1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high-quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation or view additional material from 7 00:00:13 --> 00:00:15 hundreds the MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:17 at ocw.mit.edu. 9 00:00:17 --> 00:00:23 PROFESSOR BAWENDI: Last time you talked about the first 10 00:00:23 --> 00:00:25 law of thermodynamics. 11 00:00:25 --> 00:00:31 And you talked about isothermal expansion, the Joule expansion. 12 00:00:31 --> 00:00:36 You saw a very important result. which is that for an 13 00:00:36 --> 00:00:48 ideal gas, the energy content is only dependent on the 14 00:00:48 --> 00:00:50 temperature, nothing else. 15 00:00:50 --> 00:00:53 Not the volume, not the pressure, it just cares 16 00:00:53 --> 00:00:55 about the temperature. 17 00:00:55 --> 00:01:01 So, if you have an isothermal process for an ideal gas, the 18 00:01:01 --> 00:01:06 energy doesn't change. q plus w is equal to zero for 19 00:01:06 --> 00:01:10 any isothermal process. 20 00:01:10 --> 00:01:17 And you also saw that du then could be written as Cv dT 21 00:01:17 --> 00:01:19 for an ideal gas always. 22 00:01:19 --> 00:01:21 This is not generally true. 23 00:01:21 --> 00:01:24 If you have a real gas and you write du is Cv dT, and your 24 00:01:24 --> 00:01:27 path is not a constant volume path, then you are 25 00:01:27 --> 00:01:28 making a mistake. 26 00:01:28 --> 00:01:31 But for an ideal gas, you can always write this. 27 00:01:31 --> 00:01:38 And this turns out to be very useful to remember. 28 00:01:38 --> 00:01:46 OK, now most processes that we deal with are not constant 29 00:01:46 --> 00:01:50 volume processes. 30 00:01:50 --> 00:01:54 So energy, which has this wonderful property here, du is 31 00:01:54 --> 00:01:58 Cv dt for constant volume process, which happens to be 32 00:01:58 --> 00:02:04 equal to d q, constant volume, because there's no change. 33 00:02:04 --> 00:02:07 there's no work if you've got a constant volume process. 34 00:02:07 --> 00:02:11 So du here is a very interesting quantity, because 35 00:02:11 --> 00:02:16 it's related to the heat that's going in or out of the system 36 00:02:16 --> 00:02:19 under constant volume process. 37 00:02:19 --> 00:02:23 But as I said, we're not operating usually in a 38 00:02:23 --> 00:02:24 constant volume environment. 39 00:02:24 --> 00:02:30 When I flail my arms around I generate work and heat. 40 00:02:30 --> 00:02:34 This is not a constant volume process. 41 00:02:34 --> 00:02:40 If I'm the system, what's constant when I do this? 42 00:02:40 --> 00:02:41 Anybody have an idea? 43 00:02:41 --> 00:02:46 What's the one function of state? 44 00:02:46 --> 00:02:49 I'm the system, the rest are the surrounding. 45 00:02:49 --> 00:02:52 What's the one function of state that's constant when I'm 46 00:02:52 --> 00:03:02 doing all my chemical reactions to move my arms around? 47 00:03:02 --> 00:03:03 Temperature? 48 00:03:03 --> 00:03:03 STUDENT: 49 00:03:03 --> 00:03:04 Pressure? 50 00:03:04 --> 00:03:04 PROFESSOR BAWENDI: 51 00:03:04 --> 00:03:05 Pressure, right. 52 00:03:05 --> 00:03:06 Pressure is constant. 53 00:03:06 --> 00:03:08 What is the pressure at? 54 00:03:08 --> 00:03:12 One atmosphere, one bar. 55 00:03:12 --> 00:03:14 So the most interesting processes are the processes 56 00:03:14 --> 00:03:16 where pressure is constant. 57 00:03:16 --> 00:03:20 When I had have a vial on bench top, and I do a chemical 58 00:03:20 --> 00:03:24 reaction in the vial, and it's open to the atmosphere, the 59 00:03:24 --> 00:03:26 pressure is constant at one atmosphere. 60 00:03:26 --> 00:03:29 When you've got your cells growing in your petri dish, the 61 00:03:29 --> 00:03:32 pressure is constant at one atmosphere, even if they're 62 00:03:32 --> 00:03:37 evolving gas, pressure is constant. 63 00:03:37 --> 00:03:43 So we'd really like to be able to find some sort of equation 64 00:03:43 --> 00:03:47 of state, or some sort of rather function of state that's 65 00:03:47 --> 00:03:53 going to relate the heat going in or out of the system with 66 00:03:53 --> 00:03:57 that function of state, because this isn't going to do it. du 67 00:03:57 --> 00:04:01 only relates to the heat under constant volume. 68 00:04:01 --> 00:04:03 And the heat is a really important thing to know. 69 00:04:03 --> 00:04:08 How much heat do you need to put into a system, or how much 70 00:04:08 --> 00:04:10 heat is going to come out of a system when something is 71 00:04:10 --> 00:04:11 happening in the system? 72 00:04:11 --> 00:04:15 All right, this is a really important quantity to know. 73 00:04:15 --> 00:04:19 Your boiling water or whatever, you want to know how much heat 74 00:04:19 --> 00:04:22 do you need to boil that amount of water under 75 00:04:22 --> 00:04:25 constant pressure? 76 00:04:25 --> 00:04:27 And this is where enthalpy comes in. 77 00:04:27 --> 00:04:27 You've all heard of enthalpy. 78 00:04:27 --> 00:04:33 H we're going to write it as the function of 79 00:04:33 --> 00:04:38 temperature and pressure. 80 00:04:38 --> 00:04:42 And the reason enthalpy was invented was exactly for that 81 00:04:42 --> 00:04:46 reason, because we need some way to figure out how to relate 82 00:04:46 --> 00:04:49 the heat coming in or out of a system under a constant 83 00:04:49 --> 00:04:51 pressure process. 84 00:04:51 --> 00:04:55 Because it's so important. 85 00:04:55 --> 00:04:58 And I should add and also under reversible work, where the 86 00:04:58 --> 00:05:03 external pressure is equal to the internal pressure. 87 00:05:03 --> 00:05:11 OK, so we're going to define enthalpy as u + pV, these are 88 00:05:11 --> 00:05:13 all functions of state here, So H is a function of state, and 89 00:05:13 --> 00:05:17 we're going to see now how this is, indeed, related to the heat 90 00:05:17 --> 00:05:20 flow in and out of the system. 91 00:05:20 --> 00:05:22 If you have a constant pressure, reversible 92 00:05:22 --> 00:05:23 work process. 93 00:05:23 --> 00:05:26 Let's take a system. 94 00:05:26 --> 00:05:34 Under constant pressure T1, V1, going to a second -- this 95 00:05:34 --> 00:05:39 is the system, so let me write the system here. 96 00:05:39 --> 00:05:47 And it's more dramatic if the system is a gas, p, T2, V2, And 97 00:05:47 --> 00:05:51 let's look at what happens to these functions of state, to H 98 00:05:51 --> 00:05:58 to u under this transformation. 99 00:05:58 --> 00:05:59 OK, so let's look at delta u. 100 00:05:59 --> 00:06:03 Delta u is q plus w. 101 00:06:03 --> 00:06:06 That's the first law. 102 00:06:06 --> 00:06:10 And this is a constant pressure path, so now I can write, this 103 00:06:10 --> 00:06:12 q is actually q under constant pressure. 104 00:06:12 --> 00:06:15 Little p means the path is a constant pressure path. 105 00:06:15 --> 00:06:26 And I'm doing reversible work. 106 00:06:26 --> 00:06:31 So that w is minus p, dV where p is the pressure inside the 107 00:06:31 --> 00:06:36 system, minus p delta V. 108 00:06:36 --> 00:06:44 Rearrange that, delta u is plus p delta V is equal to q p. 109 00:06:44 --> 00:06:48 All right, so this is the heat flowing in or out of the 110 00:06:48 --> 00:06:52 system, and these are all functions of state. 111 00:06:52 --> 00:06:54 This depends on the path. 112 00:06:54 --> 00:06:57 It tells you right here, the path is constant pressure. 113 00:06:57 --> 00:07:02 These don't depend on the path, right. 