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 --> 00:00:17 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:17 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:25 PROFESSOR: So last time we talked about the zeroth law, 10 00:00:25 --> 00:00:29 which is the common-sense law, which says that if you take a 11 00:00:29 --> 00:00:32 hot object next to a cold object, heat will flow from the 12 00:00:32 --> 00:00:36 hot to the cold in a way that is well defined, and it allows 13 00:00:36 --> 00:00:40 you to define temperature. 14 00:00:40 --> 00:00:43 It allows you to define the concept of a thermometer. 15 00:00:43 --> 00:00:47 You have three objects, one of them could be a thermometer. 16 00:00:47 --> 00:00:49 You have two of them separated at a distance. 17 00:00:49 --> 00:00:52 You take the third one, and you go from one to the other, and 18 00:00:52 --> 00:00:56 you see whether heat flows, when you touch one object, the 19 00:00:56 --> 00:00:59 middle object, between those two objects. 20 00:00:59 --> 00:01:01 Let me talk to you about temperature scales. 21 00:01:01 --> 00:01:04 We talked about the Celsius scale then 22 00:01:04 --> 00:01:07 the Fahrenheit scale. 23 00:01:07 --> 00:01:10 The late 1800's were a booming time for temperature scales. 24 00:01:10 --> 00:01:13 People didn't really realize how important it was to 25 00:01:13 --> 00:01:16 properly define the reference points: Fahrenheit's 26 00:01:16 --> 00:01:24 warm-blooded or 96 degrees, and Romer's 7.5 degrees. 27 00:01:24 --> 00:01:26 Romer because he didn't want to go below zero degrees measuring 28 00:01:26 --> 00:01:30 temperature outside in Denmark Those are kind of silly. 29 00:01:30 --> 00:01:34 But they're the legacy that we have today, and 30 00:01:34 --> 00:01:38 that's what we use. 31 00:01:38 --> 00:01:42 In science, we use somewhat better temperature scales. 32 00:01:42 --> 00:01:46 And the temperature scale that turns out to be well-defined 33 00:01:46 --> 00:01:52 and ends up giving us the concept of an absolute zero is 34 00:01:52 --> 00:01:55 the ideal gas thermometer. 35 00:01:55 --> 00:02:03 So, let's talk about that briefly today first. 36 00:02:03 --> 00:02:10 The ideal gas thermometer. 37 00:02:10 --> 00:02:12 It's based on Boyle's law. 38 00:02:12 --> 00:02:15 Boyle's law was an empirical law that Mr. Boyle discovered 39 00:02:15 --> 00:02:18 by doing lots of experiments, and Boyle's law says that the 40 00:02:18 --> 00:02:26 limit of the quantity pressure times the molar volume, so this 41 00:02:26 --> 00:02:30 quantity here, pressure times the molar volume, as you 42 00:02:30 --> 00:02:34 let pressure go to zero. 43 00:02:34 --> 00:02:37 So, you do this measurement, you measure with the gas, 44 00:02:37 --> 00:02:39 you measure the pressure and the molar volume. 45 00:02:39 --> 00:02:42 Then you change the pressure again, and you measure the 46 00:02:42 --> 00:02:44 pressure in the volume, and you multiply these two together, 47 00:02:44 --> 00:02:46 and you keep doing this experiment, getting the 48 00:02:46 --> 00:02:50 pressure smaller and smaller, you find that this limit turns 49 00:02:50 --> 00:02:53 out to be a constant, independent of the gas. 50 00:02:53 --> 00:02:55 It doesn't care where the gas is. 51 00:02:55 --> 00:02:58 You always get to the same constant. 52 00:02:58 --> 00:03:01 And that constant turns out to be a function 53 00:03:01 --> 00:03:04 of the temperature. 54 00:03:04 --> 00:03:06 The only function it is -- it doesn't care where the gas is. 55 00:03:06 --> 00:03:10 It only cares where the temperature is. 56 00:03:10 --> 00:03:13 All right, so now we have the makings of a good thermometer 57 00:03:13 --> 00:03:15 and a good temperature scale. 58 00:03:15 --> 00:03:19 We have a substance. 59 00:03:19 --> 00:03:22 The substance could be any gas. 60 00:03:22 --> 00:03:24 That's pretty straightforward. 61 00:03:24 --> 00:03:33 So now we have a substance, which is a gas, 62 00:03:33 --> 00:03:36 with a property. 63 00:03:36 --> 00:03:41 So now the volume of mercury, or the color of something which 64 00:03:41 --> 00:03:44 changes with temperature, or the resistivity. 65 00:03:44 --> 00:03:51 In this case here, our property is the value of the pressure 66 00:03:51 --> 00:03:52 times the volume, times the molar volume. 67 00:03:52 --> 00:03:56 That's the property. 68 00:03:56 --> 00:04:01 The property is the limit as p goes to zero of pressure 69 00:04:01 --> 00:04:03 times molar volume. 70 00:04:03 --> 00:04:04 It's a number. 71 00:04:04 --> 00:04:05 Measure it. 72 00:04:05 --> 00:04:05 It's a number. 73 00:04:05 --> 00:04:06 It's going to come out. 74 00:04:06 --> 00:04:09 That's the property that's going to give us the 75 00:04:09 --> 00:04:11 change in temperature. 76 00:04:11 --> 00:04:12 Then we need some reference points. 77 00:04:12 --> 00:04:20 And Celsius first used the boiling point of water, and 78 00:04:20 --> 00:04:25 called that 100 degrees Celsius, and the freezing point 79 00:04:25 --> 00:04:32 of water and called that zero degrees Celsius. 80 00:04:32 --> 00:04:34 And then we need an interpolation scale. 81 00:04:34 --> 00:04:38 How to go from one reference point to the other 82 00:04:38 --> 00:04:40 with this property. 83 00:04:40 --> 00:04:43 This property, which we're going to call f(t). 84 00:04:43 --> 00:04:50 There are many ways you can connect those two dots. 85 00:04:50 --> 00:04:55 If I draw a graph, and on one axis I have this temperature. 86 00:04:55 --> 00:04:57 The idea of temperature with two reference points, zero 87 00:04:57 --> 00:05:01 for the freezing point of water, 100 degrees for the 88 00:05:01 --> 00:05:05 boiling point of water. 89 00:05:05 --> 00:05:10 And on the y-axis I've got the property f(t). 90 00:05:10 --> 00:05:18 It has some value corresponding to t equals zero. 91 00:05:18 --> 00:05:20 So let's get some value right here. 92 00:05:20 --> 00:05:22 There's another value connected to this property here, 93 00:05:22 --> 00:05:26 when t is equal to 100, a reference point here. 94 00:05:26 --> 00:05:27 Now there many ways I can connect these 95 00:05:27 --> 00:05:28 two points together. 96 00:05:28 --> 00:05:31 The simplest way is to draw a straight line. 97 00:05:31 --> 00:05:34 It's called the linear interpolation. 98 00:05:34 --> 00:05:39 My line is not so straight, right here. 99 00:05:39 --> 00:05:40 You could do a different kind of line. 100 00:05:40 --> 00:05:42 You could do a quadratic, let's say. 101 00:05:42 --> 00:05:44 Something like this. 102 00:05:44 --> 00:05:45 That would be perfectly fine interpolation. 103 00:05:45 --> 00:05:58 All right, we choose to have a linear interpolation. 104 00:05:58 --> 00:06:03 That's a choice, and that choice turns out to be very 105 00:06:03 --> 00:06:06 interesting and really important, because if you 106 00:06:06 --> 00:06:09 connect these two points together, you get a straight 107 00:06:09 --> 00:06:19 line that has to intercept the x-axis at some point. 