1 00:00:00,060 --> 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 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 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 00:00:21,420 --> 00:00:25,150 PROFESSOR: This is 8.333 Statistical Mechanics. 9 00:00:25,150 --> 00:00:29,530 And I'll start by telling you a little bit about the syllabus 10 00:00:29,530 --> 00:00:32,970 before going through the structure of the course. 11 00:00:32,970 --> 00:00:37,120 So what I have written here is a, kind of, 12 00:00:37,120 --> 00:00:45,660 rough definition of statistical mechanics from my perspective. 13 00:00:45,660 --> 00:00:50,320 And the syllabus is a guide to how 14 00:00:50,320 --> 00:00:52,480 we are going to approach this object. 15 00:00:52,480 --> 00:00:56,270 So let's take a look at the syllabus over there. 16 00:00:56,270 --> 00:01:00,100 So the first thing that you're going to focus on 17 00:01:00,100 --> 00:01:03,620 is, what is it that you're trying to describe? 18 00:01:03,620 --> 00:01:06,960 And these are the equilibrium properties 19 00:01:06,960 --> 00:01:10,120 that are described best through what 20 00:01:10,120 --> 00:01:12,040 I've been doing section one, which 21 00:01:12,040 --> 00:01:14,960 has to do with thermodynamics. 22 00:01:14,960 --> 00:01:17,430 You'll start today with thermodynamics. 23 00:01:17,430 --> 00:01:20,700 Basically, it's a phenomenological approach, 24 00:01:20,700 --> 00:01:22,630 so you essentially look at something, 25 00:01:22,630 --> 00:01:25,070 as kind of a black box, without knowing 26 00:01:25,070 --> 00:01:27,160 what the ingredients are, and try 27 00:01:27,160 --> 00:01:32,560 to give some kind of description of how it's 28 00:01:32,560 --> 00:01:34,600 function and properties change. 29 00:01:34,600 --> 00:01:37,710 And these can be captured, for the case of thermal properties 30 00:01:37,710 --> 00:01:40,760 of matter through the laws of thermodynamics, which we will 31 00:01:40,760 --> 00:01:44,740 set out in this first section, which will roughly take us 32 00:01:44,740 --> 00:01:48,560 the first four lectures of the course. 33 00:01:48,560 --> 00:01:51,270 Then I said that statistical mechanics 34 00:01:51,270 --> 00:01:54,010 is a probabilistic approach. 35 00:01:54,010 --> 00:01:59,700 So we need to establish what the language of probability is. 36 00:01:59,700 --> 00:02:03,980 And that can be the topic for the second two 37 00:02:03,980 --> 00:02:06,130 and half lectures of the course. 38 00:02:06,130 --> 00:02:11,280 It is something that is less physics-y, but since the topic 39 00:02:11,280 --> 00:02:14,286 itself has to deal with probabilities, 40 00:02:14,286 --> 00:02:17,770 it is very important, from my perspective, 41 00:02:17,770 --> 00:02:23,510 to set out the language and the properties of systems 42 00:02:23,510 --> 00:02:25,890 that are described probabilistically. 43 00:02:25,890 --> 00:02:29,020 Separately, we'll devote a couple of lectures to do so. 44 00:02:29,020 --> 00:02:32,060 And in particular, it is very important 45 00:02:32,060 --> 00:02:34,750 that the laws of probability, kind of, 46 00:02:34,750 --> 00:02:39,860 simplify when you're dealing with a large number 47 00:02:39,860 --> 00:02:44,560 of variables, as captured, for example, by what we 48 00:02:44,560 --> 00:02:46,850 call the central limit theorem. 49 00:02:46,850 --> 00:02:50,000 So you can see that the third element of this course, 50 00:02:50,000 --> 00:02:52,540 which is the law of large numbers, 51 00:02:52,540 --> 00:02:57,230 is inherent also to what simplification you will 52 00:02:57,230 --> 00:03:00,630 see in the section on probability. 53 00:03:00,630 --> 00:03:03,510 And feeds back very much into how 54 00:03:03,510 --> 00:03:07,410 statistical mechanics is developed. 55 00:03:07,410 --> 00:03:11,180 But then we said large number of degrees of freedom. 56 00:03:11,180 --> 00:03:12,800 So what are these degrees of freedom? 57 00:03:15,690 --> 00:03:18,440 Well, this is now taking a different perspective. 58 00:03:18,440 --> 00:03:20,480 For us in thermodynamics, when you 59 00:03:20,480 --> 00:03:24,130 look at the system as a black box, 60 00:03:24,130 --> 00:03:29,320 and try to develop laws based on observations, 61 00:03:29,320 --> 00:03:31,970 we say that well, from the perspective of physics, 62 00:03:31,970 --> 00:03:35,270 we know that this box contains atoms and molecules. 63 00:03:35,270 --> 00:03:37,240 And these atoms and molecules are 64 00:03:37,240 --> 00:03:42,690 following very specific laws, either from Newtonian mechanics 65 00:03:42,690 --> 00:03:44,250 or quantum mechanics. 66 00:03:44,250 --> 00:03:48,550 And so if we know everything about how atoms and molecules 67 00:03:48,550 --> 00:03:51,840 behave, then we should be able to derive 68 00:03:51,840 --> 00:03:54,290 how large collections of them behave. 69 00:03:54,290 --> 00:03:58,110 And get the laws of thermodynamics 70 00:03:58,110 --> 00:04:01,750 as a consequence of these microscopic degrees of freedom 71 00:04:01,750 --> 00:04:03,540 and their dynamics. 72 00:04:03,540 --> 00:04:07,020 And so that's what we will discuss 73 00:04:07,020 --> 00:04:08,640 in the third part of the course that 74 00:04:08,640 --> 00:04:11,080 is devoted to kinetic theory. 75 00:04:11,080 --> 00:04:14,660 We will see that even at that stage, it is beneficial to, 76 00:04:14,660 --> 00:04:19,399 rather than follow individual particles in a system, 77 00:04:19,399 --> 00:04:21,670 to adopt a probabilistic approach, 78 00:04:21,670 --> 00:04:25,200 and think about densities, and how those densities evolve 79 00:04:25,200 --> 00:04:27,920 according to Liouville's Theorem. 80 00:04:27,920 --> 00:04:31,450 And what we will try to also establish 81 00:04:31,450 --> 00:04:35,000 is a very distinct difference that 82 00:04:35,000 --> 00:04:36,970 exists between thermodynamics, and where 83 00:04:36,970 --> 00:04:38,830 things are irreversible and going 84 00:04:38,830 --> 00:04:42,730 one direction, and Newtonian, or quantum mechanics, 85 00:04:42,730 --> 00:04:45,160 where things are reversible in time. 86 00:04:45,160 --> 00:04:49,150 And we'll see that really it's a matter of adapting 87 00:04:49,150 --> 00:04:53,100 the right perspective in order to see that these two 88 00:04:53,100 --> 00:04:55,270 ways of looking at the same system 89 00:04:55,270 --> 00:04:57,950 are not in contradiction. 90 00:04:57,950 --> 00:05:01,290 So having established these elements, 91 00:05:01,290 --> 00:05:07,070 we will then finally be in the place 92 00:05:07,070 --> 00:05:11,700 where we can discuss statistical mechanics in terms 93 00:05:11,700 --> 00:05:17,680 of some postulates about how probabilities behave 94 00:05:17,680 --> 00:05:20,580 for systems that are in equilibrium. 95 00:05:20,580 --> 00:05:22,490 And how based on those postulates, 96 00:05:22,490 --> 00:05:26,020 we can then derive all the laws of thermodynamics 97 00:05:26,020 --> 00:05:29,410 and all the properties of thermodynamics systems. 98 00:05:29,410 --> 00:05:31,920 That they're ordained while observations 99 00:05:31,920 --> 00:05:36,100 and phenomenological theories before. 100 00:05:36,100 --> 00:05:38,785 Now initially, in section four, we 101 00:05:38,785 --> 00:05:42,690 will do that in the context of classical systems-- description 102 00:05:42,690 --> 00:05:46,870 of particles following classical laws of motion. 103 00:05:46,870 --> 00:05:48,950 And, again, as a first simplification, 104 00:05:48,950 --> 00:05:52,930 we will typically deal with non-interacting systems, 105 00:05:52,930 --> 00:05:54,420 such as ideal gas. 106 00:05:54,420 --> 00:05:57,060 And make sure that we understand the properties 107 00:05:57,060 --> 00:05:59,570 of this important fundamental system 108 00:05:59,570 --> 00:06:03,500 from all possible perspectives. 109 00:06:03,500 --> 00:06:08,860 Then in section five, we will go on to more realistic systems 110 00:06:08,860 --> 00:06:12,210 where there are interactions among these particles. 111 00:06:12,210 --> 00:06:15,040 And there are two ways to then deal with interactions. 112 00:06:15,040 --> 00:06:18,220 You can either go by the way of perturbation theory. 113 00:06:18,220 --> 00:06:21,130 We can start with ideal system and add a little bit 114 00:06:21,130 --> 00:06:23,970 of interaction, and see how that changes things. 115 00:06:23,970 --> 00:06:28,300 And we develop some elements of graphical perturbation theories 116 00:06:28,300 --> 00:06:29,920 in this context. 117 00:06:29,920 --> 00:06:31,840 Or, you can take another perspective, 118 00:06:31,840 --> 00:06:35,020 and say that because of the presence of interactions, 119 00:06:35,020 --> 00:06:38,270 the system really adopts a totally different type 120 00:06:38,270 --> 00:06:39,410 of behavior. 121 00:06:39,410 --> 00:06:41,570 And there's a perspective known as mean field 122 00:06:41,570 --> 00:06:43,940 theory that allows you to do that. 123 00:06:43,940 --> 00:06:46,510 Then see how the same system can be 124 00:06:46,510 --> 00:06:48,740 present in different phases of matter, 125 00:06:48,740 --> 00:06:52,830 such as liquids and gas, and how this mean field 126 00:06:52,830 --> 00:06:56,110 type of prescription allows you to discuss 127 00:06:56,110 --> 00:07:00,360 the transitions between the different types of behavior. 128 00:07:00,360 --> 00:07:04,920 Eventually, you will go on, towards the last quarter 129 00:07:04,920 --> 00:07:09,430 of the course, rather than the classical description of matter 130 00:07:09,430 --> 00:07:12,890 to a quantum description of the microscopic degrees of freedom. 131 00:07:12,890 --> 00:07:17,090 And we will see how the differences and similarities 132 00:07:17,090 --> 00:07:19,210 between quantum statistical mechanics, 133 00:07:19,210 --> 00:07:21,870 classical statistical mechanics, emerge. 134 00:07:21,870 --> 00:07:23,860 And just, historically, of course, 135 00:07:23,860 --> 00:07:27,660 radius macroscopic properties of the matter, the black body 136 00:07:27,660 --> 00:07:30,100 laws, or heat capacities, have been 137 00:07:30,100 --> 00:07:33,120 very important in showing the limitations 138 00:07:33,120 --> 00:07:35,860 of classical description of matter, and the need 139 00:07:35,860 --> 00:07:40,730 to have something else, such as the quantum description. 140 00:07:40,730 --> 00:07:44,070 We will not spend too much time, more than three lectures, 141 00:07:44,070 --> 00:07:48,380 on the sort of principles of quantum statistical mechanics. 142 00:07:48,380 --> 00:07:52,040 The place where quantum statistical mechanics shows 143 00:07:52,040 --> 00:07:56,360 its power is in dealing with identical particles, which 144 00:07:56,360 --> 00:07:59,590 classically, really are kind of not 145 00:07:59,590 --> 00:08:03,090 a very well-defined concept, but quantum-mechanically, they 146 00:08:03,090 --> 00:08:06,450 are very precisely defined, what identical particles mean. 147 00:08:06,450 --> 00:08:09,270 And there are two classes-- fermions and bosons-- 148 00:08:09,270 --> 00:08:15,310 and how even if there's no interaction between them, 149 00:08:15,310 --> 00:08:20,430 quantum statistics leads to unusual behavior for quantum 150 00:08:20,430 --> 00:08:24,260 systems of identical particles, very distinct for fermions 151 00:08:24,260 --> 00:08:26,330 and for bosons. 152 00:08:26,330 --> 00:08:31,120 So that's a rough syllables of how the course will be arranged 153 00:08:31,120 --> 00:08:34,720 over the next 25, 26 lectures. 154 00:08:34,720 --> 00:08:37,320 Any questions about what we're going to cover? 155 00:08:41,669 --> 00:08:45,940 OK, then let's go here. 156 00:08:49,300 --> 00:08:52,140 So I will be teaching the course. 157 00:08:55,610 --> 00:08:58,630 My research is in condensed matter theory 158 00:08:58,630 --> 00:09:01,420 and statistical physics. 159 00:09:01,420 --> 00:09:04,510 So this is a subject that I like very much. 160 00:09:04,510 --> 00:09:09,420 And I hope to impart some of that love 161 00:09:09,420 --> 00:09:10,960 of statistical physics to you. 162 00:09:10,960 --> 00:09:14,080 And why it is an interesting topic. 163 00:09:14,080 --> 00:09:16,950 Lectures and recitations will be conducted 164 00:09:16,950 --> 00:09:20,020 in this room, Monday, Wednesday, Friday. 165 00:09:20,020 --> 00:09:22,750 And you ask, well, what does it mean that both lectures 166 00:09:22,750 --> 00:09:24,745 and recitations are here? 167 00:09:24,745 --> 00:09:28,500 Well, for that you will have to consult the timetable. 