114 00:07:02 --> 00:07:05 V doesn't care how you get there. u doesn't care 115 00:07:05 --> 00:07:08 how you get there. 116 00:07:08 --> 00:07:11 In this case, p is a constant because the path is constant. 117 00:07:11 --> 00:07:16 So we can bring the p inside, delta u plus 118 00:07:16 --> 00:07:21 delta p V it's q p. 119 00:07:21 --> 00:07:25 Take this delta outside again, delta of u plus 120 00:07:25 --> 00:07:28 p V is equal to q p. 121 00:07:28 --> 00:07:28 And there you have it. 122 00:07:28 --> 00:07:30 There is the H right there. 123 00:07:30 --> 00:07:30 The u plus p V. 124 00:07:30 --> 00:07:36 Delta H is equal to q V. 125 00:07:36 --> 00:07:40 And this is the reason why enthalpy was invented, and 126 00:07:40 --> 00:07:42 why it's so important. 127 00:07:42 --> 00:07:44 Because we want to know this. 128 00:07:44 --> 00:07:46 So this for a finite change. 129 00:07:46 --> 00:07:49 If you want to have an infinitestimally small 130 00:07:49 --> 00:07:55 change, you end up writing dh is dq sub p. 131 00:07:55 --> 00:07:57 It's not always equal to the heat. 132 00:07:57 --> 00:08:00 It's only equal to the heat if your process is constant 133 00:08:00 --> 00:08:06 pressure reversible work. 134 00:08:06 --> 00:08:09 OK, so this is the kind of, this is the kind of concept 135 00:08:09 --> 00:08:18 that needs to be branded into your brain, so that if I come 136 00:08:18 --> 00:08:20 into your bedroom in the middle of the night and I whisper to 137 00:08:20 --> 00:08:24 you delta H, you know, you should wake up and 138 00:08:24 --> 00:08:27 say q p, right? 139 00:08:27 --> 00:08:29 Heat under constant pressure reversible work. 140 00:08:29 --> 00:08:34 This should become second nature. 141 00:08:34 --> 00:08:35 This is where the intuition comes from. 142 00:08:35 --> 00:08:39 This is why people right tables and tables of delta H's. 143 00:08:39 --> 00:08:44 Why you have delta H's from all these reactions, because this 144 00:08:44 --> 00:08:47 is basically the heat, and the heat is something we can 145 00:08:47 --> 00:08:50 measure, we can control. 146 00:08:50 --> 00:08:51 We can figure out how much heat is going in and 147 00:08:51 --> 00:08:52 out of something. 148 00:08:52 --> 00:08:58 This is what we're interested in. 149 00:08:58 --> 00:09:02 OK, so last time you looked at -- any questions on this first? 150 00:09:02 --> 00:09:04 Yes. 151 00:09:04 --> 00:09:07 STUDENT: [INAUDIBLE] 152 00:09:07 --> 00:09:11 from the T delta V to the delta p here? 153 00:09:11 --> 00:09:12 What was the reasoning behind that? 154 00:09:12 --> 00:09:13 PROFESSOR BAWENDI: p is constant here. 155 00:09:13 --> 00:09:17 It's constant pressure. 156 00:09:17 --> 00:09:21 OK, so now, last time you looked at the Joule expansion 157 00:09:21 --> 00:09:28 to teach you how to relate derivatives like du/dV. 158 00:09:28 --> 00:09:34 du/dV under constant temperature. du/dT 159 00:09:34 --> 00:09:35 under constant volume. 160 00:09:35 --> 00:09:42 You use the Joule expansion to find these quantities. 161 00:09:42 --> 00:09:44 Now these quantities were useful because 162 00:09:44 --> 00:09:45 you could relate them. 163 00:09:45 --> 00:09:49 The slope of changes, with respect to volume or 164 00:09:49 --> 00:09:53 temperature of the energy with respect to quantities that 165 00:09:53 --> 00:09:55 you understood, that you could measure. 166 00:09:55 --> 00:09:57 We're going to do the same thing here. 167 00:09:57 --> 00:10:00 So if we take as our natural variables for enthalpy to be 168 00:10:00 --> 00:10:04 temperature and pressure, and we have some sort of change in 169 00:10:04 --> 00:10:09 enthalpy, dH, and it's going to be related to changes in 170 00:10:09 --> 00:10:13 temperature and pressure through the derivatives dH 171 00:10:13 --> 00:10:18 through the slope of the enthalpy in the T direction, 172 00:10:18 --> 00:10:23 keeping pressure constant, dT plus the slope of enthalpy in 173 00:10:23 --> 00:10:28 the pressure direction, keeping the temperature constant, dp, 174 00:10:28 --> 00:10:31 and these are knobs that we can turn. 175 00:10:31 --> 00:10:33 We can change of temperature. 176 00:10:33 --> 00:10:34 We can change the pressure. 177 00:10:34 --> 00:10:37 These are physical knobs that are available to 178 00:10:37 --> 00:10:38 us as experimentalists. 179 00:10:38 --> 00:10:42 And so when we turn these knobs on our system, we want to 180 00:10:42 --> 00:10:45 know how the enthalpy is changing for that system. 181 00:10:45 --> 00:10:50 Because eventually they will tell us maybe things about how 182 00:10:50 --> 00:10:52 heat is changing further on. 183 00:10:52 --> 00:10:57 OK, but in order to relate turning these physical knob to 184 00:10:57 --> 00:11:00 this quantity here, which we don't have a very good feel 185 00:11:00 --> 00:11:04 for, we've got to have a feel for the slopes. 186 00:11:04 --> 00:11:05 If I keep the pressure constant. 187 00:11:05 --> 00:11:07 I change the temperature, what does that mean? 188 00:11:07 --> 00:11:09 What is dh/dT? 189 00:11:09 --> 00:11:12 If I keep the temperature constant, and just change 190 00:11:12 --> 00:11:14 the pressure, dH is going to change, but how is 191 00:11:14 --> 00:11:15 it going to change? 192 00:11:15 --> 00:11:16 What does this mean in terms of something I can 193 00:11:16 --> 00:11:20 physically understand? 194 00:11:20 --> 00:11:23 That's the program now for the next few minutes. 195 00:11:23 --> 00:11:25 What are these quantities? 196 00:11:25 --> 00:11:28 What is dH/dT as a function, keeping pressure constant, 197 00:11:28 --> 00:11:35 what is dH/dp, keeping temperature constant? 198 00:11:35 --> 00:11:42 All right, let's start with the first one, dH/dT, keeping 199 00:11:42 --> 00:11:43 the pressure constant. 200 00:11:43 --> 00:11:53 And we're going to look at a reversible process to help us 201 00:11:53 --> 00:11:55 out, but the result is going to be independent of 202 00:11:55 --> 00:11:56 whether or not we have a reversible process or 203 00:11:56 --> 00:11:59 irreversible process. 204 00:11:59 --> 00:12:03 Constant pressure, that means dp is equal to zero. 205 00:12:03 --> 00:12:09 So for reversible process, constant pressure, 206 00:12:09 --> 00:12:11 what do we know? 207 00:12:11 --> 00:12:14 This is already branded in your brain, right? 208 00:12:14 --> 00:12:16 Reversible process, constant pressure dH=dq. 209 00:12:16 --> 00:12:19 210 00:12:19 --> 00:12:27 So we can write that down, dH=dq, constant pressure. 211 00:12:27 --> 00:12:29 That's by definition of enthalpy. 212 00:12:29 --> 00:12:30 That's why we created enthalpy. 213 00:12:30 --> 00:12:30 What else do we know? 214 00:12:30 --> 00:12:33 Well we can go look up here, looking at the 215 00:12:33 --> 00:12:37 differential, there are no approximations here. 216 00:12:37 --> 00:12:39 This is just an equality. 217 00:12:39 --> 00:12:41 I have a constant pressure process. 218 00:12:41 --> 00:12:43 This term here is equal to zero. 219 00:12:43 --> 00:12:50 That means that dH is also equal to dH/dT, 220 00:12:50 --> 00:12:55 constant pressure dT. 221 00:12:55 --> 00:12:59 All right, so now I've got more dH/dT under constant pressure. 222 00:12:59 --> 00:13:05 dH is equal to this, and it's also equal to this. 223 00:13:05 --> 00:13:08 All right, these two are equal to each other. 224 00:13:08 --> 00:13:13 Now, I know how to relate the heat flow to temperature 225 00:13:13 --> 00:13:19 change, through the heat capacity. dq constant pressure 226 00:13:19 --> 00:13:23 is that heat capacity, and I have to tell you the path 227 00:13:23 --> 00:13:24 for the heat capacity. 