1 00:00:00,000 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,840 Your support will help MIT OpenCourseWare continue to 4 00:00:06,840 --> 00:00:10,520 offer high-quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional material from 6 00:00:13,390 --> 00:00:15,990 hundreds the MIT courses, visit MIT OpenCourseWare at 7 00:00:15,990 --> 00:00:17,500 ocw.mit.edu. 8 00:00:17,500 --> 00:00:23,660 PROFESSOR BAWENDI: Last time you talked about the first law 9 00:00:23,660 --> 00:00:25,350 of thermodynamics. 10 00:00:25,350 --> 00:00:28,410 And you talked about isothermal expansion, the 11 00:00:28,410 --> 00:00:31,010 Joule expansion. 12 00:00:31,010 --> 00:00:36,370 You saw a very important result. which is that for an 13 00:00:36,370 --> 00:00:48,350 ideal gas, the energy content is only dependent on the 14 00:00:48,350 --> 00:00:50,610 temperature, nothing else. 15 00:00:50,610 --> 00:00:53,770 Not the volume, not the pressure, it just cares about 16 00:00:53,770 --> 00:00:55,400 the temperature. 17 00:00:55,400 --> 00:01:01,340 So, if you have an isothermal process for an ideal gas, the 18 00:01:01,340 --> 00:01:06,270 energy doesn't change. q plus w is equal to zero for any 19 00:01:06,270 --> 00:01:10,220 isothermal process. 20 00:01:10,220 --> 00:01:18,020 And you also saw that du then could be written as Cv dT for 21 00:01:18,020 --> 00:01:19,160 an ideal gas always. 22 00:01:19,160 --> 00:01:21,670 This is not generally true. 23 00:01:21,670 --> 00:01:24,450 If you have a real gas and you write du is Cv dT, and your 24 00:01:24,450 --> 00:01:26,960 path is not a constant volume path, then you 25 00:01:26,960 --> 00:01:28,860 are making a mistake. 26 00:01:28,860 --> 00:01:31,040 But for an ideal gas, you can always write this. 27 00:01:31,040 --> 00:01:38,670 And this turns out to be very useful to remember. 28 00:01:38,670 --> 00:01:46,640 OK, now most processes that we deal with are not constant 29 00:01:46,640 --> 00:01:50,070 volume processes. 30 00:01:50,070 --> 00:01:54,880 So energy, which has this wonderful property here, du is 31 00:01:54,880 --> 00:01:58,240 Cv dt for constant volume process, which happens to be 32 00:01:58,240 --> 00:02:04,550 equal to d q, constant volume, because there's no change. 33 00:02:04,550 --> 00:02:07,680 there's no work if you've got a constant volume process. 34 00:02:07,680 --> 00:02:11,980 So du here is a very interesting quantity, because 35 00:02:11,980 --> 00:02:15,140 it's related to the heat that's going in or out of the 36 00:02:15,140 --> 00:02:19,620 system under constant volume process. 37 00:02:19,620 --> 00:02:23,270 But as I said, we're not operating usually in a 38 00:02:23,270 --> 00:02:24,910 constant volume environment. 39 00:02:24,910 --> 00:02:30,680 When I flail my arms around I generate work and heat. 40 00:02:30,680 --> 00:02:34,040 This is not a constant volume process. 41 00:02:34,040 --> 00:02:40,550 If I'm the system, what's constant when I do this? 42 00:02:40,550 --> 00:02:41,590 Anybody have an idea? 43 00:02:41,590 --> 00:02:46,760 What's the one function of state? 44 00:02:46,760 --> 00:02:49,250 I'm the system, the rest are the surrounding. 45 00:02:49,250 --> 00:02:52,470 What's the one function of state that's constant when I'm 46 00:02:52,470 --> 00:02:55,330 doing all my chemical reactions to 47 00:02:55,330 --> 00:03:02,670 move my arms around? 48 00:03:02,670 --> 00:03:03,190 Temperature? 49 00:03:03,190 --> 00:03:04,490 STUDENT: Pressure? 50 00:03:04,490 --> 00:03:05,520 PROFESSOR BAWENDI: Pressure, right. 51 00:03:05,520 --> 00:03:06,350 Pressure is constant. 52 00:03:06,350 --> 00:03:08,920 What is the pressure at? 53 00:03:08,920 --> 00:03:12,440 One atmosphere, one bar. 54 00:03:12,440 --> 00:03:14,860 So the most interesting processes are the processes 55 00:03:14,860 --> 00:03:16,090 where pressure is constant. 56 00:03:16,090 --> 00:03:20,160 When I had have a vial on bench top, and I do a chemical 57 00:03:20,160 --> 00:03:24,860 reaction in the vial, and it's open to the atmosphere, the 58 00:03:24,860 --> 00:03:26,930 pressure is constant at one atmosphere. 59 00:03:26,930 --> 00:03:29,610 When you've got your cells growing in your petri dish, 60 00:03:29,610 --> 00:03:32,170 the pressure is constant at one atmosphere, even if 61 00:03:32,170 --> 00:03:37,610 they're evolving gas, pressure is constant. 62 00:03:37,610 --> 00:03:43,160 So we'd really like to be able to find some sort of equation 63 00:03:43,160 --> 00:03:47,160 of state, or some sort of rather function of state 64 00:03:47,160 --> 00:03:51,980 that's going to relate the heat going in or out of the 65 00:03:51,980 --> 00:03:56,360 system with that function of state, because this isn't 66 00:03:56,360 --> 00:03:59,670 going to do it. du only relates to the heat under 67 00:03:59,670 --> 00:04:01,450 constant volume. 68 00:04:01,450 --> 00:04:03,640 And the heat is a really important thing to know. 69 00:04:03,640 --> 00:04:08,250 How much heat do you need to put into a system, or how much 70 00:04:08,250 --> 00:04:10,590 heat is going to come out of a system when something is 71 00:04:10,590 --> 00:04:11,700 happening in the system? 72 00:04:11,700 --> 00:04:15,740 All right, this is a really important quantity to know. 73 00:04:15,740 --> 00:04:18,470 Your boiling water or whatever, you want to know how 74 00:04:18,470 --> 00:04:22,100 much heat do you need to boil that amount of water under 75 00:04:22,100 --> 00:04:25,700 constant pressure? 76 00:04:25,700 --> 00:04:27,190 And this is where enthalpy comes in. 77 00:04:27,190 --> 00:04:27,930 You've all heard of enthalpy. 78 00:04:27,930 --> 00:04:34,060 H we're going to write it as the function of temperature 79 00:04:34,060 --> 00:04:38,140 and pressure. 80 00:04:38,140 --> 00:04:42,360 And the reason enthalpy was invented was exactly for that 81 00:04:42,360 --> 00:04:45,180 reason, because we need some way to figure out how to 82 00:04:45,180 --> 00:04:48,890 relate the heat coming in or out of a system under a 83 00:04:48,890 --> 00:04:51,850 constant pressure process. 84 00:04:51,850 --> 00:04:55,740 Because it's so important. 85 00:04:55,740 --> 00:04:58,370 And I should add and also under reversible work, where 86 00:04:58,370 --> 00:05:03,170 the external pressure is equal to the internal pressure. 87 00:05:03,170 --> 00:05:11,110 OK, so we're going to define enthalpy as u + pV, these are 88 00:05:11,110 --> 00:05:13,620 all functions of state here, So H is a function of state, 89 00:05:13,620 --> 00:05:17,320 and we're going to see now how this is, indeed, related to 90 00:05:17,320 --> 00:05:20,170 the heat flow in and out of the system. 91 00:05:20,170 --> 00:05:21,760 If you have a constant pressure, 92 00:05:21,760 --> 00:05:23,960 reversible work process. 93 00:05:23,960 --> 00:05:26,460 Let's take a system. 94 00:05:26,460 --> 00:05:34,100 Under constant pressure T1, V1, going to a second -- this 95 00:05:34,100 --> 00:05:39,910 is the system, so let me write the system here. 96 00:05:39,910 --> 00:05:46,750 And it's more dramatic if the system is a gas, p, T2, V2, 97 00:05:46,750 --> 00:05:50,720 And let's look at what happens to these functions of state, 98 00:05:50,720 --> 00:05:58,700 to H to u under this transformation. 99 00:05:58,700 --> 00:05:59,660 OK, so let's look at delta u. 100 00:05:59,660 --> 00:06:03,490 Delta u is q plus w. 101 00:06:03,490 --> 00:06:06,870 That's the first law. 102 00:06:06,870 --> 00:06:10,100 And this is a constant pressure path, so now I can 103 00:06:10,100 --> 00:06:12,460 write, this q is actually q under constant pressure. 104 00:06:12,460 --> 00:06:15,620 Little p means the path is a constant pressure path. 105 00:06:15,620 --> 00:06:26,450 And I'm doing reversible work. 106 00:06:26,450 --> 00:06:31,520 So that w is minus p, dV where p is the pressure inside the 107 00:06:31,520 --> 00:06:39,750 system, minus p delta V. Rearrange that, delta u is 108 00:06:39,750 --> 00:06:44,990 plus p delta V is equal to q p. 109 00:06:44,990 --> 00:06:48,530 All right, so this is the heat flowing in or out of the 110 00:06:48,530 --> 00:06:52,500 system, and these are all functions of state. 111 00:06:52,500 --> 00:06:54,640 This depends on the path. 112 00:06:54,640 --> 00:06:57,700 It tells you right here, the path is constant pressure. 113 00:06:57,700 --> 00:07:02,080 These don't depend on the path, right. 