1 00:00:00,000 --> 00:00:00,650 2 00:00:00,650 --> 00:00:03,190 If you followed some of the mathematics, and some of the 3 00:00:03,190 --> 00:00:06,590 thermodynamic principles in the last several videos, what 4 00:00:06,590 --> 00:00:10,130 occurs in this video might just blow your mind. 5 00:00:10,130 --> 00:00:12,570 So not to set expectations too high, let's just 6 00:00:12,570 --> 00:00:14,506 start off with it. 7 00:00:14,506 --> 00:00:17,160 So let's say I have a container. 8 00:00:17,160 --> 00:00:19,990 And in that container, I have gas particles. 9 00:00:19,990 --> 00:00:22,650 10 00:00:22,650 --> 00:00:25,480 Inside of that container, they're bouncing around like 11 00:00:25,480 --> 00:00:29,000 gas particles tend to do, creating some pressure on the 12 00:00:29,000 --> 00:00:30,760 container of a certain volume. 13 00:00:30,760 --> 00:00:32,784 And let's say I have n particles. 14 00:00:32,784 --> 00:00:36,520 15 00:00:36,520 --> 00:00:38,920 Now, each of these particles could be 16 00:00:38,920 --> 00:00:40,860 in x different states. 17 00:00:40,860 --> 00:00:42,110 Let me write that down. 18 00:00:42,110 --> 00:00:56,420 19 00:00:56,420 --> 00:00:57,800 What do I mean by state? 20 00:00:57,800 --> 00:00:59,640 Well, let's say I take particle A. 21 00:00:59,640 --> 00:01:01,840 Let me make particle A a different color. 22 00:01:01,840 --> 00:01:05,080 Particle A could be down here in this corner, and it could 23 00:01:05,080 --> 00:01:06,810 have some velocity like that. 24 00:01:06,810 --> 00:01:08,190 It could also be in that corner and have a 25 00:01:08,190 --> 00:01:08,940 velocity like that. 26 00:01:08,940 --> 00:01:10,080 Those would be two different states. 27 00:01:10,080 --> 00:01:12,070 It could be up here, and have a velocity like that. 28 00:01:12,070 --> 00:01:14,050 It could be there and have a velocity like that. 29 00:01:14,050 --> 00:01:16,260 If you were to add up all the different states, and there 30 00:01:16,260 --> 00:01:20,340 would be a gazillion, of them, you would get x. 31 00:01:20,340 --> 00:01:22,550 That blue particle could have x different states. 32 00:01:22,550 --> 00:01:23,470 You don't know. 33 00:01:23,470 --> 00:01:24,740 We're just saying, look. 34 00:01:24,740 --> 00:01:25,545 I have this container. 35 00:01:25,545 --> 00:01:26,470 It's got n particles. 36 00:01:26,470 --> 00:01:28,670 So we just know that each of them could be 37 00:01:28,670 --> 00:01:30,330 in x different states. 38 00:01:30,330 --> 00:01:33,620 Now, if each of them can be in x different states, how many 39 00:01:33,620 --> 00:01:37,950 total configurations are there for the system as a whole? 40 00:01:37,950 --> 00:01:42,080 Well, particle A could be in x different states, and then 41 00:01:42,080 --> 00:01:44,990 particle B could be in x different places. 42 00:01:44,990 --> 00:01:46,340 So times x. 43 00:01:46,340 --> 00:01:48,880 If we just had two particles, then you would multiply all 44 00:01:48,880 --> 00:01:51,130 the different places where X could be times all the 45 00:01:51,130 --> 00:01:54,970 different places where the red particle could be, then you'd 46 00:01:54,970 --> 00:01:57,290 get all the different configurations for the system. 47 00:01:57,290 --> 00:01:58,570 But we don't have just two particles. 48 00:01:58,570 --> 00:02:00,260 We have n particles. 49 00:02:00,260 --> 00:02:04,690 So for every particle, you'd multiply it times the number 50 00:02:04,690 --> 00:02:06,840 of states it could have, and you do that 51 00:02:06,840 --> 00:02:09,729 a total of n times. 52 00:02:09,729 --> 00:02:13,290 And this is really just combinatorics here. 53 00:02:13,290 --> 00:02:14,540 You do it n times. 54 00:02:14,540 --> 00:02:16,840 This system would have n configurations. 55 00:02:16,840 --> 00:02:19,940 For example, if I had two particles, each particle had 56 00:02:19,940 --> 00:02:22,860 three different potential states, how many different 57 00:02:22,860 --> 00:02:24,330 configurations could there be? 58 00:02:24,330 --> 00:02:28,210 Well, for every three that one particle could have, the other 59 00:02:28,210 --> 00:02:30,540 one could have three different states, so you'd have nine 60 00:02:30,540 --> 00:02:31,190 different states. 61 00:02:31,190 --> 00:02:31,970 You'd multiply them. 62 00:02:31,970 --> 00:02:33,690 If you had another particle with three different states, 63 00:02:33,690 --> 00:02:35,050 you'd multiply that by three, so you have 64 00:02:35,050 --> 00:02:36,530 27 different states. 65 00:02:36,530 --> 00:02:38,660 Here we have n particles. 66 00:02:38,660 --> 00:02:41,020 Each of them could be in x different states. 