228 00:13:24 --> 00:13:28 So it's C sub p, the heat capacity under a constant 229 00:13:28 --> 00:13:32 pressure path, dT, all right? 230 00:13:32 --> 00:13:33 So these two are equal to each other. 231 00:13:33 --> 00:13:40 So, these two are equal to each other as well, which tells me 232 00:13:40 --> 00:13:49 that this derivative, dH/dT constant pressure is Cp. 233 00:13:49 --> 00:13:53 So now I have my first of my two slopes, in terms of 234 00:13:53 --> 00:13:57 something that's related to my system, the heat 235 00:13:57 --> 00:13:58 capacity of the system. 236 00:13:58 --> 00:14:04 Something I can measure and I can tabulate, and when I turn 237 00:14:04 --> 00:14:07 my dT knob here I know what's going to happen 238 00:14:07 --> 00:14:09 to the enthalpy. 239 00:14:09 --> 00:14:15 So this is the first, this is the one. 240 00:14:15 --> 00:14:19 So it's very similar to what we saw with the volume and the 241 00:14:19 --> 00:14:24 energy, where du/dV under constant temperature was equal 242 00:14:24 --> 00:14:31 to Cv in this case here, right. and you're going to find that 243 00:14:31 --> 00:14:35 there's a lot of these analogies between 244 00:14:35 --> 00:14:36 energy and enthalpy. 245 00:14:36 --> 00:14:39 You just change volume to pressure and basically you're 246 00:14:39 --> 00:14:42 looking at enthalpy under a constant -- anything that's 247 00:14:42 --> 00:14:46 done at a constant volume path with energy, there's the same 248 00:14:46 --> 00:14:49 thing happening under constant pressure path for enthalpy. 249 00:14:49 --> 00:14:53 So you can guess the answer usually that way. 250 00:14:53 --> 00:14:55 OK, so now we have the other one, dH/dp 251 00:14:55 --> 00:14:57 constant temperature. 252 00:14:57 --> 00:15:00 How do we relate this to something physical? 253 00:15:00 --> 00:15:02 Well, it's going to be an experiment, very much like 254 00:15:02 --> 00:15:03 the Joule experiment. 255 00:15:03 --> 00:15:05 The Joule experiment was a constant energy 256 00:15:05 --> 00:15:07 experiment, right. 257 00:15:07 --> 00:15:09 Here we're going to have to find a constant enthalpy 258 00:15:09 --> 00:15:13 experiment, and that is going to be the Joule-Thomson 259 00:15:13 --> 00:15:15 experiment. 260 00:15:15 --> 00:15:18 That's going to extract out a physical meaning 261 00:15:18 --> 00:15:20 to this derivative here. 262 00:15:20 --> 00:15:26 OK, the Joule-Thomson experiment. 263 00:15:26 --> 00:15:40 This is going to get us dH/dp constant temperature. 264 00:15:40 --> 00:15:41 What is this experiment? 265 00:15:41 --> 00:15:50 You take a throttle valve, which consists of some sort 266 00:15:50 --> 00:15:58 of porous plug between two cylinders that is insulated. 267 00:15:58 --> 00:16:02 There is insulation here. 268 00:16:02 --> 00:16:04 Insulation on the bottom. 269 00:16:04 --> 00:16:08 It's like a, think about a tube, a large tube, insulated 270 00:16:08 --> 00:16:15 tube, with a bunch of -- a frit inside here, which prevents 271 00:16:15 --> 00:16:18 flow of gas, which slows down the flow of gas from 272 00:16:18 --> 00:16:20 one side to the other. 273 00:16:20 --> 00:16:23 It's a blockage in this tube here. 274 00:16:23 --> 00:16:30 Then you put two pistons, one on that side here, and one on 275 00:16:30 --> 00:16:35 this side here, and the external pressure here, we're 276 00:16:35 --> 00:16:37 going to call that p1. 277 00:16:37 --> 00:16:42 The external pressure here, we're going to call that p2, 278 00:16:42 --> 00:16:48 and we're going to do it slowly enough that the pressure on 279 00:16:48 --> 00:16:54 this side of the cylinder is in equilibrium with the external 280 00:16:54 --> 00:16:59 pressure, and the pressure on this side of the cylinder is in 281 00:16:59 --> 00:17:02 equilibrium with this pressure. 282 00:17:02 --> 00:17:06 But not so slowly that these two are in equilibrium 283 00:17:06 --> 00:17:10 with each other. 284 00:17:10 --> 00:17:12 So this is restricting the flow, so there's some 285 00:17:12 --> 00:17:13 sweet spot here. 286 00:17:13 --> 00:17:17 When I'm pushing slowly enough here that the pressure here is 287 00:17:17 --> 00:17:22 equal to that one, but not so slowly that the air flow from 288 00:17:22 --> 00:17:26 here to here is fast, compared to how fast I'm pushing. 289 00:17:26 --> 00:17:27 You got the picture here? 290 00:17:27 --> 00:17:30 Any questions on that? 291 00:17:30 --> 00:17:35 All right, then as I push through, I'm going to start 292 00:17:35 --> 00:17:36 with all of my gas on this side, and at the end, I'm 293 00:17:36 --> 00:17:41 going to have all the gas on the other side. 294 00:17:41 --> 00:17:44 Let me first ask you this, is this a reversible or 295 00:17:44 --> 00:17:49 in irreversible process? 296 00:17:49 --> 00:17:52 Right, let me add one more piece of data here which I said 297 00:17:52 --> 00:17:55 in words, but which is actually important to write down 298 00:17:55 --> 00:17:57 before doing the problem. 299 00:17:57 --> 00:17:58 Is this a reverse -- any guesses? 300 00:17:58 --> 00:18:03 How many people vote for that this is a reversible process? 301 00:18:03 --> 00:18:05 I've got one vote back there, two votes, three 302 00:18:05 --> 00:18:06 votes, four votes. 303 00:18:06 --> 00:18:08 Anybody else? 304 00:18:08 --> 00:18:11 How many people think this is irreversible? 305 00:18:11 --> 00:18:16 It's about a tie, and everybody else doesn't now. 306 00:18:16 --> 00:18:20 All right, I'm going to give you ten seconds, fifteen 307 00:18:20 --> 00:18:22 seconds to make up your mind. 308 00:18:22 --> 00:18:24 You're not allowed to be on the fence here. 309 00:18:24 --> 00:18:25 You've got to decide, all right? 310 00:18:25 --> 00:18:28 This is, you can talk to your neighbors, you know, do a 311 00:18:28 --> 00:18:30 little bit of thinking. 312 00:18:30 --> 00:18:33 And I'm going to give you ten seconds to figure this out, 313 00:18:33 --> 00:19:08 what your vote is for that. 314 00:19:08 --> 00:19:10 All right, let's try again. 315 00:19:10 --> 00:19:12 How many people vote that this is reversible? 316 00:19:12 --> 00:19:17 That looks like a majority to me. 317 00:19:17 --> 00:19:18 Irreversible? 318 00:19:18 --> 00:19:21 Let's look at the show of hands? 319 00:19:21 --> 00:19:25 All right, so this is the majority here. 320 00:19:25 --> 00:19:28 Good thing physics doesn't work on the rule of the majority, 321 00:19:28 --> 00:19:31 otherwise we'd be in big trouble. 322 00:19:31 --> 00:19:34 Wow, let's walk through that. 323 00:19:34 --> 00:19:39 I'm sorry to say that this is the wrong answer. 324 00:19:39 --> 00:19:40 OK, why is that the wrong answer? 325 00:19:40 --> 00:19:42 Well, just think, you know, think about it. 326 00:19:42 --> 00:19:45 You're pushing through here. p1 is greater than p2. 327 00:19:45 --> 00:19:49 What does it mean for a process to be in 328 00:19:49 --> 00:19:51 equilibrium or reversible? 329 00:19:51 --> 00:19:59 It means at any point you can reverse the direction of time, 330 00:19:59 --> 00:20:02 and it will look fine, right. 331 00:20:02 --> 00:20:07 So now I'm pushing on this plug here, with p1 greater than p2. 332 00:20:07 --> 00:20:12 I'm pushing, I'm pushing, I'm pushing. p1 is greater than p2. 333 00:20:12 --> 00:20:13 Then I want to reverse direction of time. 334 00:20:13 --> 00:20:17 I want the arrow of time to go so that the gas goes from p2 335 00:20:17 --> 00:20:21 to p1. p2 is less than p1. 