114 00:07:02,080 --> 00:07:05,570 V doesn't care how you get there. u doesn't 115 00:07:05,570 --> 00:07:08,370 care how you get there. 116 00:07:08,370 --> 00:07:11,500 In this case, p is a constant because the path is constant. 117 00:07:11,500 --> 00:07:16,780 So we can bring the p inside, delta u plus 118 00:07:16,780 --> 00:07:21,550 delta p V it's q p. 119 00:07:21,550 --> 00:07:26,820 Take this delta outside again, delta of u plus p V 120 00:07:26,820 --> 00:07:28,090 is equal to q p. 121 00:07:28,090 --> 00:07:28,750 And there you have it. 122 00:07:28,750 --> 00:07:30,040 There is the H right there. 123 00:07:30,040 --> 00:07:36,750 The u plus p V. Delta H is equal to q V. 124 00:07:36,750 --> 00:07:40,600 And this is the reason why enthalpy was invented, and why 125 00:07:40,600 --> 00:07:42,610 it's so important. 126 00:07:42,610 --> 00:07:44,230 Because we want to know this. 127 00:07:44,230 --> 00:07:46,500 So this for a finite change. 128 00:07:46,500 --> 00:07:49,590 If you want to have an infinitestimally small change, 129 00:07:49,590 --> 00:07:55,900 you end up writing dh is dq sub p. 130 00:07:55,900 --> 00:07:57,580 It's not always equal to the heat. 131 00:07:57,580 --> 00:08:00,770 It's only equal to the heat if your process is constant 132 00:08:00,770 --> 00:08:06,260 pressure reversible work. 133 00:08:06,260 --> 00:08:09,510 OK, so this is the kind of, this is the kind of concept 134 00:08:09,510 --> 00:08:18,010 that needs to be branded into your brain, so that if I come 135 00:08:18,010 --> 00:08:19,800 into your bedroom in the middle of the night and I 136 00:08:19,800 --> 00:08:24,020 whisper to you delta H, you know, you should wake up and 137 00:08:24,020 --> 00:08:27,180 say q p, right? 138 00:08:27,180 --> 00:08:29,270 Heat under constant pressure reversible work. 139 00:08:29,270 --> 00:08:34,440 This should become second nature. 140 00:08:34,440 --> 00:08:35,760 This is where the intuition comes from. 141 00:08:35,760 --> 00:08:38,380 This is why people right tables and 142 00:08:38,380 --> 00:08:39,720 tables of delta H's. 143 00:08:39,720 --> 00:08:44,826 Why you have delta H's from all these reactions, because 144 00:08:44,826 --> 00:08:47,120 this is basically the heat, and the heat is something we 145 00:08:47,120 --> 00:08:50,130 can measure, we can control. 146 00:08:50,130 --> 00:08:51,180 We can figure out how much heat is going 147 00:08:51,180 --> 00:08:52,360 in and out of something. 148 00:08:52,360 --> 00:08:58,880 This is what we're interested in. 149 00:08:58,880 --> 00:09:01,150 OK, so last time you looked at -- any 150 00:09:01,150 --> 00:09:02,070 questions on this first? 151 00:09:02,070 --> 00:09:04,710 Yes. 152 00:09:04,710 --> 00:09:11,640 STUDENT: [INAUDIBLE] from the T delta V to the delta p here? 153 00:09:11,640 --> 00:09:12,710 What was the reasoning behind that? 154 00:09:12,710 --> 00:09:13,405 PROFESSOR BAWENDI: p is constant here. 155 00:09:13,405 --> 00:09:17,710 It's constant pressure. 156 00:09:17,710 --> 00:09:21,750 OK, so now, last time you looked at the Joule expansion 157 00:09:21,750 --> 00:09:28,020 to teach you how to relate derivatives like du/dV. 158 00:09:28,020 --> 00:09:34,610 du/dV under constant temperature. du/dT under 159 00:09:34,610 --> 00:09:35,700 constant volume. 160 00:09:35,700 --> 00:09:42,620 You use the Joule expansion to find these quantities. 161 00:09:42,620 --> 00:09:45,140 Now these quantities were useful because you could 162 00:09:45,140 --> 00:09:45,830 relate them. 163 00:09:45,830 --> 00:09:49,100 The slope of changes, with respect to volume or 164 00:09:49,100 --> 00:09:53,910 temperature of the energy with respect to quantities that you 165 00:09:53,910 --> 00:09:55,710 understood, that you could measure. 166 00:09:55,710 --> 00:09:57,200 We're going to do the same thing here. 167 00:09:57,200 --> 00:10:00,910 So if we take as our natural variables for enthalpy to be 168 00:10:00,910 --> 00:10:04,450 temperature and pressure, and we have some sort of change in 169 00:10:04,450 --> 00:10:09,790 enthalpy, dH, and it's going to be related to changes in 170 00:10:09,790 --> 00:10:13,800 temperature and pressure through the derivatives dH 171 00:10:13,800 --> 00:10:18,270 through the slope of the enthalpy in the T direction, 172 00:10:18,270 --> 00:10:23,340 keeping pressure constant, dT plus the slope of enthalpy in 173 00:10:23,340 --> 00:10:25,390 the pressure direction, keeping the temperature 174 00:10:25,390 --> 00:10:31,890 constant, dp, and these are knobs that we can turn. 175 00:10:31,890 --> 00:10:33,050 We can change of temperature. 176 00:10:33,050 --> 00:10:34,510 We can change the pressure. 177 00:10:34,510 --> 00:10:37,830 These are physical knobs that are available to us as 178 00:10:37,830 --> 00:10:38,650 experimentalists. 179 00:10:38,650 --> 00:10:42,500 And so when we turn these knobs on our system, we want 180 00:10:42,500 --> 00:10:45,760 to know how the enthalpy is changing for that system. 181 00:10:45,760 --> 00:10:50,010 Because eventually they will tell us maybe things about how 182 00:10:50,010 --> 00:10:52,130 heat is changing further on. 183 00:10:52,130 --> 00:10:57,380 OK, but in order to relate turning these physical knob to 184 00:10:57,380 --> 00:11:00,450 this quantity here, which we don't have a very good feel 185 00:11:00,450 --> 00:11:04,080 for, we've got to have a feel for the slopes. 186 00:11:04,080 --> 00:11:05,260 If I keep the pressure constant. 187 00:11:05,260 --> 00:11:07,640 I change the temperature, what does that mean? 188 00:11:07,640 --> 00:11:09,090 What is dh/dT? 189 00:11:09,090 --> 00:11:12,300 If I keep the temperature constant, and just change the 190 00:11:12,300 --> 00:11:14,535 pressure, dH is going to change, but how 191 00:11:14,535 --> 00:11:15,310 is it going to change? 192 00:11:15,310 --> 00:11:18,070 What does this mean in terms of something I can physically 193 00:11:18,070 --> 00:11:20,140 understand? 194 00:11:20,140 --> 00:11:23,740 That's the program now for the next few minutes. 195 00:11:23,740 --> 00:11:25,500 What are these quantities? 196 00:11:25,500 --> 00:11:28,870 What is dH/dT as a function, keeping pressure constant, 197 00:11:28,870 --> 00:11:35,930 what is dH/dp, keeping temperature constant? 198 00:11:35,930 --> 00:11:42,670 All right, let's start with the first one, dH/dT, keeping 199 00:11:42,670 --> 00:11:43,780 the pressure constant. 200 00:11:43,780 --> 00:11:53,170 And we're going to look at a reversible process to help us 201 00:11:53,170 --> 00:11:55,330 out, but the result is going to be independent of whether 202 00:11:55,330 --> 00:11:56,960 or not we have a reversible process or 203 00:11:56,960 --> 00:11:59,570 irreversible process. 204 00:11:59,570 --> 00:12:03,870 Constant pressure, that means dp is equal to zero. 205 00:12:03,870 --> 00:12:06,490 So for reversible process, constant 206 00:12:06,490 --> 00:12:11,560 pressure, what do we know? 207 00:12:11,560 --> 00:12:14,130 This is already branded in your brain, right? 208 00:12:14,130 --> 00:12:16,650 Reversible process, constant pressure dH=dq. 209 00:12:19,700 --> 00:12:27,770 So we can write that down, dH=dq, constant pressure. 210 00:12:27,770 --> 00:12:29,060 That's by definition of enthalpy. 211 00:12:29,060 --> 00:12:30,190 That's why we created enthalpy. 212 00:12:30,190 --> 00:12:30,870 What else do we know? 213 00:12:30,870 --> 00:12:34,880 Well we can go look up here, looking at the differential, 214 00:12:34,880 --> 00:12:37,050 there are no approximations here. 215 00:12:37,050 --> 00:12:39,870 This is just an equality. 216 00:12:39,870 --> 00:12:41,360 I have a constant pressure process. 217 00:12:41,360 --> 00:12:43,250 This term here is equal to zero. 218 00:12:43,250 --> 00:12:51,330 That means that dH is also equal to dH/dT, constant 219 00:12:51,330 --> 00:12:55,100 pressure dT. 220 00:12:55,100 --> 00:12:57,680 All right, so now I've got more dH/dT under constant 221 00:12:57,680 --> 00:13:05,190 pressure. dH is equal to this, and it's also equal to this. 222 00:13:05,190 --> 00:13:08,540 All right, these two are equal to each other. 223 00:13:08,540 --> 00:13:13,530 Now, I know how to relate the heat flow to temperature 224 00:13:13,530 --> 00:13:19,710 change, through the heat capacity. dq constant pressure 225 00:13:19,710 --> 00:13:23,780 is that heat capacity, and I have to tell you the path for 226 00:13:23,780 --> 00:13:24,610 the heat capacity. 