67 00:02:41,020 --> 00:02:43,380 So the total number of configurations we have for our 68 00:02:43,380 --> 00:02:47,580 system-- x times itself n times is just x to the n. 69 00:02:47,580 --> 00:02:53,790 So we have x to the n states in our system. 70 00:02:53,790 --> 00:02:57,530 Now, let's say that we like thinking about how many states 71 00:02:57,530 --> 00:03:00,080 a system can have. Certain states have less-- for 72 00:03:00,080 --> 00:03:02,340 example, if I had fewer particles, I would have fewer 73 00:03:02,340 --> 00:03:03,510 potential states. 74 00:03:03,510 --> 00:03:07,140 Or maybe if I had a smaller container, I would also have 75 00:03:07,140 --> 00:03:08,240 fewer potential states. 76 00:03:08,240 --> 00:03:10,510 There would be fewer potential places for our little 77 00:03:10,510 --> 00:03:12,320 particles to exist. 78 00:03:12,320 --> 00:03:17,180 So I want to create some type of state variable that tells 79 00:03:17,180 --> 00:03:21,230 me, well, how many states can my system be in? 80 00:03:21,230 --> 00:03:24,030 So this is kind of a macrostate variable. 81 00:03:24,030 --> 00:03:28,040 It tells me, how many states can my system be in? 82 00:03:28,040 --> 00:03:31,730 And let's call it s for states. 83 00:03:31,730 --> 00:03:33,640 For the first time in thermodynamics, we're actually 84 00:03:33,640 --> 00:03:38,050 using a letter that in some way is related to what we're 85 00:03:38,050 --> 00:03:39,930 actually trying to measure. 86 00:03:39,930 --> 00:03:41,650 s for states. 87 00:03:41,650 --> 00:03:44,750 And since the states, they can grow really large, let's say I 88 00:03:44,750 --> 00:03:47,350 like to take the logarithm of the number of states. 89 00:03:47,350 --> 00:03:49,860 Now this is just how I'm defining my state variable. 90 00:03:49,860 --> 00:03:51,120 I get to define it. 91 00:03:51,120 --> 00:03:53,970 So I get to put a logarithm out front. 92 00:03:53,970 --> 00:03:56,790 So let me just put a logarithm. 93 00:03:56,790 --> 00:03:59,810 So in this case, it would be the logarithm of my number of 94 00:03:59,810 --> 00:04:05,690 states-- so it would be x to the n, where this is number of 95 00:04:05,690 --> 00:04:06,940 potential states. 96 00:04:06,940 --> 00:04:10,250 97 00:04:10,250 --> 00:04:12,330 And you know, we need some kind of scaling factor. 98 00:04:12,330 --> 00:04:14,710 Maybe I'll change the units eventually. 99 00:04:14,710 --> 00:04:18,519 So let me put a little constant out front. 100 00:04:18,519 --> 00:04:21,709 Every good formula needs a constant to 101 00:04:21,709 --> 00:04:22,790 get our units right. 102 00:04:22,790 --> 00:04:24,500 I'll make that a lowercase k. 103 00:04:24,500 --> 00:04:25,580 So that's my definition. 104 00:04:25,580 --> 00:04:28,950 I call this my state variable. 105 00:04:28,950 --> 00:04:31,470 If you give me a system, I should, in theory, be able to 106 00:04:31,470 --> 00:04:35,170 tell you how many states the system can take on. 107 00:04:35,170 --> 00:04:35,760 Fair enough. 108 00:04:35,760 --> 00:04:38,760 So let me close that box right there. 109 00:04:38,760 --> 00:04:46,160 Now let's say that I were to take my box that I had-- let 110 00:04:46,160 --> 00:04:49,350 me copy and paste it. 111 00:04:49,350 --> 00:04:51,790 I take that box. 112 00:04:51,790 --> 00:04:53,860 And it just so happens that there was an 113 00:04:53,860 --> 00:04:56,450 adjacent box next to it. 114 00:04:56,450 --> 00:05:00,120 They share this wall. 115 00:05:00,120 --> 00:05:02,505 They're identical in size, although what I just drew 116 00:05:02,505 --> 00:05:03,560 isn't identical in size. 117 00:05:03,560 --> 00:05:04,290 But they're close enough. 118 00:05:04,290 --> 00:05:05,820 They're identical in size. 119 00:05:05,820 --> 00:05:09,210 And what I do, is I blow away this wall. 120 00:05:09,210 --> 00:05:11,010 I just evaporate it, all of a sudden. 121 00:05:11,010 --> 00:05:12,680 It just disappears. 122 00:05:12,680 --> 00:05:17,040 So this wall just disappears. 123 00:05:17,040 --> 00:05:18,790 Now, what's going to happen? 124 00:05:18,790 --> 00:05:22,350 Well, as soon as I blow away this wall, this is very much 125 00:05:22,350 --> 00:05:25,210 not an isostatic process. 126 00:05:25,210 --> 00:05:25,700 Right? 127 00:05:25,700 --> 00:05:27,250 All hell's going to break loose. 128 00:05:27,250 --> 00:05:29,350 I'm going to blow away this wall, and you know, the 129 00:05:29,350 --> 00:05:33,020 particles that were about to bounce off the wall are just 130 00:05:33,020 --> 00:05:34,550 going to keep going. 131 00:05:34,550 --> 00:05:34,890 Right? 132 00:05:34,890 --> 00:05:36,480 They're going to keep going until they can maybe bounce 133 00:05:36,480 --> 00:05:37,280 off of that wall. 134 00:05:37,280 --> 00:05:40,110 So right when I blow away this wall, there's no pressure 135 00:05:40,110 --> 00:05:42,580 here, because these guys have nothing to bounce off to. 136 00:05:42,580 --> 00:05:44,180 While these guys don't know anything. 