336 00:20:21 --> 00:20:25 Is that going to work out? 337 00:20:25 --> 00:20:28 If p2, the pressure in p2, is less than the pressure in p1, 338 00:20:28 --> 00:20:31 is the gas going to want to go from p2 to p1 and the 339 00:20:31 --> 00:20:36 whole thing reverse back? 340 00:20:36 --> 00:20:38 You've got to put more pressure on one side than the other if 341 00:20:38 --> 00:20:42 you want to push that gas through the throttle, right? 342 00:20:42 --> 00:20:46 So this is where the time scale issue comes into play. 343 00:20:46 --> 00:20:52 I said let's do this slowly enough that this p1 is in 344 00:20:52 --> 00:20:58 equilibrium with this p1, but not so slowly that this 345 00:20:58 --> 00:21:02 pressure is equivalent to that pressure. 346 00:21:02 --> 00:21:04 If I do it really, really slowly, so that everything is 347 00:21:04 --> 00:21:09 reversible, well I won't be able to do it, because p1 348 00:21:09 --> 00:21:11 and p2 are different. 349 00:21:11 --> 00:21:17 But suppose that I fix my pistons here, with p1 greater 350 00:21:17 --> 00:21:21 than p2, and I don't touch it, eventually p1 will 351 00:21:21 --> 00:21:21 be equal to p2. 352 00:21:21 --> 00:21:23 I'll come to some sort of equilibrium. 353 00:21:23 --> 00:21:27 So this is a system which is out of equilibrium. 354 00:21:27 --> 00:21:34 If I stop, if I move slowly, if I move more slowly, then these 355 00:21:34 --> 00:21:36 two will want equilibrium. 356 00:21:36 --> 00:21:39 So this is an irreversible process. 357 00:21:39 --> 00:21:48 The Joule-Thomson experiment is irreversible. 358 00:21:48 --> 00:21:54 OK, important -- if you are part of this group here, 359 00:21:54 --> 00:21:58 think about it and make sure you understand that. 360 00:21:58 --> 00:22:00 All right, so this is the experiment. 361 00:22:00 --> 00:22:02 Now what are we doing with that? 362 00:22:02 --> 00:22:11 The initial state, let's look at what we are doing. 363 00:22:11 --> 00:22:16 So initially, we're going to have our piston, there's 364 00:22:16 --> 00:22:18 the plug sitting here. 365 00:22:18 --> 00:22:26 Our piston on the right side here fully out, and the piston 366 00:22:26 --> 00:22:30 on my right side, your left side, fully inside. 367 00:22:30 --> 00:22:33 There's no gas on that side here. 368 00:22:33 --> 00:22:35 So there's p2 sitting here. 369 00:22:35 --> 00:22:38 There is p1 sitting here, and all of the gas is sitting 370 00:22:38 --> 00:22:42 on that side of the plug. 371 00:22:42 --> 00:22:47 Then after we're done with the experiment, we'll have 372 00:22:47 --> 00:22:55 transferred all of the gas from one side to the other. p1 373 00:22:55 --> 00:23:01 here. p2 sitting here. 374 00:23:01 --> 00:23:05 And there's going to be some volume V2 and some 375 00:23:05 --> 00:23:09 volume V1, but are not necessarily the same. 376 00:23:09 --> 00:23:12 Especially since the pressures are different. we don't know 377 00:23:12 --> 00:23:15 yet about temperature so I don't know what to say about 378 00:23:15 --> 00:23:17 these volumes because I don't know what the temperatures' 379 00:23:17 --> 00:23:20 are going to do. 380 00:23:20 --> 00:23:23 OK, so let's go through this and see what we would do 381 00:23:23 --> 00:23:27 which is to calculate the heat and the work. 382 00:23:27 --> 00:23:32 This is well insulated. 383 00:23:32 --> 00:23:40 So, what is the -- let's do the first one here. 384 00:23:40 --> 00:23:44 What's q for this process here? 385 00:23:44 --> 00:23:46 Anybody? 386 00:23:46 --> 00:23:47 STUDENT Zero. 387 00:23:47 --> 00:23:47 PROFESSOR BAWENDI: Zero, right. 388 00:23:47 --> 00:23:48 This is an adiabatic process. 389 00:23:48 --> 00:23:49 It's well insulated. 390 00:23:49 --> 00:23:56 Heat is not going in or out, adiabatic. q is equal to zero. 391 00:23:56 --> 00:23:58 So all we need to find out is the work now. 392 00:23:58 --> 00:24:02 Let's divide it up into the two sides, the work going on on the 393 00:24:02 --> 00:24:05 left hand side, my left hand side or your left hand side, 394 00:24:05 --> 00:24:07 and the work going on on the right hand side. 395 00:24:07 --> 00:24:10 So let's first look at the left hand side. 396 00:24:10 --> 00:24:15 OK, so w, first of all, work is being done to the system on 397 00:24:15 --> 00:24:17 the left hand side here. 398 00:24:17 --> 00:24:20 I'm pressing on the gas. 399 00:24:20 --> 00:24:24 So I expect that to be a positive number. 400 00:24:24 --> 00:24:26 The pressure is constant, p. 401 00:24:26 --> 00:24:29 The V goes from V1 to zero. 402 00:24:29 --> 00:24:34 So we write down p1, V1. 403 00:24:34 --> 00:24:41 On the right hand side, the work, let's call this left 404 00:24:41 --> 00:24:45 hand side, let's call this right hand side. 405 00:24:45 --> 00:24:50 Here there's an expansion going on, so the system is doing 406 00:24:50 --> 00:24:52 work to the external world. 407 00:24:52 --> 00:24:55 This piston is being brought out, so we expect the work 408 00:24:55 --> 00:24:59 to be negative, negative. 409 00:24:59 --> 00:25:01 And we start out with zero volume. 410 00:25:01 --> 00:25:03 We end up with a volume of V2, and the external pressure 411 00:25:03 --> 00:25:05 is constant to p2. 412 00:25:05 --> 00:25:08 Minus p2, V2. 413 00:25:08 --> 00:25:11 Minus p delta V. 414 00:25:11 --> 00:25:17 So the total work is the work from the left hand side plus 415 00:25:17 --> 00:25:22 the work on the right hand side, which is p1 416 00:25:22 --> 00:25:28 V1 minus p2 V2. 417 00:25:28 --> 00:25:35 Which I can rewrite as minus delta pV. 418 00:25:35 --> 00:25:39 Delta pV is p2 V2 minus p1 V1. 419 00:25:39 --> 00:25:42 It's the pressure volume multiplied together at the 420 00:25:42 --> 00:25:47 final state, minus pressure volume from the initial 421 00:25:47 --> 00:25:50 state, with the minus sign here because it's the 422 00:25:50 --> 00:25:54 negative of that. 423 00:25:54 --> 00:26:03 All right, what is delta u? delta u is q plus w. q is 424 00:26:03 --> 00:26:07 zero. delta u is just w. 425 00:26:07 --> 00:26:12 So this is also just delta u. delta u is minus delta 426 00:26:12 --> 00:26:18 pV, for the process. 427 00:26:18 --> 00:26:27 OK, delta H is delta of u plus pV. 428 00:26:27 --> 00:26:30 By definition, that's how we define enthalpy up here. 429 00:26:30 --> 00:26:32 H is u plus pV. 430 00:26:32 --> 00:26:38 Delta H is delta of u plus pV, which is equal to 431 00:26:38 --> 00:26:42 delta u, plus delta pV. 432 00:26:42 --> 00:26:44 Now delta u is minus delta pV. 433 00:26:44 --> 00:26:47 So I have minus delta pV plus delta pV. 434 00:26:47 --> 00:26:50 This is equal to zero. 435 00:26:50 --> 00:26:56 So this irreversible process, this Joule-Thomson process, is 436 00:26:56 --> 00:26:57 a constant enthalpy process. 437 00:26:57 --> 00:27:01 Delta h for this process is equal to zero. 438 00:27:01 --> 00:27:02 Adiabatic q equal to zero. 439 00:27:02 --> 00:27:06 It's also delta H which is zero. 440 00:27:06 --> 00:27:10 The two didn't necessarily follow, because remember, delta 441 00:27:10 --> 00:27:13 H is dq so p is only true for a reversible constant 442 00:27:13 --> 00:27:15 pressure process. 443 00:27:15 --> 00:27:16 This is an irreversible process. 444 00:27:16 --> 00:27:19 So a priori, this was not necessarily true. 445 00:27:19 --> 00:27:21 It turns out after you do all the math, it turns out to 446 00:27:21 --> 00:27:25 be delta H equals zero. 447 00:27:25 --> 00:27:27 All right, so this is the experiment. 