168 00:09:28,500 --> 00:09:30,660 And that's probably the most important part 169 00:09:30,660 --> 00:09:32,000 of this web page. 170 00:09:32,000 --> 00:09:35,520 And it tells you, for example, that the first five 171 00:09:35,520 --> 00:09:37,360 events for the course are all going 172 00:09:37,360 --> 00:09:40,610 to be lectures Monday, Wednesday, Friday of next week. 173 00:09:40,610 --> 00:09:44,130 And the first recitation will come up on Monday, September 174 00:09:44,130 --> 00:09:45,640 the 16th. 175 00:09:45,640 --> 00:09:48,360 And the reason for that is that the first problem 176 00:09:48,360 --> 00:09:51,620 set will be due on the 18th. 177 00:09:51,620 --> 00:09:58,320 And I will arrange for you to have six recitations on the six 178 00:09:58,320 --> 00:10:03,370 events that are before the due dates of those problem sets. 179 00:10:03,370 --> 00:10:06,770 Also indicated here is the due dates, naturally, 180 00:10:06,770 --> 00:10:08,240 of the problem sets. 181 00:10:08,240 --> 00:10:12,620 And although I had indicated that this will be handed out 182 00:10:12,620 --> 00:10:15,230 tomorrow, the first problem set is already 183 00:10:15,230 --> 00:10:17,750 available on the web, so you can start 184 00:10:17,750 --> 00:10:20,260 going to take a look at that. 185 00:10:20,260 --> 00:10:23,610 And eventually, also on the web page, 186 00:10:23,610 --> 00:10:27,600 will be posted the solutions. 187 00:10:27,600 --> 00:10:30,620 And the first one will be posted here. 188 00:10:30,620 --> 00:10:32,740 Of course, it's not available yet. 189 00:10:32,740 --> 00:10:33,970 Surprisingly. 190 00:10:33,970 --> 00:10:38,010 Once the due date has passed on the date that is indicated, 191 00:10:38,010 --> 00:10:40,560 the solutions will be posted. 192 00:10:40,560 --> 00:10:45,810 Also indicated here is that there will be various tests. 193 00:10:45,810 --> 00:10:48,620 The first test will fall on October 2. 194 00:10:48,620 --> 00:10:50,820 And another time and recitations will 195 00:10:50,820 --> 00:10:53,520 take place are prior to the tests. 196 00:10:53,520 --> 00:10:55,360 So there is actually three tests, 197 00:10:55,360 --> 00:10:57,060 and there will be three citations 198 00:10:57,060 --> 00:10:59,460 that will take place before that. 199 00:10:59,460 --> 00:11:05,220 And, ultimately, at the end, there will be a final exam. 200 00:11:05,220 --> 00:11:06,940 It's date I don't know yet. 201 00:11:06,940 --> 00:11:10,640 So I just randomly put it on the Monday of the week 202 00:11:10,640 --> 00:11:12,940 where the final exams will be held. 203 00:11:12,940 --> 00:11:16,950 And once the actual date is announced by the registrar, 204 00:11:16,950 --> 00:11:19,960 I will make sure that I put the correct date 205 00:11:19,960 --> 00:11:22,720 and place in this place. 206 00:11:22,720 --> 00:11:31,600 OK, so that's the arrangement of the various lectures 207 00:11:31,600 --> 00:11:33,660 and recitations. 208 00:11:33,660 --> 00:11:39,800 In addition to me, the teaching staff 209 00:11:39,800 --> 00:11:48,170 consists of Max [? Imachaov ?] and Anton Goloborodko. 210 00:11:48,170 --> 00:11:51,780 Anton is here, sitting at that extreme corner. 211 00:11:51,780 --> 00:11:53,465 Max, I believe, is now in Paris. 212 00:11:59,160 --> 00:12:04,340 OK, both of them work on biological systems 213 00:12:04,340 --> 00:12:07,360 that use a lot of statistical physics content to them. 214 00:12:07,360 --> 00:12:10,860 So maybe they will tell you some interesting problems 215 00:12:10,860 --> 00:12:14,200 related to that in the recitations. 216 00:12:14,200 --> 00:12:17,520 At this point in time, they have both set their office hours 217 00:12:17,520 --> 00:12:21,550 to be Thursday, 3:00 to 5:00, in their lab, which 218 00:12:21,550 --> 00:12:26,490 is close to where the medical facilities are. 219 00:12:26,490 --> 00:12:29,210 If you find that inconvenient, you could potentially 220 00:12:29,210 --> 00:12:32,280 change that, or you could get in touch 221 00:12:32,280 --> 00:12:37,130 with either the TAs, or myself, when 222 00:12:37,130 --> 00:12:40,540 you want to have specific time to meet us. 223 00:12:40,540 --> 00:12:44,420 Otherwise, I have indicated my own availability 224 00:12:44,420 --> 00:12:47,930 to be the half hours typically after lectures 225 00:12:47,930 --> 00:12:50,040 on Monday, Wednesdays, and Fridays. 226 00:12:56,340 --> 00:13:00,840 One other set of important things to note 227 00:13:00,840 --> 00:13:04,970 is how the course is organized. 228 00:13:04,970 --> 00:13:10,400 So I already mentioned to you what the syllabus of the course 229 00:13:10,400 --> 00:13:11,390 is. 230 00:13:11,390 --> 00:13:15,950 I indicated where and when the lectures and recitations 231 00:13:15,950 --> 00:13:17,900 are going to take place. 232 00:13:17,900 --> 00:13:21,450 This is the web page that I have been surfing through. 233 00:13:21,450 --> 00:13:25,300 And all of the material will be posted through the web page, 234 00:13:25,300 --> 00:13:28,260 so that's where you have to go for problem sets, solutions, 235 00:13:28,260 --> 00:13:30,710 et cetera. 236 00:13:30,710 --> 00:13:32,030 Also, grades. 237 00:13:32,030 --> 00:13:36,150 And, in particular, I have my own system 238 00:13:36,150 --> 00:13:39,750 of posting the grades, for which I 239 00:13:39,750 --> 00:13:42,010 need a pseudonym from each one of you. 240 00:13:42,010 --> 00:13:45,640 So if you could all go through this checking online, 241 00:13:45,640 --> 00:13:47,860 indicate your name, your email address, 242 00:13:47,860 --> 00:13:50,540 and choose a pseudonym, which I emphasize 243 00:13:50,540 --> 00:13:53,540 has to be different from your real name. 244 00:13:53,540 --> 00:13:58,140 And if it is your real name, I have to randomly come up 245 00:13:58,140 --> 00:14:02,580 with something like "forgot to put pseudonym" or something. 246 00:14:02,580 --> 00:14:07,250 I cannot have your real name followed by the grades. 247 00:14:07,250 --> 00:14:08,790 OK? 248 00:14:08,790 --> 00:14:14,270 I'll discuss anonymous comments, et cetera, later on. 249 00:14:14,270 --> 00:14:17,220 As you will see, I will hand out, 250 00:14:17,220 --> 00:14:21,030 through the web page, extensive lecture notes covering 251 00:14:21,030 --> 00:14:23,480 all of the material that I talk about. 252 00:14:23,480 --> 00:14:26,870 So in principle, you don't need any textbooks. 253 00:14:26,870 --> 00:14:31,830 You can refer to the notes and what you write in the lectures. 254 00:14:31,830 --> 00:14:35,480 But certainly, I-- some people like 255 00:14:35,480 --> 00:14:38,270 to have a book sitting on their bookshelf. 256 00:14:38,270 --> 00:14:41,060 So I have indicated a set of books 257 00:14:41,060 --> 00:14:43,330 that you can put on your bookshelf. 258 00:14:43,330 --> 00:14:45,840 And hopefully consult for various topics 259 00:14:45,840 --> 00:14:47,340 at different stages. 260 00:14:47,340 --> 00:14:49,770 And I will, through the problem sets, 261 00:14:49,770 --> 00:14:53,320 indicate what are good useful chapters 262 00:14:53,320 --> 00:14:55,430 or parts of these books to take a look at. 263 00:14:58,080 --> 00:15:03,330 Now how is the grade for the course constructed? 264 00:15:03,330 --> 00:15:06,900 An important part of it is through this six problem sets 265 00:15:06,900 --> 00:15:07,970 that we mentioned. 266 00:15:07,970 --> 00:15:11,520 So each one of them will count for 5%, 267 00:15:11,520 --> 00:15:15,980 for a total of 30% going towards the contribution of the problem 268 00:15:15,980 --> 00:15:16,710 sets. 269 00:15:16,710 --> 00:15:21,840 I have no problem with you forming study groups, 270 00:15:21,840 --> 00:15:26,070 as long as each person, at the end, writes their own solution. 271 00:15:26,070 --> 00:15:29,540 And I know that if you look at around sufficiently, 272 00:15:29,540 --> 00:15:33,400 you can find solutions from previous years, et cetera, 273 00:15:33,400 --> 00:15:35,540 but you will really be cheating yourself. 274 00:15:35,540 --> 00:15:37,640 And I really bring your attention 275 00:15:37,640 --> 00:15:42,520 to the code of honor that is part of the MIT integrity 276 00:15:42,520 --> 00:15:44,940 handbook. 277 00:15:44,940 --> 00:15:49,940 We have indicated, through the schedule page, 278 00:15:49,940 --> 00:15:53,680 the timeline for this six problem sets are due. 279 00:15:53,680 --> 00:15:56,410 They are typically due at 5:00 PM 280 00:15:56,410 --> 00:15:59,150 on the date that is indicated on that page, 281 00:15:59,150 --> 00:16:00,940 and also on the problem set. 282 00:16:00,940 --> 00:16:04,990 And the physics department will set up a Dropbox, 283 00:16:04,990 --> 00:16:07,350 so you can put the problem set there, 284 00:16:07,350 --> 00:16:09,410 or you can bring it to the lecture on the date 285 00:16:09,410 --> 00:16:10,200 that it is due. 286 00:16:10,200 --> 00:16:12,540 That's also acceptable. 287 00:16:12,540 --> 00:16:17,150 There is a grey area of about a day or so sometime, 288 00:16:17,150 --> 00:16:21,160 between when the problem set is due and when the solutions are 289 00:16:21,160 --> 00:16:22,360 posted. 290 00:16:22,360 --> 00:16:25,850 If problem sets are handed in during that gray area, 291 00:16:25,850 --> 00:16:28,930 they will count towards final 50%, 292 00:16:28,930 --> 00:16:32,120 rather than full towards the grade. 293 00:16:32,120 --> 00:16:35,600 Unless you sort of write to me a good excuse 294 00:16:35,600 --> 00:16:40,130 that I can give you an extension. 295 00:16:40,130 --> 00:16:43,370 Now every now and then, people encounter difficulties, 296 00:16:43,370 --> 00:16:46,030 some particular week you are overwhelmed, or whatever, 297 00:16:46,030 --> 00:16:48,750 and you can't do the particular problem set, 298 00:16:48,750 --> 00:16:52,270 and they ask me for an excuse of some form. 299 00:16:52,270 --> 00:16:55,690 And rather than doing that, I have the following metric. 300 00:16:55,690 --> 00:16:57,490 That is, each one of these problem 301 00:16:57,490 --> 00:17:00,180 sets you will find that there's a set of problems 302 00:17:00,180 --> 00:17:03,090 that are indicated as optional. 303 00:17:03,090 --> 00:17:04,349 You can do those problems. 304 00:17:04,349 --> 00:17:07,690 And they will be graded like all the other problems. 305 00:17:07,690 --> 00:17:13,180 And in case, at some later time, you 306 00:17:13,180 --> 00:17:15,290 didn't hand in some problem set, or you 307 00:17:15,290 --> 00:17:17,990 miss half of the problem set, et cetera, 308 00:17:17,990 --> 00:17:20,540 what you did on these optional problems 309 00:17:20,540 --> 00:17:24,930 can be used to make up your grade, pushing it, eventually, 310 00:17:24,930 --> 00:17:27,680 up to the 30% mark. 311 00:17:27,680 --> 00:17:30,020 If you don't do any of the optional problems, 312 00:17:30,020 --> 00:17:31,650 you just do the required problem, 313 00:17:31,650 --> 00:17:35,740 you will correctly you will reach the 30% mark. 314 00:17:35,740 --> 00:17:39,950 If you do every single problem, including optional ones, 315 00:17:39,950 --> 00:17:45,550 you will not get more than 30%, so the 30% is upper-bound. 316 00:17:45,550 --> 00:17:46,930 So there's that. 317 00:17:46,930 --> 00:17:48,600 Then you have the three tests that 318 00:17:48,600 --> 00:17:53,100 will be taking place during the lecture time, 2:30 to 4:00, 319 00:17:53,100 --> 00:17:53,980 here. 320 00:17:53,980 --> 00:17:58,980 Each one of them will count 15%, so that's another 45%. 321 00:17:58,980 --> 00:18:02,470 And the remaining 25% will be the final exam 322 00:18:02,470 --> 00:18:05,780 that will be scheduled in the finals week. 323 00:18:05,780 --> 00:18:13,000 So basically, that's the way that the grades are made up. 324 00:18:13,000 --> 00:18:18,860 And the usual definition of what grades mean-- typically, 325 00:18:18,860 --> 00:18:20,360 we have been quite generous. 326 00:18:20,360 --> 00:18:23,280 I have to also indicate that things will not 327 00:18:23,280 --> 00:18:24,960 be graded on a curve. 328 00:18:24,960 --> 00:18:27,580 So that's a MIT policy. 329 00:18:27,580 --> 00:18:31,950 And there are some links here to places 330 00:18:31,950 --> 00:18:35,330 that you can go to if you encounter difficulties 331 00:18:35,330 --> 00:18:37,880 during the semester. 332 00:18:37,880 --> 00:18:41,330 So any questions about the organization of the course? 333 00:18:46,030 --> 00:18:49,960 OK, so let's see what else we have. 334 00:18:55,040 --> 00:19:01,400 OK, course outline and schedule, we already discussed. 335 00:19:01,400 --> 00:19:09,370 They're likely to be something that are unexpected, 336 00:19:09,370 --> 00:19:11,026 or some things that have to be changed. 337 00:19:11,026 --> 00:19:12,400 Every now and then there is going 338 00:19:12,400 --> 00:19:17,900 to be a hurricane almost with probability, close to one. 339 00:19:17,900 --> 00:19:20,580 We will have a hurricane sometime during the next month 340 00:19:20,580 --> 00:19:21,320 or so. 341 00:19:21,320 --> 00:19:24,950 We may have to postpone a particular lectures 342 00:19:24,950 --> 00:19:25,760 accordingly. 