137 00:05:44,180 --> 00:05:45,690 They don't know anything until they come over here and say, 138 00:05:45,690 --> 00:05:46,680 oh, no wall. 139 00:05:46,680 --> 00:05:48,140 So the pressure is in flux. 140 00:05:48,140 --> 00:05:51,000 Even the volume is in flux, as these guys make their way 141 00:05:51,000 --> 00:05:55,990 across the entire expanse of the new volume. 142 00:05:55,990 --> 00:05:57,800 So everything is in flux. 143 00:05:57,800 --> 00:05:58,580 Right? 144 00:05:58,580 --> 00:05:59,820 And so what's our new volume? 145 00:05:59,820 --> 00:06:04,520 If we call this volume, what's this? 146 00:06:04,520 --> 00:06:06,330 This is now 2 times the volume. 147 00:06:06,330 --> 00:06:09,120 148 00:06:09,120 --> 00:06:12,230 Let's think about some of the other state variables we know. 149 00:06:12,230 --> 00:06:14,800 We know that the pressure is going to go down. 150 00:06:14,800 --> 00:06:16,450 We can even relate it, because we know that our 151 00:06:16,450 --> 00:06:17,670 volume is twice it. 152 00:06:17,670 --> 00:06:19,170 Is 2 times the volume. 153 00:06:19,170 --> 00:06:22,330 What about the temperature? 154 00:06:22,330 --> 00:06:23,890 Well, the temperature change. 155 00:06:23,890 --> 00:06:26,620 Temperature is average kinetic energy, right? 156 00:06:26,620 --> 00:06:28,790 Or it's a measure of average kinetic energy. 157 00:06:28,790 --> 00:06:31,910 So all of these molecules here, each of them have 158 00:06:31,910 --> 00:06:32,700 kinetic energy. 159 00:06:32,700 --> 00:06:35,930 They could be different amounts of kinetic energy, but 160 00:06:35,930 --> 00:06:38,410 temperature is a measure of average kinetic energy. 161 00:06:38,410 --> 00:06:41,140 Now, if I blow away this wall, does that change the kinetic 162 00:06:41,140 --> 00:06:42,680 energy of these molecules? 163 00:06:42,680 --> 00:06:43,310 No! 164 00:06:43,310 --> 00:06:44,670 It doesn't change it at all. 165 00:06:44,670 --> 00:06:46,700 So the temperature is constant. 166 00:06:46,700 --> 00:06:51,480 So if this is T1, then the temperature of this system 167 00:06:51,480 --> 00:06:53,280 here is T1. 168 00:06:53,280 --> 00:06:55,320 And you might say, hey, Sal, wait, that doesn't make sense. 169 00:06:55,320 --> 00:06:59,780 In the past, when my cylinder expanded, my 170 00:06:59,780 --> 00:07:01,450 temperature went down. 171 00:07:01,450 --> 00:07:03,450 And the reason why the temperature went down in that 172 00:07:03,450 --> 00:07:05,810 case is because your molecules were doing work. 173 00:07:05,810 --> 00:07:08,540 They expanded the container itself. 174 00:07:08,540 --> 00:07:09,890 They pushed up the cylinder. 175 00:07:09,890 --> 00:07:14,300 So they expended some of their kinetic energy to do the work. 176 00:07:14,300 --> 00:07:15,830 In this case, I just blew away that wall. 177 00:07:15,830 --> 00:07:18,350 These guys did no work whatsoever, so they didn't 178 00:07:18,350 --> 00:07:21,990 have to expend any of their kinetic energy to do any work. 179 00:07:21,990 --> 00:07:23,710 So their temperature did not change. 180 00:07:23,710 --> 00:07:24,620 So that's interesting. 181 00:07:24,620 --> 00:07:25,530 Fair enough. 182 00:07:25,530 --> 00:07:29,310 Well, in this new world, what happens? 183 00:07:29,310 --> 00:07:31,380 Eventually I get to a situation where my molecules 184 00:07:31,380 --> 00:07:33,290 fill the container. 185 00:07:33,290 --> 00:07:33,590 Right? 186 00:07:33,590 --> 00:07:34,850 We know that from common sense. 187 00:07:34,850 --> 00:07:36,500 And if you think about it on a microlevel, 188 00:07:36,500 --> 00:07:37,330 why does that happen? 189 00:07:37,330 --> 00:07:38,540 It's not a mystery. 190 00:07:38,540 --> 00:07:41,070 You know, on this direction, things were bouncing and they 191 00:07:41,070 --> 00:07:41,690 keep bouncing. 192 00:07:41,690 --> 00:07:43,750 But when they go here, there used to be a wall, and then 193 00:07:43,750 --> 00:07:44,920 they'll just keep going, and then they'll 194 00:07:44,920 --> 00:07:46,210 start bouncing here. 195 00:07:46,210 --> 00:07:49,010 So when you have gazillion particles doing a gazillion of 196 00:07:49,010 --> 00:07:51,800 these bounces, eventually, they're just as likely to be 197 00:07:51,800 --> 00:07:53,830 here as they are over there. 198 00:07:53,830 --> 00:07:54,220 Now. 199 00:07:54,220 --> 00:07:56,400 Let's do our computation again. 200 00:07:56,400 --> 00:07:59,950 In our old situation, when we just looked at this, each 201 00:07:59,950 --> 00:08:04,370 particle could be in one of x places, or in one of x states. 202 00:08:04,370 --> 00:08:09,330 Now it could be in twice as many states, right? 