448 00:27:27 --> 00:27:29 How do we go from that experiment to the terms 449 00:27:29 --> 00:27:32 that we're trying to get, these slopes. 450 00:27:32 --> 00:27:36 Remember, we're trying to get delta H, we're trying to get 451 00:27:36 --> 00:27:44 dH/dT constant pressure and dH/dp constant temperature. 452 00:27:44 --> 00:27:52 OK, these are the two things were trying to get here. 453 00:27:52 --> 00:28:08 OK, so let's write down, what we know here. 454 00:28:08 --> 00:28:10 I'm missing something. 455 00:28:10 --> 00:28:12 Oh, we already know one thing. 456 00:28:12 --> 00:28:13 We already know this guy here. 457 00:28:13 --> 00:28:14 We already did that. 458 00:28:14 --> 00:28:19 OK, dH/dT constant pressure is Cp. 459 00:28:19 --> 00:28:20 That was easy one. 460 00:28:20 --> 00:28:21 So we already know that. 461 00:28:21 --> 00:28:30 So now we can write or differential dH as Cp dT 462 00:28:30 --> 00:28:36 plus dH/dp, constant temperature, dp. 463 00:28:36 --> 00:28:41 Now we want to find out what this guy is here. 464 00:28:41 --> 00:28:45 Now for this experiment, this is a constant enthalpy 465 00:28:45 --> 00:28:47 experiment for the Joule-Thomson experiment, 466 00:28:47 --> 00:28:50 this is equal to zero. 467 00:28:50 --> 00:28:57 So I can rearrange this to get this dH/dp in terms of things 468 00:28:57 --> 00:29:01 that I can either measure, like the heat capacity, or that I 469 00:29:01 --> 00:29:05 have control of, like dT and dp. 470 00:29:05 --> 00:29:20 So in this case, dH/dp constant temperature is minus Cp dT/dp, 471 00:29:20 --> 00:29:23 and this is under constant temperature, no, not 472 00:29:23 --> 00:29:25 constant temperature. 473 00:29:25 --> 00:29:31 Whatever the experi -- for that the experiment is. 474 00:29:31 --> 00:29:39 For that experiment, the constraint, so we need a 475 00:29:39 --> 00:29:42 constraint here, right, we need a constraint here. 476 00:29:42 --> 00:29:43 Right? 477 00:29:43 --> 00:29:44 We need a constraint here. 478 00:29:44 --> 00:29:47 The constraint isn't constant temperature because the 479 00:29:47 --> 00:29:48 temperature is going to be changing. 480 00:29:48 --> 00:29:51 It's not constant pressure, because we have a 481 00:29:51 --> 00:29:52 delta p going on. 482 00:29:52 --> 00:29:55 It's not constant volume either. 483 00:29:55 --> 00:29:56 The constraint is the constraint of the experiment, 484 00:29:56 --> 00:29:59 and the constraint of the experiment is that the 485 00:29:59 --> 00:30:02 enthalpy is constant. 486 00:30:02 --> 00:30:06 So the constraints we have here, is the constant enthalpy. 487 00:30:06 --> 00:30:13 It's the constant enthalpy process that we're looking at. 488 00:30:13 --> 00:30:16 This we can do experiments on. 489 00:30:16 --> 00:30:17 It's tabulated in books, and this we can measure 490 00:30:17 --> 00:30:18 in the experiment. 491 00:30:18 --> 00:30:23 Delta p here is the change in pressure from the left side to 492 00:30:23 --> 00:30:26 the right side, and we can put a thermometer, measure the 493 00:30:26 --> 00:30:28 temperature before the experiment, and measure the 494 00:30:28 --> 00:30:31 temperature after the experiment. 495 00:30:31 --> 00:30:38 So this is something we can measure. 496 00:30:38 --> 00:30:41 So now we have this derivative, in terms of physical 497 00:30:41 --> 00:30:42 quantities, things that we can measure. 498 00:30:42 --> 00:30:45 Things that we can relate to the properties of the 499 00:30:45 --> 00:30:48 substance that we're doing the experiment on. 500 00:30:48 --> 00:30:54 So this is basically delta T and delta p and the 501 00:30:54 --> 00:30:59 Joule-Thomson experiment. 502 00:30:59 --> 00:31:02 And so Joule and Thomson did these experiments, and they 503 00:31:02 --> 00:31:05 measured lots of gases, and they found that, in fact, 504 00:31:05 --> 00:31:08 this was something that they could measure. 505 00:31:08 --> 00:31:11 Sometimes it was positive, sometimes it was negative, and 506 00:31:11 --> 00:31:13 it was an interesting number. 507 00:31:13 --> 00:31:19 And so they defined them, after many experiments, the limit of 508 00:31:19 --> 00:31:25 this, delta T delta p and the limit of delta p goes to zero 509 00:31:25 --> 00:31:27 as the Joule-Thomson coefficient. 510 00:31:27 --> 00:31:37 So, basically dT/dp, constant enthalpy is equal to mu, by 511 00:31:37 --> 00:31:41 definition, Joule-Thomson, where mu Joule-Thomson is the 512 00:31:41 --> 00:31:44 Joule-Thomson coefficient, just like you saw last time eta sub 513 00:31:44 --> 00:31:50 j was the Joule coefficient for dT/dV under constant energy. 514 00:31:50 --> 00:31:53 So there's, again, total analogy here between what we're 515 00:31:53 --> 00:31:57 doing with enthalpy to what you did last time with energy, 516 00:31:57 --> 00:32:03 replace p with T and H with u. 517 00:32:03 --> 00:32:06 Flip those two and you get the same thing. 518 00:32:06 --> 00:32:11 OK, so now we have dH/dT is equal to Cp, and we can also 519 00:32:11 --> 00:32:19 write, then, dH/dp under constant temperature is equal 520 00:32:19 --> 00:32:24 to minus Cp mu Joule-Thomson. 521 00:32:24 --> 00:32:30 We have our two derivatives in terms of physical quantities, 522 00:32:30 --> 00:32:33 which is going to allow us, then, whenever we have a change 523 00:32:33 --> 00:32:39 to go back and, when we have a change where we adjust the 524 00:32:39 --> 00:32:42 temperature and the pressure, we'll be able to know what 525 00:32:42 --> 00:32:48 the enthalpy change is. 526 00:32:48 --> 00:32:50 OK, now let's take two cases. 527 00:32:50 --> 00:32:58 Let's first start talking about ideal gases. 528 00:32:58 --> 00:33:03 The last time you saw that for an ideal gas, the energy only 529 00:33:03 --> 00:33:04 cared about the temperature. 530 00:33:04 --> 00:33:12 It didn't care what the volume was doing. du/dV under constant 531 00:33:12 --> 00:33:16 temperature was equal to zero for an ideal gas. 532 00:33:16 --> 00:33:20 And by analogy, we expect the same thing to be true here, 533 00:33:20 --> 00:33:22 because enthalpy and energy have all this analogy 534 00:33:22 --> 00:33:23 going on here. 535 00:33:23 --> 00:33:25 So let's look at an ideal gas. 536 00:33:25 --> 00:33:33 So for an ideal gas, we saw that u was only a 537 00:33:33 --> 00:33:35 function of temperature. 538 00:33:35 --> 00:33:36 We also have the equation of state for an 539 00:33:36 --> 00:33:38 ideal gas, pV = nRT. 540 00:33:38 --> 00:33:41 541 00:33:41 --> 00:33:46 We can write our definition of enthalpy, h is u plus pV. 542 00:33:46 --> 00:33:51 This only depends on the temperature. pV= nRT. 543 00:33:51 --> 00:33:57 So we have u only depends on temperature plus nRT. 544 00:33:57 --> 00:33:59 The only valuable now on this side is temperature. 545 00:33:59 --> 00:34:03 Pressure and volume have dropped out. 546 00:34:03 --> 00:34:10 So enthalpy, for an ideal gas, only cares about temperature. 547 00:34:10 --> 00:34:14 Pressure has dropped out of the picture completely here. 548 00:34:14 --> 00:34:21 So there is no p dependence here. 549 00:34:21 --> 00:34:26 H for an ideal gas is only a function of temperature. 550 00:34:26 --> 00:34:30 This is not true for a real gas, fortunately, but it's 551 00:34:30 --> 00:34:32 true for an ideal gas. 552 00:34:32 --> 00:34:41 So for an ideal gas then, dH/dp under constant temperature, 553 00:34:41 --> 00:34:42 that has to be equal to zero. 554 00:34:42 --> 00:34:45 Because temperature is constant H only cares about temperature. 