343 00:19:25,760 --> 00:19:28,600 And then the information about that will be posted here. 344 00:19:28,600 --> 00:19:31,320 Currently, the only announcement is 345 00:19:31,320 --> 00:19:33,620 what I had indicated to you before. 346 00:19:33,620 --> 00:19:35,900 Please check in online indicating 347 00:19:35,900 --> 00:19:38,130 that you are taking this course, and what 348 00:19:38,130 --> 00:19:41,260 your pseudonym is going to be. 349 00:19:41,260 --> 00:19:43,540 OK? 350 00:19:43,540 --> 00:19:47,640 I give you also the option, I would certainly 351 00:19:47,640 --> 00:19:50,490 welcome any questions that you may have for me here. 352 00:19:50,490 --> 00:19:52,310 Please feel free to interrupt. 353 00:19:52,310 --> 00:19:56,350 But sometimes people, later on, encounter questions. 354 00:19:56,350 --> 00:19:59,640 And for whatever reason, it may be 355 00:19:59,640 --> 00:20:01,920 question related to the material of the course. 356 00:20:01,920 --> 00:20:05,860 It may be related to when various things are due, 357 00:20:05,860 --> 00:20:09,910 or it may be there is some wrong notation in the problem set 358 00:20:09,910 --> 00:20:14,120 or whatever, you can certainly anonymously send 359 00:20:14,120 --> 00:20:16,360 this information to me. 360 00:20:16,360 --> 00:20:20,310 And I will try to respond. 361 00:20:20,310 --> 00:20:26,380 And anonymous responses will be posted and displayed web page 362 00:20:26,380 --> 00:20:27,190 here. 363 00:20:27,190 --> 00:20:30,100 Currently there is none, of course. 364 00:20:30,100 --> 00:20:34,220 And, finally, something that-- OK, 365 00:20:34,220 --> 00:20:40,060 so there's a web page where the problems will be posted. 366 00:20:40,060 --> 00:20:44,330 And I want to emphasize that the web page where the solutions 367 00:20:44,330 --> 00:20:48,570 are posted, you may see that you cannot get to it. 368 00:20:48,570 --> 00:20:52,120 And the reason would be that you don't have an MIT certificate. 369 00:20:52,120 --> 00:20:56,770 So MIT certificates are necessary to reach the solution 370 00:20:56,770 --> 00:20:57,950 page. 371 00:20:57,950 --> 00:21:01,640 And also they are necessary to reach 372 00:21:01,640 --> 00:21:05,770 the page that is devoted to tests. 373 00:21:05,770 --> 00:21:09,390 And actually there is something about the way 374 00:21:09,390 --> 00:21:13,490 that I do these three in-class tests that is polarizing. 375 00:21:13,490 --> 00:21:16,840 And some people very much dislike it. 376 00:21:16,840 --> 00:21:19,270 But that's the way it is, so let me tell you 377 00:21:19,270 --> 00:21:21,660 how it's going to be conducted. 378 00:21:21,660 --> 00:21:26,530 So you will have this one and a half hour [INAUDIBLE] test. 379 00:21:26,530 --> 00:21:30,850 And I can tell you, that the problems from the test 380 00:21:30,850 --> 00:21:35,800 will be out of this collection that I already posted here. 381 00:21:35,800 --> 00:21:40,940 So there is a collection of problems 382 00:21:40,940 --> 00:21:43,170 that is posted on this page. 383 00:21:43,170 --> 00:21:45,400 And furthermore, the solutions are posted. 384 00:21:45,400 --> 00:21:48,320 So there's a version of this that is with solution. 385 00:21:48,320 --> 00:21:51,320 So the problems will be taken from this, as well 386 00:21:51,320 --> 00:21:54,080 as the problem sets that you have already encountered-- 387 00:21:54,080 --> 00:21:56,530 [INAUDIBLE] solution is posted. 388 00:21:56,530 --> 00:21:59,775 So if you are familiar with this material, 389 00:21:59,775 --> 00:22:02,380 it should be no problem. 390 00:22:02,380 --> 00:22:06,930 And that's the way the first three tests this will go. 391 00:22:06,930 --> 00:22:11,500 The final will be, essentially, a collection of new problems 392 00:22:11,500 --> 00:22:14,140 that are variants of things that you've seen, but will not 393 00:22:14,140 --> 00:22:16,980 be identical to those. 394 00:22:16,980 --> 00:22:20,260 OK, so where is my cursor? 395 00:22:25,480 --> 00:22:30,760 And finally, as I indicated, the grades 396 00:22:30,760 --> 00:22:33,080 will be posted according to your pseudonym. 397 00:22:33,080 --> 00:22:36,810 So as time goes on, this table will be completed. 398 00:22:36,810 --> 00:22:42,560 And the only other thing to note is that there's actually 399 00:22:42,560 --> 00:22:47,090 going to be lecture notes for the various materials, 400 00:22:47,090 --> 00:22:49,550 starting from the first lecture, that 401 00:22:49,550 --> 00:22:52,877 will be devoted to thermodynamics. 402 00:22:52,877 --> 00:22:53,460 Any questions? 403 00:22:58,135 --> 00:22:59,105 OK. 404 00:22:59,105 --> 00:23:01,961 So let me copy that first sentence, 405 00:23:01,961 --> 00:23:03,960 and we will go on and talk about thermodynamics. 406 00:23:24,490 --> 00:23:38,016 The phenomenological description of equilibrium properties 407 00:23:38,016 --> 00:23:39,660 of microscopic systems. 408 00:23:54,570 --> 00:23:55,770 And get rid of this. 409 00:24:19,420 --> 00:24:23,490 One thing that I should have emphasized 410 00:24:23,490 --> 00:24:28,610 when I was doing the syllabus is that I expect that most of you 411 00:24:28,610 --> 00:24:31,790 have seen thermodynamics, have done 412 00:24:31,790 --> 00:24:34,040 in a certain amount of statistical mechanics, 413 00:24:34,040 --> 00:24:35,300 et cetera. 414 00:24:35,300 --> 00:24:39,230 So the idea here is really to bring 415 00:24:39,230 --> 00:24:42,180 the diversity of backgrounds that you have, 416 00:24:42,180 --> 00:24:45,090 for our graduate students, and also I know that our students 417 00:24:45,090 --> 00:24:49,030 from other departments, more or less in line with each other. 418 00:24:49,030 --> 00:24:51,020 So a lot of these things I expect 419 00:24:51,020 --> 00:24:53,680 to be, kind of, review materials, or things 420 00:24:53,680 --> 00:24:55,080 that you have seen. 421 00:24:55,080 --> 00:24:59,480 That's one reason that we kind of go through them rapidly. 422 00:24:59,480 --> 00:25:01,580 And hopefully, however, there will 423 00:25:01,580 --> 00:25:06,900 be some kind of logical systematic way of thinking 424 00:25:06,900 --> 00:25:10,380 about the entirety of them that would be useful to you. 425 00:25:10,380 --> 00:25:13,980 And in particular, thermodynamics, you say, 426 00:25:13,980 --> 00:25:16,090 is an old subject, and if you're ultimately 427 00:25:16,090 --> 00:25:18,390 going to derive the laws of thermodynamics 428 00:25:18,390 --> 00:25:23,260 from some more precise microscopic description, 429 00:25:23,260 --> 00:25:25,970 why should we go through this exercise? 430 00:25:25,970 --> 00:25:28,220 The reason is that there is really 431 00:25:28,220 --> 00:25:31,120 a beautiful example of how you can 432 00:25:31,120 --> 00:25:34,020 look at the system as a black box, 433 00:25:34,020 --> 00:25:36,860 and gradually, based on observation, 434 00:25:36,860 --> 00:25:40,040 build a consistent mathematical framework 435 00:25:40,040 --> 00:25:41,870 to describe its properties. 436 00:25:41,870 --> 00:25:45,690 And it is useful in various branches 437 00:25:45,690 --> 00:25:47,690 of science and physics. 438 00:25:47,690 --> 00:25:52,120 And kind of a more 20th century example that I can think of 439 00:25:52,120 --> 00:25:54,700 is, Landau's approach to superconductivity 440 00:25:54,700 --> 00:25:58,710 and superfluidity, where without knowing the microscopic origin 441 00:25:58,710 --> 00:26:01,490 of that, based on kind of phenomenology, 442 00:26:01,490 --> 00:26:04,690 you could write down very precise description 443 00:26:04,690 --> 00:26:07,346 of the kinds of things that superconductors 444 00:26:07,346 --> 00:26:10,410 and superfluids could manifest. 445 00:26:10,410 --> 00:26:15,680 So let's sort of put yourselves, put ourselves, 446 00:26:15,680 --> 00:26:19,500 in the perspective of how this science of thermodynamics 447 00:26:19,500 --> 00:26:20,720 was developed. 448 00:26:20,720 --> 00:26:25,520 And this is at the time where Newtonian mechanics had 449 00:26:25,520 --> 00:26:26,930 shown its power. 450 00:26:26,930 --> 00:26:30,020 It can describe orbits of things going around the sun, 451 00:26:30,020 --> 00:26:31,720 and all kinds of other things. 452 00:26:34,420 --> 00:26:36,400 But that description, clearly, does not 453 00:26:36,400 --> 00:26:42,380 apply to very simple things like how you heat up a pan of water. 454 00:26:42,380 --> 00:26:45,605 So there's some elements, including thermal properties, 455 00:26:45,605 --> 00:26:47,930 that are missing from that description. 456 00:26:47,930 --> 00:26:49,860 And you would like to complete that theory, 457 00:26:49,860 --> 00:26:53,590 or develop a theory, that is able to describe also 458 00:26:53,590 --> 00:26:55,800 heat and thermal properties. 459 00:26:55,800 --> 00:27:01,220 So how do you go about that, given that your perspective is 460 00:27:01,220 --> 00:27:04,050 the Newtonian prescription? 461 00:27:04,050 --> 00:27:08,490 So first thing to sort of a, kind of, 462 00:27:08,490 --> 00:27:12,510 parse among all of these elements, is system. 463 00:27:12,510 --> 00:27:16,420 So when describing Newtonian mechanics, 464 00:27:16,420 --> 00:27:18,730 you sort of idealize, certainly you 465 00:27:18,730 --> 00:27:22,810 realize that Newtonian mechanics does not describe things 466 00:27:22,810 --> 00:27:24,320 that we see in everyday world. 467 00:27:24,320 --> 00:27:26,420 You kind of think about point particle 468 00:27:26,420 --> 00:27:29,310 and how a point particle would move in free space. 469 00:27:29,310 --> 00:27:36,680 And so let's try to do a similar thing for our kinds of systems. 470 00:27:36,680 --> 00:27:40,080 And the thing that is causing us some difficulty 471 00:27:40,080 --> 00:27:43,370 is this issue of heat. 472 00:27:43,370 --> 00:27:48,520 And so what you can do is you can potentially 473 00:27:48,520 --> 00:27:58,440 isolate your system thermally by, what I would call, 474 00:27:58,440 --> 00:27:59,480 adiabatic laws. 475 00:28:02,830 --> 00:28:09,280 Basically say that there's these things, such as heat, 476 00:28:09,280 --> 00:28:12,370 that goes into the system that causes difficulty for me. 477 00:28:12,370 --> 00:28:14,230 So let's imagine in the same that I'm 478 00:28:14,230 --> 00:28:16,900 thinking of the point particle, that whatever 479 00:28:16,900 --> 00:28:21,350 I have is isolated from the rest of the universe, 480 00:28:21,350 --> 00:28:25,030 in some kind of box that does not allow heat transfer. 481 00:28:28,220 --> 00:28:32,080 This is to be opposed with walls that we would like 482 00:28:32,080 --> 00:28:39,590 to eventually look at, which do allow heat transfer. 483 00:28:39,590 --> 00:28:41,510 Let me choose a different color. 484 00:28:41,510 --> 00:28:43,410 Let's say green. 485 00:28:46,260 --> 00:28:52,970 So ultimately, I want to allow the exchange of whatever 486 00:28:52,970 --> 00:28:54,960 this heat is in thermal properties 487 00:28:54,960 --> 00:28:58,720 to go and take place with my system. 488 00:28:58,720 --> 00:29:01,360 OK? 489 00:29:01,360 --> 00:29:04,420 Now, this is basically isolation. 490 00:29:04,420 --> 00:29:13,460 The next element is to wait for your system 491 00:29:13,460 --> 00:29:14,690 to come to equilibrium. 492 00:29:14,690 --> 00:29:23,740 be You realize that when you, for example, start 493 00:29:23,740 --> 00:29:25,140 with something that is like this, 494 00:29:25,140 --> 00:29:28,520 you change one of the walls to allow heat to go into it. 495 00:29:28,520 --> 00:29:30,720 Then the system undergoes some changes. 496 00:29:30,720 --> 00:29:34,010 Properties that you're measuring are not 497 00:29:34,010 --> 00:29:37,300 well-defined over some period where these changes 498 00:29:37,300 --> 00:29:40,270 taking place, but if you wait sufficiently, 499 00:29:40,270 --> 00:29:42,670 then they relax to some new values. 500 00:29:42,670 --> 00:29:45,690 And then you can start making measurements. 501 00:29:45,690 --> 00:29:55,120 So this is when properties don't change. 502 00:29:55,120 --> 00:29:59,220 And the key here is observation time. 503 00:29:59,220 --> 00:30:01,900 This is part of the phenomenology, 504 00:30:01,900 --> 00:30:03,500 because it is not precise. 505 00:30:03,500 --> 00:30:05,700 I can't tell you how long you have to wait. 506 00:30:05,700 --> 00:30:08,370 It depends on the system under consideration. 507 00:30:08,370 --> 00:30:11,680 And some systems come to equilibrium easily, 508 00:30:11,680 --> 00:30:13,910 some take a long time. 509 00:30:13,910 --> 00:30:17,650 So what are the properties that you can measure? 510 00:30:17,650 --> 00:30:23,600 Once things have settled down and no longer change with time, 511 00:30:23,600 --> 00:30:28,200 you can start to measure various properties. 