203 00:08:09,330 --> 00:08:16,790 Now, each particle could be in 2x different states. 204 00:08:16,790 --> 00:08:17,990 Why do I say 2x? 205 00:08:17,990 --> 00:08:20,030 Because I have twice the area to be in. 206 00:08:20,030 --> 00:08:23,320 Now, the states aren't just, you know, position in space. 207 00:08:23,320 --> 00:08:29,610 But everything else-- so, you know, before here, maybe I had 208 00:08:29,610 --> 00:08:36,960 a positions in space times b positions, or b momentums, you 209 00:08:36,960 --> 00:08:38,789 know, where those are all the different momentums, and that 210 00:08:38,789 --> 00:08:40,620 was equal to x. 211 00:08:40,620 --> 00:08:44,490 Now I have 2a positions in volume that I could be in. 212 00:08:44,490 --> 00:08:46,270 I have twice the volume to deal with. 213 00:08:46,270 --> 00:08:50,860 So I have 2a positions in volume I can be at, but my 214 00:08:50,860 --> 00:08:53,680 momentum states are going to still be-- I just have b 215 00:08:53,680 --> 00:08:56,150 momentum states-- so this is equal to 2x. 216 00:08:56,150 --> 00:08:59,680 I now can be in 2x different states, just because I have 2 217 00:08:59,680 --> 00:09:03,450 times the volume to travel around in, right? 218 00:09:03,450 --> 00:09:05,570 So how many states are there for the system? 219 00:09:05,570 --> 00:09:08,100 Well, each particle can be in 2x states. 220 00:09:08,100 --> 00:09:11,080 So this is 2x times 2x times 2x. 221 00:09:11,080 --> 00:09:15,680 And I'm going to do that n times. 222 00:09:15,680 --> 00:09:19,500 So my new s-- so this is, you know, let's call this s 223 00:09:19,500 --> 00:09:24,540 initial-- so my s final, my new way of measuring my 224 00:09:24,540 --> 00:09:27,880 states, is going to be equal to that little constant that I 225 00:09:27,880 --> 00:09:32,000 threw in there, times the natural log of the 226 00:09:32,000 --> 00:09:33,190 new number of states. 227 00:09:33,190 --> 00:09:33,620 So what is it? 228 00:09:33,620 --> 00:09:36,030 It's 2x to the n power. 229 00:09:36,030 --> 00:09:40,880 230 00:09:40,880 --> 00:09:47,520 So my question to you is, what is my change in s when I blew 231 00:09:47,520 --> 00:09:48,770 away the wall? 232 00:09:48,770 --> 00:09:51,710 You know, there was this room here the entire time, although 233 00:09:51,710 --> 00:09:54,580 these particles really didn't care because 234 00:09:54,580 --> 00:09:55,710 this wall was there. 235 00:09:55,710 --> 00:09:58,120 So what is the change in s when I blew away this wall? 236 00:09:58,120 --> 00:09:58,760 And this should be clear. 237 00:09:58,760 --> 00:10:00,050 The temperature didn't change, because no 238 00:10:00,050 --> 00:10:01,570 kinetic energy was expended. 239 00:10:01,570 --> 00:10:02,760 And this is all in isolation. 240 00:10:02,760 --> 00:10:03,156 I should have said. 241 00:10:03,156 --> 00:10:04,330 It's adiabatic. 242 00:10:04,330 --> 00:10:05,790 There's no transfer of heat. 243 00:10:05,790 --> 00:10:07,940 So that's also why the temperature didn't change. 244 00:10:07,940 --> 00:10:10,390 So what is our change in s? 245 00:10:10,390 --> 00:10:16,700 Our change in s is equal to our s final minus our s 246 00:10:16,700 --> 00:10:19,400 initial, which is equal to-- what's our s final? 247 00:10:19,400 --> 00:10:22,410 It's this expression, right here. 248 00:10:22,410 --> 00:10:27,020 It is k times the natural log--and we can write this as 249 00:10:27,020 --> 00:10:28,980 2 to the n, x to the n. 250 00:10:28,980 --> 00:10:30,400 That's just exponent rules. 251 00:10:30,400 --> 00:10:35,470 And from that, we're going to subtract out our initial s 252 00:10:35,470 --> 00:10:38,900 value, which was this. k natural log of x to the n. 253 00:10:38,900 --> 00:10:42,590 254 00:10:42,590 --> 00:10:45,300 Now we can use our logarithm properties to say, well, you 255 00:10:45,300 --> 00:10:47,970 know, you take the logarithm of a minus the logarithm of b, 256 00:10:47,970 --> 00:10:49,400 you can just divide them. 257 00:10:49,400 --> 00:10:52,770 So this is equal to k-- you could factor that out-- times 258 00:10:52,770 --> 00:10:59,120 the logarithm of 2 to the N-- it's uppercase N, 259 00:10:59,120 --> 00:10:59,890 so let me do that. 260 00:10:59,890 --> 00:11:00,970 This is uppercase N. 261 00:11:00,970 --> 00:11:02,430 I don't want to get confused with Moles. 262 00:11:02,430 --> 00:11:06,163 Uppercase N is the number of particles we actually have. So 263 00:11:06,163 --> 00:11:11,430 it's 2 to upper case N times x to the uppercase N divided by 264 00:11:11,430 --> 00:11:13,630 x to the uppercase N. 265 00:11:13,630 --> 00:11:15,890 So these two cancel out. 266 00:11:15,890 --> 00:11:21,310 So our change in s is equal to k times the natural log of 2 267 00:11:21,310 --> 00:11:24,680 to the N-- or, if we wanted to use our logarithm properties, 268 00:11:24,680 --> 00:11:26,105 we could throw that N out front. 