108 00:06:19 --> 00:06:21 Now what does it mean to intercept the x-axis here? 109 00:06:21 --> 00:06:29 It means that the value of f(t) for this temperature is zero. 110 00:06:29 --> 00:06:32 That means that at this point right here, f(t)=0. 111 00:06:32 --> 00:06:35 112 00:06:35 --> 00:06:38 That means the pressure times the volume equals 113 00:06:38 --> 00:06:43 zero, for that gas. 114 00:06:43 --> 00:06:47 And if you're below this temperature here, this 115 00:06:47 --> 00:06:51 quantity, p times v it would be negative. 116 00:06:51 --> 00:06:54 Is that possible? 117 00:06:54 --> 00:06:55 Can we have p v negative? 118 00:06:55 --> 00:06:58 Yes? 119 00:06:58 --> 00:07:01 No, it can't be. 120 00:07:01 --> 00:07:02 Negative pressure doesn't make any sense, right? 121 00:07:02 --> 00:07:04 Negative volume doesn't make any sense. 122 00:07:04 --> 00:07:09 That means that this part here, can't happen. 123 00:07:09 --> 00:07:14 That means that this temperature right here is the 124 00:07:14 --> 00:07:18 absolute lowest temperature you can go to that 125 00:07:18 --> 00:07:19 physically makes any sense. 126 00:07:19 --> 00:07:22 That's the absolute zero. 127 00:07:22 --> 00:07:28 So the concept of an absolute zero, a temperature below which 128 00:07:28 --> 00:07:34 you just can't go, that's directly out of the scheme 129 00:07:34 --> 00:07:37 here, this linear interpolation scheme with these two 130 00:07:37 --> 00:07:39 reference points. 131 00:07:39 --> 00:07:42 If I had taken as my interpolation scheme, my white 132 00:07:42 --> 00:07:47 curve here, I could go to infinity and have the 133 00:07:47 --> 00:07:50 equivalent of absolute zero being at infinity, 134 00:07:50 --> 00:07:53 minus infinity. 135 00:07:53 --> 00:07:58 So, this temperature, this absolute zero here, which 136 00:07:58 --> 00:08:04 is absolute zero on the Kelvin scale. 137 00:08:04 --> 00:08:06 The lowest possible temperature in the Celsius scale is minus 138 00:08:06 --> 00:08:13 273.15 degrees Celsius. 139 00:08:13 --> 00:08:19 So that begs the notion of re-referencing our reference 140 00:08:19 --> 00:08:23 point, of changing our reference points. 141 00:08:23 --> 00:08:26 To change a reference point from this point here being 142 00:08:26 --> 00:08:29 zero, instead of this point here being zero. 143 00:08:29 --> 00:08:32 And so redefining then the temperature scale to the Kelvin 144 00:08:32 --> 00:08:42 scale, where t in degrees Kelvin is equal to t in 145 00:08:42 --> 00:08:50 degree Celsius, plus 273.15. 146 00:08:50 --> 00:08:55 And then you would get the Kelvin scale. 147 00:08:55 --> 00:08:59 All right, it turned out that this thermometer here wasn't 148 00:08:59 --> 00:09:02 quite perfect either. 149 00:09:02 --> 00:09:07 Just like Fahrenheit measuring 96 degrees being a 150 00:09:07 --> 00:09:12 warm-blooded, healthy man, right, that's not 151 00:09:12 --> 00:09:15 very accurate. 152 00:09:15 --> 00:09:17 Our temperature probably fluctuates during the day 153 00:09:17 --> 00:09:20 a little bit anyways, it's not very accurate. 154 00:09:20 --> 00:09:23 And similarly, the boiling point, defining that at a 100 155 00:09:23 --> 00:09:25 degrees Celsius, well that depends on the pressure. 156 00:09:25 --> 00:09:29 It depends whether you're in Denver or you're in Boston. 157 00:09:29 --> 00:09:32 Water boils at different temperatures, depending on what 158 00:09:32 --> 00:09:34 the atmospheric pressure is; same thing for the 159 00:09:34 --> 00:09:37 freezing point. 160 00:09:37 --> 00:09:38 So that means, then, you've got to define the 161 00:09:38 --> 00:09:39 pressure pretty well. 162 00:09:39 --> 00:09:41 You've got to know where the pressure is. 163 00:09:41 --> 00:09:44 It would be much better if you had a reference point that 164 00:09:44 --> 00:09:45 didn't care where the pressure was. 165 00:09:45 --> 00:09:49 Just like our substance doesn't care where the gas is. 166 00:09:49 --> 00:09:52 It's kind of universal. 167 00:09:52 --> 00:09:55 And so now, instead of using these reference points for the 168 00:09:55 --> 00:10:01 Kelvin scale, we use the absolute zero, which isn't 169 00:10:01 --> 00:10:03 going to care what the pressure is. 170 00:10:03 --> 00:10:05 It's the lowest number you can go to. 171 00:10:05 --> 00:10:09 And our other reference point is the triple point of water -- 172 00:10:09 --> 00:10:18 reference points become zero Kelvin, absolute zero, 173 00:10:18 --> 00:10:22 and the triple point. 174 00:10:22 --> 00:10:26 The triple point of water is going to be defined as 175 00:10:26 --> 00:10:30 273.16 degrees Kelvin. 176 00:10:30 --> 00:10:35 And the triple point of water is that temperature and 177 00:10:35 --> 00:10:38 pressure -- there's a unique temperature and pressure where 178 00:10:38 --> 00:10:44 water exists in equilibrium between the liquid phase, the 179 00:10:44 --> 00:10:48 vapor phase, and the solid phase. 180 00:10:48 --> 00:10:58 So the triple point is liquid, solid, gas, all in equilibrium. 181 00:10:58 --> 00:11:01 Now you may think, well I've seen that before. 182 00:11:01 --> 00:11:06 You take a glass of ice water and set it down. 183 00:11:06 --> 00:11:09 There's the water phase, there's the ice cube is the 184 00:11:09 --> 00:11:15 solid phase, and there's some water, gas, vapor, 185 00:11:15 --> 00:11:16 and that's one bar. 186 00:11:16 --> 00:11:17 Where am I going wrong here? 187 00:11:17 --> 00:11:25 The partial pressure of the water, of gaseous water, above 188 00:11:25 --> 00:11:30 that equilibrium of ice and water is not one bar, 189 00:11:30 --> 00:11:33 it's much less. 190 00:11:33 --> 00:11:40 So the partial pressure or the pressure by which you have this 191 00:11:40 --> 00:11:48 triple point, happens to be 6.1 times 10 to the minus 3 bar. 192 00:11:48 --> 00:11:52 There's hardly any vapor pressure above your 193 00:11:52 --> 00:11:55 ice water glass. 194 00:11:55 --> 00:12:00 So this unique temperature and unique pressure defines a 195 00:12:00 --> 00:12:07 triple point everywhere, and that's a great reference point. 196 00:12:07 --> 00:12:10 Any questions? 197 00:12:10 --> 00:12:11 Great. 198 00:12:11 --> 00:12:13 So now we have this ideal gas thermometer, and out of this 199 00:12:13 --> 00:12:18 ideal gas thermometer, also comes out the ideal gas law. 200 00:12:18 --> 00:12:22 Because we can take our interpolation here, our 201 00:12:22 --> 00:12:29 linear interpolation, the slope of this line. 202 00:12:29 --> 00:12:31 Let's draw it in degrees Kelvin, instead of 203 00:12:31 --> 00:12:35 in degrees Celsius. 204 00:12:35 --> 00:12:38 So we have now temperature in degrees Kelvin. 205 00:12:38 --> 00:12:43 We have the quantity f(t) here. 206 00:12:43 --> 00:12:50 We have an interpolation scheme between zero and 273.16 with 207 00:12:50 --> 00:12:53 two values for this quantity, and we have a linear 208 00:12:53 --> 00:12:57 interpolation that defines our temperature scale, our 209 00:12:57 --> 00:12:59 Kelvin temperature scale. 