1 00:00:00,080 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,570 --> 00:00:22,070 PROFESSOR: So what we've been trying 9 00:00:22,070 --> 00:00:26,730 to do is to develop a description of properties 10 00:00:26,730 --> 00:00:28,700 of some system, could be something 11 00:00:28,700 --> 00:00:34,080 like a gas, that includes both mechanical as well 12 00:00:34,080 --> 00:00:36,880 as thermal properties. 13 00:00:36,880 --> 00:00:39,620 So the first thing that we decided to do 14 00:00:39,620 --> 00:00:43,090 was to wait until this system has 15 00:00:43,090 --> 00:00:45,940 properties that do not change. 16 00:00:45,940 --> 00:00:48,340 Further, with the observation time 17 00:00:48,340 --> 00:00:51,440 that we are assigning to this system, 18 00:00:51,440 --> 00:00:54,160 and then if it is a gas we can say, 19 00:00:54,160 --> 00:00:57,000 OK the gas has some pressure and volume. 20 00:00:57,000 --> 00:01:01,190 In general, we said that we can characterize the system 21 00:01:01,190 --> 00:01:08,810 through a set of displacements and the conjugate forces which 22 00:01:08,810 --> 00:01:13,610 are used when we describe systems that are mechanical. 23 00:01:13,610 --> 00:01:14,790 Work on them is done. 24 00:01:14,790 --> 00:01:18,790 We can say that the work is Jdx, for example. 25 00:01:18,790 --> 00:01:19,420 OK? 26 00:01:19,420 --> 00:01:22,310 Then we decided that this mechanical description 27 00:01:22,310 --> 00:01:27,210 was not enough because systems could exchange heat 28 00:01:27,210 --> 00:01:32,780 and gradually started to build a more general prescription that 29 00:01:32,780 --> 00:01:35,540 included the exchange of heat. 30 00:01:35,540 --> 00:01:38,350 We saw that, for example, from the zeroth law, 31 00:01:38,350 --> 00:01:41,310 we could define temperature as a function of these quantities. 32 00:01:44,060 --> 00:01:50,920 And that we could also, from the first law define energy. 33 00:01:50,920 --> 00:01:56,740 And in particular that the change in energy, 34 00:01:56,740 --> 00:02:00,070 which is a function of where you are in equilibrium. 35 00:02:00,070 --> 00:02:03,440 So once you say that you have a system in equilibrium 36 00:02:03,440 --> 00:02:07,060 you can say what the values of the x's and J's are. 37 00:02:07,060 --> 00:02:09,720 You know somehow what the value of temperature 38 00:02:09,720 --> 00:02:11,650 is, what the value of energy is. 39 00:02:11,650 --> 00:02:15,390 And if you were to make a change in energy 40 00:02:15,390 --> 00:02:19,350 by some external means the amount of change 41 00:02:19,350 --> 00:02:21,470 would either come from mechanical work 42 00:02:21,470 --> 00:02:23,657 that you did on the system, or the heat 43 00:02:23,657 --> 00:02:24,990 that you supplied to the system. 44 00:02:28,690 --> 00:02:31,390 Now this expression is the one that you 45 00:02:31,390 --> 00:02:37,790 would like to elevate somehow and be able to compute energy 46 00:02:37,790 --> 00:02:40,660 as a function of state. 47 00:02:40,660 --> 00:02:42,760 Kind of keeping in mind what we would 48 00:02:42,760 --> 00:02:45,930 do if there was no heat around. 49 00:02:45,930 --> 00:02:49,640 And we could do things mechanically, then mechanically 50 00:02:49,640 --> 00:02:54,570 we could in principle build up the energy function by changing 51 00:02:54,570 --> 00:03:00,770 displacements lightly and calculating the amount of work 52 00:03:00,770 --> 00:03:04,451 by this formula. 53 00:03:04,451 --> 00:03:07,770 And basically we emphasized that you 54 00:03:07,770 --> 00:03:11,040 would be able to use a formula such as this 55 00:03:11,040 --> 00:03:15,800 if you were to make things sufficiently slowly 56 00:03:15,800 --> 00:03:17,960 so that at each stage in the process 57 00:03:17,960 --> 00:03:20,430 you can calculate what Ji is, which 58 00:03:20,430 --> 00:03:22,340 means that at each stage in the process 59 00:03:22,340 --> 00:03:25,941 you should be in some kind of an equilibrium state. 60 00:03:25,941 --> 00:03:26,440 OK? 61 00:03:26,440 --> 00:03:30,790 If that was the case, if you didn't have dQ then 62 00:03:30,790 --> 00:03:33,840 this would be your energy function 63 00:03:33,840 --> 00:03:36,430 and actually once you had the energy function you could, 64 00:03:36,430 --> 00:03:40,280 for example, calculate J as a derivative of energy 65 00:03:40,280 --> 00:03:42,050 with respect to x. 66 00:03:42,050 --> 00:03:44,980 So then you also start thinking about how many 67 00:03:44,980 --> 00:03:47,750 independent degrees of freedom do I have. 68 00:03:47,750 --> 00:03:51,710 Do I really need all of the set of x's and J's to describe 69 00:03:51,710 --> 00:03:53,340 my system in equilibrium. 70 00:03:53,340 --> 00:03:57,740 If additionally once I know the e, I can derive all of the J's. 71 00:03:57,740 --> 00:04:00,480 So we have to also come eventually 72 00:04:00,480 --> 00:04:04,030 to grips with how many independent degrees of freedom 73 00:04:04,030 --> 00:04:06,990 will describe our system. 74 00:04:06,990 --> 00:04:11,360 Now the first step towards completing this expression 75 00:04:11,360 --> 00:04:14,250 was to find what dQ is. 76 00:04:14,250 --> 00:04:18,329 So we needed another law of thermodynamics 77 00:04:18,329 --> 00:04:22,190 beyond zeroth law and first law that somehow related 78 00:04:22,190 --> 00:04:24,890 heat and temperature to each other. 79 00:04:24,890 --> 00:04:27,780 As we expect that somehow this expression 80 00:04:27,780 --> 00:04:30,330 will leave you temperature. 81 00:04:30,330 --> 00:04:35,930 And what we used was some version of the second law. 82 00:04:40,080 --> 00:04:46,260 That was there how heat would flow from, say, a hot air body 83 00:04:46,260 --> 00:04:48,160 to a cold air body. 84 00:04:48,160 --> 00:04:51,970 And Clausius's theorem would say that you have only heat flowing 85 00:04:51,970 --> 00:04:54,494 in one particular direction. 86 00:04:54,494 --> 00:04:57,100 Well, we sort of introduced the idea 87 00:04:57,100 --> 00:05:01,490 of engines, which are machines that 88 00:05:01,490 --> 00:05:10,690 are used to do work by taking heat from the heart air body 89 00:05:10,690 --> 00:05:12,830 to the cold air body. 90 00:05:12,830 --> 00:05:17,210 And we could define an efficiency, 91 00:05:17,210 --> 00:05:22,810 which is work divided by heat input, which since work is 92 00:05:22,810 --> 00:05:28,180 the difference between these two we could write as 1 minus Qc 93 00:05:28,180 --> 00:05:28,680 over Qh. 94 00:05:32,700 --> 00:05:35,900 And then we introduced this special class 95 00:05:35,900 --> 00:05:39,030 of engines, which were these Carnot engines. 96 00:05:43,210 --> 00:05:46,200 And the idea of this class was that you could run them 97 00:05:46,200 --> 00:05:47,390 forward and backward. 98 00:05:47,390 --> 00:05:50,450 They were kind of reversible. 99 00:05:50,450 --> 00:05:53,830 The were kind of a analogue of the frictionless type 100 00:05:53,830 --> 00:05:58,200 of processes that you would use to construct mechanical energy. 101 00:05:58,200 --> 00:06:04,650 And we found that these engines were the most efficient. 102 00:06:04,650 --> 00:06:07,310 And so the efficiency of these engines 103 00:06:07,310 --> 00:06:10,980 was marked only by the two temperatures. 104 00:06:10,980 --> 00:06:15,520 So we had the functional form potentially of efficiency 105 00:06:15,520 --> 00:06:18,520 as a function of the two temperatures involved. 106 00:06:18,520 --> 00:06:22,430 And what we also showed was that the efficiency 107 00:06:22,430 --> 00:06:25,330 of any random engine was going to be 108 00:06:25,330 --> 00:06:28,670 less than the efficiency of the Carnot engine 109 00:06:28,670 --> 00:06:32,360 that is marked by these two temperatures. 110 00:06:32,360 --> 00:06:36,290 And by putting some Carnot engines in series, 111 00:06:36,290 --> 00:06:39,740 we saw that the good way to write this 112 00:06:39,740 --> 00:06:40,930 was something of this form. 113 00:06:45,330 --> 00:06:48,110 I can in principle remove the one 114 00:06:48,110 --> 00:06:50,540 from the two sides of this equation 115 00:06:50,540 --> 00:06:54,640 and rearrange it a little bit into the form 116 00:06:54,640 --> 00:07:03,330 Qh over Th plus minus Qc over Tc because of this inequality 117 00:07:03,330 --> 00:07:06,811 writing it as less than or equal to 0. 118 00:07:06,811 --> 00:07:09,090 OK? 119 00:07:09,090 --> 00:07:11,510 What does that tell us? 120 00:07:11,510 --> 00:07:13,620 As it stands, not much. 121 00:07:13,620 --> 00:07:17,660 But then I promised you at the end of last lecture 122 00:07:17,660 --> 00:07:23,090 then this is an example of a much more powerful result, 123 00:07:23,090 --> 00:07:25,380 which is the Clausius Theorem. 124 00:07:25,380 --> 00:07:28,900 So let's start with writing what that theorem is 125 00:07:28,900 --> 00:07:30,985 and how it relates to a simple expression. 126 00:07:37,990 --> 00:07:50,970 And the theory is that for any cyclic process 127 00:07:50,970 --> 00:07:54,800 I can, cyclic, what does it mean? 128 00:07:54,800 --> 00:08:00,600 Is that in the set of coordinates in principle 129 00:08:00,600 --> 00:08:03,980 that is used to describe the system, 130 00:08:03,980 --> 00:08:06,810 I will start with the position that is equilibrium, 131 00:08:06,810 --> 00:08:11,680 so I can put that point in this state, 132 00:08:11,680 --> 00:08:15,280 but then I make a transformation. 133 00:08:15,280 --> 00:08:17,440 And ultimately return to that point. 134 00:08:17,440 --> 00:08:20,420 So the cycle is the return to that point. 135 00:08:20,420 --> 00:08:23,160 So I carry out a set of steps. 136 00:08:23,160 --> 00:08:26,550 And I haven't indicated the set of steps 137 00:08:26,550 --> 00:08:30,100 through a connected continuous curve 138 00:08:30,100 --> 00:08:33,600 in this multi-dimensional coordinate space 139 00:08:33,600 --> 00:08:36,860 because I don't want to restrict myself 140 00:08:36,860 --> 00:08:39,760 to processes that are even in equilibrium. 141 00:08:39,760 --> 00:08:42,929 So I may take a gas that I have over there, 142 00:08:42,929 --> 00:08:47,640 expand it rapidly, close it rapidly as long as I wait when 143 00:08:47,640 --> 00:08:49,720 I come back to the same point that I 144 00:08:49,720 --> 00:08:51,790 have reached my equilibrium again. 145 00:08:51,790 --> 00:08:55,200 I have done a cyclic process. 146 00:08:55,200 --> 00:08:59,550 Now at each stage of this cyclic process presumably 147 00:08:59,550 --> 00:09:02,990 the system takes in a certain amount of heat. 148 00:09:02,990 --> 00:09:06,910 So let's say that there is a dQ that 149 00:09:06,910 --> 00:09:09,860 goes at this stage of the process. 150 00:09:09,860 --> 00:09:14,030 So I use s to indicate, say, stage 0, one, two, three, 151 00:09:14,030 --> 00:09:16,110 four all the way coming back. 152 00:09:16,110 --> 00:09:19,890 So this is just a numbering process. 153 00:09:19,890 --> 00:09:22,570 The statement of Clausius's theorem 154 00:09:22,570 --> 00:09:29,220 is that for any cyclic process if I go out around the cycle, 155 00:09:29,220 --> 00:09:33,030 add up these elements of heat that are delivered 156 00:09:33,030 --> 00:09:37,660 to this system and I allow the possibility therefore 157 00:09:37,660 --> 00:09:40,520 that some of the times these elements dQ 158 00:09:40,520 --> 00:09:44,400 will be negative just as I did over here by writing this 159 00:09:44,400 --> 00:09:48,650 as minus Qc so that what the dQ is 160 00:09:48,650 --> 00:09:53,050 what has gone into whatever this black box is. 161 00:09:53,050 --> 00:09:56,020 So Qh went into this black box. 162 00:09:56,020 --> 00:09:59,460 Qc went out, which means that minus Qc went in 163 00:09:59,460 --> 00:10:01,550 so I arranged it in this fashion. 