555 00:34:45 --> 00:34:47 and that's equal to zero. 556 00:34:47 --> 00:34:50 And if that's equal to zero, that means that the 557 00:34:50 --> 00:34:54 Joule-Thomson coefficient for an ideal gas is 558 00:34:54 --> 00:34:57 also equal to zero. 559 00:34:57 --> 00:34:59 We're going to actually prove this later in the course. 560 00:34:59 --> 00:35:01 Right now, you're taking it for granted. 561 00:35:01 --> 00:35:05 Right now we told you Joule did all these experiments and he 562 00:35:05 --> 00:35:09 found out that for an ideal gas, that the limit in and 563 00:35:09 --> 00:35:12 ideal gas case was that the eta J was equal to zero. 564 00:35:12 --> 00:35:16 Therefore, from experiments, u is only a function of 565 00:35:16 --> 00:35:20 temperature for an ideal gas, and therefore from these 566 00:35:20 --> 00:35:25 experiments, we come out with delta H dH/dp is equal to zero. 567 00:35:25 --> 00:35:30 The Joule-Thomson coefficient is equal to zero. 568 00:35:30 --> 00:35:34 Later we are going to prove it exactly. but right now you're 569 00:35:34 --> 00:35:36 going to have to take it for granted. 570 00:35:36 --> 00:35:40 So, if the Joule-Thomson coefficient is equal to zero, 571 00:35:40 --> 00:35:44 just like we wrote, du = Cv dT for an ideal gas, we're going 572 00:35:44 --> 00:35:51 to have dH = Cp dT for an ideal gas as well. dH is Cp dT. 573 00:35:51 --> 00:35:55 This term goes away. dH = Cp dT. 574 00:35:55 --> 00:35:57 That's the only thing that's left behind. 575 00:35:57 --> 00:35:59 So if you know the heat capacity, you know the change 576 00:35:59 --> 00:36:02 in temperature, you know what enthalpy is doing 577 00:36:02 --> 00:36:04 for an ideal gas. 578 00:36:04 --> 00:36:11 This needs to be stressed that this is the ideal gas case. 579 00:36:11 --> 00:36:15 Now regular gases, real gases, fortunately as I said, 580 00:36:15 --> 00:36:16 don't obey this. 581 00:36:16 --> 00:36:19 This is important because we use this all the time for 582 00:36:19 --> 00:36:22 when, when we do technology. 583 00:36:22 --> 00:36:29 One example of Joule-Thomson coefficient being not equal to 584 00:36:29 --> 00:36:35 zero for a real gas that you've like experienced is if you take 585 00:36:35 --> 00:36:39 a bicycle pump, take a bicycle pump, and you're pumping 586 00:36:39 --> 00:36:41 up your tire, you're working pretty hard, 587 00:36:41 --> 00:36:42 so you're getting hot. 588 00:36:42 --> 00:36:48 But if you touch the valve going into your tire, which 589 00:36:48 --> 00:36:50 basically measures the temperature of the air 590 00:36:50 --> 00:36:53 going into your tire, that is getting hot, right. 591 00:36:53 --> 00:36:59 So if you've got to pump that tire really a lot, then you're 592 00:36:59 --> 00:37:01 going to you're going to really feel a lot of heat there. 593 00:37:01 --> 00:37:05 The compression of the, basically it's an 594 00:37:05 --> 00:37:06 adiabatic compression. 595 00:37:06 --> 00:37:09 You're taking the air inside of the pump and you're 596 00:37:09 --> 00:37:11 compressing it. 597 00:37:11 --> 00:37:14 You're doing it so fast that there's not enough time for 598 00:37:14 --> 00:37:18 heat to come out of the gas that's inside the pump towards 599 00:37:18 --> 00:37:21 the walls of the pump. 600 00:37:21 --> 00:37:24 So your time scale it just fast enough that this is basically 601 00:37:24 --> 00:37:25 an adiabatic compassion. 602 00:37:25 --> 00:37:29 Your compressing it really fast, all right, so you're 603 00:37:29 --> 00:37:32 changing the pressure. 604 00:37:32 --> 00:37:34 You're changing the pressure, and the temperature is going 605 00:37:34 --> 00:37:43 up. dT/dp is positive. dT/dp is positive. dT/dp is positive, 606 00:37:43 --> 00:37:47 well that's mu JT. dT/dp is mu JT. 607 00:37:47 --> 00:37:53 So for a real gas like air, this is a positive number. 608 00:37:53 --> 00:37:54 It's not zero. 609 00:37:54 --> 00:37:55 Air is not an ideal gas. 610 00:37:55 --> 00:38:04 It's one simple example of -- The fact that mu JT is not zero 611 00:38:04 --> 00:38:10 for real gases is how we are able to liquify things like 612 00:38:10 --> 00:38:13 hydrogen and helium, by compressing them and pushing 613 00:38:13 --> 00:38:17 them through a nozzle, and the expansion through the 614 00:38:17 --> 00:38:20 nozzle cools the gas. 615 00:38:20 --> 00:38:22 Right, in this case here it wouldn't happen if 616 00:38:22 --> 00:38:27 it was an ideal gas. 617 00:38:27 --> 00:38:32 Or in many kinds of gas refrigerators where you push a 618 00:38:32 --> 00:38:37 gas through a nozzle close to room temperature, what you find 619 00:38:37 --> 00:38:40 is that the gas coming out on the other side under lower 620 00:38:40 --> 00:38:43 pressure is cooler than the gas that went through 621 00:38:43 --> 00:38:45 on the other side. 622 00:38:45 --> 00:38:48 Real refrigerators actually work with liquids that go into 623 00:38:48 --> 00:38:50 gases so use the latent heat of the liquid, so it doesn't 624 00:38:50 --> 00:38:55 really work like the Joule-Thomson expansion. 625 00:38:55 --> 00:38:56 So this is real. 626 00:38:56 --> 00:38:59 This is real, unlike the Joule coefficient which is very 627 00:38:59 --> 00:39:01 small so that most gases have tiny Joule coefficients. 628 00:39:01 --> 00:39:03 So if you do a Joule experiment, you hardly measure 629 00:39:03 --> 00:39:04 a temperature change. 630 00:39:04 --> 00:39:07 With real gases, here you do actually measure it. 631 00:39:07 --> 00:39:14 You can feel it with your finger on your bicycle tire. 632 00:39:14 --> 00:39:19 OK, so we're going to see this using a Van der Waal's gas. 633 00:39:19 --> 00:39:25 Let's look at a Van der Waal's gas and see what happens 634 00:39:25 --> 00:39:35 in the Van der Waal's gas. 635 00:39:35 --> 00:39:36 Any questions, first? 636 00:39:36 --> 00:39:39 What we've been talking about, the Joule-Thomson experiment, 637 00:39:39 --> 00:39:47 constant enthalpy process? 638 00:39:47 --> 00:39:50 OK, so let's take our Van der Waal's gas. 639 00:39:50 --> 00:39:53 Remember the equation of state for Van der Waal's gas is not 640 00:39:53 --> 00:39:59 pV is equal to nRT, but p plus the attraction term. 641 00:39:59 --> 00:40:05 And then V minus the excluded volume term is equal to RT. 642 00:40:05 --> 00:40:08 Two parameters, this is the attraction between two atoms or 643 00:40:08 --> 00:40:10 molecules in the gas phase. 644 00:40:10 --> 00:40:11 This is the repulsion, not the repulsion. 645 00:40:11 --> 00:40:15 This is the fact that we occupy a finite volume in space, 646 00:40:15 --> 00:40:19 because they're little hard spheres in this molecule. 647 00:40:19 --> 00:40:23 OK, in a few weeks, you're going to find out that we can 648 00:40:23 --> 00:40:28 calculate dH/dp from this equation of state, and you're 649 00:40:28 --> 00:40:32 going to find out that dH/dp from that equation of state is 650 00:40:32 --> 00:40:37 proportional to b minus a over RT. 651 00:40:37 --> 00:40:39 This is going to be probably a homework at some 652 00:40:39 --> 00:40:41 point to do this. 653 00:40:41 --> 00:40:42 For now, let's take it for granted. 654 00:40:42 --> 00:40:46 Let's take it for granted that we know how to calculate this 655 00:40:46 --> 00:40:50 derivative from an equation of state like this. 656 00:40:50 --> 00:40:56 But now we're going to use that. 657 00:40:56 --> 00:40:56 OK. 658 00:40:56 --> 00:41:02 So since dH/dp under constant temperature is proportional to 659 00:41:02 --> 00:41:12 minus mu JT, then we have that dT/dp under constant enthalpy 660 00:41:12 --> 00:41:16 than it is related to the negative of this, 661 00:41:16 --> 00:41:19 a over RT minus b 662 00:41:19 --> 00:41:26 All right, this is how the temperature changes when 663 00:41:26 --> 00:41:28 you change, when you have pressure changing. 