227 00:13:24,610 --> 00:13:28,010 So it's C sub p, the heat capacity under a constant 228 00:13:28,010 --> 00:13:32,290 pressure path, dT, all right? 229 00:13:32,290 --> 00:13:33,580 So these two are equal to each other. 230 00:13:33,580 --> 00:13:40,320 So, these two are equal to each other as well, which 231 00:13:40,320 --> 00:13:49,520 tells me that this derivative, dH/dT constant pressure is Cp. 232 00:13:49,520 --> 00:13:53,960 So now I have my first of my two slopes, in terms of 233 00:13:53,960 --> 00:13:58,450 something that's related to my system, the heat capacity of 234 00:13:58,450 --> 00:13:58,880 the system. 235 00:13:58,880 --> 00:14:04,050 Something I can measure and I can tabulate, and when I turn 236 00:14:04,050 --> 00:14:06,920 my dT knob here I know what's going to 237 00:14:06,920 --> 00:14:09,710 happen to the enthalpy. 238 00:14:09,710 --> 00:14:15,600 So this is the first, this is the one. 239 00:14:15,600 --> 00:14:19,660 So it's very similar to what we saw with the volume and the 240 00:14:19,660 --> 00:14:24,640 energy, where du/dV under constant temperature was equal 241 00:14:24,640 --> 00:14:31,280 to Cv in this case here, right. and you're going to 242 00:14:31,280 --> 00:14:35,980 find that there's a lot of these analogies between energy 243 00:14:35,980 --> 00:14:36,790 and enthalpy. 244 00:14:36,790 --> 00:14:39,890 You just change volume to pressure and basically you're 245 00:14:39,890 --> 00:14:42,880 looking at enthalpy under a constant -- anything that's 246 00:14:42,880 --> 00:14:46,010 done at a constant volume path with energy, there's the same 247 00:14:46,010 --> 00:14:49,590 thing happening under constant pressure path for enthalpy. 248 00:14:49,590 --> 00:14:53,110 So you can guess the answer usually that way. 249 00:14:53,110 --> 00:14:56,240 OK, so now we have the other one, dH/dp constant 250 00:14:56,240 --> 00:14:57,850 temperature. 251 00:14:57,850 --> 00:15:00,170 How do we relate this to something physical? 252 00:15:00,170 --> 00:15:02,120 Well, it's going to be an experiment, very much like the 253 00:15:02,120 --> 00:15:03,250 Joule experiment. 254 00:15:03,250 --> 00:15:05,090 The Joule experiment was a constant 255 00:15:05,090 --> 00:15:07,750 energy experiment, right. 256 00:15:07,750 --> 00:15:09,630 Here we're going to have to find a constant enthalpy 257 00:15:09,630 --> 00:15:13,010 experiment, and that is going to be the Joule-Thomson 258 00:15:13,010 --> 00:15:15,650 experiment. 259 00:15:15,650 --> 00:15:20,530 That's going to extract out a physical meaning to this 260 00:15:20,530 --> 00:15:20,850 derivative here. 261 00:15:20,850 --> 00:15:26,810 OK, the Joule-Thomson experiment. 262 00:15:26,810 --> 00:15:40,380 This is going to get us dH/dp constant temperature. 263 00:15:40,380 --> 00:15:41,380 What is this experiment? 264 00:15:41,380 --> 00:15:50,940 You take a throttle valve, which consists of some sort of 265 00:15:50,940 --> 00:15:58,800 porous plug between two cylinders that is insulated. 266 00:15:58,800 --> 00:16:02,600 There is insulation here. 267 00:16:02,600 --> 00:16:04,030 Insulation on the bottom. 268 00:16:04,030 --> 00:16:08,190 It's like a, think about a tube, a large tube, insulated 269 00:16:08,190 --> 00:16:14,250 tube, with a bunch of -- a frit inside here, which 270 00:16:14,250 --> 00:16:18,490 prevents flow of gas, which slows down the flow of gas 271 00:16:18,490 --> 00:16:20,240 from one side to the other. 272 00:16:20,240 --> 00:16:23,190 It's a blockage in this tube here. 273 00:16:23,190 --> 00:16:30,340 Then you put two pistons, one on that side here, and one on 274 00:16:30,340 --> 00:16:35,800 this side here, and the external pressure here, we're 275 00:16:35,800 --> 00:16:37,080 going to call that p1. 276 00:16:37,080 --> 00:16:42,920 The external pressure here, we're going to call that p2, 277 00:16:42,920 --> 00:16:47,540 and we're going to do it slowly enough that the 278 00:16:47,540 --> 00:16:53,830 pressure on this side of the cylinder is in equilibrium 279 00:16:53,830 --> 00:16:57,500 with the external pressure, and the pressure on this side 280 00:16:57,500 --> 00:16:59,630 of the cylinder is in 281 00:16:59,630 --> 00:17:02,370 equilibrium with this pressure. 282 00:17:02,370 --> 00:17:06,690 But not so slowly that these two are in equilibrium with 283 00:17:06,690 --> 00:17:10,100 each other. 284 00:17:10,100 --> 00:17:12,200 So this is restricting the flow, so there's 285 00:17:12,200 --> 00:17:13,910 some sweet spot here. 286 00:17:13,910 --> 00:17:17,460 When I'm pushing slowly enough here that the pressure here is 287 00:17:17,460 --> 00:17:22,030 equal to that one, but not so slowly that the air flow from 288 00:17:22,030 --> 00:17:26,310 here to here is fast, compared to how fast I'm pushing. 289 00:17:26,310 --> 00:17:27,150 You got the picture here? 290 00:17:27,150 --> 00:17:30,820 Any questions on that? 291 00:17:30,820 --> 00:17:35,040 All right, then as I push through, I'm going to start 292 00:17:35,040 --> 00:17:36,790 with all of my gas on this side, and at the end, I'm 293 00:17:36,790 --> 00:17:41,500 going to have all the gas on the other side. 294 00:17:41,500 --> 00:17:44,420 Let me first ask you this, is this a reversible or in 295 00:17:44,420 --> 00:17:49,250 irreversible process? 296 00:17:49,250 --> 00:17:52,670 Right, let me add one more piece of data here which I 297 00:17:52,670 --> 00:17:55,150 said in words, but which is actually important to write 298 00:17:55,150 --> 00:17:57,060 down before doing the problem. 299 00:17:57,060 --> 00:17:58,590 Is this a reverse -- any guesses? 300 00:17:58,590 --> 00:18:03,030 How many people vote for that this is a reversible process? 301 00:18:03,030 --> 00:18:05,710 I've got one vote back there, two votes, 302 00:18:05,710 --> 00:18:06,780 three votes, four votes. 303 00:18:06,780 --> 00:18:08,850 Anybody else? 304 00:18:08,850 --> 00:18:11,990 How many people think this is irreversible? 305 00:18:11,990 --> 00:18:16,670 It's about a tie, and everybody else doesn't now. 306 00:18:16,670 --> 00:18:20,110 All right, I'm going to give you ten seconds, fifteen 307 00:18:20,110 --> 00:18:22,300 seconds to make up your mind. 308 00:18:22,300 --> 00:18:24,070 You're not allowed to be on the fence here. 309 00:18:24,070 --> 00:18:25,740 You've got to decide, all right? 310 00:18:25,740 --> 00:18:28,990 This is, you can talk to your neighbors, you know, do a 311 00:18:28,990 --> 00:18:30,710 little bit of thinking. 312 00:18:30,710 --> 00:18:33,730 And I'm going to give you ten seconds to figure this out, 313 00:18:33,730 --> 00:19:08,800 what your vote is for that. 314 00:19:08,800 --> 00:19:10,230 All right, let's try again. 315 00:19:10,230 --> 00:19:12,520 How many people vote that this is reversible? 316 00:19:12,520 --> 00:19:17,350 That looks like a majority to me. 317 00:19:17,350 --> 00:19:18,090 Irreversible? 318 00:19:18,090 --> 00:19:21,100 Let's look at the show of hands? 319 00:19:21,100 --> 00:19:25,660 All right, so this is the majority here. 320 00:19:25,660 --> 00:19:28,120 Good thing physics doesn't work on the rule of the 321 00:19:28,120 --> 00:19:31,630 majority, otherwise we'd be in big trouble. 322 00:19:31,630 --> 00:19:34,500 Wow, let's walk through that. 323 00:19:34,500 --> 00:19:39,230 I'm sorry to say that this is the wrong answer. 324 00:19:39,230 --> 00:19:40,970 OK, why is that the wrong answer? 325 00:19:40,970 --> 00:19:42,470 Well, just think, you know, think about it. 326 00:19:42,470 --> 00:19:45,120 You're pushing through here. p1 is greater than p2. 327 00:19:45,120 --> 00:19:50,620 What does it mean for a process to be in equilibrium 328 00:19:50,620 --> 00:19:51,620 or reversible? 329 00:19:51,620 --> 00:19:59,780 It means at any point you can reverse the direction of time, 330 00:19:59,780 --> 00:20:02,810 and it will look fine, right. 331 00:20:02,810 --> 00:20:07,310 So now I'm pushing on this plug here, with p1 332 00:20:07,310 --> 00:20:07,960 greater than p2. 333 00:20:07,960 --> 00:20:10,450 I'm pushing, I'm pushing, I'm pushing. p1 is 334 00:20:10,450 --> 00:20:12,160 greater than p2. 335 00:20:12,160 --> 00:20:13,820 Then I want to reverse direction of time. 336 00:20:13,820 --> 00:20:17,210 I want the arrow of time to go so that the gas goes from p2 337 00:20:17,210 --> 00:20:21,210 to p1. p2 is less than p1. 