512 00:30:28,200 --> 00:30:31,435 The ones that are very easy for you to identify 513 00:30:31,435 --> 00:30:35,670 are things that are associated with mechanical work, 514 00:30:35,670 --> 00:30:37,720 or mechanical properties. 515 00:30:41,060 --> 00:30:45,660 And, for example, if you have a box that contains a gas, 516 00:30:45,660 --> 00:30:49,640 you can immediately see well, what's the volume of the gas? 517 00:30:49,640 --> 00:30:51,950 You can calculate what pressure it 518 00:30:51,950 --> 00:30:54,670 is exerting on its environment. 519 00:30:54,670 --> 00:30:56,270 So this is for a gas. 520 00:30:56,270 --> 00:30:58,970 You could have, for example, a wire. 521 00:30:58,970 --> 00:31:02,090 If you have a wire, you could calculate, rather than 522 00:31:02,090 --> 00:31:04,740 its volume, its length and the force 523 00:31:04,740 --> 00:31:06,940 with which you are pulling it. 524 00:31:06,940 --> 00:31:10,980 It could be something like a magnet. 525 00:31:10,980 --> 00:31:14,830 And you could put some kind of a magnetic field on it, 526 00:31:14,830 --> 00:31:17,740 and figure out what the magnetization is. 527 00:31:17,740 --> 00:31:21,810 And this list of mechanical properties goes on. 528 00:31:21,810 --> 00:31:24,700 But you know that that's not the end of the story. 529 00:31:24,700 --> 00:31:29,380 You kind of expect that there are additional things that 530 00:31:29,380 --> 00:31:32,400 have to do with thermal aspects that you 531 00:31:32,400 --> 00:31:34,330 haven't taken into account. 532 00:31:34,330 --> 00:31:37,570 And as of yet, you don't quite know what they are. 533 00:31:37,570 --> 00:31:41,970 And you have to gradually build upon those properties. 534 00:31:41,970 --> 00:31:45,580 So you have a system. 535 00:31:45,580 --> 00:31:48,160 These are kind of the analogs of the coordinates 536 00:31:48,160 --> 00:31:49,900 and potentially velocities that you 537 00:31:49,900 --> 00:31:52,210 would have for any Newtonian particles. 538 00:31:52,210 --> 00:31:56,990 A way of describing your idealized system. 539 00:31:56,990 --> 00:32:03,330 And then you want to find rules by which these coordinates are 540 00:32:03,330 --> 00:32:05,870 coevolving, or doing things together. 541 00:32:05,870 --> 00:32:14,150 And so for that you rely on observations, 542 00:32:14,150 --> 00:32:17,039 and construct laws of thermodynamics. 543 00:32:26,150 --> 00:32:30,700 All right so this is the general approach. 544 00:32:30,700 --> 00:32:32,890 And once you follow this, let's say, 545 00:32:32,890 --> 00:32:36,290 what's the first thing that you encounter? 546 00:32:36,290 --> 00:32:40,530 You encounter what is encoded to the zeroth law. 547 00:32:43,760 --> 00:32:46,190 What's the zeroth law? 548 00:32:46,190 --> 00:32:50,540 The zeroth law is the following statement, 549 00:32:50,540 --> 00:33:02,210 if two systems-- let's call them A and B-- are separately 550 00:33:02,210 --> 00:33:18,940 in equilibrium with C-- with a third system-- then 551 00:33:18,940 --> 00:33:20,620 they are in equilibrium with each other. 552 00:33:34,710 --> 00:33:37,700 This is sort of this statement that the property 553 00:33:37,700 --> 00:33:41,765 of equilibrium has the character of transitivity. 554 00:33:50,440 --> 00:33:59,310 So what that means, pictorially, is something like this. 555 00:34:02,670 --> 00:34:15,040 Suppose I have my two boxes, A and B. And the third system 556 00:34:15,040 --> 00:34:19,960 that we're calling C. And we've established 557 00:34:19,960 --> 00:34:23,690 that A and B are separately in thermal equilibrium with C, 558 00:34:23,690 --> 00:34:28,949 which means that we have allowed exchange of heat 559 00:34:28,949 --> 00:34:33,610 to take place between A and C, between B and C, 560 00:34:33,610 --> 00:34:36,564 but currently, we assume nothing-- 561 00:34:36,564 --> 00:34:42,050 or we assume that B and C are not connected to each other. 562 00:34:42,050 --> 00:34:44,210 The statement of the law is that if I 563 00:34:44,210 --> 00:34:57,650 were to replace this red with green, so that I have also 564 00:34:57,650 --> 00:35:02,700 exchange that is going on between A and B, 565 00:35:02,700 --> 00:35:04,870 then nothing happens. 566 00:35:04,870 --> 00:35:07,450 A and B are already in equilibrium, 567 00:35:07,450 --> 00:35:11,960 and the fact that you open the possibility of exchange 568 00:35:11,960 --> 00:35:15,730 of heat between them does not change things. 569 00:35:15,730 --> 00:35:20,060 And again, this is this consequence of transitivity. 570 00:35:20,060 --> 00:35:24,030 And one of its kind of important implications 571 00:35:24,030 --> 00:35:27,230 ultimately is that we are allowed now 572 00:35:27,230 --> 00:35:32,480 based on this to add one more coordinate to this description 573 00:35:32,480 --> 00:35:34,810 that you have. 574 00:35:34,810 --> 00:35:38,730 That coordinate is the analog of temperature. 575 00:35:38,730 --> 00:35:47,460 So this transitivity really implies the existence 576 00:35:47,460 --> 00:35:49,620 of some kind of empirical temperature. 577 00:35:53,600 --> 00:35:57,160 And you may say well, this is such an obvious thing, 578 00:35:57,160 --> 00:35:59,140 there should be transitivity. 579 00:35:59,140 --> 00:36:01,580 Well, I want to usually give people 580 00:36:01,580 --> 00:36:05,960 examples that transitivity is not a universal property, 581 00:36:05,960 --> 00:36:06,960 by as follows. 582 00:36:06,960 --> 00:36:10,390 Suppose that within this room, there 583 00:36:10,390 --> 00:36:13,240 is A who wants to go on a date with C, 584 00:36:13,240 --> 00:36:15,930 and B who wants to go on a date with C. 585 00:36:15,930 --> 00:36:19,190 I'm pretty sure that it's not going to be the property that A 586 00:36:19,190 --> 00:36:25,530 wants to go through the date with B. It's 20%. 587 00:36:25,530 --> 00:36:26,040 All right. 588 00:36:30,560 --> 00:36:34,080 So let's see how we can ensure this. 589 00:36:34,080 --> 00:36:40,400 So we said that if some system has reached equilibrium, 590 00:36:40,400 --> 00:36:42,440 it has some set of coordinates. 591 00:36:42,440 --> 00:36:44,260 Let's call them A1, A2, et cetera. 592 00:36:44,260 --> 00:36:45,430 There's some number of them. 593 00:36:45,430 --> 00:36:46,660 We don't know. 594 00:36:46,660 --> 00:36:50,040 Similarly, here, we have C1, C2, et cetera. 595 00:36:50,040 --> 00:36:55,130 And for B, we have B1, B2. 596 00:36:55,130 --> 00:37:03,200 Now what does it mean that I have equilibrium of A and B? 597 00:37:03,200 --> 00:37:09,010 The implication is that if I have a system by itself, 598 00:37:09,010 --> 00:37:11,130 I have a bunch of possible coordinates. 599 00:37:11,130 --> 00:37:13,610 I can be anywhere in this coordinate space. 600 00:37:13,610 --> 00:37:16,440 And if I separately have system C, 601 00:37:16,440 --> 00:37:18,420 I have another bunch of coordinates. 602 00:37:18,420 --> 00:37:21,700 And I can be anywhere in this coordinate space. 603 00:37:21,700 --> 00:37:26,150 But if I force A and B to come together and exchange heat 604 00:37:26,150 --> 00:37:29,420 and reaches equilibrium, that is a constraint, 605 00:37:29,420 --> 00:37:32,820 which means that this set of coordinates of the two cannot 606 00:37:32,820 --> 00:37:34,660 be independently varied. 607 00:37:34,660 --> 00:37:38,980 There has to be some functional relationship between them, 608 00:37:38,980 --> 00:37:43,750 which we can, for example, cast into this form. 609 00:37:43,750 --> 00:37:46,270 So equilibrium, one constraint, one kind 610 00:37:46,270 --> 00:37:48,310 of mathematical relation. 611 00:37:48,310 --> 00:37:51,470 Similarly, equilibrium of-- this was A 612 00:37:51,470 --> 00:37:57,510 and C, B and C-- would give us some other function 613 00:37:57,510 --> 00:38:07,080 BC of coordinates of B and coordinates of C equal to 0. 614 00:38:07,080 --> 00:38:09,730 OK? 615 00:38:09,730 --> 00:38:15,540 So there is one thing that I can do. 616 00:38:15,540 --> 00:38:21,960 I can take this expression and recast it, and write it 617 00:38:21,960 --> 00:38:25,160 as let's say, the first coordinate that describes 618 00:38:25,160 --> 00:38:29,500 C is some other function, which I will call big F 619 00:38:29,500 --> 00:38:36,600 AC of coordinates of A. And all coordinates 620 00:38:36,600 --> 00:38:41,650 of C, except the first one that I have removed. 621 00:38:41,650 --> 00:38:44,130 Yes? 622 00:38:44,130 --> 00:38:47,810 AUDIENCE: When you put that function F 623 00:38:47,810 --> 00:38:50,500 AC or all the coordinates of A, and all the coordinates of C 624 00:38:50,500 --> 00:38:56,480 being zero, is that [? polynomy ?] always going 625 00:38:56,480 --> 00:39:00,170 to be true for the first law, or do you just give an example? 626 00:39:07,340 --> 00:39:10,130 PROFESSOR: It will be always true 627 00:39:10,130 --> 00:39:14,360 that I have some set of coordinates for 1, 628 00:39:14,360 --> 00:39:18,200 some set of coordinates with 2, and equilibrium 629 00:39:18,200 --> 00:39:20,460 for the two sets of coordinates is 630 00:39:20,460 --> 00:39:24,010 going to be expressible in terms of some function that 631 00:39:24,010 --> 00:39:26,230 could be arbitrarily complicated. 632 00:39:26,230 --> 00:39:29,240 It may be that I can't even write this, 633 00:39:29,240 --> 00:39:33,220 but I have to graph it, or some kind of place 634 00:39:33,220 --> 00:39:34,410 in the coordinates space. 635 00:39:34,410 --> 00:39:36,250 So what I mean is the following, you 636 00:39:36,250 --> 00:39:38,560 can imagine some higher dimensional 637 00:39:38,560 --> 00:39:41,440 space that is spanned by As and the Cs. 638 00:39:41,440 --> 00:39:44,160 And each point, where the things are in equilibrium, 639 00:39:44,160 --> 00:39:49,150 will be some-- you can put a cross in this coordinate space. 640 00:39:49,150 --> 00:39:52,000 And you can span the various places 641 00:39:52,000 --> 00:39:53,910 where equilibration takes place. 642 00:39:53,910 --> 00:39:57,490 And you will have a surface in this space. 643 00:39:57,490 --> 00:40:00,280 And that surface potential you can describe mathematically 644 00:40:00,280 --> 00:40:02,354 like this. 645 00:40:02,354 --> 00:40:03,260 OK? 646 00:40:03,260 --> 00:40:06,300 And similarly, I can do the same thing. 647 00:40:06,300 --> 00:40:09,540 And pick out coordinate C1 here, and write it 648 00:40:09,540 --> 00:40:22,020 as F BC of A1, A2, and C2, C3-- sorry, this is B1, B2. 649 00:40:22,020 --> 00:40:26,020 Actually, this brings me, this question, to maybe one point 650 00:40:26,020 --> 00:40:27,190 that I should make. 651 00:40:27,190 --> 00:40:31,270 That sometimes during this course, I will do things, 652 00:40:31,270 --> 00:40:33,130 and maybe I will even say that I do 653 00:40:33,130 --> 00:40:36,330 things that are physically rigorous, but maybe not 654 00:40:36,330 --> 00:40:38,460 so mathematically. 655 00:40:38,460 --> 00:40:42,240 And one example of this is precisely this statement. 656 00:40:42,240 --> 00:40:44,010 That is, if you give a mathematician 657 00:40:44,010 --> 00:40:45,610 and there is a function like this. 658 00:40:45,610 --> 00:40:48,329 And then you pick one, C1, and you write it 659 00:40:48,329 --> 00:40:49,870 as a function of all the others, they 660 00:40:49,870 --> 00:40:53,230 say, oh, how do know that that's even possible, that this 661 00:40:53,230 --> 00:40:55,010 exists? 662 00:40:55,010 --> 00:40:57,030 And generally, it isn't. 663 00:40:57,030 --> 00:40:59,860 A simple example would be A squared plus C 664 00:40:59,860 --> 00:41:04,590 squared equals to zero, then C's multiple valued. 665 00:41:04,590 --> 00:41:08,390 So the reason this is physically correct 666 00:41:08,390 --> 00:41:11,940 is because if we set up this situation, and really 667 00:41:11,940 --> 00:41:13,790 these very physical quantities, I 668 00:41:13,790 --> 00:41:17,540 know that I dialed my C1 to this number here, 669 00:41:17,540 --> 00:41:19,560 and all of the other things adjusted. 670 00:41:19,560 --> 00:41:22,160 So this is kind of physically OK, 671 00:41:22,160 --> 00:41:25,420 although mathematically, you would have to potentially do 672 00:41:25,420 --> 00:41:30,490 a lot of [INAUDIBLE] to arrive at this stage. 673 00:41:30,490 --> 00:41:31,870 OK. 674 00:41:31,870 --> 00:41:36,870 So now if I have to put all of those things together, 675 00:41:36,870 --> 00:41:40,540 the fact that I have equilibrium of A and B, 676 00:41:40,540 --> 00:41:46,460 plus equilibrium of B and C, then 677 00:41:46,460 --> 00:41:50,170 implies that there is eliminating 678 00:41:50,170 --> 00:41:52,890 C1 between these two, some function 679 00:41:52,890 --> 00:41:57,120 that depends on the coordinates of A and coordinates of C, 680 00:41:57,120 --> 00:42:00,870 starting from the second one, equal to some other function. 