269 00:11:26,105 --> 00:11:31,510 And we could say, our change in the s, whatever this state 270 00:11:31,510 --> 00:11:34,130 variable I've defined-- and this is a different definition 271 00:11:34,130 --> 00:11:43,270 than I did in the last video-- is equal to big N, the number 272 00:11:43,270 --> 00:11:47,070 of molecules we have, times my little constant, times the 273 00:11:47,070 --> 00:11:49,430 natural log of 2. 274 00:11:49,430 --> 00:11:55,200 So by blowing away that wall and giving my molecules twice 275 00:11:55,200 --> 00:11:58,760 as much volume to travel around in, my change in this 276 00:11:58,760 --> 00:12:02,010 little state function I defined is Nk the 277 00:12:02,010 --> 00:12:03,470 natural log of 2. 278 00:12:03,470 --> 00:12:05,400 And what really happened? 279 00:12:05,400 --> 00:12:07,990 I mean, it clearly went up, right? 280 00:12:07,990 --> 00:12:10,730 I clearly have a positive value here. 281 00:12:10,730 --> 00:12:12,890 Natural log of 2 is a positive value. 282 00:12:12,890 --> 00:12:14,540 N is positive value. 283 00:12:14,540 --> 00:12:16,360 It's going to be very large number than the number of 284 00:12:16,360 --> 00:12:18,040 molecules we had. 285 00:12:18,040 --> 00:12:20,060 And I'm assuming my constant I threw on there 286 00:12:20,060 --> 00:12:21,740 is a positive value. 287 00:12:21,740 --> 00:12:23,000 But what am I really describing? 288 00:12:23,000 --> 00:12:26,310 I'm saying that look, by blowing away that wall, my 289 00:12:26,310 --> 00:12:28,930 system can take on more states. 290 00:12:28,930 --> 00:12:32,700 There's more different things it can do. 291 00:12:32,700 --> 00:12:34,820 And I'll throw a little word out here. 292 00:12:34,820 --> 00:12:38,020 Its entropy has gone up. 293 00:12:38,020 --> 00:12:39,490 Well, actually, let's just define s 294 00:12:39,490 --> 00:12:40,670 to be the word entropy. 295 00:12:40,670 --> 00:12:43,110 We'll talk more about the word in the future. 296 00:12:43,110 --> 00:12:47,110 Its entropy has gone up, which means the number of states we 297 00:12:47,110 --> 00:12:48,040 have has gone up. 298 00:12:48,040 --> 00:12:50,090 I shouldn't use the word entropy without just saying, 299 00:12:50,090 --> 00:12:51,622 entropy I'm defining as equal to S. 300 00:12:51,622 --> 00:12:54,530 But let's just keep it with s. s for states. 301 00:12:54,530 --> 00:12:58,010 The number of states we're dealing with has gone up, and 302 00:12:58,010 --> 00:12:59,580 it's gone up by this factor. 303 00:12:59,580 --> 00:13:02,940 Actually, it's gone up by a factor of 2 to the n. 304 00:13:02,940 --> 00:13:06,100 And that's why it becomes n natural log of 2. 305 00:13:06,100 --> 00:13:07,110 Fair enough. 306 00:13:07,110 --> 00:13:07,690 Now you're saying, OK. 307 00:13:07,690 --> 00:13:09,280 This is nice, Sal. 308 00:13:09,280 --> 00:13:14,600 You have this statistical way, or I guess you could, this 309 00:13:14,600 --> 00:13:18,110 combinatoric way of measuring how many states this system 310 00:13:18,110 --> 00:13:18,860 can take on. 311 00:13:18,860 --> 00:13:20,360 And you looked at the actual molecules. 312 00:13:20,360 --> 00:13:22,450 You weren't looking at the kind of macrostates. 313 00:13:22,450 --> 00:13:23,460 And you were able to do that. 314 00:13:23,460 --> 00:13:25,840 You came up with this macrostate that says, that's 315 00:13:25,840 --> 00:13:27,840 essentially saying, how many states can I have? 316 00:13:27,840 --> 00:13:30,740 But how does that relate to that s that defined in the 317 00:13:30,740 --> 00:13:32,440 previous video? 318 00:13:32,440 --> 00:13:34,550 Remember, in the previous video, I was looking for state 319 00:13:34,550 --> 00:13:36,370 function that dealt with heat. 320 00:13:36,370 --> 00:13:42,460 And I defined s, or change in s-- I defined as change in s-- 321 00:13:42,460 --> 00:13:45,770 to be equal to the heat added to the system divided by the 322 00:13:45,770 --> 00:13:50,290 temperature that the heat was added at. 323 00:13:50,290 --> 00:13:57,020 So let's see if we can see whether these things are 324 00:13:57,020 --> 00:13:57,900 somehow related. 325 00:13:57,900 --> 00:14:01,130 So let's go back to our system, and go to a PV 326 00:14:01,130 --> 00:14:03,400 diagram, and see if we can do anything useful with that. 327 00:14:03,400 --> 00:14:08,920 328 00:14:08,920 --> 00:14:10,950 Alright. 329 00:14:10,950 --> 00:14:11,940 OK. 330 00:14:11,940 --> 00:14:14,800 So this is pressure, this is volume. 331 00:14:14,800 --> 00:14:15,300 Now. 332 00:14:15,300 --> 00:14:18,150 When we started off, before we blew away the wall, we had 333 00:14:18,150 --> 00:14:19,860 some pressure and some volume. 