338 00:20:21,210 --> 00:20:25,670 Is that going to work out? 339 00:20:25,670 --> 00:20:28,490 If p2, the pressure in p2, is less than the pressure in p1, 340 00:20:28,490 --> 00:20:31,630 is the gas going to want to go from p2 to p1 and the whole 341 00:20:31,630 --> 00:20:36,130 thing reverse back? 342 00:20:36,130 --> 00:20:38,070 You've got to put more pressure on one side than the 343 00:20:38,070 --> 00:20:40,190 other if you want to push that gas through 344 00:20:40,190 --> 00:20:42,760 the throttle, right? 345 00:20:42,760 --> 00:20:46,270 So this is where the time scale issue comes into play. 346 00:20:46,270 --> 00:20:52,730 I said let's do this slowly enough that this p1 is in 347 00:20:52,730 --> 00:20:58,040 equilibrium with this p1, but not so slowly that this 348 00:20:58,040 --> 00:21:02,200 pressure is equivalent to that pressure. 349 00:21:02,200 --> 00:21:04,790 If I do it really, really slowly, so that everything is 350 00:21:04,790 --> 00:21:09,396 reversible, well I won't be able to do it, because p1 and 351 00:21:09,396 --> 00:21:11,470 p2 are different. 352 00:21:11,470 --> 00:21:17,250 But suppose that I fix my pistons here, with p1 greater 353 00:21:17,250 --> 00:21:20,940 than p2, and I don't touch it, eventually p1 354 00:21:20,940 --> 00:21:21,810 will be equal to p2. 355 00:21:21,810 --> 00:21:23,970 I'll come to some sort of equilibrium. 356 00:21:23,970 --> 00:21:27,110 So this is a system which is out of equilibrium. 357 00:21:27,110 --> 00:21:34,040 If I stop, if I move slowly, if I move more slowly, then 358 00:21:34,040 --> 00:21:36,020 these two will want equilibrium. 359 00:21:36,020 --> 00:21:39,110 So this is an irreversible process. 360 00:21:39,110 --> 00:21:48,520 The Joule-Thomson experiment is irreversible. 361 00:21:48,520 --> 00:21:54,570 OK, important -- if you are part of this group here, think 362 00:21:54,570 --> 00:21:58,670 about it and make sure you understand that. 363 00:21:58,670 --> 00:22:00,250 All right, so this is the experiment. 364 00:22:00,250 --> 00:22:02,820 Now what are we doing with that? 365 00:22:02,820 --> 00:22:11,040 The initial state, let's look at what we are doing. 366 00:22:11,040 --> 00:22:16,270 So initially, we're going to have our piston, there's the 367 00:22:16,270 --> 00:22:18,220 plug sitting here. 368 00:22:18,220 --> 00:22:26,730 Our piston on the right side here fully out, and the piston 369 00:22:26,730 --> 00:22:30,100 on my right side, your left side, fully inside. 370 00:22:30,100 --> 00:22:33,310 There's no gas on that side here. 371 00:22:33,310 --> 00:22:35,190 So there's p2 sitting here. 372 00:22:35,190 --> 00:22:38,450 There is p1 sitting here, and all of the gas is sitting on 373 00:22:38,450 --> 00:22:42,520 that side of the plug. 374 00:22:42,520 --> 00:22:47,390 Then after we're done with the experiment, we'll have 375 00:22:47,390 --> 00:22:55,200 transferred all of the gas from one side to the other. p1 376 00:22:55,200 --> 00:23:01,900 here. p2 sitting here. 377 00:23:01,900 --> 00:23:06,220 And there's going to be some volume V2 and some volume V1, 378 00:23:06,220 --> 00:23:09,080 but are not necessarily the same. 379 00:23:09,080 --> 00:23:12,840 Especially since the pressures are different. we don't know 380 00:23:12,840 --> 00:23:15,680 yet about temperature so I don't know what to say about 381 00:23:15,680 --> 00:23:17,708 these volumes because I don't know what the temperatures' 382 00:23:17,708 --> 00:23:20,380 are going to do. 383 00:23:20,380 --> 00:23:23,860 OK, so let's go through this and see what we would do which 384 00:23:23,860 --> 00:23:27,660 is to calculate the heat and the work. 385 00:23:27,660 --> 00:23:32,950 This is well insulated. 386 00:23:32,950 --> 00:23:40,470 So, what is the -- let's do the first one here. 387 00:23:40,470 --> 00:23:44,710 What's q for this process here? 388 00:23:44,710 --> 00:23:46,820 Anybody? 389 00:23:46,820 --> 00:23:47,260 STUDENT Zero. 390 00:23:47,260 --> 00:23:47,710 PROFESSOR BAWENDI: Zero, right. 391 00:23:47,710 --> 00:23:48,910 This is an adiabatic process. 392 00:23:48,910 --> 00:23:49,760 It's well insulated. 393 00:23:49,760 --> 00:23:56,030 Heat is not going in or out, adiabatic. q is equal to zero. 394 00:23:56,030 --> 00:23:58,150 So all we need to find out is the work now. 395 00:23:58,150 --> 00:24:01,830 Let's divide it up into the two sides, the work going on 396 00:24:01,830 --> 00:24:05,160 on the left hand side, my left hand side or your left hand 397 00:24:05,160 --> 00:24:07,390 side, and the work going on on the right hand side. 398 00:24:07,390 --> 00:24:10,410 So let's first look at the left hand side. 399 00:24:10,410 --> 00:24:15,620 OK, so w, first of all, work is being done to the system on 400 00:24:15,620 --> 00:24:17,135 the left hand side here. 401 00:24:17,135 --> 00:24:20,610 I'm pressing on the gas. 402 00:24:20,610 --> 00:24:24,390 So I expect that to be a positive number. 403 00:24:24,390 --> 00:24:26,320 The pressure is constant, p. 404 00:24:26,320 --> 00:24:29,930 The V goes from V1 to zero. 405 00:24:29,930 --> 00:24:34,100 So we write down p1, V1. 406 00:24:34,100 --> 00:24:41,810 On the right hand side, the work, let's call this left 407 00:24:41,810 --> 00:24:45,720 hand side, let's call this right hand side. 408 00:24:45,720 --> 00:24:49,800 Here there's an expansion going on, so the system is 409 00:24:49,800 --> 00:24:52,320 doing work to the external world. 410 00:24:52,320 --> 00:24:56,130 This piston is being brought out, so we expect the work to 411 00:24:56,130 --> 00:24:59,170 be negative, negative. 412 00:24:59,170 --> 00:25:01,100 And we start out with zero volume. 413 00:25:01,100 --> 00:25:04,010 We end up with a volume of V2, and the external pressure is 414 00:25:04,010 --> 00:25:05,290 constant to p2. 415 00:25:05,290 --> 00:25:08,510 Minus p2, V2. 416 00:25:08,510 --> 00:25:16,660 Minus p delta V. So the total work is the work from the left 417 00:25:16,660 --> 00:25:22,240 hand side plus the work on the right hand side, which is p1 418 00:25:22,240 --> 00:25:28,130 V1 minus p2 V2. 419 00:25:28,130 --> 00:25:35,930 Which I can rewrite as minus delta pV. 420 00:25:35,930 --> 00:25:39,210 Delta pV is p2 V2 minus p1 V1. 421 00:25:39,210 --> 00:25:42,890 It's the pressure volume multiplied together at the 422 00:25:42,890 --> 00:25:48,060 final state, minus pressure volume from the initial state, 423 00:25:48,060 --> 00:25:50,500 with the minus sign here because it's 424 00:25:50,500 --> 00:25:54,030 the negative of that. 425 00:25:54,030 --> 00:26:03,620 All right, what is delta u? delta u is q plus w. q is 426 00:26:03,620 --> 00:26:07,090 zero. delta u is just w. 427 00:26:07,090 --> 00:26:14,780 So this is also just delta u. delta u is minus delta pV, for 428 00:26:14,780 --> 00:26:18,060 the process. 429 00:26:18,060 --> 00:26:27,520 OK, delta H is delta of u plus pV. 430 00:26:27,520 --> 00:26:30,900 By definition, that's how we define enthalpy up here. 431 00:26:30,900 --> 00:26:32,750 H is u plus pV. 432 00:26:32,750 --> 00:26:39,890 Delta H is delta of u plus pV, which is equal to delta u, 433 00:26:39,890 --> 00:26:42,270 plus delta pV. 434 00:26:42,270 --> 00:26:44,810 Now delta u is minus delta pV. 435 00:26:44,810 --> 00:26:47,550 So I have minus delta pV plus delta pV. 436 00:26:47,550 --> 00:26:50,700 This is equal to zero. 437 00:26:50,700 --> 00:26:56,070 So this irreversible process, this Joule-Thomson process, is 438 00:26:56,070 --> 00:26:57,590 a constant enthalpy process. 439 00:26:57,590 --> 00:27:01,220 Delta h for this process is equal to zero. 440 00:27:01,220 --> 00:27:02,680 Adiabatic q equal to zero. 441 00:27:02,680 --> 00:27:06,180 It's also delta H which is zero. 442 00:27:06,180 --> 00:27:10,280 The two didn't necessarily follow, because remember, 443 00:27:10,280 --> 00:27:13,810 delta H is dq so p is only true for a reversible constant 444 00:27:13,810 --> 00:27:15,060 pressure process. 445 00:27:15,060 --> 00:27:16,120 This is an irreversible process. 446 00:27:16,120 --> 00:27:19,270 So a priori, this was not necessarily true. 447 00:27:19,270 --> 00:27:21,670 It turns out after you do all the math, it turns out to be 448 00:27:21,670 --> 00:27:25,250 delta H equals zero. 449 00:27:25,250 --> 00:27:27,060 All right, so this is the experiment. 