681 00:42:00,870 --> 00:42:04,090 And these functions could be very different, 682 00:42:04,090 --> 00:42:07,420 coordinates of B and coordinates of C 683 00:42:07,420 --> 00:42:09,820 starting from the second one. 684 00:42:09,820 --> 00:42:10,350 Yes? 685 00:42:10,350 --> 00:42:12,562 AUDIENCE: Do you need A and C on the left there? 686 00:42:12,562 --> 00:42:14,353 PROFESSOR: A and C, and B and C. Thank you. 687 00:42:19,183 --> 00:42:20,640 OK? 688 00:42:20,640 --> 00:42:24,410 So this, everything that we have worked out here, 689 00:42:24,410 --> 00:42:27,350 really concerns putting the first part 690 00:42:27,350 --> 00:42:32,870 of this equation in mathematical form. 691 00:42:32,870 --> 00:42:35,730 But this statement-- in mathematical form-- 692 00:42:35,730 --> 00:42:38,830 but given the first part of this statement, that 693 00:42:38,830 --> 00:42:41,900 is, the second part of the statement, that 694 00:42:41,900 --> 00:42:47,710 is that I know that if I remove that red and make it green, 695 00:42:47,710 --> 00:42:51,330 so that heat can exchange, for the same values of A and B, 696 00:42:51,330 --> 00:42:54,880 A and B are in equilibrium, so I know that there is a functional 697 00:42:54,880 --> 00:42:56,200 form. 698 00:42:56,200 --> 00:43:03,150 So this is equilibrium of A and B, implies 699 00:43:03,150 --> 00:43:07,910 that there is this function that we were looking at before, 700 00:43:07,910 --> 00:43:13,400 that relates coordinates of A and B 701 00:43:13,400 --> 00:43:15,430 that constrains the equilibrium that 702 00:43:15,430 --> 00:43:19,880 should exist between A and B. OK. 703 00:43:19,880 --> 00:43:23,650 So the first part of the statement of the zeroth law, 704 00:43:23,650 --> 00:43:27,210 and the second part of the statement of the zeroth law 705 00:43:27,210 --> 00:43:31,910 can be written mathematically in these two forms. 706 00:43:31,910 --> 00:43:37,920 And the nice part about the first part 707 00:43:37,920 --> 00:43:41,300 is that it says that the equilibrium constraint that I 708 00:43:41,300 --> 00:43:44,930 have between A's and B's can mathematically 709 00:43:44,930 --> 00:43:49,730 be kind of spread out into some function on the left that only 710 00:43:49,730 --> 00:43:52,180 depends on the coordinates of A, and some 711 00:43:52,180 --> 00:43:59,300 function on the right that only depends on coordinates of B. 712 00:43:59,300 --> 00:44:02,210 So that's important, because it says 713 00:44:02,210 --> 00:44:07,760 that ultimately the equilibration between two 714 00:44:07,760 --> 00:44:11,680 objects can be cast mathematically 715 00:44:11,680 --> 00:44:13,760 as having some kind of a function-- 716 00:44:13,760 --> 00:44:16,700 and we don't know anything about the form of that function-- 717 00:44:16,700 --> 00:44:20,540 that only depends on the coordinates of A. 718 00:44:20,540 --> 00:44:25,420 And equilibration implies that there exists 719 00:44:25,420 --> 00:44:28,260 some other function that only depends 720 00:44:28,260 --> 00:44:30,420 on the coordinates of the other one. 721 00:44:30,420 --> 00:44:35,130 And in equilibrium, those two functional forms 722 00:44:35,130 --> 00:44:37,330 have to be the same. 723 00:44:37,330 --> 00:44:44,010 Now, there's two ways of getting to this statement from the two 724 00:44:44,010 --> 00:44:46,850 things that I have within up there. 725 00:44:46,850 --> 00:44:50,610 One of them is to choose some particular reference system 726 00:44:50,610 --> 00:44:54,450 for C. Let's say your C is seawater 727 00:44:54,450 --> 00:44:57,490 at some particular set of conditions. 728 00:44:57,490 --> 00:45:01,190 And so then, these are really constants 729 00:45:01,190 --> 00:45:03,795 that are appropriate to see water. 730 00:45:03,795 --> 00:45:06,710 And then you have chosen a function that 731 00:45:06,710 --> 00:45:10,340 depends on variables of A, of course of some function of B. 732 00:45:10,340 --> 00:45:12,550 Or you can say, well, I can replace 733 00:45:12,550 --> 00:45:14,710 this seawater by something else. 734 00:45:14,710 --> 00:45:18,220 And irrespective of what I choose, 735 00:45:18,220 --> 00:45:22,500 A and B, there by our definition in equilibrium with each other, 736 00:45:22,500 --> 00:45:27,650 no matter what I did with C. Or some various range of C things 737 00:45:27,650 --> 00:45:31,670 that I can do, maintaining this equilibrium between A and B. 738 00:45:31,670 --> 00:45:34,350 So in that perspective, the C variables 739 00:45:34,350 --> 00:45:35,870 are dummy coordinates. 740 00:45:35,870 --> 00:45:38,560 So you should be able to cancel them out 741 00:45:38,560 --> 00:45:40,800 from the two sides of the equation, 742 00:45:40,800 --> 00:45:44,910 and get some kind of a form like this. 743 00:45:44,910 --> 00:45:47,980 Either way, what that really means 744 00:45:47,980 --> 00:45:55,740 is that if I list all of the coordinates of, say, A, 745 00:45:55,740 --> 00:46:01,930 and I put it, say, in equilibrium with a bath that 746 00:46:01,930 --> 00:46:04,650 is at some particular temperature, 747 00:46:04,650 --> 00:46:07,580 the coordinates of A and B are constrained 748 00:46:07,580 --> 00:46:11,020 to lie in some particular surface that would correspond 749 00:46:11,020 --> 00:46:13,680 to that particular theta. 750 00:46:13,680 --> 00:46:19,670 And if I have another system for B, 751 00:46:19,670 --> 00:46:23,270 the isotherm could have completely different for that 752 00:46:23,270 --> 00:46:24,750 data. 753 00:46:24,750 --> 00:46:27,800 But any time A and B are in equilibrium, 754 00:46:27,800 --> 00:46:30,555 they would be lying on the isotherm that 755 00:46:30,555 --> 00:46:33,350 would correspond to the same fate. 756 00:46:33,350 --> 00:46:36,580 Now, again, a mechanical version of this, 757 00:46:36,580 --> 00:46:40,210 that is certainly hopefully demystifies 758 00:46:40,210 --> 00:46:43,890 any mystification that may remain, is to think a scale. 759 00:46:47,890 --> 00:46:50,050 You have something A on this scale. 760 00:46:50,050 --> 00:46:51,830 And you have something C on this scale. 761 00:46:51,830 --> 00:46:53,355 And the scale is balanced. 762 00:46:53,355 --> 00:46:59,390 You replace A with B, and B and C are in balance, 763 00:46:59,390 --> 00:47:02,220 then you know that A and B are in balance with each other. 764 00:47:02,220 --> 00:47:04,450 And that implies that, indeed, there 765 00:47:04,450 --> 00:47:08,552 is a property of the objects that you put on the balance. 766 00:47:08,552 --> 00:47:10,010 You can either think of it as mass, 767 00:47:10,010 --> 00:47:12,980 or more appropriately, the gravitational force 768 00:47:12,980 --> 00:47:16,190 that they experience, that they need. 769 00:47:16,190 --> 00:47:17,290 The thing is balanced. 770 00:47:17,290 --> 00:47:19,090 The forces are equal. 771 00:47:19,090 --> 00:47:21,230 So it's essentially the same thing. 772 00:47:21,230 --> 00:47:23,520 Now having established this, then you 773 00:47:23,520 --> 00:47:27,150 want to go and figure out what the formula is 774 00:47:27,150 --> 00:47:31,320 that relates the property that needs to be balanced, 775 00:47:31,320 --> 00:47:34,120 which is maybe the mass or the gravitational force in terms 776 00:47:34,120 --> 00:47:36,530 of density, volume, et cetera. 777 00:47:36,530 --> 00:47:38,400 So that's what you would like to do here. 778 00:47:38,400 --> 00:47:43,520 We would like to be able to relate this temperature 779 00:47:43,520 --> 00:47:46,510 function to all the other coordinates of the system, 780 00:47:46,510 --> 00:47:48,087 as we're going to. 781 00:47:48,087 --> 00:47:48,670 Any questions? 782 00:47:53,450 --> 00:47:54,858 Yes? 783 00:47:54,858 --> 00:47:58,211 AUDIENCE: So how did [INAUDIBLE] that there is the isotherm? 784 00:47:58,211 --> 00:48:01,570 Is it coming from the second conclusion? 785 00:48:01,570 --> 00:48:02,310 PROFESSOR: OK. 786 00:48:02,310 --> 00:48:08,320 So what we have said is that when two objects are 787 00:48:08,320 --> 00:48:13,170 in equilibrium, this function is the same between. 788 00:48:13,170 --> 00:48:17,270 So let's say that we pick some kind of a vat-- 789 00:48:17,270 --> 00:48:19,190 like, it could be a lake. 790 00:48:19,190 --> 00:48:21,930 And we know that the lake is so big 791 00:48:21,930 --> 00:48:24,710 that if you put something in equilibrium with that, 792 00:48:24,710 --> 00:48:28,520 it's not extracting too much heat or whatever from the lake 793 00:48:28,520 --> 00:48:33,390 can change its temperature function. 794 00:48:33,390 --> 00:48:37,300 So we put our system in equilibrium 795 00:48:37,300 --> 00:48:40,420 with the lake, which is at some particular fixed 796 00:48:40,420 --> 00:48:41,130 value of theta. 797 00:48:41,130 --> 00:48:45,520 And we don't know what theta is, but it's a constant. 798 00:48:45,520 --> 00:48:48,560 So if I were to fiddle around with the system 799 00:48:48,560 --> 00:48:51,080 that I put in the lake-- I change its volume, 800 00:48:51,080 --> 00:48:53,600 I change its length, I do something-- 801 00:48:53,600 --> 00:48:55,850 and it stays in equilibrium with the length, 802 00:48:55,850 --> 00:48:58,210 there is some function of the coordinates 803 00:48:58,210 --> 00:49:01,160 of that system that is equal to theta. 804 00:49:01,160 --> 00:49:07,320 So again, in general, I can make a diagram 805 00:49:07,320 --> 00:49:12,790 that has various coordinates of the system, A1, A2, A3. 806 00:49:12,790 --> 00:49:18,460 And for every combination that is that this theta, 807 00:49:18,460 --> 00:49:20,700 I will put a point here. 808 00:49:20,700 --> 00:49:24,410 And in principle, I can vary these coordinates. 809 00:49:24,410 --> 00:49:27,120 And this amounts to one constraint 810 00:49:27,120 --> 00:49:30,160 in however many dimensional space I have. 811 00:49:30,160 --> 00:49:34,170 So if we span some kind of a surface-- 812 00:49:34,170 --> 00:49:35,925 so if you're in three dimension, there 813 00:49:35,925 --> 00:49:37,605 would be a two dimensional surface. 814 00:49:37,605 --> 00:49:38,980 If you're in two dimension, there 815 00:49:38,980 --> 00:49:42,950 would be a line that would correspond to this constraint. 816 00:49:42,950 --> 00:49:45,480 Presumably, if I change the lake with something 817 00:49:45,480 --> 00:49:47,860 else, so that theta changes, I will 818 00:49:47,860 --> 00:49:51,685 be prescribing some other curve and some other surface in this. 819 00:49:51,685 --> 00:49:58,415 Now these are surfaces in coordinate space of A. 820 00:49:58,415 --> 00:50:02,880 In order to be in equilibrium with an entity at the fixed 821 00:50:02,880 --> 00:50:07,320 value of theta, they prescribe some particular surface 822 00:50:07,320 --> 00:50:10,765 in the entire coordinate space and they're called isotherms. 823 00:50:16,080 --> 00:50:18,560 OK? 824 00:50:18,560 --> 00:50:20,740 Actually, let's state that a little 825 00:50:20,740 --> 00:50:26,840 bit further, because you would like 826 00:50:26,840 --> 00:50:31,020 to give a number to temperature, so many degrees 827 00:50:31,020 --> 00:50:33,820 Celsius, or Fahrenheit, or whatever. 828 00:50:33,820 --> 00:50:36,700 So how do you do that? 829 00:50:36,700 --> 00:50:45,670 And one way to do that is to use what is called the ideal gas 830 00:50:45,670 --> 00:50:49,330 temperature space-- the ideal gas scale. 831 00:50:54,520 --> 00:50:58,040 So you need some property at this stage 832 00:50:58,040 --> 00:51:02,870 in order to construct a temperature scale. 833 00:51:02,870 --> 00:51:07,810 And it turns out that a gas is a system that we 834 00:51:07,810 --> 00:51:11,550 keep coming back to again and again. 835 00:51:11,550 --> 00:51:15,670 So as I go through the various laws of thermodynamics, 836 00:51:15,670 --> 00:51:18,260 I will mention something special that 837 00:51:18,260 --> 00:51:23,100 happens to this law for the case of this ideal gas. 838 00:51:23,100 --> 00:51:25,160 And actually, right now, define also 839 00:51:25,160 --> 00:51:27,940 what I mean by the ideal gas. 840 00:51:27,940 --> 00:51:31,740 So we said that a gas, in general, you 841 00:51:31,740 --> 00:51:37,710 can define through coordinates P and V. 842 00:51:37,710 --> 00:51:42,140 So if I put this gas in a piston, 843 00:51:42,140 --> 00:51:47,990 and I submerge this piston, let's say in a lake, 844 00:51:47,990 --> 00:51:53,270 so that it is always at whatever temperature this lake is. 845 00:51:53,270 --> 00:51:57,500 Then I can change the volume of this, 846 00:51:57,500 --> 00:52:00,080 and measure what the pressure is, or change the pressure, 847 00:52:00,080 --> 00:52:02,180 and figure out what the volume is. 848 00:52:02,180 --> 00:52:08,180 And I find out that there is a surface in this space where 849 00:52:08,180 --> 00:52:11,850 this curve that corresponds to being equilibrium 850 00:52:11,850 --> 00:52:15,780 with this [? leaves-- ?] this is the measure of the isotherm. 