664 00:41:28 --> 00:41:31 So when you do an expansion or compression, my 665 00:41:31 --> 00:41:38 adiabatic compression of my bicycle pump is dT/dp. 666 00:41:38 --> 00:41:43 All right. 667 00:41:43 --> 00:41:46 Or when I do an expansion of hydrogen or helium at low 668 00:41:46 --> 00:41:52 temperature, through a Joule-Thomson experiment, 669 00:41:52 --> 00:41:53 when I want to liquify hydrogen or helium. 670 00:41:53 --> 00:41:58 I want to cool a gas with a Joule-Thomson experiment, what 671 00:41:58 --> 00:42:00 temperature do I have to be at? 672 00:42:00 --> 00:42:03 So this tells you that you have to be careful what temperature 673 00:42:03 --> 00:42:06 you're at, because depending on how high, how big this 674 00:42:06 --> 00:42:10 temperature here is, you could either be, have a negative 675 00:42:10 --> 00:42:15 dT/dp, if this first term is small enough, meaning if 676 00:42:15 --> 00:42:18 temperature is very high, then you end up with a 677 00:42:18 --> 00:42:19 negative term. 678 00:42:19 --> 00:42:23 If the temperature is very small, then one over the 679 00:42:23 --> 00:42:26 temperature is large, and the first term wins and you 680 00:42:26 --> 00:42:27 have a positive number. 681 00:42:27 --> 00:42:30 So there's some special temperature which is going to 682 00:42:30 --> 00:42:34 depend on the gas where the first term is going to be equal 683 00:42:34 --> 00:42:38 to the second term, where Joule-Thomson is zero, or 684 00:42:38 --> 00:42:42 it's going to behave like an ideal gas. 685 00:42:42 --> 00:42:48 So when that is the case, we're going to call that 686 00:42:48 --> 00:42:51 temperature the inversion temperature or T inv. 687 00:42:51 --> 00:42:54 We call that inversion because on one side you end up 688 00:42:54 --> 00:42:55 cooling if you compress. 689 00:42:55 --> 00:42:58 And on the other side of that temperature you end up 690 00:42:58 --> 00:43:03 heating if you compress. 691 00:43:03 --> 00:43:08 OK, so there's some temperature, t inversion minus 692 00:43:08 --> 00:43:12 b where the gas behaves like an ideal gas. 693 00:43:12 --> 00:43:15 The Joule-Thomson coefficient is zero and that inversion 694 00:43:15 --> 00:43:19 temperature, you can solve for it. 695 00:43:19 --> 00:43:29 Conversion temperature is equal to a over R times b. 696 00:43:29 --> 00:43:32 OK, so when you are at that temperature, everything looks 697 00:43:32 --> 00:43:35 like an ideal gas, as far as the enthalpy changes 698 00:43:35 --> 00:43:37 are concerned. 699 00:43:37 --> 00:43:41 Now if you're at the temperature which is higher 700 00:43:41 --> 00:43:47 than the inversion temperature, in that case here, a over RT is 701 00:43:47 --> 00:43:51 small compared to b, and this is going to turn out 702 00:43:51 --> 00:43:52 to be negative. 703 00:43:52 --> 00:43:55 So if you had a high temperature, this a 704 00:43:55 --> 00:43:57 small compared to b. 705 00:43:57 --> 00:44:06 If you're negative, which means that dT/dp at constant 706 00:44:06 --> 00:44:14 H is less than zero. 707 00:44:14 --> 00:44:20 So that means that if you compress something 708 00:44:20 --> 00:44:21 it's going to cool. 709 00:44:21 --> 00:44:27 The temperature rises when the pressure drops, right? 710 00:44:27 --> 00:44:32 Or in this case here, if I do my Joule experiment delta p is 711 00:44:32 --> 00:44:36 negative, p2 is less than p1, that means that delta 712 00:44:36 --> 00:44:37 T is positive, right? 713 00:44:37 --> 00:44:40 So in this experiment here, this side is going to heat up. 714 00:44:40 --> 00:44:49 So for materials where T inversion is low, lower than 715 00:44:49 --> 00:44:52 room temperature, then you would end up heating 716 00:44:52 --> 00:44:54 up in this expansion. 717 00:44:54 --> 00:44:57 You're basically expanding the gas from one side to the other 718 00:44:57 --> 00:45:01 and the expansion causes the temperature to rise. 719 00:45:01 --> 00:45:08 If T is less than T inversion, you have the opposite case, and 720 00:45:08 --> 00:45:13 dT/dp is greater than zero. 721 00:45:13 --> 00:45:17 So in this experiment here, delta p is less than zero. 722 00:45:17 --> 00:45:19 You need to have this whole thing greater than zero. 723 00:45:19 --> 00:45:21 So delta T is less than zero as well. 724 00:45:21 --> 00:45:23 So if you're below the inversion temperature and 725 00:45:23 --> 00:45:26 you do the Joule-Thomson experiment, you're going to end 726 00:45:26 --> 00:45:29 up with something that's colder on this side than that side. 727 00:45:29 --> 00:45:32 Ideal gas would be the same temperature. 728 00:45:32 --> 00:45:37 But now, so this is where the refrigeration comes in. 729 00:45:37 --> 00:45:40 So if you take a gas, and you're below the inversion 730 00:45:40 --> 00:45:43 temperature and you make it go through this irreversible 731 00:45:43 --> 00:45:46 process, the gas comes out colder from that side 732 00:45:46 --> 00:45:47 than that side. 733 00:45:47 --> 00:45:52 So the work that you're doing to expand, to go through this 734 00:45:52 --> 00:45:59 experiment, ends up cooling the gas. 735 00:45:59 --> 00:46:08 OK, for most gases, T inversion is much greater than 736 00:46:08 --> 00:46:10 300 degrees Kelvin. 737 00:46:10 --> 00:46:11 Much greater than room temperature. 738 00:46:11 --> 00:46:13 For more most gases, if you do this experiment at room 739 00:46:13 --> 00:46:16 temperature, you end up cooling the gas, and you can cool it 740 00:46:16 --> 00:46:21 measurably, which is why also, your bicycle pump, you know, 741 00:46:21 --> 00:46:24 you push down, you compress, this is an expansion here. 742 00:46:24 --> 00:46:27 You're expanding from this side to that side. 743 00:46:27 --> 00:46:29 Bicycle pump you are compressing the gas, which is 744 00:46:29 --> 00:46:32 the opposite, you end up heating up the air in 745 00:46:32 --> 00:46:38 the bicycle pump in your compression. 746 00:46:38 --> 00:46:41 There are two exceptions to this rule that most gases have 747 00:46:41 --> 00:46:44 a T inversion which is greater than 300 degrees Kelvin, 748 00:46:44 --> 00:46:45 and that's hydrogen. 749 00:46:45 --> 00:46:52 The T inversion for hydrogen turns out to be 193 degrees 750 00:46:52 --> 00:46:57 Kelvin, and the T inversion for helium turns out to 751 00:46:57 --> 00:47:04 be 53 degrees Kelvin. 752 00:47:04 --> 00:47:07 And so when, there's a lot of liquid helium that's 753 00:47:07 --> 00:47:09 being used on campus. 754 00:47:09 --> 00:47:10 We use a liquid helium. 755 00:47:10 --> 00:47:14 And so in order to make a liquid helium, you can't take 756 00:47:14 --> 00:47:17 helium at room temperature and do this, because if you did, 757 00:47:17 --> 00:47:19 you would just heat it up, because the room temperature 758 00:47:19 --> 00:47:22 is above the inversion temperature, so Joule-Thomson 759 00:47:22 --> 00:47:23 would heat up the helium. 760 00:47:23 --> 00:47:27 So you need first to take the liquid helium and cool it below 761 00:47:27 --> 00:47:32 53 degrees Kelvin before you can do the Joule-Thomson to 762 00:47:32 --> 00:47:34 cool it even further to make liquid helium. 763 00:47:34 --> 00:47:35 So you have to do it in stages. 