure the heat capacities, the things 522 00:40:27 --> 00:40:28 in the calorimeter. 523 00:40:28 --> 00:40:30 You'll see how much heat is needed to raise the 524 00:40:30 --> 00:40:31 temperature a degree. 525 00:40:31 --> 00:40:33 That give you your heat capacity for the gas or 526 00:40:33 --> 00:40:34 the liquid or the solid. 527 00:40:34 --> 00:40:37 So in fact, it's extremely practical to make all those 528 00:40:37 --> 00:40:40 measurements, and you can easily find those values of the 529 00:40:40 --> 00:40:44 heat capacities and the delta H's tabulated for a huge 530 00:40:44 --> 00:40:47 number of substances. 531 00:40:47 --> 00:40:50 So this is, in fact, the practical procedure then of 532 00:40:50 --> 00:40:54 protocol for calculating absolute entropies of all sorts 533 00:40:54 --> 00:41:00 of materials at ordinary temperatures and pressures. 534 00:41:00 --> 00:41:03 Very important. 535 00:41:03 --> 00:41:12 Now, one of those corollaries to this law is that in fact 536 00:41:12 --> 00:41:16 it's impossible to reduce the temperature of any substance, 537 00:41:16 --> 00:41:20 any system, all the way to absolute, exact, no 538 00:41:20 --> 00:41:23 approximations, zero Kelvin. 539 00:41:23 --> 00:41:29 Because you can't quite get down to zero Kelvin. 540 00:41:29 --> 00:41:31 And there are various ways that you can see that 541 00:41:31 --> 00:41:43 this must be the case. 542 00:41:43 --> 00:41:53 But here's one way to think about it. 543 00:41:53 --> 00:42:08 So, let's just write that first. 544 00:42:08 --> 00:42:16 All right, can't get quite down to zero Kelvin. 545 00:42:16 --> 00:42:29 All right, let's consider a mole of an ideal gas. 546 00:42:29 --> 00:42:32 So p is RT over V. 547 00:42:32 --> 00:42:44 And let's start at T1 and V1, and now let's bring it down to 548 00:42:44 --> 00:42:48 some lower temperature, T2 in some spontaneous process. 549 00:42:48 --> 00:42:53 We'll make it adiabatic so it's like an irreversible expansion. 550 00:42:53 --> 00:42:55 I just want to calculate what delta S would be 551 00:42:55 --> 00:42:59 there in terms of T and V. 552 00:42:59 --> 00:43:21 So, well, du is T dS minus p dV. 553 00:43:21 --> 00:43:25 554 00:43:25 --> 00:43:33 dS is du over T plus p over T dV. 555 00:43:33 --> 00:43:42 But we can also write du is Cv dT in this case, right? 556 00:43:42 --> 00:43:55 So that says that p over T, that's R over V, and we can 557 00:43:55 --> 00:44:25 write, dS is Cv dT over T plus R dV over V. 558 00:44:25 --> 00:44:45 It's Cv, I'll write it as d(log T) plus R d(log V), and so 559 00:44:45 --> 00:44:55 delta S, Cv log of T2 minus log of T1, if T2 is my final 560 00:44:55 --> 00:45:07 temperature, plus R log of V2 minus log of V1, okay. 561 00:45:07 --> 00:45:16 Or Cv log of T2 over T1 plus R log of V2 over V1. 562 00:45:16 --> 00:45:35 Well, what does it mean when T2 is zero? 563 00:45:35 --> 00:45:41 Well, I don't know what it means. 564 00:45:41 --> 00:45:51 This turns into negative infinity. 565 00:45:51 --> 00:45:56 So we're going to write it again as our either evil or 566 00:45:56 --> 00:46:00 at least unattainable color. 567 00:46:00 --> 00:46:02 What's going to happen? 568 00:46:02 --> 00:46:06 I mean you could say, well, we can counteract it 569 00:46:06 --> 00:46:08 by having this go to plus infinity, right. 570 00:46:08 --> 00:46:11 Make the volume infinite. 571 00:46:11 --> 00:46:13 In other words, have the expansion be in through 572 00:46:13 --> 00:46:14 an infinite volume. 573 00:46:14 --> 00:46:18 Of course that's impossible. 574 00:46:18 --> 00:46:20 That would be the only way to counteract the 575 00:46:20 --> 00:46:24 divergence of this term. 576 00:46:24 --> 00:46:29 In practice, you really cannot get to absolute zero, but it 577 00:46:29 --> 00:46:32 is possible to get extremely close. 578 00:46:32 --> 00:46:34 That's doable experimentally. 579 00:46:34 --> 00:46:38 It's possible to get down to some micro or nano Kelvin 580 00:46:38 --> 00:46:41 temperatures, right. 581 00:46:41 --> 00:46:49 In fact, our MIT physicist, Wolfgang Ketterle, by bringing 582 00:46:49 --> 00:46:53 atoms and molecules down to extremely low temperatures, was 583 00:46:53 --> 00:46:57 able to see them all reach the very lowest possible quantum 584 00:46:57 --> 00:46:59 state available to them. 585 00:46:59 --> 00:47:03 All sorts of unusual properties emerge in that kind of state, 586 00:47:03 --> 00:47:06 where the atoms and molecules behave coherently. 