851 00:52:15,780 --> 00:52:21,850 Now the ideal gas law is that when I go to the limit there, 852 00:52:21,850 --> 00:52:27,430 V goes to infinity, or P goes to zero. 853 00:52:27,430 --> 00:52:31,130 So that the whole thing becomes very dilute. 854 00:52:31,130 --> 00:52:33,290 No matter what gas you put here-- 855 00:52:33,290 --> 00:52:35,400 whether it's argon, oxygen, krypton, 856 00:52:35,400 --> 00:52:43,020 whatever-- you find that in this limit, this shape of this curve 857 00:52:43,020 --> 00:52:47,220 is special in the sense that if you were to increase 858 00:52:47,220 --> 00:52:49,310 the volume by a factor of two, the pressure 859 00:52:49,310 --> 00:52:51,670 will go down by a factor of two, such 860 00:52:51,670 --> 00:52:53,740 that the product PV is a constant. 861 00:52:58,160 --> 00:53:01,650 And again, this is only true in the limit 862 00:53:01,650 --> 00:53:04,980 where either V goes to infinity or P goes to 0. 863 00:53:04,980 --> 00:53:07,770 And this is the property that all gases have. 864 00:53:10,480 --> 00:53:13,390 So you say OK, I will use that. 865 00:53:13,390 --> 00:53:17,000 Maybe I will define what the value of this product 866 00:53:17,000 --> 00:53:20,260 is to be the temperature. 867 00:53:20,260 --> 00:53:24,290 So if I were to replace this bath with a bath that 868 00:53:24,290 --> 00:53:28,430 was hotter than this product for the same amount of gas, 869 00:53:28,430 --> 00:53:30,590 for the same wire, would be different. 870 00:53:30,590 --> 00:53:33,726 And I would get a different constant. 871 00:53:33,726 --> 00:53:35,520 You maybe still want a number. 872 00:53:35,520 --> 00:53:40,400 So what you say is that the temperature 873 00:53:40,400 --> 00:53:51,340 of the system in degrees Kelvin is 273 times 16. 874 00:53:51,340 --> 00:54:04,030 The limit of PV, as V goes to infinity of your system, 875 00:54:04,030 --> 00:54:10,300 divided by the limit of the same thing 876 00:54:10,300 --> 00:54:18,240 at the triple point of water [? iced. ?] 877 00:54:22,070 --> 00:54:24,620 So what does that mean? 878 00:54:24,620 --> 00:54:29,090 So you have this thing by which you 879 00:54:29,090 --> 00:54:31,330 want to measure temperature. 880 00:54:31,330 --> 00:54:33,860 You put it in contact with the system-- could 881 00:54:33,860 --> 00:54:37,510 be a bath of water, could be anything, yes? 882 00:54:37,510 --> 00:54:39,310 You go through this exercise and you 883 00:54:39,310 --> 00:54:42,510 calculate what this product is. 884 00:54:42,510 --> 00:54:44,830 So you have this product. 885 00:54:44,830 --> 00:54:48,870 Then what you do is you take your system, 886 00:54:48,870 --> 00:54:52,800 and you put it in a case that there are icebergs, 887 00:54:52,800 --> 00:54:54,320 and there's water, and there will 888 00:54:54,320 --> 00:54:58,610 be some steam that will naturally evaporate. 889 00:54:58,610 --> 00:55:02,080 So you calculate the same product of PV 890 00:55:02,080 --> 00:55:05,380 in this system that is the triple point 891 00:55:05,380 --> 00:55:07,720 of ice water, et cetera. 892 00:55:07,720 --> 00:55:11,810 So you've measured one product appropriate to your system, 893 00:55:11,810 --> 00:55:14,380 one product appropriate to this reference 894 00:55:14,380 --> 00:55:19,290 point that people have set, and then the ratio of those things 895 00:55:19,290 --> 00:55:22,340 is going to give you the temperature of the system 896 00:55:22,340 --> 00:55:24,130 that you want to measure. 897 00:55:24,130 --> 00:55:27,740 This is clearly a very convoluted way of doing things, 898 00:55:27,740 --> 00:55:31,150 but it's a kind of rigorous definition of what 899 00:55:31,150 --> 00:55:34,470 the ideal gas temperature scale is. 900 00:55:34,470 --> 00:55:38,300 And it depends on this particular property 901 00:55:38,300 --> 00:55:44,290 of the diluted gases that the production of PV is a constant. 902 00:55:44,290 --> 00:55:49,880 And again, this number is set by definition 903 00:55:49,880 --> 00:55:53,930 to be the temperature of the triple point of the ice water 904 00:55:53,930 --> 00:55:54,430 gas. 905 00:56:08,670 --> 00:56:10,634 OK. 906 00:56:10,634 --> 00:56:11,300 Other questions? 907 00:56:13,930 --> 00:56:14,820 All right. 908 00:56:14,820 --> 00:56:20,360 So this is now time to go through the first law. 909 00:56:24,070 --> 00:56:28,470 Now I'll write again this statement, 910 00:56:28,470 --> 00:56:31,690 and then we'll start to discuss what it really means. 911 00:56:31,690 --> 00:56:56,250 So if the state of an otherwise adiabatically isolated system 912 00:56:56,250 --> 00:57:14,750 is changed by work, the amount of work 913 00:57:14,750 --> 00:57:22,900 is only function of initial and final points. 914 00:57:32,240 --> 00:57:32,740 OK. 915 00:57:32,740 --> 00:57:36,270 So let's parse what that means and think 916 00:57:36,270 --> 00:57:38,000 about some particular example. 917 00:57:38,000 --> 00:57:42,575 So let's imagine that we have isolated some system. 918 00:57:46,570 --> 00:57:49,740 So that it's not completely boring, 919 00:57:49,740 --> 00:57:52,380 let's imagine that maybe it's a gas. 920 00:57:52,380 --> 00:57:56,590 So it has P and V as some set of coordinates. 921 00:57:56,590 --> 00:58:00,470 Let's say that we put some kind of a spring or wire in it, 922 00:58:00,470 --> 00:58:01,730 so we can pull on it. 923 00:58:01,730 --> 00:58:04,030 And we can ask how much we pulled, 924 00:58:04,030 --> 00:58:07,410 and what is the length of this system. 925 00:58:07,410 --> 00:58:09,530 Maybe we even put a magnet in it, 926 00:58:09,530 --> 00:58:15,080 so we have magnetization that we can measure 927 00:58:15,080 --> 00:58:19,070 if we were to pass some kind of a current 928 00:58:19,070 --> 00:58:21,630 and exert some kind of magnetic field. 929 00:58:21,630 --> 00:58:24,520 So there's a whole bunch of coordinates 930 00:58:24,520 --> 00:58:30,420 that I'm completely familiar with from doing my mechanics 931 00:58:30,420 --> 00:58:32,760 courses and electromagnetic courses. 932 00:58:32,760 --> 00:58:38,880 So I know various ways to do mechanical work on this system. 933 00:58:38,880 --> 00:58:43,630 So the system is otherwise isolated, 934 00:58:43,630 --> 00:58:48,640 because I don't really know how to handle this concept of heat 935 00:58:48,640 --> 00:58:52,380 yet, but I certainly have no problems with mechanical work. 936 00:58:52,380 --> 00:58:57,320 And so what I do is, I imagine that it is initially isolated. 937 00:58:57,320 --> 00:59:01,570 What I do is therefore, I have some-- in this case, 938 00:59:01,570 --> 00:59:03,780 six dimensional coordinate space. 939 00:59:03,780 --> 00:59:06,940 I'm will only draw two out of the six. 940 00:59:06,940 --> 00:59:12,640 And I start some initial point, let's call it I. 941 00:59:12,640 --> 00:59:16,380 And then I start doing various types of things to this. 942 00:59:16,380 --> 00:59:19,270 I could, for example, first pull on this, 943 00:59:19,270 --> 00:59:21,110 so that the length changes, changes 944 00:59:21,110 --> 00:59:24,810 the current, put pressure so that 945 00:59:24,810 --> 00:59:26,840 the volume changes, et cetera. 946 00:59:26,840 --> 00:59:29,385 At the end of this story, I'm at some other point 947 00:59:29,385 --> 00:59:32,250 that I will call F. 948 00:59:32,250 --> 00:59:36,980 Now I could have performed this change from the initial 949 00:59:36,980 --> 00:59:41,610 to the final state through one set of changes 950 00:59:41,610 --> 00:59:44,640 taking place one after the other. 951 00:59:44,640 --> 00:59:48,350 But maybe I will change that, and I will perform things 952 00:59:48,350 --> 00:59:50,880 in a different way. 953 00:59:50,880 --> 00:59:52,380 So there's path number 1. 954 00:59:52,380 --> 00:59:53,820 Then there's path number 2. 955 00:59:53,820 --> 00:59:56,260 And there's huge number of different paths 956 00:59:56,260 --> 00:59:59,990 that I can, in principle, take in order 957 00:59:59,990 --> 01:00:04,720 to change between the initial and final points 958 01:00:04,720 --> 01:00:08,540 by playing around with the mechanical coordinates that 959 01:00:08,540 --> 01:00:10,190 describe the system. 960 01:00:10,190 --> 01:00:14,290 I always ensure that, initially, I was in equilibrium, 961 01:00:14,290 --> 01:00:18,230 so I could know exactly what the value of these parameters are. 962 01:00:18,230 --> 01:00:21,830 And finally, I wait until I have reached equilibrium. 963 01:00:21,830 --> 01:00:25,050 So again, I know what this things are. 964 01:00:25,050 --> 01:00:29,000 And I know what mechanical work is. 965 01:00:29,000 --> 01:00:31,250 And I can calculate along each one 966 01:00:31,250 --> 01:00:35,520 of these paths, the net amount of work. 967 01:00:35,520 --> 01:00:37,392 The work is delivered in different ways-- 968 01:00:37,392 --> 01:00:39,350 through the magnetic field, through the pulling 969 01:00:39,350 --> 01:00:41,530 of the spring, to the hydrostatic pressure, 970 01:00:41,530 --> 01:00:44,180 et cetera-- but ultimately, when I add up 971 01:00:44,180 --> 01:00:46,450 all of the increments of the work, 972 01:00:46,450 --> 01:00:51,150 I will find that all of them will give you the same delta 973 01:00:51,150 --> 01:00:56,180 W, irrespective of the path. 974 01:00:56,180 --> 01:00:57,410 OK. 975 01:00:57,410 --> 01:00:59,140 Now this reminds me of the following, 976 01:00:59,140 --> 01:01:07,310 that if I'm rolling a ball on top of a hill. 977 01:01:07,310 --> 01:01:09,410 And there is no friction. 978 01:01:09,410 --> 01:01:11,550 The amount of work that I do in order 979 01:01:11,550 --> 01:01:15,450 to take it from here to here, irrespective of the path 980 01:01:15,450 --> 01:01:17,540 that I take on the hill, it really 981 01:01:17,540 --> 01:01:20,380 is a function of the difference in potential energy 982 01:01:20,380 --> 01:01:23,240 between the final and initial points. 983 01:01:23,240 --> 01:01:26,610 So it's the same type of thing. 984 01:01:26,610 --> 01:01:31,520 Rather than moving these coordinates on a hill, 985 01:01:31,520 --> 01:01:35,740 I am moving them in this set of parameters 986 01:01:35,740 --> 01:01:38,100 that thermodynamically describe the system, 987 01:01:38,100 --> 01:01:40,120 but I see the same thing that I would 988 01:01:40,120 --> 01:01:45,080 see in the absence of friction for rolling a ball of the hill. 989 01:01:45,080 --> 01:01:47,290 And immediately, I would deduce here 990 01:01:47,290 --> 01:01:49,690 that there is this potential energy, 991 01:01:49,690 --> 01:01:51,720 and the amount of work that I have 992 01:01:51,720 --> 01:01:53,820 to do to roll this off the hill is the difference 993 01:01:53,820 --> 01:01:57,330 between the potential energy between the two points. 994 01:01:57,330 --> 01:02:02,260 So here, similarly, I would say that this delta W-- 995 01:02:02,260 --> 01:02:07,890 I define it to be the difference between some function that 996 01:02:07,890 --> 01:02:11,430 is called the internal energy-- that depends 997 01:02:11,430 --> 01:02:13,440 on the final set of coordinates. 998 01:02:13,440 --> 01:02:15,980 And there's a whole bunch of them. 999 01:02:15,980 --> 01:02:18,390 Think of them in some pictorial context. 1000 01:02:18,390 --> 01:02:21,560 Minus what you have initially. 1001 01:02:26,320 --> 01:02:31,790 So in the same sense that the zeroth law allowed 1002 01:02:31,790 --> 01:02:36,940 me to construct some function of coordinates that 1003 01:02:36,940 --> 01:02:39,910 was relevant to thermal equilibrium, 1004 01:02:39,910 --> 01:02:43,910 the first law allows me to define another function 1005 01:02:43,910 --> 01:02:46,165 of coordinates, which is this internal energy. 1006 01:02:51,560 --> 01:02:56,870 Of course, the internal energy is the remnant of the energy 1007 01:02:56,870 --> 01:02:59,030 that we know in mechanical systems 1008 01:02:59,030 --> 01:03:00,830 to be conserved quantity. 1009 01:03:00,830 --> 01:03:02,750 And this is the statement of fact. 1010 01:03:02,750 --> 01:03:07,110 So far, nothing surprising. 1011 01:03:07,110 --> 01:03:10,920 Now the real content of the first law 1012 01:03:10,920 --> 01:03:13,620 is when we violate this condition. 1013 01:03:13,620 --> 01:03:20,290 So essentially, what we do is we replace the adiabatic walls 1014 01:03:20,290 --> 01:03:30,850 that were surrounding our very same system with walls 1015 01:03:30,850 --> 01:03:34,640 that allow the exchange of heat. 1016 01:03:34,640 --> 01:03:38,690 And I do exactly the same set of changes. 1017 01:03:38,690 --> 01:03:41,410 So maybe in one case I stretch this, 1018 01:03:41,410 --> 01:03:43,310 and then I change the volume, et cetera. 