334 00:14:19,860 --> 00:14:23,010 So this is V1. 335 00:14:23,010 --> 00:14:26,060 And then we blew away the wall, and we got to-- 336 00:14:26,060 --> 00:14:27,350 Actually, let me do it a little bit differently. 337 00:14:27,350 --> 00:14:31,030 338 00:14:31,030 --> 00:14:35,290 I want that to be just right there. 339 00:14:35,290 --> 00:14:38,270 Let me make it right there. 340 00:14:38,270 --> 00:14:41,770 So that is our V1. 341 00:14:41,770 --> 00:14:44,680 This is our original state that we're in. 342 00:14:44,680 --> 00:14:46,990 So state initial, or however we want it. 343 00:14:46,990 --> 00:14:47,970 That's our initial pressure. 344 00:14:47,970 --> 00:14:49,260 And then we blew away the wall, and our 345 00:14:49,260 --> 00:14:51,300 volume doubled, right? 346 00:14:51,300 --> 00:14:54,860 So we could call this 2V1. 347 00:14:54,860 --> 00:14:58,180 Our volume doubled, our pressure would have gone down, 348 00:14:58,180 --> 00:15:00,010 and we're here. 349 00:15:00,010 --> 00:15:00,330 Right? 350 00:15:00,330 --> 00:15:02,310 That's our state 2. 351 00:15:02,310 --> 00:15:04,660 That's this scenario right here, after we 352 00:15:04,660 --> 00:15:05,750 blew away the wall. 353 00:15:05,750 --> 00:15:10,030 Now, what we did was not a quasistatic process. 354 00:15:10,030 --> 00:15:13,170 I can't draw the path here, because right when I blew away 355 00:15:13,170 --> 00:15:15,260 the wall, all hell broke loose, and things like 356 00:15:15,260 --> 00:15:17,380 pressure and volume weren't well defined. 357 00:15:17,380 --> 00:15:20,680 Eventually it got back to an equilibrium where this filled 358 00:15:20,680 --> 00:15:22,840 the container, and nothing else was in flux. 359 00:15:22,840 --> 00:15:25,100 And we could go back to here, and we could say, OK, now the 360 00:15:25,100 --> 00:15:26,270 pressure and the volume is this. 361 00:15:26,270 --> 00:15:28,680 But we don't know what happened in between that. 362 00:15:28,680 --> 00:15:33,200 So if we wanted to figure out our Q/T, or the heat into the 363 00:15:33,200 --> 00:15:35,760 system, we learned in the last video, the heat added to the 364 00:15:35,760 --> 00:15:38,590 system is equal to the work done by the system. 365 00:15:38,590 --> 00:15:41,790 We'd be at a loss, because the work done by the system is the 366 00:15:41,790 --> 00:15:44,450 area under some curve, but there's no curve to speak of 367 00:15:44,450 --> 00:15:47,710 here, because our system wasn't defined while all the 368 00:15:47,710 --> 00:15:49,270 hell had broke loose. 369 00:15:49,270 --> 00:15:50,610 So what can we do? 370 00:15:50,610 --> 00:15:55,350 Well, remember, this is a state function. 371 00:15:55,350 --> 00:15:56,610 And this is a state function. 372 00:15:56,610 --> 00:15:58,040 And I showed that in the last video. 373 00:15:58,040 --> 00:16:00,530 So it shouldn't be dependent on how we got 374 00:16:00,530 --> 00:16:02,090 from there to there. 375 00:16:02,090 --> 00:16:02,840 Right? 376 00:16:02,840 --> 00:16:06,375 So this change in entropy-- actually, let me be careful 377 00:16:06,375 --> 00:16:07,130 with my words. 378 00:16:07,130 --> 00:16:16,030 This change in s, so s2 minus s1, should be independent of 379 00:16:16,030 --> 00:16:20,090 the process that got me from s1 to s2. 380 00:16:20,090 --> 00:16:24,410 So this is independent of whatever crazy path-- I mean, 381 00:16:24,410 --> 00:16:27,160 I could have taken some crazy, quasistatic 382 00:16:27,160 --> 00:16:29,340 path like that, right? 383 00:16:29,340 --> 00:16:33,910 So any path that goes from this s1 to this s2 will have 384 00:16:33,910 --> 00:16:37,370 the same heat going into the system, or should have the 385 00:16:37,370 --> 00:16:39,180 same-- let me take that-- 386 00:16:39,180 --> 00:16:43,480 Any system that goes from s1 to s2, regardless of its path, 387 00:16:43,480 --> 00:16:46,550 will have the same change in entropy, or their 388 00:16:46,550 --> 00:16:47,680 same change in s. 389 00:16:47,680 --> 00:16:50,250 Because their s was something here, and it's something 390 00:16:50,250 --> 00:16:51,480 different over here. 391 00:16:51,480 --> 00:16:53,500 And you just take the difference between the two. 392 00:16:53,500 --> 00:16:56,670 So what's a system that we know that can do that? 393 00:16:56,670 --> 00:17:00,790 Well, let's say that we did an isothermal. 394 00:17:00,790 --> 00:17:03,440 And we know that these are all the same isotherm, right? 395 00:17:03,440 --> 00:17:06,400 We know that the temperature didn't change. 