164 00:10:01,550 --> 00:10:03,140 You can see that the elements of heat 165 00:10:03,140 --> 00:10:05,530 are divided by some temperature. 166 00:10:05,530 --> 00:10:10,430 I generalize that expression over here by writing it T of S 167 00:10:10,430 --> 00:10:12,170 here. 168 00:10:12,170 --> 00:10:15,710 And the statement of Clausius's theorem 169 00:10:15,710 --> 00:10:23,270 is that this quantity has the same sign constraint as here 170 00:10:23,270 --> 00:10:32,190 where I told you that dQ is heat delivered 171 00:10:32,190 --> 00:10:49,880 to system at temperature T sub s. 172 00:10:52,630 --> 00:10:56,350 Now this particular thing requires a certain amount 173 00:10:56,350 --> 00:10:59,080 of thinking about. 174 00:10:59,080 --> 00:11:03,920 Because I told you that your system is not in equilibrium. 175 00:11:03,920 --> 00:11:06,250 So what is this T sub S if you have 176 00:11:06,250 --> 00:11:08,020 a system that is not in equilibrium? 177 00:11:08,020 --> 00:11:11,480 You can't define its temperature. 178 00:11:11,480 --> 00:11:15,110 However, you can imagine that whatever machine 179 00:11:15,110 --> 00:11:20,400 was delivering it had sump particular temperature. 180 00:11:20,400 --> 00:11:23,580 And so that's the temperature of the device 181 00:11:23,580 --> 00:11:28,290 or whatever is delivering this heat to the system. 182 00:11:28,290 --> 00:11:30,760 And you would be justified to say, well, 183 00:11:30,760 --> 00:11:33,570 why is that even useful? 184 00:11:33,570 --> 00:11:38,990 Because certainly if I were to have exactly the same cycle, 185 00:11:38,990 --> 00:11:43,370 but deliver the heat elements through a different process 186 00:11:43,370 --> 00:11:46,830 this Ts would be different. 187 00:11:46,830 --> 00:11:50,140 So this is really also a function 188 00:11:50,140 --> 00:11:54,160 of the method through which I want to carry out this cycle. 189 00:11:54,160 --> 00:11:58,390 So given that it doesn't sound like a particularly useful 190 00:11:58,390 --> 00:11:58,890 theorem. 191 00:11:58,890 --> 00:12:00,990 It seems very arbitrary. 192 00:12:00,990 --> 00:12:02,420 So let's first prove it. 193 00:12:02,420 --> 00:12:03,630 The proof is simple. 194 00:12:03,630 --> 00:12:06,700 And then see whether it's useful or not. 195 00:12:06,700 --> 00:12:10,490 Any questions at this point? 196 00:12:10,490 --> 00:12:10,990 OK. 197 00:12:10,990 --> 00:12:13,150 So how do we go proving it? 198 00:12:13,150 --> 00:12:16,080 We are going to use these Carnot engines. 199 00:12:16,080 --> 00:12:24,340 So what I will imagine is that I have some big vat that 200 00:12:24,340 --> 00:12:27,860 is at some, let's say, for purposes 201 00:12:27,860 --> 00:12:34,560 of, use a high temperature, a vat of hot water. 202 00:12:34,560 --> 00:12:38,710 And I use this vat as the process 203 00:12:38,710 --> 00:12:42,560 that provides heat to this entity. 204 00:12:42,560 --> 00:12:44,790 Exactly what do I mean by that? 205 00:12:44,790 --> 00:12:53,340 I will use a Carnot engine to take heat. 206 00:12:53,340 --> 00:12:57,620 Let's call it Qh, although its sign is not that particularly 207 00:12:57,620 --> 00:12:59,730 important. 208 00:12:59,730 --> 00:13:07,530 And convert this into the heat element 209 00:13:07,530 --> 00:13:10,050 dQs that is delivered here. 210 00:13:10,050 --> 00:13:16,460 So since I'm using s, an infinitesimal element here, 211 00:13:16,460 --> 00:13:18,890 let me call this dQ. 212 00:13:18,890 --> 00:13:22,200 And let's call it dQ0 because I'm picking it 213 00:13:22,200 --> 00:13:26,080 up from the temperature of T0. 214 00:13:26,080 --> 00:13:31,040 So I have as a result of this process 215 00:13:31,040 --> 00:13:33,340 a certain amount of work. 216 00:13:33,340 --> 00:13:37,420 All of this corresponds to the s-th element 217 00:13:37,420 --> 00:13:40,300 through this particular cycle. 218 00:13:40,300 --> 00:13:44,210 So I divide my cycle to lots of elements. 219 00:13:44,210 --> 00:13:47,420 Each element takes in this amount of heat 220 00:13:47,420 --> 00:13:54,900 and I use a Carnot engine to deliver that heat and always 221 00:13:54,900 --> 00:14:01,620 taking the Carnot engine from the input from T0, 222 00:14:01,620 --> 00:14:07,080 but what it delivers is where this randomness in temperature 223 00:14:07,080 --> 00:14:09,210 T of S comes about. 224 00:14:09,210 --> 00:14:11,360 Why did I use a Carnot engine? 225 00:14:11,360 --> 00:14:14,320 Because I said that sometimes this cycle 226 00:14:14,320 --> 00:14:15,930 is going to take in heat. 227 00:14:15,930 --> 00:14:18,590 Sometimes it is going to give out heat. 228 00:14:18,590 --> 00:14:22,790 So I want to be able to run this both forward and backwards. 229 00:14:22,790 --> 00:14:23,472 Yes? 230 00:14:23,472 --> 00:14:26,670 AUDIENCE: Are you delivering the work to the cycle too? 231 00:14:26,670 --> 00:14:27,329 PROFESSOR: No. 232 00:14:27,329 --> 00:14:27,870 AUDIENCE: OK. 233 00:14:27,870 --> 00:14:29,933 PROFESSOR: The work goes whenever it goes. 234 00:14:29,933 --> 00:14:30,474 AUDIENCE: OK. 235 00:14:30,474 --> 00:14:31,250 PROFESSOR: OK? 236 00:14:31,250 --> 00:14:33,900 And indeed that's the next step of the argument, which 237 00:14:33,900 --> 00:14:36,310 is what happened to the work? 238 00:14:36,310 --> 00:14:37,240 OK? 239 00:14:37,240 --> 00:14:39,240 So we do this. 240 00:14:39,240 --> 00:14:44,724 At the end of this story I am back to where I started. 241 00:14:44,724 --> 00:14:45,650 AUDIENCE: Maybe I'm-- 242 00:14:45,650 --> 00:14:45,960 PROFESSOR: Yes? 243 00:14:45,960 --> 00:14:47,668 AUDIENCE: Maybe I missed something basic, 244 00:14:47,668 --> 00:14:50,580 but I don't understand why you can't just 245 00:14:50,580 --> 00:14:57,060 define the function dQ not of s and ignore the Carnot engine. 246 00:14:57,060 --> 00:14:58,886 Just deliver it the key. 247 00:14:58,886 --> 00:15:01,980 What was wrong with just the blue arrow? 248 00:15:01,980 --> 00:15:02,770 PROFESSOR: OK. 249 00:15:02,770 --> 00:15:06,720 Why can't I just take heat from here to here? 250 00:15:06,720 --> 00:15:09,650 Because in principle where I put it 251 00:15:09,650 --> 00:15:12,710 is at a different temperature when I eventually 252 00:15:12,710 --> 00:15:14,180 go to equilibrium. 253 00:15:14,180 --> 00:15:16,960 And then I would run into problems of second law, 254 00:15:16,960 --> 00:15:19,030 as to the ability to transfer heat 255 00:15:19,030 --> 00:15:21,060 from a hotter to colder engine. 256 00:15:21,060 --> 00:15:23,860 So if you like this intermediate stage, 257 00:15:23,860 --> 00:15:28,290 allows me ultimately two vary this temperature Ts. 258 00:15:28,290 --> 00:15:32,825 What you say is correct as long as the T of S is uniformly T0. 259 00:15:32,825 --> 00:15:33,950 It kind of becomes useless. 260 00:15:36,571 --> 00:15:37,070 OK? 261 00:15:39,661 --> 00:15:40,160 All right. 262 00:15:40,160 --> 00:15:47,360 So we enclose our cycle, the Carnot engine, everything 263 00:15:47,360 --> 00:15:49,540 into a box. 264 00:15:49,540 --> 00:15:52,450 And we see what this box is doing. 265 00:15:52,450 --> 00:15:54,670 So there is the box. 266 00:15:54,670 --> 00:16:07,330 The box takes in elements of heat, dQ from the reservoir. 267 00:16:07,330 --> 00:16:12,490 And does elements of work, dW at different stages. 268 00:16:15,105 --> 00:16:18,780 Now once the cycle has been completed, 269 00:16:18,780 --> 00:16:27,020 the net result is that I have extracted an amount of heat, 270 00:16:27,020 --> 00:16:31,510 so this is the net, extracted heat is 271 00:16:31,510 --> 00:16:37,620 the integral over the cycle of dQ0 of s. 272 00:16:37,620 --> 00:16:41,640 And converted it to work. 273 00:16:41,640 --> 00:16:44,810 Oh, one other step. 274 00:16:44,810 --> 00:16:46,870 Once I'm using the Carnot engine, 275 00:16:46,870 --> 00:16:49,290 and I think I emphasized this, there 276 00:16:49,290 --> 00:16:52,570 is a relationship between heat and temperature. 277 00:16:52,570 --> 00:16:54,920 It is this proportionally-- Qh is 278 00:16:54,920 --> 00:16:57,610 proportional to h for the Carnot engine. 279 00:16:57,610 --> 00:17:00,050 Qc is proportional to Tc. 280 00:17:00,050 --> 00:17:04,740 So there is a proportionality between the heat that I take 281 00:17:04,740 --> 00:17:09,819 and the heat that I deliver that is related to the temperatures. 282 00:17:09,819 --> 00:17:21,880 So dQ0 is in fact T0 dQ that goes into the engine divided 283 00:17:21,880 --> 00:17:27,168 by the temperature at which it is delivered to the engine. 284 00:17:27,168 --> 00:17:29,460 OK? 285 00:17:29,460 --> 00:17:33,120 Now I know also something about this 286 00:17:33,120 --> 00:17:35,180 because the second law of thermodynamics 287 00:17:35,180 --> 00:17:38,180 says that no process is possible whose soul result 288 00:17:38,180 --> 00:17:42,250 is the conversion of heat to work. 289 00:17:42,250 --> 00:17:43,370 I can do the opposite. 290 00:17:43,370 --> 00:17:46,040 I can convert work to heat, but that 291 00:17:46,040 --> 00:17:50,884 means that the integral here has to be negative. 292 00:17:50,884 --> 00:17:53,140 OK? 293 00:17:53,140 --> 00:17:56,870 Now T0 is just a positive constant. 294 00:17:56,870 --> 00:17:59,310 So once I divide through by that I 295 00:17:59,310 --> 00:18:05,251 have gotten the proof of the statement [INAUDIBLE]. 296 00:18:05,251 --> 00:18:05,750 OK? 297 00:18:14,680 --> 00:18:15,180 Questions? 298 00:18:19,330 --> 00:18:19,984 Yes? 299 00:18:19,984 --> 00:18:21,400 AUDIENCE: So how is this is really 300 00:18:21,400 --> 00:18:25,330 going to help with the definition of T of s 301 00:18:25,330 --> 00:18:31,510 because get theoretically even your extraction of heat 302 00:18:31,510 --> 00:18:36,470 and your doing the work in that black box or the opposite 303 00:18:36,470 --> 00:18:38,900 could happen irreversibly, right? 304 00:18:38,900 --> 00:18:39,400 So you-- 305 00:18:39,400 --> 00:18:40,316 PROFESSOR: Exactly. 306 00:18:40,316 --> 00:18:41,690 AUDIENCE: --still have a problem. 307 00:18:41,690 --> 00:18:42,356 PROFESSOR: Yeah. 308 00:18:42,356 --> 00:18:45,490 So as I have written it here I would agree. 309 00:18:45,490 --> 00:18:47,720 It looks completely useless. 310 00:18:47,720 --> 00:18:50,420 So let's make it useful. 311 00:18:50,420 --> 00:18:53,420 How am I going to make it useful? 312 00:18:53,420 --> 00:18:58,300 So let's start some set of corollaries to this theorem. 313 00:18:58,300 --> 00:19:02,830 For the first thing is we get rid of this definition 314 00:19:02,830 --> 00:19:06,320 by considering a reversible transformation. 315 00:19:14,150 --> 00:19:15,800 OK? 316 00:19:15,800 --> 00:19:20,560 So same picture that I was drawing before with multiple 317 00:19:20,560 --> 00:19:21,880 coordinates. 318 00:19:21,880 --> 00:19:24,830 Start some point and come back. 319 00:19:24,830 --> 00:19:30,010 But now I will draw is solid curve in this space, 320 00:19:30,010 --> 00:19:33,410 meaning that at each stage of the process 321 00:19:33,410 --> 00:19:37,890 I will do it sufficiently slowly so that my system comes 322 00:19:37,890 --> 00:19:41,440 to thermal equilibrium and I can identify it 323 00:19:41,440 --> 00:19:45,500 as a point in this generalized phase space. 