764 00:47:35 --> 00:47:39 You take your room temperature liquid helium, and you cool it 765 00:47:39 --> 00:47:43 with liquid nitrogen to 77 degrees Kelvin, the new, you're 766 00:47:43 --> 00:47:46 not quite there yet, unfortunately right? 767 00:47:46 --> 00:47:50 Then you take hydrogen you cool it would liquid nitrogen to 768 00:47:50 --> 00:47:53 77, then you can use your hydrogen gas. 769 00:47:53 --> 00:47:57 Do a Joule-Thomson hydrogen gas which you first cool 770 00:47:57 --> 00:47:58 with liquid nitrogen. 771 00:47:58 --> 00:48:01 Liquid nitrogen, 77, that's below 193, so you can do 772 00:48:01 --> 00:48:05 Joule-Thomson on hydrogen, cool the hydrogen to below 53, then 773 00:48:05 --> 00:48:09 use your cold hydrogen to cool the helium, and then you can do 774 00:48:09 --> 00:48:12 the Joule-Thomson on the helium to cool it further 775 00:48:12 --> 00:48:14 until it liquifies. 776 00:48:14 --> 00:48:19 So that's what you do when you make liquid helium. 777 00:48:19 --> 00:48:24 All right, any questions on this lecture or any of the 778 00:48:24 --> 00:48:34 concepts that we talked about? 779 00:48:34 --> 00:48:39 The last thing that we're going to do today then is to look at 780 00:48:39 --> 00:48:46 a relationship which is going to turn out to be very useful. 781 00:48:46 --> 00:48:48 It's a relationship for ideal gases which relates the heat 782 00:48:48 --> 00:48:59 capacities at constant pressure and constant volume. 783 00:48:59 --> 00:49:00 Cp = Cv + R. 784 00:49:00 --> 00:49:06 785 00:49:06 --> 00:49:08 This is very useful because often you just have lots of 786 00:49:08 --> 00:49:12 tables of Cp's but sometimes you want to know what the 787 00:49:12 --> 00:49:16 energy change is going to be for the ideal gas, and you know 788 00:49:16 --> 00:49:19 that du is Cv dT not Cp dT. 789 00:49:19 --> 00:49:22 So, you need to know what Cv is, and if this is true always, 790 00:49:22 --> 00:49:25 then there's a very easy way to go from one to the other. 791 00:49:25 --> 00:49:28 We're going to do it two ways. 792 00:49:28 --> 00:49:30 Today we'll do the first way, and then next time we'll 793 00:49:30 --> 00:49:31 do the second way. 794 00:49:31 --> 00:49:35 The first way is just to turn the crank on the math, and the 795 00:49:35 --> 00:49:38 second way is to do a little bit more imaginative 796 00:49:38 --> 00:49:41 about the process. 797 00:49:41 --> 00:49:43 OK, let's just turn the crank on the maths. 798 00:49:43 --> 00:49:45 What do we know about the Cp and Cv? 799 00:49:45 --> 00:49:49 Well Cp, we already know how to relate it to dH/dT through the 800 00:49:49 --> 00:49:54 slope of the enthalpy, and we also related Cv to the slope of 801 00:49:54 --> 00:49:57 the energy with respect to temperature under 802 00:49:57 --> 00:49:59 constant volume. 803 00:49:59 --> 00:50:04 And we also know that H is u plus pV. 804 00:50:04 --> 00:50:07 Right, H is u plus pV. 805 00:50:07 --> 00:50:09 So we're going to take the derivatives of both sides of 806 00:50:09 --> 00:50:13 this equation here, by, with respect to temperature, 807 00:50:13 --> 00:50:16 keeping pressure constant. 808 00:50:16 --> 00:50:22 So we have dH/dT keeping pressure constant, is du/dT 809 00:50:22 --> 00:50:23 keeping pressure constant. 810 00:50:23 --> 00:50:26 Got to keep track of what's being constant, kept constant, 811 00:50:26 --> 00:50:35 plus dPV/dT, keeping pressure constant. dH/dT keeping 812 00:50:35 --> 00:50:42 pressure constant, that's just Cp, and then we have du/dT 813 00:50:42 --> 00:50:43 keeping pressure constant. 814 00:50:43 --> 00:50:47 The p is constant here, comes out of the equation, so we have 815 00:50:47 --> 00:50:52 p and then we have dV/dT well it's an ideal gas, an ideal gas 816 00:50:52 --> 00:51:01 for dV/dT for an ideal gas is equal to R over p because 817 00:51:01 --> 00:51:06 pV is equal to RT, right. 818 00:51:06 --> 00:51:11 So dV/dT is Rp. 819 00:51:11 --> 00:51:16 So dV/dT is times R over p, the p's are going 820 00:51:16 --> 00:51:19 to cancel out here. 821 00:51:19 --> 00:51:23 OK, so now it's very tempting at this state to say, oh 822 00:51:23 --> 00:51:26 there's the answer right here. du/dT is Cv. 823 00:51:26 --> 00:51:33 There I have it Cp is equal to Cv plus R, right? 824 00:51:33 --> 00:51:35 But even though it looks like that's the right way to do it, 825 00:51:35 --> 00:51:39 it's actually not right because it turns out to be right 826 00:51:39 --> 00:51:42 by sort of by accident. 827 00:51:42 --> 00:51:52 But here you've got pressure constant. du, this is du, not H 828 00:51:52 --> 00:51:58 here. du/dT is only equal to Cv when the volume is constant, 829 00:51:58 --> 00:52:00 not when the pressure is constant. 830 00:52:00 --> 00:52:03 So if you're going to turn the crank on the math correctly, 831 00:52:03 --> 00:52:08 you're going to have to change this p into a V somehow. 832 00:52:08 --> 00:52:10 Because this isn't Cv mathematically speaking. 833 00:52:10 --> 00:52:13 We don't know what it is yet. 834 00:52:13 --> 00:52:15 In order to change this from a p to a V, you have 835 00:52:15 --> 00:52:19 to use the chain rule. 836 00:52:19 --> 00:52:25 So let's use the chain rule. 837 00:52:25 --> 00:52:33 And then we'll be done. 838 00:52:33 --> 00:52:38 OK, so u is actually a function of temperature and volume, 839 00:52:38 --> 00:52:40 which in this case here could be a function of pressure 840 00:52:40 --> 00:52:41 and temperature. 841 00:52:41 --> 00:52:49 So if we want du/dT under constant pressure, you have 842 00:52:49 --> 00:52:50 to use the chain rule. 843 00:52:50 --> 00:52:52 There's the pressure sitting right here. 844 00:52:52 --> 00:53:03 It's going to be du/dT under constant volume, plus du/dV 845 00:53:03 --> 00:53:08 dV/dT under constant pressure. 846 00:53:08 --> 00:53:22 All right, chain rule. du/dT constant pressure is the direct 847 00:53:22 --> 00:53:24 derivative with respect to temperature here, which is 848 00:53:24 --> 00:53:26 sitting by itself under constant volume, keeping this 849 00:53:26 --> 00:53:28 constant but there is temperature sitting 850 00:53:28 --> 00:53:30 right here too. 851 00:53:30 --> 00:53:33 That's where that term comes from, du/dV dV/dT. 852 00:53:33 --> 00:53:36 853 00:53:36 --> 00:53:40 Now, for an ideal gas, du/dV under constant temperature 854 00:53:40 --> 00:53:40 is equal to zero. 855 00:53:40 --> 00:53:45 It doesn't care what the volume is doing. 856 00:53:45 --> 00:53:46 It only cares what temperature is. 857 00:53:46 --> 00:53:49 If temperature is constant, there's no change in energy. 858 00:53:49 --> 00:53:51 For an ideal gas, this is zero. 859 00:53:51 --> 00:53:55 It's not zero for a real gas. 860 00:53:55 --> 00:53:58 Right, so this whole term disappears and for an ideal 861 00:53:58 --> 00:54:02 gas, it turns out that du/dT constant pressure is equal to 862 00:54:02 --> 00:54:06 du/dT constant volume, but this is equal to Cv 863 00:54:06 --> 00:54:07 for an ideal gas. 864 00:54:07 --> 00:54:10 It wouldn't be true for a real gas, and this is a common 865 00:54:10 --> 00:54:13 mistake that people make for real gases to equate this. 866 00:54:13 --> 00:54:14 This is only true for an ideal gas. 867 00:54:14 --> 00:54:19 Since it's true for an ideal gas, then we can go ahead and 868 00:54:19 --> 00:54:27 replace this with Cv, and then we have Cp with Cv plus R, 869 00:54:27 --> 00:54:30 which is what we were after. 870 00:54:30 --> 00:54:34 OK, next time we'll do the other way of getting 871 00:54:34 --> 00:54:35 to the same answer. 872 00:54:35 --> 00:54:36