396 00:17:06,400 --> 00:17:07,108 I told you that. 397 00:17:07,108 --> 00:17:09,759 Because no kinetic energy was expended, and none of the 398 00:17:09,760 --> 00:17:11,280 particles did any work. 399 00:17:11,280 --> 00:17:15,390 So we can say, we can think of a theoretical process in which 400 00:17:15,390 --> 00:17:20,290 instead of doing something like that, we could have had a 401 00:17:20,290 --> 00:17:24,050 situation where we started off with our original container 402 00:17:24,050 --> 00:17:26,260 with our molecules in it. 403 00:17:26,260 --> 00:17:29,890 We could have put a reservoir here that's equivalent to the 404 00:17:29,890 --> 00:17:31,370 temperature that we're at. 405 00:17:31,370 --> 00:17:34,360 And then this could have been a piston that was maybe, we 406 00:17:34,360 --> 00:17:37,640 were pushing on it with some rocks that are pushing in the 407 00:17:37,640 --> 00:17:38,850 left-wards direction. 408 00:17:38,850 --> 00:17:42,840 And we slowly and slowly remove the rocks, so that 409 00:17:42,840 --> 00:17:46,140 these gases could push the piston and do some work, and 410 00:17:46,140 --> 00:17:49,760 fill this entire volume, or twice the volume. 411 00:17:49,760 --> 00:17:52,650 And then the temperature would have been kept constant by 412 00:17:52,650 --> 00:17:54,240 this heat reservoir. 413 00:17:54,240 --> 00:17:57,470 So this type of a process is kind of a sideways version of 414 00:17:57,470 --> 00:18:00,250 what I've done in the Carnot diagrams. That would be 415 00:18:00,250 --> 00:18:02,660 described like this. 416 00:18:02,660 --> 00:18:05,820 You'd go from this state to that state, and it would be a 417 00:18:05,820 --> 00:18:09,106 quasistatic static process along an isotherm. 418 00:18:09,106 --> 00:18:10,400 So it would look like that. 419 00:18:10,400 --> 00:18:13,280 So you could have a curve there. 420 00:18:13,280 --> 00:18:17,430 Now, for that process, what is the area 421 00:18:17,430 --> 00:18:22,330 under the curve there? 422 00:18:22,330 --> 00:18:27,240 Well, the area under the curve is just the integral-- and 423 00:18:27,240 --> 00:18:29,950 we've done this multiple times-- from our initial 424 00:18:29,950 --> 00:18:35,280 volume to our second volume, which is twice it, of p times 425 00:18:35,280 --> 00:18:36,840 our change in volume, right? 426 00:18:36,840 --> 00:18:40,100 p is our height, times our little changes in volume, give 427 00:18:40,100 --> 00:18:41,030 us each rectangle. 428 00:18:41,030 --> 00:18:43,750 And then the integral is just the sum along all of these. 429 00:18:43,750 --> 00:18:48,580 So that's essentially the work that this system does. 430 00:18:48,580 --> 00:18:49,110 Right? 431 00:18:49,110 --> 00:18:51,690 And the work that this system does, since we are on an 432 00:18:51,690 --> 00:18:55,570 isotherm, it is equal to the heat added to the system. 433 00:18:55,570 --> 00:18:57,610 Because our internal energy didn't change. 434 00:18:57,610 --> 00:18:58,490 So what is this? 435 00:18:58,490 --> 00:19:00,860 We've done this multiple times, but I'll redo it. 436 00:19:00,860 --> 00:19:05,140 So this is equal to the integral of V1 to 2V1. 437 00:19:05,140 --> 00:19:07,820 PV equals NRT, right? 438 00:19:07,820 --> 00:19:09,010 NRT. 439 00:19:09,010 --> 00:19:11,970 So P is equal to NRT/V. 440 00:19:11,970 --> 00:19:16,150 NRT over V dv. 441 00:19:16,150 --> 00:19:18,690 And the t is T1. 442 00:19:18,690 --> 00:19:21,340 Now, all of this is happening along an isotherm, so all of 443 00:19:21,340 --> 00:19:23,770 these terms are constant. 444 00:19:23,770 --> 00:19:27,920 So this is equal to the integral from V1 to 2V1 of 445 00:19:27,920 --> 00:19:31,610 NRT1 times 1 over V dv. 446 00:19:31,610 --> 00:19:34,130 I've done this integral multiple times. 447 00:19:34,130 --> 00:19:37,870 And so this is equal to-- I'll skip a couple of steps here, 448 00:19:37,870 --> 00:19:40,140 because I've done it in several videos already-- the 449 00:19:40,140 --> 00:19:45,180 natural log of 2V1 over V1, right? 450 00:19:45,180 --> 00:19:47,100 The antiderivative of this is the natural log. 451 00:19:47,100 --> 00:19:49,620 Take the natural log of that minus the natural log of that, 452 00:19:49,620 --> 00:19:53,100 which is equal to the natural log of 2V1 over V1. 453 00:19:53,100 --> 00:19:57,630 Which is just the same thing as NRT1 times the 454 00:19:57,630 --> 00:20:00,190 natural log of 2. 455 00:20:00,190 --> 00:20:01,470 Interesting. 456 00:20:01,470 --> 00:20:06,530 Now, let's add one little one little interesting thing to 457 00:20:06,530 --> 00:20:08,280 this to this equation. 458 00:20:08,280 --> 00:20:11,460 So this is NRT, but if I wanted to write in terms of 459 00:20:11,460 --> 00:20:14,930 the number of molecules, N is the number of moles. 460 00:20:14,930 --> 00:20:21,470 So I could rewrite N as the number of molecules we have 461 00:20:21,470 --> 00:20:25,870 divided by, 6 times 10 to the 23 power. 