324 00:19:45,500 --> 00:19:47,550 OK? 325 00:19:47,550 --> 00:19:50,960 So now I have points in this generalized phase space. 326 00:19:50,960 --> 00:19:54,320 To each point I can therefore assign the equilibrium 327 00:19:54,320 --> 00:19:55,060 temperature. 328 00:19:55,060 --> 00:19:57,702 That I am at that stage of the cycle. 329 00:19:57,702 --> 00:19:59,130 OK? 330 00:19:59,130 --> 00:20:03,180 So now I can request my Carnot engines 331 00:20:03,180 --> 00:20:08,750 to deliver the heat at the temperature of the system 332 00:20:08,750 --> 00:20:12,690 to be maintaining the whole system plus the Carnot engines 333 00:20:12,690 --> 00:20:15,960 at thermal equilibrium as I switch from one cycle 334 00:20:15,960 --> 00:20:17,540 to another cycle. 335 00:20:17,540 --> 00:20:18,790 OK? 336 00:20:18,790 --> 00:20:23,120 So now I have a relationship that 337 00:20:23,120 --> 00:20:26,670 is pertaining to things that are in equilibrium. 338 00:20:26,670 --> 00:20:29,530 As long as I follow the equilibrium temperature 339 00:20:29,530 --> 00:20:35,700 of the system and do reversible transformations 340 00:20:35,700 --> 00:20:40,585 I have the result that dQ reversible over Ts 341 00:20:40,585 --> 00:20:41,817 is less than 0. 342 00:20:44,619 --> 00:20:47,649 But the things about-- yes? 343 00:20:47,649 --> 00:20:49,098 AUDIENCE: Could you explain again 344 00:20:49,098 --> 00:20:54,894 why the path integral of dQ not over s 345 00:20:54,894 --> 00:20:56,360 is smaller than [INAUDIBLE]? 346 00:20:56,360 --> 00:20:58,130 PROFESSOR: OK. 347 00:20:58,130 --> 00:21:03,450 Clausius's theorem says that if I took heat from a vat 348 00:21:03,450 --> 00:21:09,560 and converted it to useful work I have violated the second law. 349 00:21:09,560 --> 00:21:11,680 So the sign of the net amount of heat 350 00:21:11,680 --> 00:21:16,370 that I extracted better be negative, right? 351 00:21:16,370 --> 00:21:16,870 OK. 352 00:21:16,870 --> 00:21:21,200 So you are not going to be able to have a vat 353 00:21:21,200 --> 00:21:26,250 and extract energy out of the lake and run your engine. 354 00:21:26,250 --> 00:21:28,370 It's not allowed. 355 00:21:28,370 --> 00:21:31,030 OK? 356 00:21:31,030 --> 00:21:37,240 Now my path is reversible, which means that I could go this way, 357 00:21:37,240 --> 00:21:40,660 or I could go the opposite way. 358 00:21:40,660 --> 00:21:42,270 And if I go the opposite way, what 359 00:21:42,270 --> 00:21:43,840 was the definition of reversible? 360 00:21:43,840 --> 00:21:48,910 I reverse all of a heat exchanges inputs and outputs. 361 00:21:48,910 --> 00:21:49,980 OK? 362 00:21:49,980 --> 00:21:52,340 Which means that what I was calling previously 363 00:21:52,340 --> 00:21:56,060 dQ becomes minus dQ. 364 00:21:56,060 --> 00:22:02,510 So the inequality holds for both minus or plus, 365 00:22:02,510 --> 00:22:07,340 which means that as long as I do reversible transformations 366 00:22:07,340 --> 00:22:11,721 I must have the integral to be equal to 0. 367 00:22:11,721 --> 00:22:14,530 OK? 368 00:22:14,530 --> 00:22:15,490 Good. 369 00:22:15,490 --> 00:22:19,660 So we constructed in a sense analog 370 00:22:19,660 --> 00:22:24,340 of doing frictionless ways of doing work 371 00:22:24,340 --> 00:22:26,650 and increasing the energy of the system. 372 00:22:26,650 --> 00:22:29,980 We have an idea about reversible ways 373 00:22:29,980 --> 00:22:32,450 and their relationship to T, well, 374 00:22:32,450 --> 00:22:36,200 one step further I need to go. 375 00:22:36,200 --> 00:22:43,800 Which is that I go from some point A 376 00:22:43,800 --> 00:22:49,024 to some point B in my coordinate space. 377 00:22:49,024 --> 00:22:49,910 OK? 378 00:22:49,910 --> 00:22:51,560 I can do it multiple ways. 379 00:22:51,560 --> 00:22:55,640 I can go this way and then come back. 380 00:22:55,640 --> 00:22:59,150 Reversibly, I can go this way and come back 381 00:22:59,150 --> 00:23:00,280 one way or the other. 382 00:23:00,280 --> 00:23:03,060 So every way I have a number of choices. 383 00:23:03,060 --> 00:23:06,830 Essentially what it says is that I can go from say A 384 00:23:06,830 --> 00:23:13,790 to B, dQ, reversible over T, over some path, 385 00:23:13,790 --> 00:23:19,362 and the answer is going to be the same for doing it 386 00:23:19,362 --> 00:23:20,320 along a different path. 387 00:23:27,780 --> 00:23:32,390 Because if I went along this path 388 00:23:32,390 --> 00:23:36,250 I can close the reversible cycle either by returning that way 389 00:23:36,250 --> 00:23:37,410 or by going the other way. 390 00:23:37,410 --> 00:23:39,780 So going this way or going that way, 391 00:23:39,780 --> 00:23:42,480 the result of the integration along reversible path 392 00:23:42,480 --> 00:23:43,920 should be the same. 393 00:23:43,920 --> 00:23:47,300 And it should be the same also for any other path 394 00:23:47,300 --> 00:23:49,140 that I take between these two points. 395 00:23:52,290 --> 00:23:57,850 So it kind of again reminds us of pushing a particle 396 00:23:57,850 --> 00:23:59,780 up a hill. 397 00:23:59,780 --> 00:24:02,510 And as long as you are doing things frictionlessly 398 00:24:02,510 --> 00:24:05,410 the amount of work that you do in this process 399 00:24:05,410 --> 00:24:07,210 does not depend on how you go there. 400 00:24:07,210 --> 00:24:09,010 It's just the potential energy difference 401 00:24:09,010 --> 00:24:11,090 between the two points. 402 00:24:11,090 --> 00:24:15,020 So this entity being independent of the path 403 00:24:15,020 --> 00:24:18,185 implies immediately that there is 404 00:24:18,185 --> 00:24:21,950 some function like a potential energy that only depends 405 00:24:21,950 --> 00:24:24,330 on the two end points and this is 406 00:24:24,330 --> 00:24:27,367 how you would decline the entropy. 407 00:24:36,169 --> 00:24:39,110 OK? 408 00:24:39,110 --> 00:24:44,200 So what we have essentially said is 409 00:24:44,200 --> 00:24:49,000 that if I take an infinitesimal step there's 410 00:24:49,000 --> 00:24:53,510 an infinitesimal amount of heat that I can do reversibly, that 411 00:24:53,510 --> 00:24:57,130 is related to the change of the state function 412 00:24:57,130 --> 00:25:04,750 through let's say dS is dQ reversible over T. 413 00:25:04,750 --> 00:25:08,975 Or vice versa, dQ [INAUDIBLE] TdS. 414 00:25:08,975 --> 00:25:09,475 OK? 415 00:25:13,300 --> 00:25:14,630 We've done it finally. 416 00:25:14,630 --> 00:25:17,700 We can get our expression because we 417 00:25:17,700 --> 00:25:21,430 have now the possibility to go from this point 418 00:25:21,430 --> 00:25:26,300 to this point reversibly using a reversible transformation. 419 00:25:32,840 --> 00:25:38,780 And calculate the change in energy. 420 00:25:38,780 --> 00:25:42,500 Now quite generally the change in energy conservation lay, 421 00:25:42,500 --> 00:25:46,390 it's dW plus dQ. 422 00:25:46,390 --> 00:25:50,527 If I do this reversible change so that each element 423 00:25:50,527 --> 00:25:51,360 I am in equilibrium. 424 00:25:51,360 --> 00:25:53,140 I know what J is. 425 00:25:53,140 --> 00:25:55,490 The dW is sum over iJidxi. 426 00:25:59,400 --> 00:26:01,440 And I have established that as long 427 00:26:01,440 --> 00:26:06,860 as I do the reversible transformations dQ is TdS. 428 00:26:06,860 --> 00:26:08,568 And so we have this formula. 429 00:26:12,120 --> 00:26:12,620 OK? 430 00:26:15,280 --> 00:26:19,090 So I will write that again because really this 431 00:26:19,090 --> 00:26:21,780 is the most important relationship 432 00:26:21,780 --> 00:26:25,640 that you need to know from thermal dynamics. 433 00:26:28,508 --> 00:26:32,310 And we have to put all kinds of colorful things 434 00:26:32,310 --> 00:26:34,074 around it so you remember. 435 00:26:36,940 --> 00:26:38,810 OK? 436 00:26:38,810 --> 00:26:44,330 Now in particular there is a very common misconception 437 00:26:44,330 --> 00:26:50,300 which is that results are relevant to transformations. 438 00:26:50,300 --> 00:26:54,600 And you derive this result for a reversible transformation. 439 00:26:54,600 --> 00:26:58,380 So this formula is only valid for reversible transformation. 440 00:26:58,380 --> 00:26:58,930 No. 441 00:26:58,930 --> 00:27:00,340 That's not the case. 442 00:27:00,340 --> 00:27:03,890 It is like saying, I calculated the potential energy 443 00:27:03,890 --> 00:27:08,230 of pushing something up the hill through a frictionless process. 444 00:27:08,230 --> 00:27:10,330 Therefore, the potential energy is 445 00:27:10,330 --> 00:27:13,310 only relevant for frictionless transformations. 446 00:27:13,310 --> 00:27:14,220 No. 447 00:27:14,220 --> 00:27:15,910 Potential energy exists. 448 00:27:15,910 --> 00:27:17,580 It's a function of states. 449 00:27:17,580 --> 00:27:19,410 Energy is a function of state. 450 00:27:19,410 --> 00:27:23,540 And we have been able to relate energy as a function of state 451 00:27:23,540 --> 00:27:26,940 to all the other equilibrium quantities that we 452 00:27:26,940 --> 00:27:29,026 have over here. 453 00:27:29,026 --> 00:27:29,797 OK? 454 00:27:29,797 --> 00:27:30,380 Any questions? 455 00:27:34,700 --> 00:27:36,240 OK. 456 00:27:36,240 --> 00:27:41,360 So now we can answer another side question 457 00:27:41,360 --> 00:27:44,250 that I raised at the beginning, which 458 00:27:44,250 --> 00:27:49,760 is I started with xi and Ji, as describing 459 00:27:49,760 --> 00:27:51,760 the system in equilibrium. 460 00:27:51,760 --> 00:27:55,480 Then I gradually added T. I added E, 461 00:27:55,480 --> 00:27:58,720 now I've added S. How many things 462 00:27:58,720 --> 00:28:01,920 do I need to describe the system in equilibrium? 463 00:28:01,920 --> 00:28:05,100 A lot of these are dependent on each other. 464 00:28:05,100 --> 00:28:07,490 And in particular saw that mechanically 465 00:28:07,490 --> 00:28:11,250 if there was the only thing that was happening in the story, 466 00:28:11,250 --> 00:28:14,440 J's could be obtained from the derivatives of energy 467 00:28:14,440 --> 00:28:16,390 with respect to x. 468 00:28:16,390 --> 00:28:20,550 So J's, in some sense, once you had E were given to you. 469 00:28:20,550 --> 00:28:22,290 You didn't have to list all of the J's. 470 00:28:22,290 --> 00:28:26,650 You just needed to add E as a function of x's. 471 00:28:26,650 --> 00:28:30,120 But when you have thermal transformations around, 472 00:28:30,120 --> 00:28:31,540 that's not enough. 473 00:28:31,540 --> 00:28:36,030 This equation says that if there are n ways of doing work 474 00:28:36,030 --> 00:28:40,130 on the system, there is one of the doing heat. 475 00:28:40,130 --> 00:28:44,590 You have n plus 1 independent degrees of freedom.. 476 00:28:44,590 --> 00:28:54,320 So n ways of sum over i, Ji, dxi, one way of SdT, 477 00:28:54,320 --> 00:29:04,420 you have n plus one independent variables 478 00:29:04,420 --> 00:29:05,580 to describe your system. 479 00:29:12,920 --> 00:29:17,000 And once you realize that you have 480 00:29:17,000 --> 00:29:20,160 a certain amount of freedom in choosing precisely 481 00:29:20,160 --> 00:29:23,060 which n plus one you want to select. 482 00:29:23,060 --> 00:29:31,810 So I could select all of the x's and one temperature. 483 00:29:31,810 --> 00:29:35,270 I could select all of the J's and one energy. 484 00:29:35,270 --> 00:29:37,870 So I have a number of choices. 485 00:29:37,870 --> 00:29:42,140 And once I have made my choice I can rearrange things 486 00:29:42,140 --> 00:29:43,000 accordingly. 