210 00:12:59 --> 00:13:07 And so the slope of this thing is f(t) at the triple point, 211 00:13:07 --> 00:13:10 which is this point here, this is the temperature of the 212 00:13:10 --> 00:13:16 triple point of water, divided by 273.16. 213 00:13:16 --> 00:13:18 That's the slope of that line. 214 00:13:18 --> 00:13:25 The quantity here, which is f (t of the triple point), 215 00:13:25 --> 00:13:33 divided by the value of the x-axis here. 216 00:13:33 --> 00:13:37 So that's the slope, and the intercept is zero, so the 217 00:13:37 --> 00:13:45 function f(t), you just multiply by t here. 218 00:13:45 --> 00:13:50 This is the slope. f(t) is just the limit. 219 00:13:50 --> 00:13:56 As p goes to zero of p times v bar. 220 00:13:56 --> 00:14:00 And so now we have this quantity, p times v bar, and 221 00:14:00 --> 00:14:03 the limit of p goes to zero is equal to a constant 222 00:14:03 --> 00:14:07 times the temperature. 223 00:14:07 --> 00:14:09 That's a universal statement. 224 00:14:09 --> 00:14:11 It's true of every gas. 225 00:14:11 --> 00:14:14 I didn't say this is only true of hydrogen or nitrogen, This 226 00:14:14 --> 00:14:20 is any gas because I'm taking this limit p equals to zero. 227 00:14:20 --> 00:14:22 Now this constant is just a constant. 228 00:14:22 --> 00:14:23 I'm going to call it r. 229 00:14:23 --> 00:14:25 I'm going to call it r. 230 00:14:25 --> 00:14:31 It's going to be the gas constant, and now I have r 231 00:14:31 --> 00:14:37 times t is equal to the limit, p goes to zero of p r. 232 00:14:37 --> 00:14:44 It's true for any gas, and if I remove this limit here, r t is 233 00:14:44 --> 00:14:52 equal to p v bar, I'm going to call that an ideal gas. 234 00:14:52 --> 00:14:55 See, this is the property of an ideal gas. 235 00:14:55 --> 00:14:55 What does it mean, ideal gas? 236 00:14:55 --> 00:14:58 It means that the molecules or the atoms and the gas don't 237 00:14:58 --> 00:15:01 know about each other. 238 00:15:01 --> 00:15:02 They effectively have no volume. 239 00:15:02 --> 00:15:04 They have no interactions with each other. 240 00:15:04 --> 00:15:06 They occupy the same volume in space. 241 00:15:06 --> 00:15:08 They don't care that there are other atoms and 242 00:15:08 --> 00:15:09 molecules around. 243 00:15:09 --> 00:15:12 So that's basically what you do when you take p goes to zero. 244 00:15:12 --> 00:15:15 You make the volume infinitely large, the density of the 245 00:15:15 --> 00:15:17 gas infinitely small. 246 00:15:17 --> 00:15:19 The atoms or molecules in the gas don't know that there are 247 00:15:19 --> 00:15:22 other atoms and molecules in the gas, and then you end up 248 00:15:22 --> 00:15:26 with this universal property. 249 00:15:26 --> 00:15:29 All right, so gases that have this universal property, even 250 00:15:29 --> 00:15:32 when the pressure is not zero, those are the ideal gases. 251 00:15:32 --> 00:15:35 And for the sake of this class, we're going to consider most 252 00:15:35 --> 00:15:40 gases to be ideal gases. 253 00:15:40 --> 00:15:44 Questions? 254 00:15:44 --> 00:15:50 So now, this equation here relates three state functions 255 00:15:50 --> 00:15:53 together: the pressure, the volume, and the temperature. 256 00:15:53 --> 00:15:56 Now, if you remember, we said that if you had a substance, if 257 00:15:56 --> 00:16:00 you knew the number of moles and two properties, you knew 258 00:16:00 --> 00:16:02 everything about the gas. 259 00:16:02 --> 00:16:11 Which means that you can re-write this in the form, 260 00:16:11 --> 00:16:16 volume, for instance, is equal to the function of n, p, t.. 261 00:16:16 --> 00:16:21 In this case, V = (nRT)/P. 262 00:16:21 --> 00:16:27 Have two quantities and the number of moles gives 263 00:16:27 --> 00:16:29 you another property. 264 00:16:29 --> 00:16:31 You don't need to know the volume. 265 00:16:31 --> 00:16:32 All you need to know is the pressure and temperature and 266 00:16:32 --> 00:16:34 the number of moles to get the volume. 267 00:16:34 --> 00:16:36 This is called an equation of state. 268 00:16:36 --> 00:16:43 It relate state properties to each other. 269 00:16:43 --> 00:16:46 In this case it relates the volume to the pressure 270 00:16:46 --> 00:16:49 and the temperature. 271 00:16:49 --> 00:16:55 Now, if you're an engineer, and you use the ideal gas law to 272 00:16:55 --> 00:17:01 design a chemical plant or a boiler or an electrical plant, 273 00:17:01 --> 00:17:08 you know, a steam plant, you're going to be in big trouble. 274 00:17:08 --> 00:17:14 Your plant is going to blow up, because the ideal gas law works 275 00:17:14 --> 00:17:16 only in very small range of pressures and temperatures 276 00:17:16 --> 00:17:18 for most gases. 277 00:17:18 --> 00:17:25 So, we have other equations of states for real gases. 278 00:17:25 --> 00:17:27 This is an equation of state for an ideal gases. 279 00:17:27 --> 00:17:30 For real gases, there's a whole bunch of equation the states 280 00:17:30 --> 00:17:34 that you can find in textbooks, and I'm just going to go 281 00:17:34 --> 00:17:37 through a few of them. 282 00:17:37 --> 00:17:40 The first one uses something called a compressibility 283 00:17:40 --> 00:17:45 factor, z. 284 00:17:45 --> 00:17:46 Compressibility factor, z. 285 00:17:46 --> 00:17:52 And instead of writing PV = RT, which would be the ideal gas 286 00:17:52 --> 00:17:55 law, we put a fudge factor in there. 287 00:17:55 --> 00:17:59 And the fudge factor is called z. 288 00:17:59 --> 00:18:04 Now we can put real instead of ideal for our volume. z is the 289 00:18:04 --> 00:18:11 compressibility factor, and z is the ratio of the volume of 290 00:18:11 --> 00:18:21 the real gas divided by what it would be were it an ideal gas. 291 00:18:21 --> 00:18:26 So, if z is less than 1, then the real gas is more 292 00:18:26 --> 00:18:28 compact then the ideal gas. 293 00:18:28 --> 00:18:30 It's a smaller volume. 294 00:18:30 --> 00:18:33 If z is greater than 1, then the real gas means that the 295 00:18:33 --> 00:18:36 atoms and molecules in the real gas are repelling each other 296 00:18:36 --> 00:18:40 and wants to have a bigger volume. 297 00:18:40 --> 00:18:42 And you can find these compressibility 298 00:18:42 --> 00:18:43 factors in tables. 299 00:18:43 --> 00:18:46 If you want to know the compressibility factors for 300 00:18:46 --> 00:18:49 water, for steam, at a certain pressure and temperature, you 301 00:18:49 --> 00:18:52 go to a table and you find it. 302 00:18:52 --> 00:18:58 So that's one example of a real equation of state. 303 00:18:58 --> 00:19:02 Not a very useful one for our purposes in this class here. 304 00:19:02 --> 00:19:06 Another one is the virial expansion. 305 00:19:06 --> 00:19:08 It's a little bit more useful. 