450 00:27:27,060 --> 00:27:29,260 How do we go from that experiment to the terms that 451 00:27:29,260 --> 00:27:32,220 we're trying to get, these slopes. 452 00:27:32,220 --> 00:27:36,470 Remember, we're trying to get delta H, we're trying to get 453 00:27:36,470 --> 00:27:44,480 dH/dT constant pressure and dH/dp constant temperature. 454 00:27:44,480 --> 00:27:52,170 OK, these are the two things were trying to get here. 455 00:27:52,170 --> 00:28:08,900 OK, so let's write down, what we know here. 456 00:28:08,900 --> 00:28:10,580 I'm missing something. 457 00:28:10,580 --> 00:28:12,410 Oh, we already know one thing. 458 00:28:12,410 --> 00:28:13,510 We already know this guy here. 459 00:28:13,510 --> 00:28:14,640 We already did that. 460 00:28:14,640 --> 00:28:19,280 OK, dH/dT constant pressure is Cp. 461 00:28:19,280 --> 00:28:20,270 That was easy one. 462 00:28:20,270 --> 00:28:21,280 So we already know that. 463 00:28:21,280 --> 00:28:31,190 So now we can write or differential dH as Cp dT plus 464 00:28:31,190 --> 00:28:36,820 dH/dp, constant temperature, dp. 465 00:28:36,820 --> 00:28:41,520 Now we want to find out what this guy is here. 466 00:28:41,520 --> 00:28:45,050 Now for this experiment, this is a constant enthalpy 467 00:28:45,050 --> 00:28:47,755 experiment for the Joule-Thomson experiment, this 468 00:28:47,755 --> 00:28:50,930 is equal to zero. 469 00:28:50,930 --> 00:28:57,770 So I can rearrange this to get this dH/dp in terms of things 470 00:28:57,770 --> 00:29:01,560 that I can either measure, like the heat capacity, or 471 00:29:01,560 --> 00:29:05,320 that I have control of, like dT and dp. 472 00:29:05,320 --> 00:29:11,940 So in this case, dH/dp constant temperature is minus 473 00:29:11,940 --> 00:29:23,265 Cp dT/dp, and this is under constant temperature, no, not 474 00:29:23,265 --> 00:29:25,790 constant temperature. 475 00:29:25,790 --> 00:29:31,780 Whatever the experi -- for that the experiment is. 476 00:29:31,780 --> 00:29:39,810 For that experiment, the constraint, so we need a 477 00:29:39,810 --> 00:29:42,280 constraint here, right, we need a constraint here. 478 00:29:42,280 --> 00:29:43,080 Right? 479 00:29:43,080 --> 00:29:44,620 We need a constraint here. 480 00:29:44,620 --> 00:29:47,910 The constraint isn't constant temperature because the 481 00:29:47,910 --> 00:29:48,980 temperature is going to be changing. 482 00:29:48,980 --> 00:29:51,070 It's not constant pressure, because we have a 483 00:29:51,070 --> 00:29:52,190 delta p going on. 484 00:29:52,190 --> 00:29:55,030 It's not constant volume either. 485 00:29:55,030 --> 00:29:56,620 The constraint is the constraint of the experiment, 486 00:29:56,620 --> 00:29:59,860 and the constraint of the experiment is that the 487 00:29:59,860 --> 00:30:02,420 enthalpy is constant. 488 00:30:02,420 --> 00:30:05,520 So the constraints we have here, is 489 00:30:05,520 --> 00:30:06,890 the constant enthalpy. 490 00:30:06,890 --> 00:30:13,290 It's the constant enthalpy process that we're looking at. 491 00:30:13,290 --> 00:30:16,200 This we can do experiments on. 492 00:30:16,200 --> 00:30:18,090 It's tabulated in books, and this we can measure in the 493 00:30:18,090 --> 00:30:18,620 experiment. 494 00:30:18,620 --> 00:30:23,150 Delta p here is the change in pressure from the left side to 495 00:30:23,150 --> 00:30:26,010 the right side, and we can put a thermometer, measure the 496 00:30:26,010 --> 00:30:28,320 temperature before the experiment, and measure the 497 00:30:28,320 --> 00:30:31,240 temperature after the experiment. 498 00:30:31,240 --> 00:30:38,640 So this is something we can measure. 499 00:30:38,640 --> 00:30:40,830 So now we have this derivative, in terms of 500 00:30:40,830 --> 00:30:42,870 physical quantities, things that we can measure. 501 00:30:42,870 --> 00:30:45,630 Things that we can relate to the properties of the 502 00:30:45,630 --> 00:30:48,730 substance that we're doing the experiment on. 503 00:30:48,730 --> 00:30:54,810 So this is basically delta T and delta p and the 504 00:30:54,810 --> 00:30:59,640 Joule-Thomson experiment. 505 00:30:59,640 --> 00:31:02,650 And so Joule and Thomson did these experiments, and they 506 00:31:02,650 --> 00:31:05,800 measured lots of gases, and they found that, in fact, this 507 00:31:05,800 --> 00:31:08,500 was something that they could measure. 508 00:31:08,500 --> 00:31:11,430 Sometimes it was positive, sometimes it was negative, and 509 00:31:11,430 --> 00:31:13,170 it was an interesting number. 510 00:31:13,170 --> 00:31:17,760 And so they defined them, after many experiments, the 511 00:31:17,760 --> 00:31:23,670 limit of this, delta T delta p and the limit of delta p goes 512 00:31:23,670 --> 00:31:27,000 to zero as the Joule-Thomson coefficient. 513 00:31:27,000 --> 00:31:37,360 So, basically dT/dp, constant enthalpy is equal to mu, by 514 00:31:37,360 --> 00:31:41,330 definition, Joule-Thomson, where mu Joule-Thomson is the 515 00:31:41,330 --> 00:31:44,686 Joule-Thomson coefficient, just like you saw last time 516 00:31:44,686 --> 00:31:49,470 eta sub j was the Joule coefficient for dT/dV under 517 00:31:49,470 --> 00:31:50,910 constant energy. 518 00:31:50,910 --> 00:31:53,620 So there's, again, total analogy here between what 519 00:31:53,620 --> 00:31:56,680 we're doing with enthalpy to what you did last time with 520 00:31:56,680 --> 00:32:03,300 energy, replace p with T and H with u. 521 00:32:03,300 --> 00:32:06,730 Flip those two and you get the same thing. 522 00:32:06,730 --> 00:32:11,020 OK, so now we have dH/dT is equal to Cp, and we can also 523 00:32:11,020 --> 00:32:19,090 write, then, dH/dp under constant temperature is equal 524 00:32:19,090 --> 00:32:24,700 to minus Cp mu Joule-Thomson. 525 00:32:24,700 --> 00:32:30,590 We have our two derivatives in terms of physical quantities, 526 00:32:30,590 --> 00:32:33,010 which is going to allow us, then, whenever we have a 527 00:32:33,010 --> 00:32:39,180 change to go back and, when we have a change where we adjust 528 00:32:39,180 --> 00:32:41,300 the temperature and the pressure, we'll be able to 529 00:32:41,300 --> 00:32:48,670 know what the enthalpy change is. 530 00:32:48,670 --> 00:32:50,510 OK, now let's take two cases. 531 00:32:50,510 --> 00:32:58,170 Let's first start talking about ideal gases. 532 00:32:58,170 --> 00:33:03,170 The last time you saw that for an ideal gas, the energy only 533 00:33:03,170 --> 00:33:04,800 cared about the temperature. 534 00:33:04,800 --> 00:33:11,610 It didn't care what the volume was doing. du/dV under 535 00:33:11,610 --> 00:33:16,240 constant temperature was equal to zero for an ideal gas. 536 00:33:16,240 --> 00:33:20,360 And by analogy, we expect the same thing to be true here, 537 00:33:20,360 --> 00:33:22,210 because enthalpy and energy have all this 538 00:33:22,210 --> 00:33:23,910 analogy going on here. 539 00:33:23,910 --> 00:33:25,900 So let's look at an ideal gas. 540 00:33:25,900 --> 00:33:34,110 So for an ideal gas, we saw that u was only a function of 541 00:33:34,110 --> 00:33:35,420 temperature. 542 00:33:35,420 --> 00:33:36,970 We also have the equation of state for an 543 00:33:36,970 --> 00:33:38,300 ideal gas, pV = nRT. 544 00:33:41,140 --> 00:33:46,720 We can write our definition of enthalpy, h is u plus pV. 545 00:33:46,720 --> 00:33:51,610 This only depends on the temperature. pV= nRT. 546 00:33:51,610 --> 00:33:57,320 So we have u only depends on temperature plus nRT. 547 00:33:57,320 --> 00:33:59,980 The only valuable now on this side is temperature. 548 00:33:59,980 --> 00:34:03,320 Pressure and volume have dropped out. 549 00:34:03,320 --> 00:34:10,220 So enthalpy, for an ideal gas, only cares about temperature. 550 00:34:10,220 --> 00:34:14,680 Pressure has dropped out of the picture completely here. 551 00:34:14,680 --> 00:34:21,760 So there is no p dependence here. 552 00:34:21,760 --> 00:34:26,680 H for an ideal gas is only a function of temperature. 