1019 01:03:43,310 --> 01:03:46,500 I do exactly the same set of changes 1020 01:03:46,500 --> 01:03:50,160 that I did in this case, I try to repeat them 1021 01:03:50,160 --> 01:03:52,840 in the presence of diathermic walls, 1022 01:03:52,840 --> 01:03:57,430 go from the same initial state to the same final state. 1023 01:03:57,430 --> 01:03:59,020 So the initial state is the same. 1024 01:03:59,020 --> 01:04:02,220 The final state, I postulate to be the same. 1025 01:04:02,220 --> 01:04:05,990 And what is observed is that, in this case, 1026 01:04:05,990 --> 01:04:14,860 the diathermic walls-- which allow heat exchange-- 1027 01:04:14,860 --> 01:04:18,170 that the amount of work that you have to do 1028 01:04:18,170 --> 01:04:22,280 is not equal to the change in internal energy. 1029 01:04:25,684 --> 01:04:26,184 OK? 1030 01:04:30,100 --> 01:04:37,460 Now you really believe that energy is a good quantity. 1031 01:04:37,460 --> 01:04:42,260 And so at this stage, you make a postulate, if you like, 1032 01:04:42,260 --> 01:04:45,690 it's the part of the corollary of the first law that 1033 01:04:45,690 --> 01:04:48,850 allows you to define exactly what heat is. 1034 01:04:48,850 --> 01:04:51,740 So you're gradually defining all of the things that 1035 01:04:51,740 --> 01:04:55,310 were missing in the original formulation. 1036 01:04:55,310 --> 01:05:03,340 You define this heat input to the system 1037 01:05:03,340 --> 01:05:13,270 to be the difference in energy that you expected, 1038 01:05:13,270 --> 01:05:15,570 minus the amount of work that you 1039 01:05:15,570 --> 01:05:19,920 did in the presence of these walls. 1040 01:05:19,920 --> 01:05:21,820 OK? 1041 01:05:21,820 --> 01:05:22,560 Yes? 1042 01:05:22,560 --> 01:05:26,506 AUDIENCE: If we need the first law to the point of heat-- 1043 01:05:26,506 --> 01:05:27,130 PROFESSOR: Yes. 1044 01:05:27,130 --> 01:05:31,542 AUDIENCE: --how did we define [? adiabatic ?] used in-- 1045 01:05:31,542 --> 01:05:32,250 PROFESSOR: Right. 1046 01:05:32,250 --> 01:05:35,270 AUDIENCE: --first law and the zeroth law? 1047 01:05:35,270 --> 01:05:37,860 PROFESSOR: As I said, it's an idealization. 1048 01:05:37,860 --> 01:05:40,270 So it's, in the same sense that you can say well, 1049 01:05:40,270 --> 01:05:43,600 how would you define Newtonian mechanics 1050 01:05:43,600 --> 01:05:47,310 that force is proportional to mass times acceleration, what 1051 01:05:47,310 --> 01:05:50,150 is the experimental evidence for that? 1052 01:05:50,150 --> 01:05:52,980 You can only do that really when you go to vacuum. 1053 01:05:52,980 --> 01:05:55,920 So what you can do is you can gradually 1054 01:05:55,920 --> 01:06:00,620 immerse your particle into more and more dilute systems, 1055 01:06:00,620 --> 01:06:03,910 and see what is the limiting behavior in that sense. 1056 01:06:03,910 --> 01:06:06,350 You can try to do something similar here. 1057 01:06:06,350 --> 01:06:08,610 You can imagine that you put your system 1058 01:06:08,610 --> 01:06:12,760 in some kind of a glass, double glass, container, 1059 01:06:12,760 --> 01:06:15,980 and you gradually pump out all of the gas 1060 01:06:15,980 --> 01:06:17,910 this is between the two of them. 1061 01:06:17,910 --> 01:06:20,280 So that, ultimately, you arrive at vacuum. 1062 01:06:20,280 --> 01:06:22,080 You also have to mirror things, so there 1063 01:06:22,080 --> 01:06:24,760 is no radiation exchange, et cetera. 1064 01:06:24,760 --> 01:06:27,800 So gradually, you can try to experimentally reach 1065 01:06:27,800 --> 01:06:30,710 that idealization and see what happens. 1066 01:06:30,710 --> 01:06:33,220 But essentially, it is certainly correct 1067 01:06:33,220 --> 01:06:36,630 that any statement that I make about adiabatic walls 1068 01:06:36,630 --> 01:06:37,900 is an isolation. 1069 01:06:37,900 --> 01:06:40,202 But it's an isolation in the same sense 1070 01:06:40,202 --> 01:06:41,660 that when you think about the point 1071 01:06:41,660 --> 01:06:42,910 particle in Newton's laws. 1072 01:06:48,881 --> 01:06:49,380 OK? 1073 01:06:57,540 --> 01:06:59,830 Let's go a little bit further with this. 1074 01:07:09,490 --> 01:07:23,260 So we have that in differential form, 1075 01:07:23,260 --> 01:07:29,310 if I go from one point in coordinate space 1076 01:07:29,310 --> 01:07:31,870 that describes the system in equilibrium, 1077 01:07:31,870 --> 01:07:34,660 where energy's defined to a nearby point, 1078 01:07:34,660 --> 01:07:38,700 I can calculate what the value of the change in this energy 1079 01:07:38,700 --> 01:07:40,070 function is. 1080 01:07:40,070 --> 01:07:44,230 And I can say that there is a quantity, dE, 1081 01:07:44,230 --> 01:07:46,730 that depends on a whole bunch of coordinates that 1082 01:07:46,730 --> 01:07:47,530 define the system. 1083 01:07:50,140 --> 01:07:54,730 And what the first law says is that if you 1084 01:07:54,730 --> 01:07:57,840 try to operationally make this change from one point 1085 01:07:57,840 --> 01:08:02,250 to another point, you have to supply work through the system, 1086 01:08:02,250 --> 01:08:05,820 or you have to supply heat through the system. 1087 01:08:05,820 --> 01:08:10,760 And we write them in this form. 1088 01:08:10,760 --> 01:08:13,435 And what this D and d bar-- and we 1089 01:08:13,435 --> 01:08:19,000 will encounter this many times in future also and define 1090 01:08:19,000 --> 01:08:22,020 them better-- is that E is a function of state. 1091 01:08:22,020 --> 01:08:25,380 It depends on where you are in this parameter space. 1092 01:08:25,380 --> 01:08:30,260 So in the same sense that maybe you have a function of x and y, 1093 01:08:30,260 --> 01:08:33,520 could be like x squared plus 3x y, 1094 01:08:33,520 --> 01:08:37,149 I can define what dE is in terms of the x and y. 1095 01:08:37,149 --> 01:08:40,890 So that's where I have this quantity. 1096 01:08:40,890 --> 01:08:45,979 But dW and dQ depend on precisely how this system 1097 01:08:45,979 --> 01:08:48,410 was made to go from here to here. 1098 01:08:48,410 --> 01:08:52,430 And you can sort of go between how much contribution to dE 1099 01:08:52,430 --> 01:08:55,569 comes from here or from there, by certainly say, 1100 01:08:55,569 --> 01:08:57,529 changing the properties of the walls 1101 01:08:57,529 --> 01:09:01,319 from being adiabatic to being diathermal, et cetera. 1102 01:09:01,319 --> 01:09:06,630 So these quantities really, as opposed to this quantity that 1103 01:09:06,630 --> 01:09:09,109 depends on stage, these quantities 1104 01:09:09,109 --> 01:09:12,084 depend on path, the conditions by which you 1105 01:09:12,084 --> 01:09:13,819 implement a particular change. 1106 01:09:18,819 --> 01:09:23,870 Now there is a desire, and it's very important for us 1107 01:09:23,870 --> 01:09:27,569 to actually construct what this function is. 1108 01:09:27,569 --> 01:09:31,540 You want to, sort of, know what the energy function 1109 01:09:31,540 --> 01:09:33,600 for, let's say, a mechanical system is. 1110 01:09:33,600 --> 01:09:36,744 In a potential, you want to know what the form of E 1111 01:09:36,744 --> 01:09:39,439 is and then you can do a lot of things with it. 1112 01:09:39,439 --> 01:09:45,090 So how do we construct that for our thermodynamics system? 1113 01:09:45,090 --> 01:09:47,689 Again, we can idealize things and go 1114 01:09:47,689 --> 01:09:52,434 to processes that are so-called, quasi-static. 1115 01:09:59,690 --> 01:10:07,640 Which effectively means slow, or slow enough, 1116 01:10:07,640 --> 01:10:12,750 to maintain equilibrium. 1117 01:10:15,850 --> 01:10:18,240 And the general idea is, suppose I 1118 01:10:18,240 --> 01:10:23,110 wanted to calculate what the potential energy of a spring 1119 01:10:23,110 --> 01:10:28,960 is, or what the potential energy of a particle rolling on a hill 1120 01:10:28,960 --> 01:10:34,290 is, well, one way that I could do that is I could, let's say, 1121 01:10:34,290 --> 01:10:39,180 pull on this sufficiently slowly, so that as I'm pulling 1122 01:10:39,180 --> 01:10:43,630 this by a certain amount, the spring does not 1123 01:10:43,630 --> 01:10:45,570 start to vibrate. 1124 01:10:45,570 --> 01:10:48,890 And so the force that I'm exerting externally, 1125 01:10:48,890 --> 01:10:53,430 is really the force that is internal to the spring. 1126 01:10:53,430 --> 01:10:56,360 If I really push it to rapidly, the spring 1127 01:10:56,360 --> 01:10:58,090 will start to oscillate. 1128 01:10:58,090 --> 01:11:01,370 There's really no relation now between the external force 1129 01:11:01,370 --> 01:11:04,300 that, let's say, is in uniform value and the internal force 1130 01:11:04,300 --> 01:11:05,810 that is oscillating. 1131 01:11:05,810 --> 01:11:08,190 I don't want to do that because when that happens 1132 01:11:08,190 --> 01:11:11,760 I don't know where I am in this coordinate space. 1133 01:11:11,760 --> 01:11:13,970 I want to do things sufficiently slowly 1134 01:11:13,970 --> 01:11:17,500 so that I go from here to here here-- every time 1135 01:11:17,500 --> 01:11:23,220 I am on this plane that defines my properties in equilibrium. 1136 01:11:23,220 --> 01:11:26,630 If I do that, then I can calculate the amount of work, 1137 01:11:26,630 --> 01:11:32,850 and for the case of the spring, it 1138 01:11:32,850 --> 01:11:35,450 would be the force times extension. 1139 01:11:35,450 --> 01:11:37,370 And so we generalize that, and we 1140 01:11:37,370 --> 01:11:42,620 say that if I have multiple ways of doing work on the system, 1141 01:11:42,620 --> 01:11:45,560 there is the analog of the change in length 1142 01:11:45,560 --> 01:11:49,820 of the spring-- that is the displacement of the spring, 1143 01:11:49,820 --> 01:12:03,790 if you like-- multiplied by some generalized force that 1144 01:12:03,790 --> 01:12:05,179 is conjugate to that. 1145 01:12:11,050 --> 01:12:15,850 And indeed, mechanically, you would 1146 01:12:15,850 --> 01:12:21,300 define the conjugate variables by differentiation. 1147 01:12:21,300 --> 01:12:23,830 So if you, for example, know the potential energy 1148 01:12:23,830 --> 01:12:26,250 of the spring as a function of its length, 1149 01:12:26,250 --> 01:12:29,420 you take a derivative and you know what the force is. 1150 01:12:29,420 --> 01:12:32,570 So this is essentially writing that relationship 1151 01:12:32,570 --> 01:12:35,230 for the case of the spring, generalize 1152 01:12:35,230 --> 01:12:37,280 to multiple coordinates. 1153 01:12:37,280 --> 01:12:43,330 And we can make a table of what these displacements are 1154 01:12:43,330 --> 01:12:47,450 and the corresponding coordinates for the types 1155 01:12:47,450 --> 01:12:49,580 of systems that we are likely to encounter. 1156 01:13:01,150 --> 01:13:03,250 So what is x? 1157 01:13:03,250 --> 01:13:05,220 What is j? 1158 01:13:05,220 --> 01:13:08,010 And the first thing that we mentioned 1159 01:13:08,010 --> 01:13:10,620 was the case of the wire. 1160 01:13:10,620 --> 01:13:14,860 And for the wire we can-- the displacement of the length 1161 01:13:14,860 --> 01:13:16,860 is important, and the conjugate variable 1162 01:13:16,860 --> 01:13:20,450 is the force with which you are pulling on this. 1163 01:13:20,450 --> 01:13:25,380 In the first problem set, you will deal-- in the first test 1164 01:13:25,380 --> 01:13:29,880 preparation, you will deal with the case of film. 1165 01:13:29,880 --> 01:13:33,856 And for the film, what you do is you change the area, 1166 01:13:33,856 --> 01:13:35,230 or if you have a balloon, you can 1167 01:13:35,230 --> 01:13:37,870 blow on the balloon, and the surface area of the balloon 1168 01:13:37,870 --> 01:13:38,860 changes. 1169 01:13:38,860 --> 01:13:41,730 And there's a corresponding conjugate variable 1170 01:13:41,730 --> 01:13:44,405 to that which is called the surface tension. 1171 01:13:48,880 --> 01:13:52,150 This is essentially the same thing, going from one dimension 1172 01:13:52,150 --> 01:13:53,900 to two dimension. 1173 01:13:53,900 --> 01:13:56,690 And if I were to go one dimension higher, 1174 01:13:56,690 --> 01:14:02,410 in the case of the gas, I have the volume of the box. 1175 01:14:02,410 --> 01:14:06,330 And I went through this just for the notation 1176 01:14:06,330 --> 01:14:09,620 that the hydrostatic pressure of the work 1177 01:14:09,620 --> 01:14:13,840 is defined to be minus p w-- minus PBV. 1178 01:14:13,840 --> 01:14:14,480 Sorry. 1179 01:14:14,480 --> 01:14:20,050 pW is minus PVV, as opposed to, say, the force of the spring, 1180 01:14:20,050 --> 01:14:21,570 that is FDL. 1181 01:14:21,570 --> 01:14:23,600 It's just, again, matter of definition 1182 01:14:23,600 --> 01:14:27,050 and how we define the sign of the pressure. 1183 01:14:27,050 --> 01:14:31,720 And for the case of the magnet that we also briefly mentioned, 1184 01:14:31,720 --> 01:14:34,850 we have here MDB. 1185 01:14:34,850 --> 01:14:39,720 The one thing that you can see is the trend 1186 01:14:39,720 --> 01:14:43,020 that all of these quantities, if you 1187 01:14:43,020 --> 01:14:46,920 make the size of your system twice as big, these quantities 1188 01:14:46,920 --> 01:14:49,240 will get proportionately bigger. 