487 00:29:43,000 --> 00:29:46,750 So suppose I had chosen the energy in xi, 488 00:29:46,750 --> 00:29:54,870 then I could write dS to be dE over T minus sum 489 00:29:54,870 --> 00:29:59,200 over i Ji over T xi. 490 00:29:59,200 --> 00:30:04,166 So I just rearrange this expression and solve for dS. 491 00:30:04,166 --> 00:30:07,650 So this amounts to my having chosen 492 00:30:07,650 --> 00:30:13,710 x's and E's as independent variables and prescribed 493 00:30:13,710 --> 00:30:20,820 through this important law s as a function of E and x's. 494 00:30:20,820 --> 00:30:22,540 What about everything else? 495 00:30:22,540 --> 00:30:25,970 Well, everything else you can calculate by derivatives. 496 00:30:25,970 --> 00:30:31,740 So 1 over T would be dS by dE at constant x. 497 00:30:31,740 --> 00:30:44,000 While Ji over T would be minus dS by dxi at constant E and xJ 498 00:30:44,000 --> 00:30:45,362 that is not equal to i. 499 00:30:53,298 --> 00:30:55,290 OK? 500 00:30:55,290 --> 00:30:55,920 Fine. 501 00:30:55,920 --> 00:31:00,810 So that's sort of kind of extract 502 00:31:00,810 --> 00:31:06,690 the mathematical content that we can get out of the second law 503 00:31:06,690 --> 00:31:09,020 through this Clausius's theorem. 504 00:31:09,020 --> 00:31:12,730 There is one important corollary that 505 00:31:12,730 --> 00:31:16,210 has to do be irreversible transformations. 506 00:31:24,220 --> 00:31:29,180 After all, in setting up the Clausius's theorem 507 00:31:29,180 --> 00:31:33,080 I did not say anything about the necessity 508 00:31:33,080 --> 00:31:38,140 of being in equilibrium in order to achieve the inequality. 509 00:31:38,140 --> 00:31:42,340 Reversibility allowed me to get it as an equality. 510 00:31:42,340 --> 00:31:44,010 So what can I do? 511 00:31:44,010 --> 00:31:47,950 I can take any complicated space. 512 00:31:47,950 --> 00:31:54,340 I can pick a point A and in principle, 513 00:31:54,340 --> 00:31:59,690 make an irreversible transformation to some point B. 514 00:31:59,690 --> 00:32:02,860 And maybe in that process I will get 515 00:32:02,860 --> 00:32:05,940 some heat inputted through the system dQ. 516 00:32:08,590 --> 00:32:18,310 And what I would have is that the integral going from A to B, 517 00:32:18,310 --> 00:32:28,550 dQ divided by T along the path that I have described here 518 00:32:28,550 --> 00:32:31,830 which is irreversible. 519 00:32:31,830 --> 00:32:34,960 But I can't say anything about it at this point. 520 00:32:34,960 --> 00:32:38,560 Clausius's theorem has to do with cycles. 521 00:32:38,560 --> 00:32:45,470 So I can, in principle connect back from B to A reversibly. 522 00:32:45,470 --> 00:32:55,920 So I go integral from B to A, dQ reversible over T. 523 00:32:55,920 --> 00:33:00,310 And having completed the cycle I know for this cycle 524 00:33:00,310 --> 00:33:02,020 that this is negative. 525 00:33:05,190 --> 00:33:10,470 If I were to take this to the other side of the equation 526 00:33:10,470 --> 00:33:15,240 it becomes essentially the difference of the entropies, 527 00:33:15,240 --> 00:33:17,850 or here it's also the difference of entropies. 528 00:33:17,850 --> 00:33:20,530 But I prefer it to be on the other side. 529 00:33:20,530 --> 00:33:23,470 And then to make these two points very 530 00:33:23,470 --> 00:33:26,310 close to each other, in which case, 531 00:33:26,310 --> 00:33:33,747 I can write that dQ is less than or equal to TdS. 532 00:33:39,400 --> 00:33:46,230 This is the change in entropy in going from A to B. OK? 533 00:33:46,230 --> 00:33:51,530 Again, there's some question here as to what this T is. 534 00:33:51,530 --> 00:33:55,170 So let's get rid of that uncertainty 535 00:33:55,170 --> 00:33:57,470 by looking at processes that are adiabatic. 536 00:34:02,410 --> 00:34:04,800 So I make sure that there is no heat 537 00:34:04,800 --> 00:34:10,639 exchange in going from A to B. What do I conclude then? 538 00:34:10,639 --> 00:34:14,560 Is that for these processes the change in entropy 539 00:34:14,560 --> 00:34:15,380 has to be positive. 540 00:34:18,121 --> 00:34:18,620 OK? 541 00:34:18,620 --> 00:34:22,900 Because T's are generally positive quantities. 542 00:34:22,900 --> 00:34:24,370 OK? 543 00:34:24,370 --> 00:34:28,280 So a consequence of Clausius's theorem 544 00:34:28,280 --> 00:34:33,730 is that entropy can only increase in any transformation. 545 00:34:33,730 --> 00:34:37,080 Its change in entropy would be 0 for these adiabatic 546 00:34:37,080 --> 00:34:44,360 transformations if you have reversible processes. 547 00:34:46,989 --> 00:34:49,630 So this is actually another one of those statements 548 00:34:49,630 --> 00:34:51,480 that you have to think a little bit 549 00:34:51,480 --> 00:34:55,480 to see whether it has any use or not. 550 00:34:55,480 --> 00:35:00,250 Because we all say that entropy has to increase, 551 00:35:00,250 --> 00:35:02,670 but is that what we have proven here? 552 00:35:02,670 --> 00:35:04,080 Not quite. 553 00:35:04,080 --> 00:35:09,900 Because what we've proven is that if you have a system that 554 00:35:09,900 --> 00:35:16,090 is isolated, so there is a system that is isolated 555 00:35:16,090 --> 00:35:19,790 because I want dQ to be 0. 556 00:35:19,790 --> 00:35:24,060 And I want entropy to increase. 557 00:35:24,060 --> 00:35:25,290 OK? 558 00:35:25,290 --> 00:35:30,160 So let's say we start with some initial condition 559 00:35:30,160 --> 00:35:32,850 and we wait and go to some final condition 560 00:35:32,850 --> 00:35:35,650 of whatever is inside the box. 561 00:35:35,650 --> 00:35:39,470 I should be able to calculate entropy 562 00:35:39,470 --> 00:35:46,160 before and after in order to see that this change is positive. 563 00:35:46,160 --> 00:35:46,660 OK? 564 00:35:46,660 --> 00:35:50,320 But we say entropy is a function of state. 565 00:35:50,320 --> 00:35:57,780 And what have I done here to change this state? 566 00:35:57,780 --> 00:36:00,810 I cannot calculate entropy if the system is not 567 00:36:00,810 --> 00:36:02,520 in equilibrium. 568 00:36:02,520 --> 00:36:04,890 If the system is in equilibrium it presumably 569 00:36:04,890 --> 00:36:07,470 does not change as a function of time. 570 00:36:07,470 --> 00:36:10,440 So what is this expression relevant to? 571 00:36:10,440 --> 00:36:12,390 The expression is relevant to when 572 00:36:12,390 --> 00:36:16,100 you have some kind of a constraint that you remove. 573 00:36:16,100 --> 00:36:18,730 So imagine that let's say there is 574 00:36:18,730 --> 00:36:23,930 a gas in this room and some kind of piston that is initially 575 00:36:23,930 --> 00:36:26,910 clamped to some position. 576 00:36:26,910 --> 00:36:29,120 Then I can calculate the entropy of what's 577 00:36:29,120 --> 00:36:31,450 on the left side, what's on the right hand side, 578 00:36:31,450 --> 00:36:33,930 and that would be my initial entropy. 579 00:36:33,930 --> 00:36:35,720 I remove the clamp. 580 00:36:35,720 --> 00:36:39,940 I remove the constraint so this can slide back and forth. 581 00:36:39,940 --> 00:36:43,370 And eventually it comes to another position. 582 00:36:43,370 --> 00:36:46,510 So then I can again calculate the final entropy 583 00:36:46,510 --> 00:36:48,380 from the sum of what's on the left side 584 00:36:48,380 --> 00:36:50,100 and what's on the right hand side. 585 00:36:50,100 --> 00:36:52,800 And I can see that entropy is increased. 586 00:36:52,800 --> 00:36:56,410 So basically this statement really 587 00:36:56,410 --> 00:37:01,200 is useful when there is some internal constraint 588 00:37:01,200 --> 00:37:03,400 that you can remove. 589 00:37:03,400 --> 00:37:05,910 And as you remove the internal constraint, 590 00:37:05,910 --> 00:37:07,270 the entropy increases. 591 00:37:07,270 --> 00:37:09,810 So I guess that image that we all have 592 00:37:09,810 --> 00:37:13,410 is that if I give you two pictures of a room, one 593 00:37:13,410 --> 00:37:17,120 with books nicely arranged on a shelf and one 594 00:37:17,120 --> 00:37:19,680 with books randomly distributed you 595 00:37:19,680 --> 00:37:22,740 would have no problem to time order them. 596 00:37:22,740 --> 00:37:25,210 And actually presumably there was some constraint 597 00:37:25,210 --> 00:37:27,730 from the shelf that was removed and then 598 00:37:27,730 --> 00:37:29,800 these things fell down. 599 00:37:29,800 --> 00:37:32,410 So that's the kind of process for which 600 00:37:32,410 --> 00:37:36,037 this statement of the second law is appropriate. 601 00:37:36,037 --> 00:37:36,620 Any questions? 602 00:37:40,010 --> 00:37:41,320 OK. 603 00:37:41,320 --> 00:37:46,350 So that statement is something about approach to equilibrium. 604 00:37:49,940 --> 00:37:54,500 But as I said that there was some initial configuration 605 00:37:54,500 --> 00:37:59,750 with some constraint and as a function of removing 606 00:37:59,750 --> 00:38:01,680 that constraint the system would go 607 00:38:01,680 --> 00:38:05,960 from one sort of equilibrium, constrained equilibrium 608 00:38:05,960 --> 00:38:11,410 to an unconstrained equilibrium and this increase of entropy 609 00:38:11,410 --> 00:38:14,080 tells us in which direction it would go. 610 00:38:14,080 --> 00:38:17,470 You can time order the various processes. 611 00:38:17,470 --> 00:38:23,050 Now sometimes this adiabatic way of doing things 612 00:38:23,050 --> 00:38:25,180 is not the best way. 613 00:38:25,180 --> 00:38:30,400 And so depending on what it is that you are looking at, 614 00:38:30,400 --> 00:38:32,840 rather than looking at entropy you 615 00:38:32,840 --> 00:38:36,540 will be looking at different functions. 616 00:38:36,540 --> 00:38:39,690 And so the next step is to construct different types 617 00:38:39,690 --> 00:38:42,310 of functions that are useful for telling us 618 00:38:42,310 --> 00:38:44,820 something about equilibrium. 619 00:38:44,820 --> 00:38:49,040 Now I'll start doing that by some function with which you 620 00:38:49,040 --> 00:38:51,290 are very familiar. 621 00:38:51,290 --> 00:38:57,170 And has practically nothing to do 622 00:38:57,170 --> 00:38:59,300 with entropy and temperature. 623 00:38:59,300 --> 00:39:01,970 So we are going to look at mechanical equilibrium. 624 00:39:07,327 --> 00:39:08,790 OK. 625 00:39:08,790 --> 00:39:11,720 And so the kind of prototype of this, 626 00:39:11,720 --> 00:39:16,980 since we've been thinking in terms of springs or wires, 627 00:39:16,980 --> 00:39:20,690 is let's imagine that I have a wire that 628 00:39:20,690 --> 00:39:23,490 has some natural lent. 629 00:39:23,490 --> 00:39:26,970 And it is sitting there happily in equilibrium. 630 00:39:26,970 --> 00:39:32,090 And then I attach, let's say a mast to it and because 631 00:39:32,090 --> 00:39:36,030 of that I'm pulling it with a force 632 00:39:36,030 --> 00:39:37,690 that I will indicate by J. 633 00:39:37,690 --> 00:39:39,740 So J, if you are in gravity, would 634 00:39:39,740 --> 00:39:43,780 be mass of this times J. OK? 635 00:39:43,780 --> 00:39:47,480 Now if you were doing things in vacuum, 636 00:39:47,480 --> 00:39:51,890 what would happen to your x, which 637 00:39:51,890 --> 00:39:53,740 is the distortion that you would have 638 00:39:53,740 --> 00:39:56,440 from the equilibrium as a function of time? 639 00:40:00,590 --> 00:40:03,240 What would happen is presumably the thing 640 00:40:03,240 --> 00:40:07,930 would start to oscillate and you would get something like this. 641 00:40:07,930 --> 00:40:11,210 But certainly not equilibrium, but in real world 642 00:40:11,210 --> 00:40:14,380 there is always friction that is operating. 