553 00:34:26,680 --> 00:34:30,290 This is not true for a real gas, fortunately, but it's 554 00:34:30,290 --> 00:34:32,510 true for an ideal gas. 555 00:34:32,510 --> 00:34:39,890 So for an ideal gas then, dH/dp under constant 556 00:34:39,890 --> 00:34:42,800 temperature, that has to be equal to zero. 557 00:34:42,800 --> 00:34:45,170 Because temperature is constant H only cares about 558 00:34:45,170 --> 00:34:47,030 temperature. and that's equal to zero. 559 00:34:47,030 --> 00:34:50,980 And if that's equal to zero, that means that the 560 00:34:50,980 --> 00:34:54,410 Joule-Thomson coefficient for an ideal gas is 561 00:34:54,410 --> 00:34:57,440 also equal to zero. 562 00:34:57,440 --> 00:34:59,470 We're going to actually prove this later in the course. 563 00:34:59,470 --> 00:35:01,800 Right now, you're taking it for granted. 564 00:35:01,800 --> 00:35:04,980 Right now we told you Joule did all these experiments and 565 00:35:04,980 --> 00:35:09,020 he found out that for an ideal gas, that the limit in and 566 00:35:09,020 --> 00:35:12,260 ideal gas case was that the eta J was equal to zero. 567 00:35:12,260 --> 00:35:16,590 Therefore, from experiments, u is only a function of 568 00:35:16,590 --> 00:35:20,140 temperature for an ideal gas, and therefore from these 569 00:35:20,140 --> 00:35:23,240 experiments, we come out with delta H 570 00:35:23,240 --> 00:35:25,870 dH/dp is equal to zero. 571 00:35:25,870 --> 00:35:30,140 The Joule-Thomson coefficient is equal to zero. 572 00:35:30,140 --> 00:35:34,050 Later we are going to prove it exactly. but right now you're 573 00:35:34,050 --> 00:35:36,930 going to have to take it for granted. 574 00:35:36,930 --> 00:35:40,470 So, if the Joule-Thomson coefficient is equal to zero, 575 00:35:40,470 --> 00:35:44,815 just like we wrote, du = Cv dT for an ideal gas, we're going 576 00:35:44,815 --> 00:35:50,670 to have dH = Cp dT for an ideal gas as well. 577 00:35:50,670 --> 00:35:51,900 dH is Cp dT. 578 00:35:51,900 --> 00:35:55,240 This term goes away. dH = Cp dT. 579 00:35:55,240 --> 00:35:57,990 That's the only thing that's left behind. 580 00:35:57,990 --> 00:35:59,890 So if you know the heat capacity, you know the change 581 00:35:59,890 --> 00:36:02,300 in temperature, you know what enthalpy is doing 582 00:36:02,300 --> 00:36:04,430 for an ideal gas. 583 00:36:04,430 --> 00:36:11,615 This needs to be stressed that this is the ideal gas case. 584 00:36:11,615 --> 00:36:14,250 Now regular gases, real gases, fortunately as I 585 00:36:14,250 --> 00:36:16,680 said, don't obey this. 586 00:36:16,680 --> 00:36:19,240 This is important because we use this all the time for 587 00:36:19,240 --> 00:36:22,380 when, when we do technology. 588 00:36:22,380 --> 00:36:29,590 One example of Joule-Thomson coefficient being not equal to 589 00:36:29,590 --> 00:36:34,700 zero for a real gas that you've like experienced is if 590 00:36:34,700 --> 00:36:38,630 you take a bicycle pump, take a bicycle pump, and you're 591 00:36:38,630 --> 00:36:41,910 pumping up your tire, you're working pretty hard, so you're 592 00:36:41,910 --> 00:36:42,930 getting hot. 593 00:36:42,930 --> 00:36:48,310 But if you touch the valve going into your tire, which 594 00:36:48,310 --> 00:36:50,270 basically measures the temperature of the air going 595 00:36:50,270 --> 00:36:53,440 into your tire, that is getting hot, right. 596 00:36:53,440 --> 00:36:59,490 So if you've got to pump that tire really a lot, then you're 597 00:36:59,490 --> 00:37:00,460 going to you're going to really feel 598 00:37:00,460 --> 00:37:01,610 a lot of heat there. 599 00:37:01,610 --> 00:37:06,420 The compression of the, basically it's an adiabatic 600 00:37:06,420 --> 00:37:06,940 compression. 601 00:37:06,940 --> 00:37:09,650 You're taking the air inside of the pump and you're 602 00:37:09,650 --> 00:37:11,510 compressing it. 603 00:37:11,510 --> 00:37:14,040 You're doing it so fast that there's not enough time for 604 00:37:14,040 --> 00:37:18,710 heat to come out of the gas that's inside the pump towards 605 00:37:18,710 --> 00:37:21,160 the walls of the pump. 606 00:37:21,160 --> 00:37:23,560 So your time scale it just fast enough that this is 607 00:37:23,560 --> 00:37:25,150 basically an adiabatic compassion. 608 00:37:25,150 --> 00:37:29,290 Your compressing it really fast, all right, so you're 609 00:37:29,290 --> 00:37:32,160 changing the pressure. 610 00:37:32,160 --> 00:37:34,740 You're changing the pressure, and the temperature is going 611 00:37:34,740 --> 00:37:42,960 up. dT/dp is positive. dT/dp is positive. dT/dp is 612 00:37:42,960 --> 00:37:47,330 positive, well that's mu JT. dT/dp is mu JT. 613 00:37:47,330 --> 00:37:53,610 So for a real gas like air, this is a positive number. 614 00:37:53,610 --> 00:37:54,540 It's not zero. 615 00:37:54,540 --> 00:37:55,930 Air is not an ideal gas. 616 00:37:55,930 --> 00:38:02,080 It's one simple example of -- 617 00:38:02,080 --> 00:38:08,340 The fact that mu JT is not zero for real gases is how we 618 00:38:08,340 --> 00:38:11,680 are able to liquify things like hydrogen and helium, by 619 00:38:11,680 --> 00:38:16,660 compressing them and pushing them through a nozzle, and the 620 00:38:16,660 --> 00:38:20,760 expansion through the nozzle cools the gas. 621 00:38:20,760 --> 00:38:22,540 Right, in this case here it wouldn't happen if it 622 00:38:22,540 --> 00:38:27,170 was an ideal gas. 623 00:38:27,170 --> 00:38:32,610 Or in many kinds of gas refrigerators where you push a 624 00:38:32,610 --> 00:38:36,790 gas through a nozzle close to room temperature, what you 625 00:38:36,790 --> 00:38:39,800 find is that the gas coming out on the other side under 626 00:38:39,800 --> 00:38:43,850 lower pressure is cooler than the gas that went through on 627 00:38:43,850 --> 00:38:45,240 the other side. 628 00:38:45,240 --> 00:38:48,170 Real refrigerators actually work with liquids that go into 629 00:38:48,170 --> 00:38:50,970 gases so use the latent heat of the liquid, so it doesn't 630 00:38:50,970 --> 00:38:55,850 really work like the Joule-Thomson expansion. 631 00:38:55,850 --> 00:38:56,600 So this is real. 632 00:38:56,600 --> 00:38:59,130 This is real, unlike the Joule coefficient which is very 633 00:38:59,130 --> 00:39:01,890 small so that most gases have tiny Joule coefficients. 634 00:39:01,890 --> 00:39:03,755 So if you do a Joule experiment, you hardly measure 635 00:39:03,755 --> 00:39:04,620 a temperature change. 636 00:39:04,620 --> 00:39:07,850 With real gases, here you do actually measure it. 637 00:39:07,850 --> 00:39:14,670 You can feel it with your finger on your bicycle tire. 638 00:39:14,670 --> 00:39:19,190 OK, so we're going to see this using a Van der Waal's gas. 639 00:39:19,190 --> 00:39:25,660 Let's look at a Van der Waal's gas and see what happens in 640 00:39:25,660 --> 00:39:35,340 the Van der Waal's gas. 641 00:39:35,340 --> 00:39:36,360 Any questions, first? 642 00:39:36,360 --> 00:39:39,520 What we've been talking about, the Joule-Thomson experiment, 643 00:39:39,520 --> 00:39:47,530 constant enthalpy process? 644 00:39:47,530 --> 00:39:50,410 OK, so let's take our Van der Waal's gas. 645 00:39:50,410 --> 00:39:53,320 Remember the equation of state for Van der Waal's gas is not 646 00:39:53,320 --> 00:39:59,250 pV is equal to nRT, but p plus the attraction term. 647 00:39:59,250 --> 00:40:05,840 And then V minus the excluded volume term is equal to RT. 648 00:40:05,840 --> 00:40:08,390 Two parameters, this is the attraction between two atoms 649 00:40:08,390 --> 00:40:10,060 or molecules in the gas phase. 650 00:40:10,060 --> 00:40:11,970 This is the repulsion, not the repulsion. 651 00:40:11,970 --> 00:40:14,950 This is the fact that we occupy a finite volume in 652 00:40:14,950 --> 00:40:19,290 space, because they're little hard spheres in this molecule. 653 00:40:19,290 --> 00:40:23,310 OK, in a few weeks, you're going to find out that we can 654 00:40:23,310 --> 00:40:28,550 calculate dH/dp from this equation of state, and you're 655 00:40:28,550 --> 00:40:32,290 going to find out that dH/dp from that equation of state is 656 00:40:32,290 --> 00:40:37,800 proportional to b minus a over RT. 657 00:40:37,800 --> 00:40:39,560 This is going to be probably a homework at 658 00:40:39,560 --> 00:40:41,080 some point to do this. 659 00:40:41,080 --> 00:40:42,420 For now, let's take it for granted. 