306 00:19:08 --> 00:19:13 What you do is you take that fudge factor, and you expand 307 00:19:13 --> 00:19:15 it out into a Taylor series. 308 00:19:15 --> 00:19:23 So, we have the p v real over r t is equal to z. 309 00:19:23 --> 00:19:28 Now, we're going to take z and say all right, under most 310 00:19:28 --> 00:19:32 conditions, it's pretty close to 1, when it's an ideal gas. 311 00:19:32 --> 00:19:37 And then we have to add corrections to that, and the 312 00:19:37 --> 00:19:41 corrections are going to be more important, the 313 00:19:41 --> 00:19:44 larger the volume is. 314 00:19:44 --> 00:19:47 Remember, it's the limit of p times v goes to zero, so if you 315 00:19:47 --> 00:19:50 have a large volume with a large pressure, then you're 316 00:19:50 --> 00:19:52 out of the ideal gas regime. 317 00:19:52 --> 00:20:00 So let's take Taylor series in one over the volume, it's going 318 00:20:00 --> 00:20:04 to be one over the volume squared, etcetera. 319 00:20:04 --> 00:20:07 And these factors on top, which are going to depend on the 320 00:20:07 --> 00:20:11 temperature, are the virial coefficients, and those 321 00:20:11 --> 00:20:14 depend on the substance. 322 00:20:14 --> 00:20:17 So you have this p B(t) here. 323 00:20:17 --> 00:20:25 This is called a second virial coefficient. 324 00:20:25 --> 00:20:29 And then, so you can get, you can actually find 325 00:20:29 --> 00:20:31 a graph of this B(t). 326 00:20:31 --> 00:20:32 It's going to look something like this. 327 00:20:32 --> 00:20:39 It's the function of temperature, as B(t). 328 00:20:39 --> 00:20:41 There's going to be some temperature where B(t) 329 00:20:41 --> 00:20:42 is equal to zero. 330 00:20:42 --> 00:20:45 In that case, your gas is going to look awfully 331 00:20:45 --> 00:20:46 like an ideal gas. 332 00:20:46 --> 00:20:51 Above some temperature is going to be positive, below some 333 00:20:51 --> 00:20:53 temperature is going to be negative. 334 00:20:53 --> 00:20:57 Generally, we ignore the high order terms here. 335 00:20:57 --> 00:21:00 So again, if you do a calculation where you're close 336 00:21:00 --> 00:21:04 enough to the ideal gas, and you need to design your, if you 337 00:21:04 --> 00:21:07 have an engineer designing something that's got a bunch of 338 00:21:07 --> 00:21:13 gases around, this is a useful thing to use. 339 00:21:13 --> 00:21:18 Now, the most interesting one for our class, the equation of 340 00:21:18 --> 00:21:20 state that's the most interesting, is the Van der 341 00:21:20 --> 00:21:24 Waals equation of state, developed by Mr. Van 342 00:21:24 --> 00:21:29 der Waals in 1873. 343 00:21:29 --> 00:21:31 And the beauty of that equation of state is that it only 344 00:21:31 --> 00:21:37 relies on two parameters. 345 00:21:37 --> 00:21:38 So let's build it up. 346 00:21:38 --> 00:21:43 Let's see where it comes from. 347 00:21:43 --> 00:21:44 Let me just first write it down, the Van der Waals 348 00:21:44 --> 00:21:53 equation of state. p plus a over v bar squared times v 349 00:21:53 --> 00:21:57 bar minus b equals r t. 350 00:21:57 --> 00:22:01 All right, if you take a equal to zero, these are the 351 00:22:01 --> 00:22:03 two parameters, a and b. 352 00:22:03 --> 00:22:04 If you take those two equal to zero, you have 353 00:22:04 --> 00:22:05 p v is equal to r t. 354 00:22:05 --> 00:22:08 That's the ideal gas law. 355 00:22:08 --> 00:22:09 Let's build this up. 356 00:22:09 --> 00:22:11 Let's see where this comes from, where these parameters 357 00:22:11 --> 00:22:14 a and b comes from. 358 00:22:14 --> 00:22:15 So, the first thing we're going to do is we're going to take 359 00:22:15 --> 00:22:21 our gas in our box, let's build a box full of gases here. 360 00:22:21 --> 00:22:24 We've got a bunch of gas molecules or atoms. 361 00:22:24 --> 00:22:30 OK, there's the volume of a box here. 362 00:22:30 --> 00:22:34 While these gas molecules or atoms through first 363 00:22:34 --> 00:22:37 approximation, are like hard spheres. 364 00:22:37 --> 00:22:39 They occupy a certain volume. 365 00:22:39 --> 00:22:45 Each atom or molecule occupies a particular volume. 366 00:22:45 --> 00:22:58 And so, we can call b is the volume per mole of the hard 367 00:22:58 --> 00:23:05 spheres, volume per mole that is the little sphere 368 00:23:05 --> 00:23:07 that the molecules are. 369 00:23:07 --> 00:23:11 So that the volume that is available to any one 370 00:23:11 --> 00:23:16 of those spheres is actually smaller than v. 371 00:23:16 --> 00:23:18 Because you've got all these other little spheres around, so 372 00:23:18 --> 00:23:21 the actual volume seen by any one of those spheres 373 00:23:21 --> 00:23:23 is smaller than v. 374 00:23:23 --> 00:23:29 So when we take our ideal gas law, p v bar is equal to r t we 375 00:23:29 --> 00:23:32 have to replace v bar by the actual volume available 376 00:23:32 --> 00:23:34 to this hard sphere. 377 00:23:34 --> 00:23:42 So instead of v bar, we write p v bar minus b, equal r t. 378 00:23:42 --> 00:23:47 OK, that's the hard sphere volume of the spheres. 379 00:23:47 --> 00:23:51 Now, those molecules or atoms that are in here, 380 00:23:51 --> 00:23:53 also feel each other. 381 00:23:53 --> 00:23:55 There are a whole bunch of forces that you learn in 382 00:23:55 --> 00:23:57 5.112, 5.111 like with Van der Waals' attractions 383 00:23:57 --> 00:23:58 and things like this. 384 00:23:58 --> 00:24:07 So there are attractive forces, or repulsive forces that these 385 00:24:07 --> 00:24:14 molecules feel, and that's going to change the pressure 386 00:24:14 --> 00:24:16 that the molecules feel. 387 00:24:16 --> 00:24:19 For instance, if I have, what is pressure? 388 00:24:19 --> 00:24:21 Pressure is when you have one of these hard spheres 389 00:24:21 --> 00:24:23 colliding against the wall. 390 00:24:23 --> 00:24:24 There's the hard sphere. 391 00:24:24 --> 00:24:27 It wants to collide against the wall to create a force on the 392 00:24:27 --> 00:24:30 wall, and I have a couple of the hard spheres that are 393 00:24:30 --> 00:24:34 nearby, right, and in the absence of any interactions, 394 00:24:34 --> 00:24:34 I get a certain pressure. 395 00:24:34 --> 00:24:37 This thing would but careen into the wall, kaboom! 396 00:24:37 --> 00:24:41 You'd have this little force, but in the presence of these 397 00:24:41 --> 00:24:45 interactions, you've got these other molecules here that are 398 00:24:45 --> 00:24:51 watching this, you know, their partner sort of wants to do 399 00:24:51 --> 00:24:54 damage to themselves, like hitting that wall, 400 00:24:54 --> 00:24:54 and they say, no! 401 00:24:54 --> 00:24:56 Come back, come back, right? 402 00:24:56 --> 00:24:59 There is an attractive force. 