643 00:40:14,380 --> 00:40:16,420 And so if there is a lot of friction 644 00:40:16,420 --> 00:40:19,950 what is going to happen is that your x 645 00:40:19,950 --> 00:40:24,220 will come to some final value. 646 00:40:24,220 --> 00:40:29,460 So again, this is a constraint, which I allowed this to move. 647 00:40:29,460 --> 00:40:31,700 And because of that constraint I went 648 00:40:31,700 --> 00:40:34,530 from the initial equilibrium that corresponded to x 649 00:40:34,530 --> 00:40:38,460 equals to 0 to some final equilibrium. 650 00:40:38,460 --> 00:40:40,350 And you say, well, if I want to calculate 651 00:40:40,350 --> 00:40:42,820 that final equilibrium, what I need 652 00:40:42,820 --> 00:40:45,950 to do is to remove all of the kinetic energy 653 00:40:45,950 --> 00:40:48,350 so that the thing does not move anymore, 654 00:40:48,350 --> 00:40:52,020 which means that I have to minimize the potential energy. 655 00:40:52,020 --> 00:40:54,070 I'll give it the symbol H for reasons 656 00:40:54,070 --> 00:40:55,790 to become apparent shortly. 657 00:40:55,790 --> 00:40:58,980 And that is the sum of the energy 658 00:40:58,980 --> 00:41:00,990 that I have in the spring. 659 00:41:00,990 --> 00:41:03,180 Let's imagine that it is a hook end spring. 660 00:41:03,180 --> 00:41:06,920 We would have a formula such as this, plus the potential energy 661 00:41:06,920 --> 00:41:09,580 that I have in my mass, which is related 662 00:41:09,580 --> 00:41:11,780 to how far I go up and down. 663 00:41:11,780 --> 00:41:18,140 And say an appropriate description is minus Jx. 664 00:41:18,140 --> 00:41:18,640 OK? 665 00:41:18,640 --> 00:41:21,990 So this is the net potential energy of the spring plus 666 00:41:21,990 --> 00:41:23,840 whatever is supplying the mass that 667 00:41:23,840 --> 00:41:26,290 is applying in the external force. 668 00:41:26,290 --> 00:41:31,280 And so in this case, what you have is that minimizing 669 00:41:31,280 --> 00:41:36,590 H with respect to the constraint x 670 00:41:36,590 --> 00:41:41,600 will give you that the x equilibrium is J over K. OK? 671 00:41:41,600 --> 00:41:47,460 So basically it comes down to J over K. 672 00:41:47,460 --> 00:41:51,350 And that the value of this function, potential energy 673 00:41:51,350 --> 00:41:55,310 function in equilibrium substituting this over here 674 00:41:55,310 --> 00:42:01,796 is minus J squared over K. 2K. 675 00:42:01,796 --> 00:42:02,296 OK? 676 00:42:06,670 --> 00:42:14,170 So in many processes that involve thermodynamics 677 00:42:14,170 --> 00:42:17,490 we are going to do essentially the same thing. 678 00:42:17,490 --> 00:42:21,470 And we are going to call this net potential central energy, 679 00:42:21,470 --> 00:42:23,954 in fact, an enthalpy. 680 00:42:23,954 --> 00:42:25,360 OK? 681 00:42:25,360 --> 00:42:29,090 And the generalized version of what I wrote down for you 682 00:42:29,090 --> 00:42:35,880 is the following, in the process that I described 683 00:42:35,880 --> 00:42:39,630 which was related to the increase in entropy 684 00:42:39,630 --> 00:42:42,085 we were dealing with a system that was isolated. 685 00:42:45,090 --> 00:42:49,220 dQ was 0 and there was no work because the coordinates were 686 00:42:49,220 --> 00:42:52,040 not changing. dW was 0. 687 00:42:52,040 --> 00:42:54,890 Whereas here, I am looking at a process 688 00:42:54,890 --> 00:42:57,820 where there is a certain amount of work 689 00:42:57,820 --> 00:43:04,020 because this external J that I am putting 690 00:43:04,020 --> 00:43:06,010 is doing work on the spring. 691 00:43:06,010 --> 00:43:10,020 It is adding energy to the spring. 692 00:43:10,020 --> 00:43:15,080 And the thing is that because of the friction 693 00:43:15,080 --> 00:43:17,540 it doesn't continue to oscillate. 694 00:43:17,540 --> 00:43:24,410 So some of the energy that I would have put in the system, 695 00:43:24,410 --> 00:43:32,670 if I really were using the formula Jidxi is lost somehow. 696 00:43:32,670 --> 00:43:37,460 We can see those oscillations have gone away. 697 00:43:37,460 --> 00:43:45,310 Now I'm always doing this process at constant J. Right? 698 00:43:45,310 --> 00:43:55,134 So if I prescribe for you some path of x as a function of T, 699 00:43:55,134 --> 00:44:01,760 Jdx is the same thing as d of Jx because J does not change. 700 00:44:01,760 --> 00:44:03,596 x changes. 701 00:44:03,596 --> 00:44:06,560 OK? 702 00:44:06,560 --> 00:44:10,830 I'm going to also imagine processes 703 00:44:10,830 --> 00:44:20,760 where mechanical equilibrium, but with no heat so that dQ 704 00:44:20,760 --> 00:44:21,370 is 0. 705 00:44:28,220 --> 00:44:31,590 If I were to add these two together, 706 00:44:31,590 --> 00:44:34,265 then I will get an inequality involving 707 00:44:34,265 --> 00:44:44,110 dE, which is less than or equal to d of Jixi, which I can write 708 00:44:44,110 --> 00:44:54,250 as the change in e minus Jixi has to be negative. 709 00:44:54,250 --> 00:44:57,330 So appropriate to processes, which 710 00:44:57,330 --> 00:45:00,310 are conducted so that there is mechanical work 711 00:45:00,310 --> 00:45:04,410 at constant force, is this function 712 00:45:04,410 --> 00:45:08,960 that is usually called H for enthalpy. 713 00:45:08,960 --> 00:45:14,310 And what we find is that the dH is going to be negative. 714 00:45:14,310 --> 00:45:16,140 And again, if you think about it this 715 00:45:16,140 --> 00:45:19,410 is precisely this enthalpy is none other 716 00:45:19,410 --> 00:45:21,095 than the potential energy that is 717 00:45:21,095 --> 00:45:23,270 in the mechanical degrees of freedom. 718 00:45:23,270 --> 00:45:27,110 And there's always loss in the universe. 719 00:45:27,110 --> 00:45:32,050 And so the direction naturally is for this potential energy 720 00:45:32,050 --> 00:45:34,700 to decrease. 721 00:45:34,700 --> 00:45:42,390 One thing to note is that dH is dE. 722 00:45:44,950 --> 00:45:52,860 Thinking of it not now as a process going on at constant J, 723 00:45:52,860 --> 00:45:56,620 but as a change in a function of state 724 00:45:56,620 --> 00:46:00,320 because we said that once I prescribe where 725 00:46:00,320 --> 00:46:04,590 I am in the coordinate space I know what, say e is. 726 00:46:04,590 --> 00:46:05,830 I know what J is. 727 00:46:05,830 --> 00:46:11,130 I can certainly construct a function, which is e minus Jx. 728 00:46:11,130 --> 00:46:12,950 It's another function of state. 729 00:46:12,950 --> 00:46:18,670 And I can ask what happens if I make a change from one 730 00:46:18,670 --> 00:46:21,800 point in space to another point. 731 00:46:21,800 --> 00:46:26,250 Along the path that potentially allows variations in J dE 732 00:46:26,250 --> 00:46:27,370 I know, is Jidxi. 733 00:46:31,980 --> 00:46:34,760 And then, actually I'm using summation conversion. 734 00:46:34,760 --> 00:46:39,320 So let's remove that, plus TdS and here 735 00:46:39,320 --> 00:46:43,470 it becomes minus Jidxi minus xidJi. 736 00:46:47,510 --> 00:46:51,040 Jidxi is cancelled. 737 00:46:51,040 --> 00:46:59,570 And we that if I construct the function H as e minus Jx, 738 00:46:59,570 --> 00:47:04,420 the natural variations of it are TdS, just 739 00:47:04,420 --> 00:47:11,215 like dE but rather than Jdx I have minus xdJ. 740 00:47:13,965 --> 00:47:16,760 OK? 741 00:47:16,760 --> 00:47:21,970 So H is naturally expressed as a function of variables that are 742 00:47:21,970 --> 00:47:25,720 s and the Ji's. 743 00:47:25,720 --> 00:47:29,750 And remember that we said we are doing things at constant J. 744 00:47:29,750 --> 00:47:32,460 So it's nice that it happens that way. 745 00:47:32,460 --> 00:47:36,550 And in particular if I do partial derivatives 746 00:47:36,550 --> 00:47:48,610 I find that my xi is minus dH by dJi at constant s. 747 00:47:48,610 --> 00:47:50,890 So that in the same way before I said 748 00:47:50,890 --> 00:47:55,160 that if you have the energy function dE by dx will give J, 749 00:47:55,160 --> 00:48:00,212 if you have the H functions, dH by dJ would give you x. 750 00:48:00,212 --> 00:48:00,920 Let's check. 751 00:48:00,920 --> 00:48:04,310 Here we had an H function. 752 00:48:04,310 --> 00:48:11,020 If I do a dH by dJ with a minus sign, what do I get? 753 00:48:11,020 --> 00:48:14,810 I will get J over K. What was my equilibrium x? 754 00:48:14,810 --> 00:48:18,415 It was J over K. OK? 755 00:48:32,860 --> 00:48:37,030 Now both of these functions that we have encountered so far 756 00:48:37,030 --> 00:48:41,670 were expressing the system energy 757 00:48:41,670 --> 00:48:47,535 or enthalpy have as their argument entropy. 758 00:48:47,535 --> 00:48:52,190 Now if I give you a box you will have a hard time potentially 759 00:48:52,190 --> 00:48:57,110 figuring out what entropy is, but varying temperature 760 00:48:57,110 --> 00:48:59,580 is something that we do all the time. 761 00:48:59,580 --> 00:49:02,930 So it would be great if we could start expressing things 762 00:49:02,930 --> 00:49:05,930 not in terms of entropy, in terms of anything else. 763 00:49:05,930 --> 00:49:09,020 Well, what's the natural thing if not entropy? 764 00:49:09,020 --> 00:49:11,840 Here we went from x to the conjugate variable 765 00:49:11,840 --> 00:49:17,670 J. I can make a transformation that goes from s 766 00:49:17,670 --> 00:49:22,780 to the conjugate variable, which is temperature. 767 00:49:22,780 --> 00:49:25,760 And that would be relevant, therefore, 768 00:49:25,760 --> 00:49:30,000 when I'm thinking about isothermal processes. 769 00:49:30,000 --> 00:49:34,720 And so what I'm going to do is to reverse the role of the two 770 00:49:34,720 --> 00:49:41,420 elements of energy that I had previously 771 00:49:41,420 --> 00:49:43,340 for mechanical processes. 772 00:49:43,340 --> 00:49:48,710 So what we had there was that dQ was 0, 773 00:49:48,710 --> 00:49:50,480 but I want dQ to be non-0. 774 00:49:53,170 --> 00:49:56,720 What we had there was that dW was non-0. 775 00:49:56,720 --> 00:50:00,880 So let's figure out processes where dW is 0, 776 00:50:00,880 --> 00:50:05,230 but dQ, the same way that previously we were doing dW 777 00:50:05,230 --> 00:50:13,780 at constant J, let's do dQ not equals to 0 at constant T. 778 00:50:13,780 --> 00:50:18,570 And constant T is where the isothermal expression comes 779 00:50:18,570 --> 00:50:19,900 into play. 780 00:50:19,900 --> 00:50:21,920 OK? 781 00:50:21,920 --> 00:50:32,760 So if dW is 0, I have a natural inequality that involves dQ. 782 00:50:32,760 --> 00:50:35,680 I just erased it, I think. 783 00:50:35,680 --> 00:50:36,180 Yeah. 784 00:50:36,180 --> 00:50:37,190 It's not here. 785 00:50:37,190 --> 00:50:40,226 dQ is less than or equal to TdS. 786 00:50:43,410 --> 00:50:46,800 And if T is a constant throughout the process, 787 00:50:46,800 --> 00:50:50,356 I can write this as d of TS. 788 00:50:50,356 --> 00:50:55,890 Add these two together to get that dE less than 789 00:50:55,890 --> 00:51:03,820 or equal to d of TS, or taking it to the other side, 790 00:51:03,820 --> 00:51:06,895 I can define a function which is the Helmholtz free energy. 791 00:51:13,200 --> 00:51:18,070 F, which is E minus TS. 792 00:51:18,070 --> 00:51:21,270 And what we have is that the natural direction 793 00:51:21,270 --> 00:51:24,590 of flow for free energy when we remove 794 00:51:24,590 --> 00:51:27,810 some kind of a constraint at constant temperature 795 00:51:27,810 --> 00:51:33,540 is that we will tend to minimize the free energy. 