660 00:40:42,420 --> 00:40:46,850 Let's take it for granted that we know how to calculate this 661 00:40:46,850 --> 00:40:50,060 derivative from an equation of state like this. 662 00:40:50,060 --> 00:40:56,010 But now we're going to use that. 663 00:40:56,010 --> 00:40:56,330 OK. 664 00:40:56,330 --> 00:41:02,980 So since dH/dp under constant temperature is proportional to 665 00:41:02,980 --> 00:41:12,120 minus mu JT, then we have that dT/dp under constant enthalpy 666 00:41:12,120 --> 00:41:18,490 than it is related to the negative of this, a over RT 667 00:41:18,490 --> 00:41:19,330 minus b 668 00:41:19,330 --> 00:41:26,230 All right, this is how the temperature changes when you 669 00:41:26,230 --> 00:41:28,180 change, when you have pressure changing. 670 00:41:28,180 --> 00:41:31,750 So when you do an expansion or compression, my adiabatic 671 00:41:31,750 --> 00:41:38,880 compression of my bicycle pump is dT/dp. 672 00:41:38,880 --> 00:41:43,150 All right. 673 00:41:43,150 --> 00:41:46,900 Or when I do an expansion of hydrogen or helium at low 674 00:41:46,900 --> 00:41:52,630 temperature, through a Joule-Thomson experiment, when 675 00:41:52,630 --> 00:41:53,720 I want to liquify hydrogen or helium. 676 00:41:53,720 --> 00:41:58,360 I want to cool a gas with a Joule-Thomson experiment, what 677 00:41:58,360 --> 00:42:00,770 temperature do I have to be at? 678 00:42:00,770 --> 00:42:02,910 So this tells you that you have to be careful what 679 00:42:02,910 --> 00:42:06,240 temperature you're at, because depending on how high, how big 680 00:42:06,240 --> 00:42:09,780 this temperature here is, you could either be, have a 681 00:42:09,780 --> 00:42:14,770 negative dT/dp, if this first term is small enough, meaning 682 00:42:14,770 --> 00:42:18,890 if temperature is very high, then you end up with a 683 00:42:18,890 --> 00:42:19,940 negative term. 684 00:42:19,940 --> 00:42:23,260 If the temperature is very small, then one over the 685 00:42:23,260 --> 00:42:26,450 temperature is large, and the first term wins and you have a 686 00:42:26,450 --> 00:42:27,920 positive number. 687 00:42:27,920 --> 00:42:30,300 So there's some special temperature which is going to 688 00:42:30,300 --> 00:42:34,440 depend on the gas where the first term is going to be 689 00:42:34,440 --> 00:42:38,670 equal to the second term, where Joule-Thomson is zero, 690 00:42:38,670 --> 00:42:42,730 or it's going to behave like an ideal gas. 691 00:42:42,730 --> 00:42:48,470 So when that is the case, we're going to call that 692 00:42:48,470 --> 00:42:51,590 temperature the inversion temperature or T inv. 693 00:42:51,590 --> 00:42:55,080 We call that inversion because on one side you end up cooling 694 00:42:55,080 --> 00:42:55,450 if you compress. 695 00:42:55,450 --> 00:42:58,450 And on the other side of that temperature you end up heating 696 00:42:58,450 --> 00:43:03,740 if you compress. 697 00:43:03,740 --> 00:43:08,480 OK, so there's some temperature, t inversion minus 698 00:43:08,480 --> 00:43:12,650 b where the gas behaves like an ideal gas. 699 00:43:12,650 --> 00:43:15,880 The Joule-Thomson coefficient is zero and that inversion 700 00:43:15,880 --> 00:43:19,470 temperature, you can solve for it. 701 00:43:19,470 --> 00:43:29,480 Conversion temperature is equal to a over R times b. 702 00:43:29,480 --> 00:43:32,790 OK, so when you are at that temperature, everything looks 703 00:43:32,790 --> 00:43:34,420 like an ideal gas, as far as the 704 00:43:34,420 --> 00:43:37,480 enthalpy changes are concerned. 705 00:43:37,480 --> 00:43:41,470 Now if you're at the temperature which is higher 706 00:43:41,470 --> 00:43:44,580 than the inversion temperature, in that case 707 00:43:44,580 --> 00:43:50,900 here, a over RT is small compared to b, and this is 708 00:43:50,900 --> 00:43:52,860 going to turn out to be negative. 709 00:43:52,860 --> 00:43:56,260 So if you had a high temperature, this a small 710 00:43:56,260 --> 00:43:57,280 compared to b. 711 00:43:57,280 --> 00:44:06,960 If you're negative, which means that dT/dp at constant H 712 00:44:06,960 --> 00:44:14,770 is less than zero. 713 00:44:14,770 --> 00:44:20,930 So that means that if you compress something 714 00:44:20,930 --> 00:44:21,820 it's going to cool. 715 00:44:21,820 --> 00:44:27,860 The temperature rises when the pressure drops, right? 716 00:44:27,860 --> 00:44:32,760 Or in this case here, if I do my Joule experiment delta p is 717 00:44:32,760 --> 00:44:36,540 negative, p2 is less than p1, that means that delta T is 718 00:44:36,540 --> 00:44:37,650 positive, right? 719 00:44:37,650 --> 00:44:40,650 So in this experiment here, this side is going to heat up. 720 00:44:40,650 --> 00:44:49,870 So for materials where T inversion is low, lower than 721 00:44:49,870 --> 00:44:53,870 room temperature, then you would end up heating up in 722 00:44:53,870 --> 00:44:54,660 this expansion. 723 00:44:54,660 --> 00:44:57,120 You're basically expanding the gas from one side to the other 724 00:44:57,120 --> 00:45:01,550 and the expansion causes the temperature to rise. 725 00:45:01,550 --> 00:45:08,310 If T is less than T inversion, you have the opposite case, 726 00:45:08,310 --> 00:45:13,450 and dT/dp is greater than zero. 727 00:45:13,450 --> 00:45:17,850 So in this experiment here, delta p is less than zero. 728 00:45:17,850 --> 00:45:19,530 You need to have this whole thing greater than zero. 729 00:45:19,530 --> 00:45:21,990 So delta T is less than zero as well. 730 00:45:21,990 --> 00:45:23,970 So if you're below the inversion temperature and you 731 00:45:23,970 --> 00:45:26,320 do the Joule-Thomson experiment, you're going to 732 00:45:26,320 --> 00:45:28,040 end up with something that's colder on this 733 00:45:28,040 --> 00:45:29,920 side than that side. 734 00:45:29,920 --> 00:45:32,240 Ideal gas would be the same temperature. 735 00:45:32,240 --> 00:45:37,480 But now, so this is where the refrigeration comes in. 736 00:45:37,480 --> 00:45:40,700 So if you take a gas, and you're below the inversion 737 00:45:40,700 --> 00:45:43,040 temperature and you make it go through this irreversible 738 00:45:43,040 --> 00:45:46,260 process, the gas comes out colder from that 739 00:45:46,260 --> 00:45:47,930 side than that side. 740 00:45:47,930 --> 00:45:52,890 So the work that you're doing to expand, to go through this 741 00:45:52,890 --> 00:45:59,970 experiment, ends up cooling the gas. 742 00:45:59,970 --> 00:46:08,000 OK, for most gases, T inversion is much greater than 743 00:46:08,000 --> 00:46:10,280 300 degrees Kelvin. 744 00:46:10,280 --> 00:46:11,450 Much greater than room temperature. 745 00:46:11,450 --> 00:46:13,600 For more most gases, if you do this experiment at room 746 00:46:13,600 --> 00:46:16,640 temperature, you end up cooling the gas, and you can 747 00:46:16,640 --> 00:46:21,060 cool it measurably, which is why also, your bicycle pump, 748 00:46:21,060 --> 00:46:23,680 you know, you push down, you compress, this is 749 00:46:23,680 --> 00:46:24,900 an expansion here. 750 00:46:24,900 --> 00:46:27,060 You're expanding from this side to that side. 751 00:46:27,060 --> 00:46:29,530 Bicycle pump you are compressing the gas, which is 752 00:46:29,530 --> 00:46:32,840 the opposite, you end up heating up the air in the 753 00:46:32,840 --> 00:46:38,070 bicycle pump in your compression. 754 00:46:38,070 --> 00:46:41,733 There are two exceptions to this rule that most gases have 755 00:46:41,733 --> 00:46:44,830 a T inversion which is greater than 300 degrees Kelvin, and 756 00:46:44,830 --> 00:46:45,720 that's hydrogen. 757 00:46:45,720 --> 00:46:52,010 The T inversion for hydrogen turns out to be 193 degrees 758 00:46:52,010 --> 00:46:58,810 Kelvin, and the T inversion for helium turns out to be 53 759 00:46:58,810 --> 00:47:04,960 degrees Kelvin. 760 00:47:04,960 --> 00:47:08,030 And so when, there's a lot of liquid helium that's being 761 00:47:08,030 --> 00:47:09,060 used on campus. 762 00:47:09,060 --> 00:47:10,810 We use a liquid helium. 763 00:47:10,810 --> 00:47:14,170 And so in order to make a liquid helium, you can't take 764 00:47:14,170 --> 00:47:17,100 helium at room temperature and do this, because if you did, 765 00:47:17,100 --> 00:47:19,770 you would just heat it up, because the room temperature 766 00:47:19,770 --> 00:47:22,780 is above the inversion temperature, so Joule-Thomson 767 00:47:22,780 --> 00:47:23,920 would heat up the helium. 768 00:47:23,920 --> 00:47:26,510 So you need first to take the liquid helium and cool it 769 00:47:26,510 --> 00:47:31,880 below 53 degrees Kelvin before you can do the Joule-Thomson 770 00:47:31,880 --> 00:47:34,180 to cool it even further to make liquid helium. 771 00:47:34,180 --> 00:47:35,860 So you have to do it in stages. 