403 00:24:59 --> 00:25:01 There are no other molecules on that side of the wall. 404 00:25:01 --> 00:25:04 So there's an attractive force that makes the velocity 405 00:25:04 --> 00:25:05 within not quite as fast. 406 00:25:05 --> 00:25:08 The force is not quite as strong as it was without 407 00:25:08 --> 00:25:09 this attractive force. 408 00:25:09 --> 00:25:13 So the real pressure is not quite the same because of this 409 00:25:13 --> 00:25:16 attractive force as it was, as it would be without the 410 00:25:16 --> 00:25:17 attractive forces. 411 00:25:17 --> 00:25:20 The pressure is a little bit less in this case here. 412 00:25:20 --> 00:25:25 So instead of this p here. 413 00:25:25 --> 00:25:31 Now if I re-write this equation here as p is equal to r t 414 00:25:31 --> 00:25:35 divided by v bar minus b, just re-writing this 415 00:25:35 --> 00:25:39 equation as it is. 416 00:25:39 --> 00:25:41 So the pressure is going to depend on how strong this 417 00:25:41 --> 00:25:46 attractive force is. 418 00:25:46 --> 00:25:49 So the pressure is going to be less if there's a 419 00:25:49 --> 00:25:51 strong attractive force. 420 00:25:51 --> 00:25:55 And the 1 over v squared is a statistical, is basically a 421 00:25:55 --> 00:26:00 probability of having another molecule, a second molecule 422 00:26:00 --> 00:26:02 in the volume of space. 423 00:26:02 --> 00:26:06 So, if the molar volume is small, then one over v bar 424 00:26:06 --> 00:26:10 is large, there's a large probability of having two 425 00:26:10 --> 00:26:13 spheres together in the same volume. 426 00:26:13 --> 00:26:16 If the molar volume is large, that means that there's a lot 427 00:26:16 --> 00:26:19 of room for the molecules, and they're now going to be close 428 00:26:19 --> 00:26:22 to each other, and so this isn't going to be as important. 429 00:26:22 --> 00:26:26 So, a is the strength of the interaction, v bar is how 430 00:26:26 --> 00:26:29 likely they are to be close to each other. 431 00:26:29 --> 00:26:31 And that's going to affect the actual pressure 432 00:26:31 --> 00:26:35 seen by the gas. 433 00:26:35 --> 00:26:43 And a is greater than zero when you have the attraction. 434 00:26:43 --> 00:26:47 And that gives use the Van der Waals' equation of state, with 435 00:26:47 --> 00:26:52 two parameters, the hard sphere volume and the attraction. 436 00:26:52 --> 00:26:54 You don't have to go look up in tables or books. 437 00:26:54 --> 00:26:57 You don't have to have all the values of the second virial 438 00:26:57 --> 00:27:01 coefficient, or the fudge factor, just two variables that 439 00:27:01 --> 00:27:04 make physical sense, and you get an equation of state which 440 00:27:04 --> 00:27:07 is a reasonable equation of state, and that's the power of 441 00:27:07 --> 00:27:09 the Van der Waals' equation of state, and that's the one we're 442 00:27:09 --> 00:27:14 going to be using later on this class to describe real gases. 443 00:27:14 --> 00:27:19 Question? 444 00:27:19 --> 00:27:22 OK, so we've done the zeroth law. 445 00:27:22 --> 00:27:25 We've done temperature, equations of state. 446 00:27:25 --> 00:27:27 We're ready for the first law. 447 00:27:27 --> 00:27:28 We're just going to go to through these laws 448 00:27:28 --> 00:27:31 pretty quickly here. 449 00:27:31 --> 00:27:34 Remember, the first law is the upbeat law. 450 00:27:34 --> 00:27:36 It's the one that says, hey, you know, life 451 00:27:36 --> 00:27:37 is all rosy here. 452 00:27:37 --> 00:27:42 We can take energy from fossil fuels and burn it up and 453 00:27:42 --> 00:27:46 make it heat, and change that energy into work. 454 00:27:46 --> 00:27:49 And it's the same energy, and we probably can do 455 00:27:49 --> 00:27:51 that with 100% efficiency. 456 00:27:51 --> 00:27:54 We can take heat from the air surrounding us and run our car 457 00:27:54 --> 00:27:56 on it with 100% efficiency. 458 00:27:56 --> 00:27:59 Is this possible? 459 00:27:59 --> 00:28:01 That's what the first law says, it's possible; work is 460 00:28:01 --> 00:28:05 heat, and heat is work, and they're the same thing. 461 00:28:05 --> 00:28:08 You can break even, maybe. 462 00:28:08 --> 00:28:10 So let's go back and see what work is. 463 00:28:10 --> 00:28:17 Let's go back to our freshman physics. 464 00:28:17 --> 00:28:24 Work, work is if you take a force, and you push something 465 00:28:24 --> 00:28:27 a certain distance, you do work on it. 466 00:28:27 --> 00:28:30 So if I take my chalk here and I push on it, I'm doing 467 00:28:30 --> 00:28:33 work to push that chalk. 468 00:28:33 --> 00:28:37 Force times distance is work. 469 00:28:37 --> 00:28:40 The applied force times the distance. 470 00:28:40 --> 00:28:42 There are many kinds of work. 471 00:28:42 --> 00:28:46 There's electrical work, take the motor, you plug it into the 472 00:28:46 --> 00:28:48 wall, electricity makes the fan go around, that's 473 00:28:48 --> 00:28:49 electrical work. 474 00:28:49 --> 00:28:50 There's magnetic work. 475 00:28:50 --> 00:28:55 There is work due to gravity. 476 00:28:55 --> 00:28:58 In this class here, we're going to stick to one kind of work 477 00:28:58 --> 00:29:03 which is expansion work. 478 00:29:03 --> 00:29:06 So expansion work, for instance, or compression work, 479 00:29:06 --> 00:29:11 is if you have a piston with a gas in it. 480 00:29:11 --> 00:29:14 All right, you put a pressure on this piston here, and 481 00:29:14 --> 00:29:18 you compress the gas down. 482 00:29:18 --> 00:29:20 This is compression work. 483 00:29:20 --> 00:29:26 Now the volume gets smaller. p external here. 484 00:29:26 --> 00:29:30 Pressure, the piston goes down by some volume l. 485 00:29:30 --> 00:29:35 The piston has a cross-sectional area, a, and 486 00:29:35 --> 00:29:41 the force -- pressure is force per volume area. 487 00:29:41 --> 00:29:48 So the force that you're pushing down on here is 488 00:29:48 --> 00:29:51 the external pressure times the area. 489 00:29:51 --> 00:29:55 Pressure is force per volume area. 490 00:29:55 --> 00:29:57 That's the force you're using to push down. 491 00:29:57 --> 00:30:03 Now the work that's it is calculated when you push down 492 00:30:03 --> 00:30:09 with the pressure on this piston here, that work is force 493 00:30:09 --> 00:30:19 times distance, f times I. f is p external times a, 494 00:30:19 --> 00:30:20 times the distance l. 495 00:30:20 --> 00:30:27 So that's p external times the change in the volume. 496 00:30:27 --> 00:30:32 The area times this distance is a volume, and that is the 497 00:30:32 --> 00:30:34 change in volume from going to the initial state to 498 00:30:34 --> 00:30:37 the final state. 499 00:30:37 --> 00:30:39 Now we need to have a convention. 500 00:30:39 --> 00:30:40 We've got force. 501 00:30:40 --> 00:30:44 Work is force times distance, it's p external times delta v, 502 00:30:44 --> 00:30:47 and I'm going to be stressing a lot that this is the 503 00:30:47 --> 00:30:47 external pressure. 