796 00:51:33,540 --> 00:51:38,860 And then again doing the manipulation of dF 797 00:51:38,860 --> 00:51:47,760 is dE minus d of TS in general will have minus TdS minus SdT. 798 00:51:47,760 --> 00:51:51,720 The TdS part of the E will vanish and what I will have 799 00:51:51,720 --> 00:51:59,410 is Jidxi minus SdT. 800 00:51:59,410 --> 00:52:02,560 So now we can see that finally for the free energy, 801 00:52:02,560 --> 00:52:06,490 the Helmholtz free energy, and natural variable 802 00:52:06,490 --> 00:52:12,540 rather than being entropy is the temperature and all 803 00:52:12,540 --> 00:52:14,857 set of displacements. 804 00:52:14,857 --> 00:52:17,190 And then if you ask, well, what happened to the entropy? 805 00:52:17,190 --> 00:52:21,560 You say, well, I can get the entropy as minus dF 806 00:52:21,560 --> 00:52:23,805 by dT at constant x. 807 00:52:51,030 --> 00:52:52,740 If I take this picture of the gas 808 00:52:52,740 --> 00:52:57,740 that I had put at the beginning, which was just erased, 809 00:52:57,740 --> 00:53:00,980 picture of a piston that can slide. 810 00:53:00,980 --> 00:53:02,770 And I think of that as a box that I 811 00:53:02,770 --> 00:53:06,070 bring into this room, well, then if you'll quickly 812 00:53:06,070 --> 00:53:09,560 adjust to the temperature of the room, 813 00:53:09,560 --> 00:53:12,220 as well as the pressure of the room. 814 00:53:12,220 --> 00:53:16,120 So this is going to be a general transformation in which 815 00:53:16,120 --> 00:53:20,500 both heat and the work are exchanged. 816 00:53:20,500 --> 00:53:25,410 If is at external temperature and pressure that is fixed. 817 00:53:25,410 --> 00:53:33,801 So it's a transformation that is isothermal and constant J. 818 00:53:33,801 --> 00:53:35,550 In pressure you would call it [INAUDIBLE]. 819 00:53:39,230 --> 00:53:40,610 OK? 820 00:53:40,610 --> 00:53:42,300 And then quite generally you would 821 00:53:42,300 --> 00:53:45,940 say extending what we had before, 822 00:53:45,940 --> 00:53:53,660 in this case dQ is, again, less than TdS because of Clausius. 823 00:53:53,660 --> 00:54:00,490 dW is less than Jdx because you always 824 00:54:00,490 --> 00:54:07,300 lose something to friction. 825 00:54:07,300 --> 00:54:09,670 You always lose something to friction. 826 00:54:09,670 --> 00:54:12,100 You add the two things together, and you 827 00:54:12,100 --> 00:54:18,630 get that dE is less than or equal to TdS, plus Jidxi. 828 00:54:21,360 --> 00:54:23,940 For the transformations that we are considering 829 00:54:23,940 --> 00:54:29,100 that are at constant t and J I can 830 00:54:29,100 --> 00:54:33,860 take these expressions to mean the same thing as d 831 00:54:33,860 --> 00:54:43,450 of ts plus Jixi, which means that more generally, 832 00:54:43,450 --> 00:54:53,060 I can define a function which is the Gibbs free energy, G, 833 00:54:53,060 --> 00:54:57,270 which is the energy minus ts. 834 00:54:57,270 --> 00:55:02,890 This part is the Helmholtz free energy, minus Jixi. 835 00:55:09,760 --> 00:55:16,010 And that if I regard this as a function of state, which 836 00:55:16,010 --> 00:55:21,445 certainly will be minimized under conditions of constant J 837 00:55:21,445 --> 00:55:25,280 and T for some constraint that is removed. 838 00:55:25,280 --> 00:55:28,130 Quite generally as a state function 839 00:55:28,130 --> 00:55:30,420 depending on all of these variables, 840 00:55:30,420 --> 00:55:35,000 it's variations would be able to be 841 00:55:35,000 --> 00:55:41,188 expressed as minus SdT minus Jidxi. 842 00:55:44,394 --> 00:55:50,162 ie, our G is naturally a function of temperature 843 00:55:50,162 --> 00:55:51,370 and the set of displacements. 844 00:55:59,610 --> 00:56:00,110 OK. 845 00:56:00,110 --> 00:56:02,520 I don't think music was appropriate at this point. 846 00:56:02,520 --> 00:56:06,090 I would have put it earlier, but such is life. 847 00:56:10,110 --> 00:56:11,930 OK. 848 00:56:11,930 --> 00:56:15,550 Now there is something that I kind of 849 00:56:15,550 --> 00:56:19,087 did not pay sufficient-- yes? 850 00:56:19,087 --> 00:56:19,753 AUDIENCE: Sorry. 851 00:56:19,753 --> 00:56:24,500 Didn't you get it backwards between the J's and the x's? 852 00:56:24,500 --> 00:56:26,400 PROFESSOR: I did. 853 00:56:26,400 --> 00:56:26,990 I did. 854 00:56:26,990 --> 00:56:28,690 I wrote it wrong here. 855 00:56:31,730 --> 00:56:34,027 So maybe that's what the music was for. 856 00:56:38,600 --> 00:56:39,790 OK. 857 00:56:39,790 --> 00:56:42,082 Is it fine? 858 00:56:42,082 --> 00:56:42,930 Anything else? 859 00:56:48,210 --> 00:56:49,340 OK. 860 00:56:49,340 --> 00:56:53,140 The thing that I wasn't sufficiently careful with 861 00:56:53,140 --> 00:56:57,470 was the list of things that goes into mechanical work. 862 00:57:01,080 --> 00:57:08,210 And one additional care that what needs to have, 863 00:57:08,210 --> 00:57:13,460 which is that suppose I were to tell you, 864 00:57:13,460 --> 00:57:17,710 let's say, the pressure and temperature as my two 865 00:57:17,710 --> 00:57:21,070 variables, and I guess this is what I would have here 866 00:57:21,070 --> 00:57:25,800 if I go and look at pressure as my force. 867 00:57:25,800 --> 00:57:28,600 Then have I told you everything that I 868 00:57:28,600 --> 00:57:30,016 need to know about the system? 869 00:57:34,190 --> 00:57:37,670 So I have a box, let's say, in this room 870 00:57:37,670 --> 00:57:39,450 that is at the pressure of this room 871 00:57:39,450 --> 00:57:41,320 and at the temperature of this room. 872 00:57:41,320 --> 00:57:45,810 Have I completely specified the box? 873 00:57:45,810 --> 00:57:47,807 What have I left? 874 00:57:47,807 --> 00:57:49,140 AUDIENCE: The size. [INAUDIBLE]. 875 00:57:49,140 --> 00:57:51,956 PROFESSOR: How big is it, yes. 876 00:57:51,956 --> 00:57:57,150 So what I have left, previously maybe I had volume, 877 00:57:57,150 --> 00:57:58,970 but I sort of discarded the volume 878 00:57:58,970 --> 00:58:02,020 in terms of the pressure. 879 00:58:02,020 --> 00:58:05,510 So what I'd really need is to have the number of particles. 880 00:58:05,510 --> 00:58:07,065 I have to specify something. 881 00:58:07,065 --> 00:58:09,190 Maybe the mass, something that has 882 00:58:09,190 --> 00:58:11,640 still is going to distinguish these boxes 883 00:58:11,640 --> 00:58:14,630 in terms of their size. 884 00:58:14,630 --> 00:58:18,140 Now it is most useful to think in terms 885 00:58:18,140 --> 00:58:20,350 of the number of particles. 886 00:58:20,350 --> 00:58:24,170 So I give you a box and I can say that within this box 887 00:58:24,170 --> 00:58:26,790 I have so many molecules of oxygen and nitrogen, 888 00:58:26,790 --> 00:58:32,870 or whatever is the composition, but then we sort of start 889 00:58:32,870 --> 00:58:35,140 to get into the realm of chemistry and the fact 890 00:58:35,140 --> 00:58:38,000 that the different chemical components can start 891 00:58:38,000 --> 00:58:42,040 to react with each other, maybe even 892 00:58:42,040 --> 00:58:44,840 if you have a box that is one component, 893 00:58:44,840 --> 00:58:48,720 some of the molecules are going to get absorbed on the surface. 894 00:58:48,720 --> 00:58:52,460 So precisely what the number is is potentially 895 00:58:52,460 --> 00:58:54,590 something that is a variable. 896 00:58:54,590 --> 00:58:59,850 And in order to specify exactly what the energy 897 00:58:59,850 --> 00:59:03,210 content of a system is you have to specify 898 00:59:03,210 --> 00:59:05,750 how many particles and the energy 899 00:59:05,750 --> 00:59:08,010 carried by those particles. 900 00:59:08,010 --> 00:59:12,540 So the list of things that appears here, 901 00:59:12,540 --> 00:59:16,970 especially when you think in terms of chemical systems, 902 00:59:16,970 --> 00:59:20,045 has an additional element that is typically 903 00:59:20,045 --> 00:59:20,920 called chemical work. 904 00:59:25,110 --> 00:59:31,470 And so let's separate mechanical work from the chemical work. 905 00:59:31,470 --> 00:59:38,090 So mechanical work was sum over iJidxi 906 00:59:38,090 --> 00:59:40,750 where these were real displacements 907 00:59:40,750 --> 00:59:44,100 and we can actually write an expression that 908 00:59:44,100 --> 00:59:49,880 is very similar here where the displacements are 909 00:59:49,880 --> 00:59:52,520 the number of particles of different species, 910 00:59:52,520 --> 00:59:56,740 allowing potentially there going to other species 911 00:59:56,740 --> 01:00:00,550 through chemical reactions provided we multiply these 912 01:00:00,550 --> 01:00:04,300 by some appropriate chemical potential. 913 01:00:04,300 --> 01:00:07,220 So these mu's are a chemical potential. 914 01:00:12,930 --> 01:00:18,180 So for the reason that I stated before 915 01:00:18,180 --> 01:00:28,670 that if I were to just-- this ambiguity 916 01:00:28,670 --> 01:00:31,410 that I had about the size of the box 917 01:00:31,410 --> 01:00:35,770 tells me that I should have at least one thing 918 01:00:35,770 --> 01:00:40,270 left in my system describing the variables 919 01:00:40,270 --> 01:00:43,290 that is proportional to the size of the system. 920 01:00:43,290 --> 01:00:45,660 So typically what you do when you construct 921 01:00:45,660 --> 01:00:50,630 a Gibbs free energy is that you subtract the mechanical work, 922 01:00:50,630 --> 01:00:53,460 but you don't subtract the chemical work. 923 01:00:53,460 --> 01:00:57,460 So the ends will remain and can tell you 924 01:00:57,460 --> 01:01:00,490 how big your system is. 925 01:01:00,490 --> 01:01:02,380 OK? 926 01:01:02,380 --> 01:01:06,610 Now there is a conjugate way I can certainly do the other way. 927 01:01:06,610 --> 01:01:13,500 So here I constructed something in which mechanical work was 928 01:01:13,500 --> 01:01:16,340 removed, but chemical work remained. 929 01:01:16,340 --> 01:01:18,840 I could do it the other way around. 930 01:01:18,840 --> 01:01:23,230 I can define a function that is obtained 931 01:01:23,230 --> 01:01:31,150 by subtracting from the energy the chemical work, 932 01:01:31,150 --> 01:01:38,720 but leaving the mechanical component aside. 933 01:01:38,720 --> 01:01:45,940 So this function dG, OK, I should go one step before. 934 01:01:45,940 --> 01:01:52,310 So once we separate out the contributions of Jidxi 935 01:01:52,310 --> 01:01:55,790 into mechanical and chemical components, 936 01:01:55,790 --> 01:02:02,800 I would write dE as TdS plus sum over iJidxi, 937 01:02:02,800 --> 01:02:07,159 plus sum over alpha mu alpha dN alpha. 938 01:02:07,159 --> 01:02:08,550 OK? 939 01:02:08,550 --> 01:02:13,900 I just explicitly separating out those two sources of work. 940 01:02:13,900 --> 01:02:18,840 Now when I constructed G, which subtracts from e, the mu, 941 01:02:18,840 --> 01:02:20,870 but not the J, what happens? 942 01:02:20,870 --> 01:02:24,160 I will get a minus TdS. 943 01:02:24,160 --> 01:02:25,595 This will remain unchanged. 944 01:02:28,290 --> 01:02:33,230 This will get transformed to N alpha, d mu alpha. 945 01:02:36,400 --> 01:02:37,430 Just hold on a second. 946 01:02:37,430 --> 01:02:43,870 So this entity that is called the grand potential actually 947 01:02:43,870 --> 01:02:47,340 completes the list of these functions of state 948 01:02:47,340 --> 01:03:00,820 that I wanted to tell you is naturally 949 01:03:00,820 --> 01:03:06,050 a function of T, the chemical potentials 950 01:03:06,050 --> 01:03:07,900 as well as the displacement. 