772 00:47:35,860 --> 00:47:39,200 You take your room temperature liquid helium, and you cool it 773 00:47:39,200 --> 00:47:43,550 with liquid nitrogen to 77 degrees Kelvin, the new, 774 00:47:43,550 --> 00:47:46,380 you're not quite there yet, unfortunately right? 775 00:47:46,380 --> 00:47:50,130 Then you take hydrogen you cool it would liquid nitrogen 776 00:47:50,130 --> 00:47:53,720 to 77, then you can use your hydrogen gas. 777 00:47:53,720 --> 00:47:58,260 Do a Joule-Thomson hydrogen gas which you first cool with 778 00:47:58,260 --> 00:47:58,690 liquid nitrogen. 779 00:47:58,690 --> 00:48:01,160 Liquid nitrogen, 77, that's below 193, so you can do 780 00:48:01,160 --> 00:48:05,740 Joule-Thomson on hydrogen, cool the hydrogen to below 53, 781 00:48:05,740 --> 00:48:09,120 then use your cold hydrogen to cool the helium, and then you 782 00:48:09,120 --> 00:48:12,340 can do the Joule-Thomson on the helium to cool it further 783 00:48:12,340 --> 00:48:14,910 until it liquifies. 784 00:48:14,910 --> 00:48:19,810 So that's what you do when you make liquid helium. 785 00:48:19,810 --> 00:48:24,640 All right, any questions on this lecture or any of the 786 00:48:24,640 --> 00:48:34,210 concepts that we talked about? 787 00:48:34,210 --> 00:48:39,610 The last thing that we're going to do today then is to 788 00:48:39,610 --> 00:48:43,120 look at a relationship which is going to turn out to be 789 00:48:43,120 --> 00:48:46,160 very useful. 790 00:48:46,160 --> 00:48:48,630 It's a relationship for ideal gases which relates the heat 791 00:48:48,630 --> 00:48:59,840 capacities at constant pressure and constant volume. 792 00:48:59,840 --> 00:49:08,050 Cp = Cv + R. This is very useful because often you just 793 00:49:08,050 --> 00:49:11,670 have lots of tables of Cp's but sometimes you want to know 794 00:49:11,670 --> 00:49:15,350 what the energy change is going to be for the ideal gas, 795 00:49:15,350 --> 00:49:19,780 and you know that du is Cv dT not Cp dT. 796 00:49:19,780 --> 00:49:22,350 So, you need to know what Cv is, and if this is true 797 00:49:22,350 --> 00:49:24,110 always, then there's a very easy way to go 798 00:49:24,110 --> 00:49:25,200 from one to the other. 799 00:49:25,200 --> 00:49:28,190 We're going to do it two ways. 800 00:49:28,190 --> 00:49:30,820 Today we'll do the first way, and then next time we'll do 801 00:49:30,820 --> 00:49:31,770 the second way. 802 00:49:31,770 --> 00:49:35,350 The first way is just to turn the crank on the math, and the 803 00:49:35,350 --> 00:49:37,930 second way is to do a little bit more 804 00:49:37,930 --> 00:49:41,720 imaginative about the process. 805 00:49:41,720 --> 00:49:43,360 OK, let's just turn the crank on the maths. 806 00:49:43,360 --> 00:49:45,070 What do we know about the Cp and Cv? 807 00:49:45,070 --> 00:49:49,800 Well Cp, we already know how to relate it to dH/dT through 808 00:49:49,800 --> 00:49:54,480 the slope of the enthalpy, and we also related Cv to the 809 00:49:54,480 --> 00:49:57,480 slope of the energy with respect to temperature under 810 00:49:57,480 --> 00:49:59,840 constant volume. 811 00:49:59,840 --> 00:50:04,050 And we also know that H is u plus pV. 812 00:50:04,050 --> 00:50:07,520 Right, H is u plus pV. 813 00:50:07,520 --> 00:50:09,630 So we're going to take the derivatives of both sides of 814 00:50:09,630 --> 00:50:13,680 this equation here, by, with respect to temperature, 815 00:50:13,680 --> 00:50:16,070 keeping pressure constant. 816 00:50:16,070 --> 00:50:22,540 So we have dH/dT keeping pressure constant, is du/dT 817 00:50:22,540 --> 00:50:23,710 keeping pressure constant. 818 00:50:23,710 --> 00:50:26,810 Got to keep track of what's being constant, kept constant, 819 00:50:26,810 --> 00:50:35,860 plus dPV/dT, keeping pressure constant. dH/dT keeping 820 00:50:35,860 --> 00:50:42,740 pressure constant, that's just Cp, and then we have du/dT 821 00:50:42,740 --> 00:50:43,950 keeping pressure constant. 822 00:50:43,950 --> 00:50:47,760 The p is constant here, comes out of the equation, so we 823 00:50:47,760 --> 00:50:51,460 have p and then we have dV/dT well it's an ideal gas, an 824 00:50:51,460 --> 00:51:00,420 ideal gas for dV/dT for an ideal gas is equal to R over p 825 00:51:00,420 --> 00:51:06,730 because pV is equal to RT, right. 826 00:51:06,730 --> 00:51:11,210 So dV/dT is Rp. 827 00:51:11,210 --> 00:51:16,672 So dV/dT is times R over p, the p's are going 828 00:51:16,672 --> 00:51:19,380 to cancel out here. 829 00:51:19,380 --> 00:51:23,000 OK, so now it's very tempting at this state to say, oh 830 00:51:23,000 --> 00:51:26,550 there's the answer right here. du/dT is Cv. 831 00:51:26,550 --> 00:51:33,320 There I have it Cp is equal to Cv plus R, right? 832 00:51:33,320 --> 00:51:35,750 But even though it looks like that's the right way to do it, 833 00:51:35,750 --> 00:51:38,850 it's actually not right because it turns out to be 834 00:51:38,850 --> 00:51:42,230 right by sort of by accident. 835 00:51:42,230 --> 00:51:50,750 But here you've got pressure constant. du, this is du, not 836 00:51:50,750 --> 00:51:58,160 H here. du/dT is only equal to Cv when the volume is 837 00:51:58,160 --> 00:52:00,010 constant, not when the pressure is constant. 838 00:52:00,010 --> 00:52:03,130 So if you're going to turn the crank on the math correctly, 839 00:52:03,130 --> 00:52:08,660 you're going to have to change this p into a V somehow. 840 00:52:08,660 --> 00:52:10,895 Because this isn't Cv mathematically speaking. 841 00:52:10,895 --> 00:52:13,610 We don't know what it is yet. 842 00:52:13,610 --> 00:52:16,060 In order to change this from a p to a V, you have to use the 843 00:52:16,060 --> 00:52:19,040 chain rule. 844 00:52:19,040 --> 00:52:25,070 So let's use the chain rule. 845 00:52:25,070 --> 00:52:33,670 And then we'll be done. 846 00:52:33,670 --> 00:52:37,040 OK, so u is actually a function of temperature and 847 00:52:37,040 --> 00:52:40,560 volume, which in this case here could be a function of 848 00:52:40,560 --> 00:52:41,900 pressure and temperature. 849 00:52:41,900 --> 00:52:49,480 So if we want du/dT under constant pressure, you have to 850 00:52:49,480 --> 00:52:50,230 use the chain rule. 851 00:52:50,230 --> 00:52:52,510 There's the pressure sitting right here. 852 00:52:52,510 --> 00:53:03,650 It's going to be du/dT under constant volume, plus du/dV 853 00:53:03,650 --> 00:53:08,860 dV/dT under constant pressure. 854 00:53:08,860 --> 00:53:22,140 All right, chain rule. du/dT constant pressure is the 855 00:53:22,140 --> 00:53:24,710 direct derivative with respect to temperature here, which is 856 00:53:24,710 --> 00:53:26,860 sitting by itself under constant volume, keeping this 857 00:53:26,860 --> 00:53:28,350 constant but there is temperature 858 00:53:28,350 --> 00:53:30,950 sitting right here too. 859 00:53:30,950 --> 00:53:33,260 That's where that term comes from, du/dV dV/dT. 860 00:53:36,560 --> 00:53:40,160 Now, for an ideal gas, du/dV under constant temperature is 861 00:53:40,160 --> 00:53:40,800 equal to zero. 862 00:53:40,800 --> 00:53:45,360 It doesn't care what the volume is doing. 863 00:53:45,360 --> 00:53:46,960 It only cares what temperature is. 864 00:53:46,960 --> 00:53:49,880 If temperature is constant, there's no change in energy. 865 00:53:49,880 --> 00:53:51,920 For an ideal gas, this is zero. 866 00:53:51,920 --> 00:53:55,160 It's not zero for a real gas. 867 00:53:55,160 --> 00:53:58,080 Right, so this whole term disappears and for an ideal 868 00:53:58,080 --> 00:54:02,130 gas, it turns out that du/dT constant pressure is equal to 869 00:54:02,130 --> 00:54:06,550 du/dT constant volume, but this is equal to Cv 870 00:54:06,550 --> 00:54:07,490 for an ideal gas. 871 00:54:07,490 --> 00:54:10,210 It wouldn't be true for a real gas, and this is a common 872 00:54:10,210 --> 00:54:13,320 mistake that people make for real gases to equate this. 873 00:54:13,320 --> 00:54:14,990 This is only true for an ideal gas. 874 00:54:14,990 --> 00:54:19,835 Since it's true for an ideal gas, then we can go ahead and 875 00:54:19,835 --> 00:54:27,790 replace this with Cv, and then we have Cp with Cv plus R, 876 00:54:27,790 --> 00:54:30,780 which is what we were after. 877 00:54:30,780 --> 00:54:34,890 OK, next time we'll do the other way of getting to the 878 00:54:34,890 --> 00:54:36,440 same answer.