504 00:30:47 --> 00:30:50 This is the pressure that you're applying against 505 00:30:50 --> 00:30:54 the piston, not the pressure of the gas. 506 00:30:54 --> 00:30:57 It's the pressure the external world is applying on 507 00:30:57 --> 00:31:01 this poor system here. 508 00:31:01 --> 00:31:02 OK, but we need a convention here. 509 00:31:02 --> 00:31:06 The convention, and then we need to stick to it. 510 00:31:06 --> 00:31:07 And this convention, unfortunately, has 511 00:31:07 --> 00:31:08 changed over the ages. 512 00:31:08 --> 00:31:12 But we're going to pick one, and we're going to stick to 513 00:31:12 --> 00:31:17 it, which is that if the environment does work on the 514 00:31:17 --> 00:31:23 system, if we push down on this thing and do work on it, to 515 00:31:23 --> 00:31:32 compress it, then we call that work negative work. 516 00:31:32 --> 00:31:38 No, we call that work positive work. 517 00:31:38 --> 00:31:40 All right, so that means we need to put a negative sign 518 00:31:40 --> 00:31:48 right here, by convention. 519 00:31:48 --> 00:31:56 So if delta v is negative, in this case delta v is negative, 520 00:31:56 --> 00:31:58 OK, delta v is negative, pressure is a positive number, 521 00:31:58 --> 00:32:02 negative times negative is positive, work is 522 00:32:02 --> 00:32:04 greater than zero. 523 00:32:04 --> 00:32:09 We're doing work on the system, to the system. 524 00:32:09 --> 00:32:13 In this case here, work is positive. 525 00:32:13 --> 00:32:16 If you have expansion on the other side, if the system is 526 00:32:16 --> 00:32:19 expanding in the other direction, if you're going this 527 00:32:19 --> 00:32:27 way, right, you're going to do work to the environment. 528 00:32:27 --> 00:32:29 There might be a mass here. 529 00:32:29 --> 00:32:30 This could be a car. 530 00:32:30 --> 00:32:33 Pistons in the car, right, so the piston goes up. 531 00:32:33 --> 00:32:34 That's going to drive the wheels. 532 00:32:34 --> 00:32:36 The car is going to go forward. 533 00:32:36 --> 00:32:38 You're doing work on the environment. 534 00:32:38 --> 00:32:41 Delta v is going to be negative. w is going 535 00:32:41 --> 00:32:43 to be negative. 536 00:32:43 --> 00:32:45 Sorry, I got it backwards again. 537 00:32:45 --> 00:32:47 Delta v is positive in this direction here, 538 00:32:47 --> 00:32:50 the work is negative. 539 00:32:50 --> 00:32:55 So work on the system is positive. 540 00:32:55 --> 00:32:57 Work done by the system is negative. 541 00:32:57 --> 00:33:01 Convention, OK, this negative sign is just a pure convention. 542 00:33:01 --> 00:33:02 You just got to use it all the time. 543 00:33:02 --> 00:33:07 If you use an old textbook, written when I was taking 544 00:33:07 --> 00:33:10 thermodynamics, they have the opposite convention, and 545 00:33:10 --> 00:33:11 it's very confusing. 546 00:33:11 --> 00:33:15 But now we've all agreed on this convention, and work is 547 00:33:15 --> 00:33:20 going to be with the negative sign here. 548 00:33:20 --> 00:33:25 OK, any questions? 549 00:33:25 --> 00:33:28 This is an example where the external pressure here is kept 550 00:33:28 --> 00:33:31 fixed as the volume changes, but it doesn't have 551 00:33:31 --> 00:33:33 to be kept fixed. 552 00:33:33 --> 00:33:36 I could change my external pressure through the whole 553 00:33:36 --> 00:33:37 process, and that's the path. 554 00:33:37 --> 00:33:40 We talked about the path last time being very important. 555 00:33:40 --> 00:33:41 Defining the path. 556 00:33:41 --> 00:33:46 So if I have a path where my pressure is changing, then I 557 00:33:46 --> 00:33:50 can't go directly from this large volume to 558 00:33:50 --> 00:33:50 this small volume. 559 00:33:50 --> 00:33:55 I have to go in little steps, infinitely small steps. 560 00:33:55 --> 00:34:00 So, instead of writing work is the negative of p external 561 00:34:00 --> 00:34:07 times delta v, I'm going to write a differential. dw is 562 00:34:07 --> 00:34:14 minus p external dv, where this depends on the path, it depends 563 00:34:14 --> 00:34:18 on path and is changing as v and p change. 564 00:34:18 --> 00:34:22 Now I'm going to add a little thing here. 565 00:34:22 --> 00:34:26 I'm going to put a little bar right here. 566 00:34:26 --> 00:34:31 And the little bar here means that this dw that I'm putting 567 00:34:31 --> 00:34:37 here is not an exact differential. 568 00:34:37 --> 00:34:42 What do I mean by that? 569 00:34:42 --> 00:34:45 I mean that if I take the integral of this to find out 570 00:34:45 --> 00:34:50 how much work I've done on the system, I need 571 00:34:50 --> 00:34:52 to know the path. 572 00:34:52 --> 00:34:53 That's what this means here. 573 00:34:53 --> 00:34:58 It's not enough to know the initial state and the final 574 00:34:58 --> 00:35:00 state to find what w is. 575 00:35:00 --> 00:35:04 You also need to know how you got there. 576 00:35:04 --> 00:35:07 This is very different from the functions of state, like 577 00:35:07 --> 00:35:10 pressure and temperature. 578 00:35:10 --> 00:35:11 There's a volume, there's a temperature, than 579 00:35:11 --> 00:35:13 the pressure here. 580 00:35:13 --> 00:35:14 There's other volume, temperature and pressure 581 00:35:14 --> 00:35:17 here, corresponding to this system here. 582 00:35:17 --> 00:35:19 And this volume, temperature and pressure doesn't 583 00:35:19 --> 00:35:21 care how you got there. 584 00:35:21 --> 00:35:23 It is what it is. 585 00:35:23 --> 00:35:26 It defines the state of the system. 586 00:35:26 --> 00:35:29 The amount of work you've put in to get here 587 00:35:29 --> 00:35:30 depends on the path. 588 00:35:30 --> 00:35:32 It's not a function of state. 589 00:35:32 --> 00:35:34 It's not an exact differential. 590 00:35:34 --> 00:35:37 So the delta v here is an exact differential, 591 00:35:37 --> 00:35:40 but this dw is not. 592 00:35:40 --> 00:35:42 That's going to be really important. 593 00:35:42 --> 00:35:45 So if you want to find out how much work you've done, you take 594 00:35:45 --> 00:35:51 the integral from the initial state to the final state of 595 00:35:51 --> 00:36:01 dw minus from one to two p external dv, and you've got 596 00:36:01 --> 00:36:05 to know what the path is. 597 00:36:05 --> 00:36:15 So let's look at this path dependence briefly here. 598 00:36:15 --> 00:36:20 We're going to do two different paths, and see how they're 599 00:36:20 --> 00:36:24 different in terms of the work that comes out. 600 00:36:24 --> 00:36:27 So we're going to take an ideal gas, we can 601 00:36:27 --> 00:36:28 assume that it's ideal. 602 00:36:28 --> 00:36:33 Let's take argon, for instance, a nice, non-interacting gas. 603 00:36:33 --> 00:36:35 We're going to do a compression. 604 00:36:35 --> 00:36:42 We're going to take argon, with a certain gas, certain pressure 605 00:36:42 --> 00:36:45 p1, volume V1, and we're going to a final state 606 00:36:45 --> 00:36:50 argon, gas, p2, V2. 607 00:36:50 --> 00:36:58 Where V1 is greater than V2, and p1 is less than p2. 