1189 01:14:49,240 --> 01:14:50,420 So they're called extensive. 1190 01:14:54,000 --> 01:14:58,830 Whereas the force and all the other quantities are intensive. 1191 01:15:04,070 --> 01:15:08,540 So what we've established is that if we focus only 1192 01:15:08,540 --> 01:15:11,990 on these types of transformations, that 1193 01:15:11,990 --> 01:15:18,430 don't include heat, we can relate the change in energy 1194 01:15:18,430 --> 01:15:20,500 directly to dW. 1195 01:15:20,500 --> 01:15:24,590 And the dW, we can write as the sum over i Ji dxi. 1196 01:15:28,290 --> 01:15:33,520 Now in order to really get dE, in general, 1197 01:15:33,520 --> 01:15:35,280 I have to add to this dQ. 1198 01:15:38,320 --> 01:15:45,366 And so really the question that we have is, 1199 01:15:45,366 --> 01:15:50,480 is there some kind of analog of dW 1200 01:15:50,480 --> 01:15:54,130 that we can write down for dQ? 1201 01:15:54,130 --> 01:15:58,250 And if you think about it, if you have two springs that 1202 01:15:58,250 --> 01:16:01,360 are in equilibrium, so that the thing does not 1203 01:16:01,360 --> 01:16:04,435 go one way or the other, the force exerted from one 1204 01:16:04,435 --> 01:16:07,190 is the same as the force exerted on the other. 1205 01:16:07,190 --> 01:16:10,490 So in equilibrium, mechanical equilibrium forces, 1206 01:16:10,490 --> 01:16:14,100 pressures, et cetera, are generally the same. 1207 01:16:14,100 --> 01:16:17,340 And we've established that when you are in thermal equilibrium, 1208 01:16:17,340 --> 01:16:19,550 temperatures are the same. 1209 01:16:19,550 --> 01:16:23,840 So we have a very good guess that we should really 1210 01:16:23,840 --> 01:16:27,640 have temperature appearing here. 1211 01:16:27,640 --> 01:16:31,020 And then the question that we'll sort of build on 1212 01:16:31,020 --> 01:16:33,720 is what is the analog of the quantity 1213 01:16:33,720 --> 01:16:34,998 that we have to put here. 1214 01:16:44,090 --> 01:16:50,760 Let me finish by telling you one other story that 1215 01:16:50,760 --> 01:16:53,600 is related to the ideal gas. 1216 01:16:53,600 --> 01:16:56,840 I said that for every one of these laws, 1217 01:16:56,840 --> 01:17:02,460 we can kind of come up with a peculiar feature that 1218 01:17:02,460 --> 01:17:05,330 is unique to the case of the ideal gas. 1219 01:17:05,330 --> 01:17:08,460 And there is a property related to its energy 1220 01:17:08,460 --> 01:17:11,610 that we will shortly explore. 1221 01:17:11,610 --> 01:17:16,970 But let me first say what the consequence of these kinds 1222 01:17:16,970 --> 01:17:20,450 of relations for measurable quantities 1223 01:17:20,450 --> 01:17:24,650 that we can already try to deduce are. 1224 01:17:24,650 --> 01:17:30,800 So one thing that you can relate to heat, 1225 01:17:30,800 --> 01:17:35,855 and properties of particular material, is the heat capacity. 1226 01:17:38,560 --> 01:17:43,720 So what you can do is you can take your material 1227 01:17:43,720 --> 01:17:48,670 and put some amount of heat into it. 1228 01:17:48,670 --> 01:17:52,050 And ask, what is the corresponding change 1229 01:17:52,050 --> 01:17:54,087 in temperature? 1230 01:17:54,087 --> 01:17:55,420 That would be the heat capacity. 1231 01:17:58,050 --> 01:18:02,050 Now this bar that we have over here, 1232 01:18:02,050 --> 01:18:09,420 tells me that this quantity-- which we will denote on C-- 1233 01:18:09,420 --> 01:18:16,460 will depend on the path through which the heat is 1234 01:18:16,460 --> 01:18:17,910 added to the system. 1235 01:18:17,910 --> 01:18:22,760 Because we've established that, depending on how you add heat 1236 01:18:22,760 --> 01:18:25,140 to the system, the change that you 1237 01:18:25,140 --> 01:18:28,410 have in the coordinate space is going 1238 01:18:28,410 --> 01:18:30,220 to be distinct potentially. 1239 01:18:30,220 --> 01:18:32,220 Actually, we establish the other way 1240 01:18:32,220 --> 01:18:35,300 that for a given change in the coordinate system, 1241 01:18:35,300 --> 01:18:37,340 the amount of the Q depends on path, 1242 01:18:37,340 --> 01:18:40,370 so they're kind of equivalent statements. 1243 01:18:40,370 --> 01:18:49,880 So if I'm dealing with a gas, I can add the heat to it, 1244 01:18:49,880 --> 01:18:52,780 at least among many other possibilities, 1245 01:18:52,780 --> 01:18:54,590 in two distinct ways. 1246 01:18:54,590 --> 01:19:01,190 I can do this at constant volume or at constant pressure. 1247 01:19:01,190 --> 01:19:05,030 So if I think about the coordinate space 1248 01:19:05,030 --> 01:19:08,800 of the gas, which is PV, and I start 1249 01:19:08,800 --> 01:19:11,720 from some particular point, I can either 1250 01:19:11,720 --> 01:19:14,420 go along a path like this, or I can go along 1251 01:19:14,420 --> 01:19:17,630 the path like this, depending on which one of these quantities 1252 01:19:17,630 --> 01:19:20,310 I keep fixed. 1253 01:19:20,310 --> 01:19:25,060 So then in one case what I need is 1254 01:19:25,060 --> 01:19:29,650 the change in Q at constant v, dT. 1255 01:19:29,650 --> 01:19:33,078 In the other case, this change in Q, a constant P, dT. 1256 01:19:36,360 --> 01:19:39,220 Now a consequence of the first law 1257 01:19:39,220 --> 01:19:48,735 is that I know that dQ's are related to dE minus dW. 1258 01:19:48,735 --> 01:19:52,870 And dW for the gas is minus PdV. 1259 01:19:52,870 --> 01:19:56,030 So I can write it in this fashion. 1260 01:19:56,030 --> 01:20:00,760 Divided by dT, and in one case, done at constant V, 1261 01:20:00,760 --> 01:20:05,680 and in the other case, done at constant p. 1262 01:20:10,110 --> 01:20:12,620 The distinction between these two paths 1263 01:20:12,620 --> 01:20:18,090 immediately becomes clear, because along the paths where 1264 01:20:18,090 --> 01:20:21,500 the volume is kept constant, there 1265 01:20:21,500 --> 01:20:29,430 is no mechanical work that is done along the this path, yes? 1266 01:20:29,430 --> 01:20:32,470 And so this contribution is 0. 1267 01:20:32,470 --> 01:20:35,360 And you have the result that this 1268 01:20:35,360 --> 01:20:39,960 is going to be related to the change in energy 1269 01:20:39,960 --> 01:20:43,270 with temperature at constant V, whereas here you 1270 01:20:43,270 --> 01:20:48,010 have the change in energy, in temperature at constant P, 1271 01:20:48,010 --> 01:20:53,426 plus P dV by dT at constant P. 1272 01:20:53,426 --> 01:20:58,220 So there is some additional manipulations of derivatives 1273 01:20:58,220 --> 01:21:04,420 that is involved, but rather than looking 1274 01:21:04,420 --> 01:21:13,460 at that in the general case, I will follow the consequence 1275 01:21:13,460 --> 01:21:17,410 of that for a particular example, which 1276 01:21:17,410 --> 01:21:20,130 is ideal gas expansion. 1277 01:21:27,070 --> 01:21:31,020 So as this is an observation called Joule's experiment. 1278 01:21:31,020 --> 01:21:36,140 I take a gas that is adiabatically isolated 1279 01:21:36,140 --> 01:21:38,010 from its environment. 1280 01:21:38,010 --> 01:21:43,880 I connect it, also maintaining adiabatic isolation 1281 01:21:43,880 --> 01:21:45,880 to another chamber. 1282 01:21:45,880 --> 01:21:49,070 And initially all of the gas is here, 1283 01:21:49,070 --> 01:21:51,920 and this chamber is empty. 1284 01:21:51,920 --> 01:21:59,247 And then I remove this, and the gas goes and occupies 1285 01:21:59,247 --> 01:21:59,830 both chambers. 1286 01:22:03,170 --> 01:22:06,570 Observation, there is some temperature 1287 01:22:06,570 --> 01:22:11,090 initially in this system that I can measure. 1288 01:22:11,090 --> 01:22:15,240 Finally, I can measure the temperature of this system. 1289 01:22:15,240 --> 01:22:17,522 And I find that the two temperatures are the same. 1290 01:22:24,610 --> 01:22:27,180 In this example, since the whole thing 1291 01:22:27,180 --> 01:22:32,080 was adiabatically isolated, that the Q is 0. 1292 01:22:32,080 --> 01:22:36,130 There is no mechanical work that done on the system, 1293 01:22:36,130 --> 01:22:41,260 so delta W is also 0. 1294 01:22:41,260 --> 01:22:45,880 And since delta is the same, I conclude 1295 01:22:45,880 --> 01:22:53,880 that delta E is the same for this two cases. 1296 01:22:53,880 --> 01:22:56,840 Now, in principle, E is a function 1297 01:22:56,840 --> 01:23:00,500 of pressure and volume. 1298 01:23:00,500 --> 01:23:04,590 And pressure and volume certainly changed very much 1299 01:23:04,590 --> 01:23:09,010 by going from that place to another place. 1300 01:23:09,010 --> 01:23:10,290 OK, so let's follow that. 1301 01:23:10,290 --> 01:23:15,630 So E we said is a function of pressure and volume. 1302 01:23:15,630 --> 01:23:18,760 Now since I know that for the ideal gas, 1303 01:23:18,760 --> 01:23:21,530 the product of pressure and volume is temperature, 1304 01:23:21,530 --> 01:23:25,040 I can certainly exchange one of these variables 1305 01:23:25,040 --> 01:23:26,240 for temperature. 1306 01:23:26,240 --> 01:23:28,000 So I can write, let's say, pressure 1307 01:23:28,000 --> 01:23:31,790 to be proportional to temperature over volume. 1308 01:23:31,790 --> 01:23:34,720 And then rewrite this as a function 1309 01:23:34,720 --> 01:23:36,280 of temperature and volume. 1310 01:23:39,400 --> 01:23:45,410 Now I know that in this process the volume changed, 1311 01:23:45,410 --> 01:23:48,500 but the temperature did not change. 1312 01:23:48,500 --> 01:23:52,390 And therefore, the internal energy that did not change 1313 01:23:52,390 --> 01:23:55,540 can only be a function of temperature. 1314 01:23:55,540 --> 01:23:58,840 So this Joule expansion experiment immediately 1315 01:23:58,840 --> 01:24:02,390 tells me that while internal energy, in principle, 1316 01:24:02,390 --> 01:24:05,570 is a function of P and V, it is really 1317 01:24:05,570 --> 01:24:07,590 a function of the product of P and V, 1318 01:24:07,590 --> 01:24:09,050 because the product of P and V, we 1319 01:24:09,050 --> 01:24:12,380 established to be proportional to temperature. 1320 01:24:12,380 --> 01:24:13,420 OK? 1321 01:24:13,420 --> 01:24:17,340 So now if I go and look at these expressions, 1322 01:24:17,340 --> 01:24:21,830 I can see that these, dE by dT, and irrespective of V and P 1323 01:24:21,830 --> 01:24:25,660 is the same thing, because E only depends on temperature. 1324 01:24:25,660 --> 01:24:31,075 And since V is-- we know that PV is proportional to temperature, 1325 01:24:31,075 --> 01:24:37,950 a constant P, dV by dT is the same as V over T. 1326 01:24:37,950 --> 01:24:41,550 And so if I look at the difference between those two 1327 01:24:41,550 --> 01:24:46,470 expressions, what I find is that this part cancels. 1328 01:24:46,470 --> 01:24:52,520 This part gives me the value of this product, PV over T, 1329 01:24:52,520 --> 01:24:55,900 which we said is a constant. 1330 01:24:55,900 --> 01:25:00,450 And it is certainly depends on the amount of gas 1331 01:25:00,450 --> 01:25:02,630 that you have. 1332 01:25:02,630 --> 01:25:05,880 And so you can pick a particular amount of gas. 1333 01:25:05,880 --> 01:25:08,175 You can experimentally verify this phenomenon, 1334 01:25:08,175 --> 01:25:11,710 that the difference of heat capacity along these two paths 1335 01:25:11,710 --> 01:25:13,160 is a constant. 1336 01:25:13,160 --> 01:25:17,880 That constant is the same for the same amount of gas 1337 01:25:17,880 --> 01:25:21,210 for different types of argon, krypton, et cetera. 1338 01:25:21,210 --> 01:25:24,030 And since it is proportional, eventually 1339 01:25:24,030 --> 01:25:28,000 to the amount of matter, we will ultimately 1340 01:25:28,000 --> 01:25:32,520 see that it can be set to be the number of particles 1341 01:25:32,520 --> 01:25:36,770 making up the gas in some constant of proportionality 1342 01:25:36,770 --> 01:25:41,070 that we will identify later in statistical physics 1343 01:25:41,070 --> 01:25:42,920 is the Boltzmann parameter, which 1344 01:25:42,920 --> 01:25:49,257 is 1.43 times 10 to the minus 23 or whatever it is. 1345 01:25:53,240 --> 01:25:56,065 All of this depends partly on the definition 1346 01:25:56,065 --> 01:25:58,890 that you made of temperature. 1347 01:25:58,890 --> 01:26:03,630 What we will do next time is to review 1348 01:26:03,630 --> 01:26:06,310 all of this, because I've went through them a little bit more 1349 01:26:06,310 --> 01:26:14,390 rapidly, and try to identify what the conjugate variable is 1350 01:26:14,390 --> 01:26:17,790 that we have to put for temperature, 1351 01:26:17,790 --> 01:26:22,380 so that we can write the form of dE in a more symmetric fashion. 1352 01:26:22,380 --> 01:26:23,889 Thank you.