587 00:47:06 --> 00:47:10 You can make matter waves, right, and see interferences 588 00:47:10 --> 00:47:12 among them because of the fact that they're all in this 589 00:47:12 --> 00:47:14 lowest quantum state. 590 00:47:14 --> 00:47:17 So it is possible to get extremely low temperatures, 591 00:47:17 --> 00:47:23 but never absolute zero. 592 00:47:23 --> 00:47:34 Here's another way to think about it. 593 00:47:34 --> 00:47:38 You could consider what happens to the absolute entropy, 594 00:47:38 --> 00:47:56 starting at T equals zero, right. 595 00:47:56 --> 00:48:00 So we already saw that, you know, that we can go, the first 596 00:48:00 --> 00:48:04 step will go from zero to some temperature of Cp over T for 597 00:48:04 --> 00:48:11 the solid dT if we start at zero Kelvin and warm up. 598 00:48:11 --> 00:48:16 Now, already we've got a problem. 599 00:48:16 --> 00:48:23 If this initial temperature really is zero, what happens? 600 00:48:23 --> 00:48:26 This diverge, right. 601 00:48:26 --> 00:48:31 Well, in fact, what that suggests is as you approach 602 00:48:31 --> 00:48:40 zero Kelvin, the heat capacity also approaches zero. 603 00:48:40 --> 00:48:54 So, does this go to infinity as T approaches zero Kelvin? 604 00:48:54 --> 00:49:05 Well, not if Cp of the solid approaches zero as 605 00:49:05 --> 00:49:09 T approaches zero Kelvin. 606 00:49:09 --> 00:49:16 And in fact, we can measure Cp, heat capacities at very low 607 00:49:16 --> 00:49:20 temperatures, and what we find is, they do. 608 00:49:20 --> 00:49:27 They do go to zero as you approach zero Kelvin. 609 00:49:27 --> 00:49:36 So in fact, Cp of T approaches zero as T approaches 610 00:49:36 --> 00:49:38 zero Kelvin. 611 00:49:38 --> 00:49:41 Good! 612 00:49:41 --> 00:49:49 But one thing that this says is, remember, that dT is dq at 613 00:49:49 --> 00:49:51 the heat in divided by Cp. 614 00:49:51 --> 00:49:55 615 00:49:55 --> 00:49:58 And this is getting really, really small. 616 00:49:58 --> 00:50:00 It's going to zero as temperature goes 617 00:50:00 --> 00:50:01 to zero, right. 618 00:50:01 --> 00:50:03 Which means that this is enormous. 619 00:50:03 --> 00:50:09 What it says is even the tiniest amount the heat input 620 00:50:09 --> 00:50:15 leads to a significant change increase in the temperature. 621 00:50:15 --> 00:50:17 So, this is another way of understanding why it just 622 00:50:17 --> 00:50:21 becomes impossible to lower the temperature of a system to 623 00:50:21 --> 00:50:25 absolute zero, because any kind of contact, and I mean any 624 00:50:25 --> 00:50:29 kind, like let's say you've got the system in some cryostat. 625 00:50:29 --> 00:50:31 Of course the walls of the cryostat aren't at zero 626 00:50:31 --> 00:50:33 Kelvin, but somewhere in here it's at zero Kelvin. 627 00:50:33 --> 00:50:36 You can say, okay I won't won't make it in 628 00:50:36 --> 00:50:37 contact with the walls. 629 00:50:37 --> 00:50:44 You can try, but actually light, photons that emit from 630 00:50:44 --> 00:50:47 the walls go into the sample. 631 00:50:47 --> 00:50:50 You wouldn't think that heats something up very much, and it 632 00:50:50 --> 00:50:53 doesn't heat it up very much, but we're not talking 633 00:50:53 --> 00:50:55 about very much. 634 00:50:55 --> 00:50:57 It does heat it up by enough that you can't 635 00:50:57 --> 00:50:58 get the absolute zero. 636 00:50:58 --> 00:51:02 In other words, somehow it will be in contact with 637 00:51:02 --> 00:51:06 stuff around that is not at absolute zero Kelvin. 638 00:51:06 --> 00:51:12 And even that kind of radiative contact is enough, in fact, to 639 00:51:12 --> 00:51:16 make it not reach absolute zero. 640 00:51:16 --> 00:51:19 So the point is, one way or another, you can easily see 641 00:51:19 --> 00:51:23 that it becomes impossible to keep pulling heat out of 642 00:51:23 --> 00:51:26 something and keep it down at the temperature that's right 643 00:51:26 --> 00:51:30 there at zero, but again it's possible to get very close. 644 00:51:30 --> 00:51:33 All right, what we're going to be able to do next time is take 645 00:51:33 --> 00:51:37 what we've seen so far, and develop the conditions for 646 00:51:37 --> 00:51:38 reaching equilibrium. 647 00:51:38 --> 00:51:41 So in a general sense, we'll be able to tell which way the 648 00:51:41 --> 00:51:45 processes go left unto themselves to move 649 00:51:45 --> 00:51:46 toward equilibrium.