951 01:03:07,900 --> 01:03:10,230 For example, if it's a gas the displacements 952 01:03:10,230 --> 01:03:12,470 will include the volume which tell you 953 01:03:12,470 --> 01:03:14,000 how much material you have. 954 01:03:14,000 --> 01:03:14,882 Yes? 955 01:03:14,882 --> 01:03:16,340 AUDIENCE: In that case, do you need 956 01:03:16,340 --> 01:03:23,635 to subtract an additional TS from E and [INAUDIBLE]? 957 01:03:23,635 --> 01:03:24,510 PROFESSOR: Very good. 958 01:03:28,580 --> 01:03:31,340 These are all examples of the energies 959 01:03:31,340 --> 01:03:34,610 so that you define them as a function of temperature 960 01:03:34,610 --> 01:03:36,210 rather than entropy. 961 01:03:36,210 --> 01:03:38,310 The only things that really remain, 962 01:03:38,310 --> 01:03:40,730 the one function that you sort of keep 963 01:03:40,730 --> 01:03:43,630 as a function of entropy is energy itself. 964 01:03:47,030 --> 01:03:48,040 Enthalpy, too. 965 01:03:48,040 --> 01:03:49,815 Yes, it has its uses. 966 01:03:52,760 --> 01:03:53,260 Yes? 967 01:03:55,660 --> 01:03:58,610 AUDIENCE: So I have many questions about [INAUDIBLE]. 968 01:03:58,610 --> 01:03:59,500 PROFESSOR: OK. 969 01:03:59,500 --> 01:04:00,416 AUDIENCE: [INAUDIBLE]. 970 01:04:03,020 --> 01:04:07,790 Right now I wanted to ask you say that so a particle is added 971 01:04:07,790 --> 01:04:11,126 to the system, a particle of some species. 972 01:04:11,126 --> 01:04:15,305 And this is going to change the energy of the system. 973 01:04:15,305 --> 01:04:18,470 So my question is while this particle can bring energy 974 01:04:18,470 --> 01:04:21,620 in several ways, like kinetic energy, 975 01:04:21,620 --> 01:04:25,413 or maybe it's this chemical bound that's been broken-- 976 01:04:25,413 --> 01:04:26,038 PROFESSOR: Yes. 977 01:04:26,038 --> 01:04:27,010 AUDIENCE: between energy from there. 978 01:04:27,010 --> 01:04:27,660 PROFESSOR: Absolutely. 979 01:04:27,660 --> 01:04:29,060 AUDIENCE: Or also it's rest energy 980 01:04:29,060 --> 01:04:29,670 from special [INAUDIBLE]. 981 01:04:29,670 --> 01:04:30,410 PROFESSOR: Yes. 982 01:04:30,410 --> 01:04:32,526 AUDIENCE: So which ones are we counting? 983 01:04:32,526 --> 01:04:34,200 Which ones are we not counting? 984 01:04:34,200 --> 01:04:35,220 PROFESSOR: OK. 985 01:04:35,220 --> 01:04:40,600 So what you see is that when we do all of these calculations 986 01:04:40,600 --> 01:04:43,290 we definitely need to include when 987 01:04:43,290 --> 01:04:46,130 we are talking about a gas, the kinetic energy component. 988 01:04:48,650 --> 01:04:52,120 The change in the covalent bond energy definitely 989 01:04:52,120 --> 01:04:55,860 has to be there if I take an oxygen molecule 990 01:04:55,860 --> 01:04:58,240 and separate it to two oxygen atoms. 991 01:04:58,240 --> 01:05:00,080 There is that change in the energy that 992 01:05:00,080 --> 01:05:06,780 has to be included corresponding kinetic energies. 993 01:05:06,780 --> 01:05:10,850 Typically the rest mass is not included. 994 01:05:10,850 --> 01:05:15,280 And so if I were to include that it will be a shift of chemical 995 01:05:15,280 --> 01:05:18,040 potential by an amount that is mc squared. 996 01:05:18,040 --> 01:05:20,220 Why do people not bother? 997 01:05:20,220 --> 01:05:23,940 Because typically what we will be trying to look at 998 01:05:23,940 --> 01:05:26,640 is the change when something happens. 999 01:05:26,640 --> 01:05:29,500 And you have to bring those particles presumably 1000 01:05:29,500 --> 01:05:31,250 from somewhere. 1001 01:05:31,250 --> 01:05:35,430 So as long as you bring in particles from somewhere then 1002 01:05:35,430 --> 01:05:37,650 there's the difference between the rest mass 1003 01:05:37,650 --> 01:05:39,510 that you had outside and rest mass 1004 01:05:39,510 --> 01:05:41,250 that you have here, that's 0. 1005 01:05:41,250 --> 01:05:44,560 It's really all of the other things that 1006 01:05:44,560 --> 01:05:48,580 contribute to useful quantities that you would be involved 1007 01:05:48,580 --> 01:05:52,870 with like temperature, pressure. 1008 01:05:52,870 --> 01:05:54,570 The rest mass is irrelevant. 1009 01:05:54,570 --> 01:05:57,400 Now I can't rule out that you will come up it 1010 01:05:57,400 --> 01:06:02,670 some process that is going on in [INAUDIBLE] for whatever 1011 01:06:02,670 --> 01:06:06,040 where the rest mass is an important contribution. 1012 01:06:06,040 --> 01:06:11,600 So it may, to some extent, depend on the circumstance 1013 01:06:11,600 --> 01:06:13,330 that you are looking at. 1014 01:06:13,330 --> 01:06:16,170 But if you are asking truly what should I 1015 01:06:16,170 --> 01:06:18,940 include in the chemical potential, 1016 01:06:18,940 --> 01:06:22,095 it will include the rest mass. 1017 01:06:22,095 --> 01:06:26,700 AUDIENCE: So for example in-- I don't know, in cosmology 1018 01:06:26,700 --> 01:06:29,665 they often talk of the chemical potential 1019 01:06:29,665 --> 01:06:32,516 of the different species, like elementary particles 1020 01:06:32,516 --> 01:06:33,462 in the cosmos. 1021 01:06:33,462 --> 01:06:34,170 PROFESSOR: Right. 1022 01:06:34,170 --> 01:06:36,480 AUDIENCE: They say that the chemical potential of photons 1023 01:06:36,480 --> 01:06:36,880 is 0. 1024 01:06:36,880 --> 01:06:37,505 PROFESSOR: Yes. 1025 01:06:37,505 --> 01:06:41,057 AUDIENCE: So here they have fixed some kind convention? 1026 01:06:41,057 --> 01:06:41,640 PROFESSOR: No. 1027 01:06:41,640 --> 01:06:44,200 You see there is a difference between photons 1028 01:06:44,200 --> 01:06:48,470 and other particles such as electrons or whatever, which 1029 01:06:48,470 --> 01:06:52,810 is that you have value in number or some other number 1030 01:06:52,810 --> 01:06:55,340 conservation that is applicable. 1031 01:06:55,340 --> 01:06:59,060 So any process that you have will not 1032 01:06:59,060 --> 01:07:01,840 change the number of [INAUDIBLE]. 1033 01:07:01,840 --> 01:07:04,950 Whereas there are processes in which there is something that 1034 01:07:04,950 --> 01:07:07,500 is heated and will give you whatever number of 1035 01:07:07,500 --> 01:07:09,180 photons that you wish. 1036 01:07:09,180 --> 01:07:11,720 So there is a distinction between things 1037 01:07:11,720 --> 01:07:15,830 that are conserved and things that are not conserved. 1038 01:07:15,830 --> 01:07:19,830 So We will get to that maybe later on as 1039 01:07:19,830 --> 01:07:24,510 to why it is appropriate to set chemical potential to 0 1040 01:07:24,510 --> 01:07:26,110 for non-conserved things. 1041 01:07:31,140 --> 01:07:33,310 OK? 1042 01:07:33,310 --> 01:07:35,290 Yeah, but it's certainly something 1043 01:07:35,290 --> 01:07:40,370 that we have immediately a feel for what the pressure is 1044 01:07:40,370 --> 01:07:45,390 for what the temperature is, so we 1045 01:07:45,390 --> 01:07:49,000 have our senses seem to sort of tell you 1046 01:07:49,000 --> 01:07:55,870 the reality of temperature and pressure, 1047 01:07:55,870 --> 01:07:59,530 but maybe not so much the chemical potential. 1048 01:07:59,530 --> 01:08:04,220 So I tried to see whether we have a sense that actually 1049 01:08:04,220 --> 01:08:06,080 is sensitive to chemical potential 1050 01:08:06,080 --> 01:08:08,680 and we do to some extent. 1051 01:08:08,680 --> 01:08:10,830 As you drink something and you say 1052 01:08:10,830 --> 01:08:13,545 it's too salty, or whatever, so somehow you 1053 01:08:13,545 --> 01:08:15,960 are measuring the chemical potential of sort. 1054 01:08:15,960 --> 01:08:19,534 So if you like that, that would be your sensual equivalent 1055 01:08:19,534 --> 01:08:20,450 of chemical potential. 1056 01:08:23,630 --> 01:08:25,270 OK? 1057 01:08:25,270 --> 01:08:26,068 Yes? 1058 01:08:26,068 --> 01:08:27,568 AUDIENCE: Is there a potential where 1059 01:08:27,568 --> 01:08:30,519 you have both chemical and mechanical work? 1060 01:08:30,519 --> 01:08:31,717 [INAUDIBLE] 1061 01:08:31,717 --> 01:08:32,300 PROFESSOR: OK. 1062 01:08:32,300 --> 01:08:35,000 So that will run into the problem 1063 01:08:35,000 --> 01:08:38,740 of if I do that, then my parameter is 1064 01:08:38,740 --> 01:08:43,210 going to be T, J, and mu. 1065 01:08:43,210 --> 01:08:48,260 And then I can ask you, how much do I have? 1066 01:08:48,260 --> 01:08:51,330 Because all of these quantities are intensive. 1067 01:08:51,330 --> 01:08:52,250 Right? 1068 01:08:52,250 --> 01:08:55,680 So it could be a box this is one cubic centimeters 1069 01:08:55,680 --> 01:08:57,569 or miles long. 1070 01:08:57,569 --> 01:09:00,724 So I am not allowed to do that. 1071 01:09:00,724 --> 01:09:03,569 And there is a mathematical reason for that 1072 01:09:03,569 --> 01:09:05,319 that I was going to come to shortly. 1073 01:09:08,043 --> 01:09:08,709 Other questions? 1074 01:09:11,480 --> 01:09:12,939 OK. 1075 01:09:12,939 --> 01:09:19,220 So We are going to take an interlude. 1076 01:09:19,220 --> 01:09:23,569 Actually, the last question is quite relevant to what 1077 01:09:23,569 --> 01:09:26,380 we are going to do next, which is that we have 1078 01:09:26,380 --> 01:09:30,429 these functions defined, many coordinates, 1079 01:09:30,429 --> 01:09:34,220 and one of the things that thermodynamics allows you to do 1080 01:09:34,220 --> 01:09:38,620 is to relate measurements of one set of quantities, 1081 01:09:38,620 --> 01:09:40,520 another set of quantities. 1082 01:09:40,520 --> 01:09:44,200 And it does so through developing 1083 01:09:44,200 --> 01:09:46,399 mathematical relationships that you 1084 01:09:46,399 --> 01:09:49,790 can have between these different functions of state. 1085 01:09:49,790 --> 01:09:56,900 So the next segment has to do with mathematical results, 1086 01:09:56,900 --> 01:10:02,430 which I will subdivide into two sets of statements. 1087 01:10:02,430 --> 01:10:06,020 One set of statement follow from the discussion 1088 01:10:06,020 --> 01:10:09,095 that we have had so far, which has to do with extensivity. 1089 01:10:14,190 --> 01:10:15,570 What do I mean? 1090 01:10:15,570 --> 01:10:21,060 Let's take a look at our most fundamental expression, which 1091 01:10:21,060 --> 01:10:30,150 is dE is TdS plus sum over iJdxi, 1092 01:10:30,150 --> 01:10:34,517 plus sum over alpha mu alpha dN alpha. 1093 01:10:37,720 --> 01:10:46,420 Now you recognize that certainly as the amount of material 1094 01:10:46,420 --> 01:10:50,020 gets increased, the number of particles increases. 1095 01:10:50,020 --> 01:10:53,400 The typical sizes and displacements get bigger. 1096 01:10:53,400 --> 01:10:55,750 The energy content gets bigger. 1097 01:10:55,750 --> 01:10:58,915 The entropy content is related to the heat content 1098 01:10:58,915 --> 01:11:00,910 and so that gets bigger. 1099 01:11:00,910 --> 01:11:06,490 So these differential forms that I have on this expression, 1100 01:11:06,490 --> 01:11:10,430 they're all proportional to the size of the system. 1101 01:11:10,430 --> 01:11:12,060 What does that mean? 1102 01:11:12,060 --> 01:11:14,700 It means that e, the way that I have it 1103 01:11:14,700 --> 01:11:22,480 here has as its natural variables S, x, N. 1104 01:11:22,480 --> 01:11:27,790 And extensivity means that if I were to make my system twice as 1105 01:11:27,790 --> 01:11:31,710 large, so that all of these quantities 1106 01:11:31,710 --> 01:11:34,460 would get multiplied by a factor of two, 1107 01:11:34,460 --> 01:11:38,187 the energy content would get multiplied by a factor of two. 1108 01:11:43,660 --> 01:11:46,420 OK? 1109 01:11:46,420 --> 01:11:53,592 Now it is important to state that this is not a requirement. 