608 00:36:58 --> 00:37:07 So if I draw this on a p v diagram, so there is 609 00:37:07 --> 00:37:09 volume on this axis. 610 00:37:09 --> 00:37:11 There's pressure on this axis. 611 00:37:11 --> 00:37:13 There is V1 here. 612 00:37:13 --> 00:37:15 There's V2 here. 613 00:37:15 --> 00:37:18 There's p1 here, and p2 here. 614 00:37:18 --> 00:37:20 So I'm starting at p1, V1. 615 00:37:20 --> 00:37:25 I'm starting right here. 616 00:37:25 --> 00:37:30 And I'm going to end right here. 617 00:37:30 --> 00:37:33 Initial find -- there are many ways I can get from 618 00:37:33 --> 00:37:35 one state to the other. 619 00:37:35 --> 00:37:40 Draw any sort of line to go here, right? 620 00:37:40 --> 00:37:42 There are a couple obvious ones, which we're going 621 00:37:42 --> 00:37:45 to -- we can calculate, which we're going to do. 622 00:37:45 --> 00:37:52 So, the first obvious one is to take V1 to V2 623 00:37:52 --> 00:38:01 first with p constant. 624 00:38:01 --> 00:38:03 So take this path here. 625 00:38:03 --> 00:38:08 I take V1 to V2 first, keeping the pressure constant at p1, 626 00:38:08 --> 00:38:12 then I take p1 to p2 keeping the volume constant at V2. 627 00:38:12 --> 00:38:14 Let's call this path 1. 628 00:38:14 --> 00:38:21 Then you take p1 to p2 with V constant. 629 00:38:21 --> 00:38:31 An isobaric process followed by a constant volume process. 630 00:38:31 --> 00:38:33 You could also do a different path. 631 00:38:33 --> 00:38:42 You could do, let me draw p v, there's my initial state. 632 00:38:42 --> 00:38:47 My final state here, I could take, first, I could change 633 00:38:47 --> 00:38:51 the pressure, and then change the volume. 634 00:38:51 --> 00:38:59 So the second process, if you take p1 to p2, V constant, and 635 00:38:59 --> 00:39:04 then you take V1 to V2 with p constant. 636 00:39:04 --> 00:39:10 This is path number two. 637 00:39:10 --> 00:39:12 Both are perfectly fine paths, and I'm going to assume 638 00:39:12 --> 00:39:15 that these paths are also reversible. 639 00:39:15 --> 00:39:19 Let's assume that both are reversible, meaning that I'm 640 00:39:19 --> 00:39:25 doing this pretty slowly, so as I change, let's say I'm 641 00:39:25 --> 00:39:30 changing my volumes here, V1 to V2, it's happening, I'm 642 00:39:30 --> 00:39:33 compressing it slowly, slowly, slowly so that at any point I 643 00:39:33 --> 00:39:39 could reverse the process without losing energy, right? 644 00:39:39 --> 00:39:47 It's always an equilibrium. 645 00:39:47 --> 00:39:51 All right, let's calculate the work that's involved with 646 00:39:51 --> 00:39:55 these two processes. 647 00:39:55 --> 00:39:58 Remember it's the external pressure that's important. 648 00:39:58 --> 00:40:02 In this case, because it's a reversible process, the 649 00:40:02 --> 00:40:05 external pressure turns out to be always the same as 650 00:40:05 --> 00:40:07 the internal pressure. 651 00:40:07 --> 00:40:13 It's reversible, that means that p external, equals p. 652 00:40:13 --> 00:40:16 I'm doing it very slowly so that I'm always in equilibrium 653 00:40:16 --> 00:40:18 between the external pressure and the internal pressure so 654 00:40:18 --> 00:40:23 I can go back and forth. 655 00:40:23 --> 00:40:26 So, let's calculate w1. 656 00:40:26 --> 00:40:28 The work for path one. 657 00:40:28 --> 00:40:31 First thing is I change the volume from V1 to V2 The 658 00:40:31 --> 00:40:34 external pressure is kept constant, p1, so it's minus 659 00:40:34 --> 00:40:39 the integral from 1, V1 to V2, p1, dv. 660 00:40:39 --> 00:40:49 And then the next step here is I'm going from -- the 661 00:40:49 --> 00:40:50 pressure is changing. 662 00:40:50 --> 00:40:58 I'm going from V2 to V2 dv -- what do you think 663 00:40:58 --> 00:40:58 this integral is? 664 00:40:58 --> 00:41:03 Right, so this is easy part, zero here. 665 00:41:03 --> 00:41:04 This one is also pretty easy. 666 00:41:04 --> 00:41:12 That's minus p1 times V2 minus V1. p1 times V2 minus V1. 667 00:41:12 --> 00:41:18 What that turns out to be, this area right here. 668 00:41:18 --> 00:41:19 It's V1 minus V2 times p1. 669 00:41:19 --> 00:41:23 This is w1 here. 670 00:41:23 --> 00:41:34 OK, I can re-write this as p1 time V1 minus V2 and get rid 671 00:41:34 --> 00:41:36 of this negative sign here. 672 00:41:36 --> 00:41:44 Now V1 is bigger than V2, so this is positive. 673 00:41:44 --> 00:41:50 So I am compressing, I'm doing work to the system, positive 674 00:41:50 --> 00:41:54 work everything follows our convention. 675 00:41:54 --> 00:42:01 Number two here, OK, the first thing I do is I change the 676 00:42:01 --> 00:42:06 pressure under constant volume, V1, V1 minus p dv, and then I 677 00:42:06 --> 00:42:14 change the volume from V1 to V2 and then this is p2, dv. 678 00:42:14 --> 00:42:18 This first integral is zero V1 to V1, then I get minus 679 00:42:18 --> 00:42:25 p2 times V2 minus V1 or p2 times V1 minus V2. 680 00:42:25 --> 00:42:27 Again, a positive number. 681 00:42:27 --> 00:42:29 I'm doing work to the system to go from the initial 682 00:42:29 --> 00:42:33 state to the final state. 683 00:42:33 --> 00:42:37 But it's not the same as w1. 684 00:42:37 --> 00:42:40 In this case, I have p1 times delta V. 685 00:42:40 --> 00:42:44 In this case here, I have p2 times delta V. 686 00:42:44 --> 00:42:56 And p2 is bigger than p1. w2 is bigger than w1. 687 00:42:56 --> 00:43:00 The amount of work that you're doing on the system depends 688 00:43:00 --> 00:43:04 on the path that you take. 689 00:43:04 --> 00:43:08 All right, how do I, practically speaking, 690 00:43:08 --> 00:43:10 how do I do this? 691 00:43:10 --> 00:43:12 Anybody have an idea? 692 00:43:12 --> 00:43:23 How do I keep p1 constant while I'm lowering the volume? 693 00:43:23 --> 00:43:24 STUDENT: Change the temperature? 694 00:43:24 --> 00:43:24 PROFESSOR: Change the temperature, right. 695 00:43:24 --> 00:43:31 So what I'm doing here is I'm cooling, and then when I'm 696 00:43:31 --> 00:43:34 sitting at a fixed volume and I'm increasing the 697 00:43:34 --> 00:43:37 pressure, what am I doing? 698 00:43:37 --> 00:43:38 I'm heating, right? 699 00:43:38 --> 00:43:40 So I'm doing cooling and heating cycles. 700 00:43:40 --> 00:43:45 So in this case here, I cool and then I heat. 701 00:43:45 --> 00:43:47 In this case here, I heat and then I cool. 702 00:43:47 --> 00:43:52 All right, so I'm burning some energy, I'm burning some 703 00:43:52 --> 00:43:59 fuel to do this somehow, to get that work to happen. 704 00:43:59 --> 00:44:04 All right, now suppose that I took these two paths, 705 00:44:04 --> 00:44:11 and coupled them together. 706 00:44:11 --> 00:44:12 So in this case, it's the amount of work is the 707 00:44:12 --> 00:44:14 area under that curve. 708 00:44:14 --> 00:44:16 And in this case here, the amount of work is bigger, 709 00:44:16 --> 00:44:21 w2 is bigger, and it's the area under this curve. 