1110 01:11:53,592 --> 01:11:54,092 OK? 1111 01:11:57,400 --> 01:12:00,900 This is a statement about most things 1112 01:12:00,900 --> 01:12:05,550 that we encounter around us, but once you go, 1113 01:12:05,550 --> 01:12:08,890 let's say, to the cosmos, and you have a star, 1114 01:12:08,890 --> 01:12:12,050 the gravitational energy of the star 1115 01:12:12,050 --> 01:12:14,280 is not proportional to its volume. 1116 01:12:14,280 --> 01:12:18,260 It goes the size to some fractional power. 1117 01:12:18,260 --> 01:12:22,910 So that is an example of a system because of the way 1118 01:12:22,910 --> 01:12:25,620 gravity works as a long range force, that 1119 01:12:25,620 --> 01:12:28,300 is not extensive system. 1120 01:12:28,300 --> 01:12:31,510 Typically this would work as long 1121 01:12:31,510 --> 01:12:34,110 as you have interactions among your elements 1122 01:12:34,110 --> 01:12:36,200 that are sufficiently short-ranged 1123 01:12:36,200 --> 01:12:40,720 so that you don't get non-extensivity. 1124 01:12:40,720 --> 01:12:44,130 If that's the case then I can take a derivative 1125 01:12:44,130 --> 01:12:48,860 of this expression with respect to lambda, evaluated at lambda 1126 01:12:48,860 --> 01:12:51,900 equals 2, 1. 1127 01:12:51,900 --> 01:12:54,440 If I take a derivative with respect to lambda 1128 01:12:54,440 --> 01:13:02,960 here I will have dE by dS, times S. I mean, 1129 01:13:02,960 --> 01:13:04,800 the argument here is lambda S. I am 1130 01:13:04,800 --> 01:13:08,090 taking a derivative with respect to the first argument. 1131 01:13:08,090 --> 01:13:12,810 So I bring out a factor of S. dE by dS at constant x 1132 01:13:12,810 --> 01:13:18,885 and N. And then the next term is xi dE with respect 1133 01:13:18,885 --> 01:13:23,000 to all the different xi's at constant S and N. 1134 01:13:23,000 --> 01:13:30,380 Next one is going to be N alpha, dE by dN alpha 1135 01:13:30,380 --> 01:13:36,480 at constant x and S. And there's only one lambda 1136 01:13:36,480 --> 01:13:40,350 on this side, which is going to give me E of Sx. 1137 01:13:43,154 --> 01:13:43,654 OK? 1138 01:13:46,530 --> 01:13:52,110 Now if I look at my initial expression 1139 01:13:52,110 --> 01:13:56,000 I can immediately see that dE by dS at constant x 1140 01:13:56,000 --> 01:13:59,830 and N is none other than T. So this 1141 01:13:59,830 --> 01:14:05,010 is the same thing as T. dE by dxi at constant s 1142 01:14:05,010 --> 01:14:07,970 and n is none other than Ji. 1143 01:14:07,970 --> 01:14:15,120 dE by dN alpha at constant x and Js is none other than mu alpha. 1144 01:14:15,120 --> 01:14:20,130 So once I set the argument to be 1, so that all of these things 1145 01:14:20,130 --> 01:14:22,850 are evaluated at lambda equals to 1, 1146 01:14:22,850 --> 01:14:30,450 I have the result that e is equal to TS plus Jixi, plus mu 1147 01:14:30,450 --> 01:14:33,010 alpha N alpha. 1148 01:14:33,010 --> 01:14:41,430 So in some sense all I did was I took the more fundamental 1149 01:14:41,430 --> 01:14:45,070 expression and removed all of the d's. 1150 01:14:45,070 --> 01:14:48,440 So some places, this is called the fundamental relation, 1151 01:14:48,440 --> 01:14:54,455 but I don't like that because it is only valued for systems that 1152 01:14:54,455 --> 01:14:57,020 are extensive, whereas the initial formula 1153 01:14:57,020 --> 01:15:00,300 is valued irrespective of that. 1154 01:15:00,300 --> 01:15:01,800 OK? 1155 01:15:01,800 --> 01:15:05,610 Now once I have this what I can do 1156 01:15:05,610 --> 01:15:09,230 is I can think of this as a relationship 1157 01:15:09,230 --> 01:15:12,550 among different functions of state. 1158 01:15:12,550 --> 01:15:18,540 Take a derivative and write it as dE is d of TS. 1159 01:15:18,540 --> 01:15:26,530 D of TS is TdS plus SdT, plus d of Jixi, which gives me 1160 01:15:26,530 --> 01:15:34,340 Jidxi plus xidJi, plus d of mu alpha. 1161 01:15:34,340 --> 01:15:37,340 And alpha will give me mu alpha dn alpha, 1162 01:15:37,340 --> 01:15:40,895 plus N alpha d mu alpha. 1163 01:15:40,895 --> 01:15:41,870 OK? 1164 01:15:41,870 --> 01:15:45,140 This is just a rewriting of this expression 1165 01:15:45,140 --> 01:15:47,290 in differential form. 1166 01:15:47,290 --> 01:15:54,430 But I know that dE is the same thing as TdS plus Jdx 1167 01:15:54,430 --> 01:15:55,590 plus mu n alpha. 1168 01:15:59,610 --> 01:16:05,820 So immediately what it tells me is that the intensive variables 1169 01:16:05,820 --> 01:16:23,580 T, J, and mu our constrained to satisfy this relation, which 1170 01:16:23,580 --> 01:16:33,261 is called a Gibbs-Duhem relation. 1171 01:16:33,261 --> 01:16:33,760 OK? 1172 01:16:36,510 --> 01:16:43,530 So this is, if you like, the mathematical reason 1173 01:16:43,530 --> 01:16:46,700 for my answer before. 1174 01:16:46,700 --> 01:16:56,040 That I cannot choose this set of variables to describe my system 1175 01:16:56,040 --> 01:17:00,040 because this set of variables are not independent. 1176 01:17:00,040 --> 01:17:03,790 If I vary two of them the variation of the other one 1177 01:17:03,790 --> 01:17:05,090 is fixed. 1178 01:17:05,090 --> 01:17:09,880 And I said I need N plus 1 independent degrees of freedom. 1179 01:17:09,880 --> 01:17:14,070 If I choose all of the intensive ones 1180 01:17:14,070 --> 01:17:20,110 I'm really using one additional relationship that relates them, 1181 01:17:20,110 --> 01:17:21,660 makes them dependent. 1182 01:17:21,660 --> 01:17:23,580 And of course this also goes to the fact 1183 01:17:23,580 --> 01:17:27,640 that I won't know what the size of the system is, both of them 1184 01:17:27,640 --> 01:17:28,900 reflections of extensivity. 1185 01:17:31,500 --> 01:17:37,620 Just as an example, let's see. 1186 01:17:37,620 --> 01:17:43,840 Chemical potential along isotherm. 1187 01:17:51,920 --> 01:17:57,610 So for a gas, what would I have? 1188 01:17:57,610 --> 01:18:05,310 I would write SdT minus Vdp. 1189 01:18:05,310 --> 01:18:10,310 That is my contribution from the work here remembering 1190 01:18:10,310 --> 01:18:14,150 that hydrostatic work at the wrong side. 1191 01:18:14,150 --> 01:18:20,670 And then I have N d mu equals to 0. 1192 01:18:20,670 --> 01:18:26,170 I said how do these vary along an isotherm? 1193 01:18:26,170 --> 01:18:30,700 Isotherm means that I have to set dT equals to 0. 1194 01:18:30,700 --> 01:18:35,040 If I'm along an isotherm then I have that d mu 1195 01:18:35,040 --> 01:18:38,070 is rearranging things a little bit. 1196 01:18:38,070 --> 01:18:40,530 Mu over NdP. 1197 01:18:40,530 --> 01:18:45,010 So this is the formula that I will have to use. 1198 01:18:45,010 --> 01:18:48,660 Now let's specialize to the case of an ideal gas. 1199 01:18:48,660 --> 01:18:52,870 So I'm going to use here an ideal gas. 1200 01:18:52,870 --> 01:18:56,640 For the case of the ideal gas we had the relationship 1201 01:18:56,640 --> 01:18:59,700 that PV was NKT. 1202 01:18:59,700 --> 01:19:03,580 So v over n is the same thing as kT over p. 1203 01:19:07,060 --> 01:19:09,660 Remember that we are dealing with an isotherm, 1204 01:19:09,660 --> 01:19:11,540 so T is constant. 1205 01:19:11,540 --> 01:19:15,040 dp over pi can integrate, and therefore 1206 01:19:15,040 --> 01:19:20,930 conclude that mu at a constant along an isotherm, 1207 01:19:20,930 --> 01:19:26,930 as a function of the other two intensive variables T and P, 1208 01:19:26,930 --> 01:19:33,060 is some reference that comes from a constant of integration. 1209 01:19:33,060 --> 01:19:37,170 And then I have KT, integral of dP over P 1210 01:19:37,170 --> 01:19:41,530 is log of P divided by sum. 1211 01:19:41,530 --> 01:19:43,945 Actually, I already put the constant. 1212 01:19:47,276 --> 01:19:47,776 OK? 1213 01:20:07,210 --> 01:20:11,520 Very briefly to be expanded upon next lecture, 1214 01:20:11,520 --> 01:20:15,420 the other set of mathematical relations 1215 01:20:15,420 --> 01:20:18,150 go under the name of Maxwell relations. 1216 01:20:26,430 --> 01:20:29,123 And they follow from the following observation. 1217 01:20:33,960 --> 01:20:37,450 If I have a function of more than one variable, 1218 01:20:37,450 --> 01:20:44,840 let's say, f of x and y, two variables, then the natural way 1219 01:20:44,840 --> 01:20:47,240 to write this consistent with what 1220 01:20:47,240 --> 01:20:54,860 we had before is that df is df by dx partial at constant ydx, 1221 01:20:54,860 --> 01:20:58,900 plus dfy, the y at constant x dy. 1222 01:21:01,820 --> 01:21:05,190 But one thing that we know from calculus 1223 01:21:05,190 --> 01:21:08,920 is that these are first derivatives, 1224 01:21:08,920 --> 01:21:12,430 and there are a bunch of second derivatives. 1225 01:21:12,430 --> 01:21:18,100 In particular, the second derivative d2 f, dx, dy 1226 01:21:18,100 --> 01:21:24,660 is independent of the order of taking the derivatives. 1227 01:21:24,660 --> 01:21:27,630 Which means that if I take a derivative of this object 1228 01:21:27,630 --> 01:21:30,200 with respect to y, I will get the same thing 1229 01:21:30,200 --> 01:21:33,570 if I were to take a derivative of the second object 1230 01:21:33,570 --> 01:21:36,310 with respect to x. 1231 01:21:36,310 --> 01:21:42,200 So in particular if I go back to my fundamental equation, 1232 01:21:42,200 --> 01:21:45,290 dE, let's forget about the chemical potential 1233 01:21:45,290 --> 01:21:52,630 because time is running short, Jidxi plus TdS. 1234 01:21:52,630 --> 01:21:57,810 I identify T and Ji as the first derivatives. 1235 01:21:57,810 --> 01:22:02,070 That is, Ji is the first derivative of e 1236 01:22:02,070 --> 01:22:05,895 with respect to xi at constant S. T 1237 01:22:05,895 --> 01:22:08,560 is the first derivative of E, with respect 1238 01:22:08,560 --> 01:22:11,420 to S at constant x. 1239 01:22:11,420 --> 01:22:16,760 Then if I were to take a second derivative to construct d2E 1240 01:22:16,760 --> 01:22:26,270 with respect to x and T, and S, I could do it two ways. 1241 01:22:26,270 --> 01:22:31,060 I already have a derivative with respect to x that gave me Ji, 1242 01:22:31,060 --> 01:22:33,590 so I take another derivative of J 1243 01:22:33,590 --> 01:22:38,020 with respect to S, now at constant x. 1244 01:22:38,020 --> 01:22:40,820 Or I can take a derivative here, which 1245 01:22:40,820 --> 01:22:52,240 is dT by dxi at constant S. And depending 1246 01:22:52,240 --> 01:22:58,990 on which one of the many functions of state that we 1247 01:22:58,990 --> 01:23:04,250 introduced, E, F, G, H, you can certainly 1248 01:23:04,250 --> 01:23:08,500 make corresponding second derivative inequalities. 1249 01:23:08,500 --> 01:23:13,040 And the key to sort of understanding all of them 1250 01:23:13,040 --> 01:23:16,020 is this equality of second derivatives. 1251 01:23:16,020 --> 01:23:18,850 So what I will start next time around 1252 01:23:18,850 --> 01:23:23,270 is to just give you a set of things on the left hand 1253 01:23:23,270 --> 01:23:24,680 side, the analog of this. 1254 01:23:24,680 --> 01:23:29,860 So for example, I could choose dS by dxi at constant T 1255 01:23:29,860 --> 01:23:33,070 and show that just by looking at the form of this 1256 01:23:33,070 --> 01:23:37,840 how we will be able to construct what this thing is related 1257 01:23:37,840 --> 01:23:41,300 through a Maxwell construction. 1258 01:23:41,300 --> 01:23:41,900 OK? 1259 01:23:41,900 --> 01:23:43,346 Thank you.