710 00:44:21 --> 00:44:26 Now, suppose I took this two paths, and I took -- couple 711 00:44:26 --> 00:44:28 them together with one the reverse of the other. 712 00:44:28 --> 00:44:32 So I have my initial state, my final state, my initial 713 00:44:32 --> 00:44:35 state, my final state here. 714 00:44:35 --> 00:44:40 And I start by taking my first path here. 715 00:44:40 --> 00:44:44 I cool, I heat. 716 00:44:44 --> 00:44:46 So there's w1. 717 00:44:46 --> 00:44:52 So the w total that I'm going to get, is w1, and then instead 718 00:44:52 --> 00:44:57 of the path from V1 to, from 1 to 2 going like this as we had 719 00:44:57 --> 00:45:08 before, I'm going to take it backwards. 720 00:45:08 --> 00:45:10 If I go backwards, to work -- everything is symmetric, the 721 00:45:10 --> 00:45:13 work becomes the negative from what I had calculated before, 722 00:45:13 --> 00:45:21 so this becomes minus what I calculated before for w2. 723 00:45:21 --> 00:45:29 The total work, in this case here, is p1 times V1 minus p2 724 00:45:29 --> 00:45:38 times V1 minus V2, it's p1 minus p2 times V1 minus V2. 725 00:45:38 --> 00:45:41 This is a positive number, p1 is smaller than p2. 726 00:45:41 --> 00:45:43 This is a negative number. 727 00:45:43 --> 00:45:48 The total work is less than zero. 728 00:45:48 --> 00:45:53 That's the work that the system is doing to the environment. 729 00:45:53 --> 00:45:54 I'm doing work to the environment. 730 00:45:54 --> 00:45:56 The work is negative, which means that work is being 731 00:45:56 --> 00:45:57 done to the environment. 732 00:45:57 --> 00:46:07 And that work is the area inside the rectangle. 733 00:46:07 --> 00:46:10 What you've built is an engine. 734 00:46:10 --> 00:46:17 You cool, you heat, you heat, you cool, you get back to the 735 00:46:17 --> 00:46:20 same place, but you've just done work to the environment. 736 00:46:20 --> 00:46:23 You've just built a heat engine. 737 00:46:23 --> 00:46:28 You take fuel, rather you take something that's warm, and you 738 00:46:28 --> 00:46:31 put it in contact with the atmosphere, it cools down. 739 00:46:31 --> 00:46:34 You take your fuel, you heat it up again. 740 00:46:34 --> 00:46:35 It expands. 741 00:46:35 --> 00:46:39 You change your constraints on your system, you heat it up 742 00:46:39 --> 00:46:43 some more, then you take the heat source away, and you 743 00:46:43 --> 00:46:46 put it back in contact with the atmosphere. 744 00:46:46 --> 00:46:48 And you cool it a little bit, change the constraints, cool 745 00:46:48 --> 00:46:50 it a little bit more, and heat, and you've got a 746 00:46:50 --> 00:46:53 closed cycle engine. 747 00:46:53 --> 00:46:55 We're going to work with some more complicated 748 00:46:55 --> 00:46:57 engines before. 749 00:46:57 --> 00:47:00 But the important part here is that the work is not zero. 750 00:47:00 --> 00:47:01 You're starting at one point. 751 00:47:01 --> 00:47:04 You're going around a cycle and you're going back 752 00:47:04 --> 00:47:05 to the same point. 753 00:47:05 --> 00:47:07 The pressure, temperature, and volume are exactly the same 754 00:47:07 --> 00:47:08 here as when you started out. 755 00:47:08 --> 00:47:10 But the w is not zero. 756 00:47:10 --> 00:47:15 The w, for the closed path, and when I put a circle there on my 757 00:47:15 --> 00:47:18 integral that means a closed path, when you start and end at 758 00:47:18 --> 00:47:26 the same point, right, this is not zero. 759 00:47:26 --> 00:47:29 If you had an exact differential, the exact 760 00:47:29 --> 00:47:31 differential around a closed path, you would get zero. 761 00:47:31 --> 00:47:36 It wouldn't care where the path is. 762 00:47:36 --> 00:47:37 Here this cares where the path is. 763 00:47:37 --> 00:47:44 So, work is not a function of state. 764 00:47:44 --> 00:47:53 Any questions on work before we move on to heat, briefly? 765 00:47:53 --> 00:48:05 So heat is a quantity that flows into a substance, 766 00:48:05 --> 00:48:08 something that flows into a substance that changes it's 767 00:48:08 --> 00:48:13 temperature, very broadly defined. 768 00:48:13 --> 00:48:16 And, again, we have a sign convention for heat. 769 00:48:16 --> 00:48:20 So heat, we're going to call that q. 770 00:48:20 --> 00:48:24 And our sign convention is that if we change our temperature 771 00:48:24 --> 00:48:33 from T1 to T2, where T2 it's greater than T1 then heat 772 00:48:33 --> 00:48:36 is going to be positive. 773 00:48:36 --> 00:48:39 Heat needs to go into the system to change the 774 00:48:39 --> 00:48:43 temperature and make it go up. 775 00:48:43 --> 00:48:45 If the temperature of the system goes down, heat flows 776 00:48:45 --> 00:48:47 down heat flows out of the system, and we call 777 00:48:47 --> 00:48:50 that negative q. 778 00:48:50 --> 00:48:54 Same convention is for w, basically. 779 00:48:54 --> 00:48:56 Now, you can have a change of temperature without 780 00:48:56 --> 00:48:58 any heat being involved. 781 00:48:58 --> 00:49:05 I can take an insulated box, and I can have a chemical 782 00:49:05 --> 00:49:08 reaction in that insulated box. 783 00:49:08 --> 00:49:11 I can take a heat pack, like the kind you buy at a pharmacy. 784 00:49:11 --> 00:49:15 Break it up. 785 00:49:15 --> 00:49:17 It gets hot. 786 00:49:17 --> 00:49:20 There's no heat flowing from the environment to the system. 787 00:49:20 --> 00:49:22 I have to define my terms. 788 00:49:22 --> 00:49:24 My system is whatever's inside the box. 789 00:49:24 --> 00:49:26 It's insulated. 790 00:49:26 --> 00:49:28 It's a closed system. 791 00:49:28 --> 00:49:30 In fact, it's an isolated system. 792 00:49:30 --> 00:49:32 There's no energy or matter that can go through 793 00:49:32 --> 00:49:33 that boundary. 794 00:49:33 --> 00:49:37 Yet, the temperature goes up. 795 00:49:37 --> 00:49:44 So, I can have a temperature change which is an adiabatic 796 00:49:44 --> 00:49:45 temperature change. 797 00:49:45 --> 00:49:52 Adiabatic means without heat. 798 00:49:52 --> 00:49:55 Or I could have a non-adiabatic, I could take the 799 00:49:55 --> 00:49:59 same temperature change, by taking a flame, or a heat 800 00:49:59 --> 00:50:04 source and heating up my substance. 801 00:50:04 --> 00:50:09 So, clearly q is going to depend on the path. 802 00:50:09 --> 00:50:13 I'm going from T1 to T2, and I have two ways to go here. 803 00:50:13 --> 00:50:15 One is non-adiabatic. 804 00:50:15 --> 00:50:21 One is adiabatic. 805 00:50:21 --> 00:50:24 All right, now what we're going to learn next time, and Bob 806 00:50:24 --> 00:50:29 Field is going to teach the lecture next time, is how heat 807 00:50:29 --> 00:50:32 and work are related, and how they're really the same thing, 808 00:50:32 --> 00:50:36 and how they're related through the first law, through 809 00:50:36 --> 00:50:38 energy conservation. 810 00:50:38 --> 00:50:42 OK, I'll see you on Wednesday then.