462 00:20:25,870 --> 00:20:26,660 Right? 463 00:20:26,660 --> 00:20:28,710 That's what n could be written as. 464 00:20:28,710 --> 00:20:33,260 So if we do it that way, then what is our-- remember, all of 465 00:20:33,260 --> 00:20:36,040 this, we were trying to find the amount of 466 00:20:36,040 --> 00:20:38,470 work done by our system. 467 00:20:38,470 --> 00:20:38,840 Right? 468 00:20:38,840 --> 00:20:41,290 But if we do it this way, this equation will turn 469 00:20:41,290 --> 00:20:43,050 into-- so let's see. 470 00:20:43,050 --> 00:20:46,810 The work done by our system-- this is our quasistatic 471 00:20:46,810 --> 00:20:51,500 processes that's going from this state to that state, but 472 00:20:51,500 --> 00:20:54,800 it's doing it in a quasistatic way, so that we can get an 473 00:20:54,800 --> 00:20:57,020 area under the curve. 474 00:20:57,020 --> 00:20:59,810 So the work done by this system is equal to-- 475 00:20:59,810 --> 00:21:00,560 I'll just write it. 476 00:21:00,560 --> 00:21:08,190 N times R over 6 times 10 to the 23, times T1 477 00:21:08,190 --> 00:21:10,460 natural log of 2. 478 00:21:10,460 --> 00:21:11,560 Fair enough. 479 00:21:11,560 --> 00:21:14,140 Let's make this into some new constant. 480 00:21:14,140 --> 00:21:18,000 For convenience, let me call it a lowercase k. 481 00:21:18,000 --> 00:21:20,830 So the work we did is equal to the number of particles we 482 00:21:20,830 --> 00:21:24,170 had, times some new constant-- we'll call that Boltzmann 483 00:21:24,170 --> 00:21:27,920 constant, so it's really just 8 divided by that. 484 00:21:27,920 --> 00:21:32,270 Times T1, times the natural log of 2. 485 00:21:32,270 --> 00:21:33,990 Fair enough. 486 00:21:33,990 --> 00:21:35,740 Now, that's only in this situation. 487 00:21:35,740 --> 00:21:38,230 The other situation did no work, right? 488 00:21:38,230 --> 00:21:41,930 So I can't talk about this system doing any work. 489 00:21:41,930 --> 00:21:43,860 But this system did do some work. 490 00:21:43,860 --> 00:21:48,270 And since it did it along an isotherm, delta-- the change 491 00:21:48,270 --> 00:21:52,410 in internal energy is equal to 0, so the change in internal 492 00:21:52,410 --> 00:21:56,580 energy, which is equal to the heat applied to the system 493 00:21:56,580 --> 00:21:59,390 minus the work done by the system-- this is going to be 494 00:21:59,390 --> 00:22:02,760 equal to 0, since our temperature didn't change. 495 00:22:02,760 --> 00:22:05,440 So the work is going to be equal to the heat 496 00:22:05,440 --> 00:22:06,970 added to the system. 497 00:22:06,970 --> 00:22:12,550 So the heat added to the system by our little reservoir 498 00:22:12,550 --> 00:22:16,050 there is going to be-- so the heat is going to be the number 499 00:22:16,050 --> 00:22:19,310 of particles we had times Boltzmann constant, times our 500 00:22:19,310 --> 00:22:21,200 temperature that we're on the isotherm, times the 501 00:22:21,200 --> 00:22:23,090 natural log of 2. 502 00:22:23,090 --> 00:22:25,060 And all this is a byproduct of the fact that 503 00:22:25,060 --> 00:22:26,900 we doubled our volume. 504 00:22:26,900 --> 00:22:32,560 Now, in the last video, I defined change in s as equal 505 00:22:32,560 --> 00:22:35,550 to Q divided by the heat added, divided by the 506 00:22:35,550 --> 00:22:37,570 temperature at which I'm adding it. 507 00:22:37,570 --> 00:22:42,790 So for this system, this quasistatic system, what was 508 00:22:42,790 --> 00:22:44,160 the change in s? 509 00:22:44,160 --> 00:22:47,750 How much did our s-term, our s-state, change by? 510 00:22:47,750 --> 00:22:52,060 So change in s is equal to heat added divided by our 511 00:22:52,060 --> 00:22:52,340 temperature. 512 00:22:52,340 --> 00:22:55,590 Our temperature is T1, so that's equal to this thing. 513 00:22:55,590 --> 00:23:02,920 Nk T1 times the natural log of 2, all of that over T1. 514 00:23:02,920 --> 00:23:05,050 So our delta-- these cancel out. 515 00:23:05,050 --> 00:23:12,170 And our change in our s-quantity is equal to Nk 516 00:23:12,170 --> 00:23:14,360 times the natural log of 2. 517 00:23:14,360 --> 00:23:18,720 Now, you should be starting to experience an a-ha moment. 518 00:23:18,720 --> 00:23:21,500 When we defined in the previous video, we were just 519 00:23:21,500 --> 00:23:23,720 playing with thermodynamics, and we said, gee, we'd really 520 00:23:23,720 --> 00:23:26,380 like to have a state variable that deals with heat, and we 521 00:23:26,380 --> 00:23:30,040 just made up this thing right here that said, change in that 522 00:23:30,040 --> 00:23:33,350 state variable is equal to the heat applied to the system 523 00:23:33,350 --> 00:23:36,320 divided by the temperature at which the heat was applied. 524 00:23:36,320 --> 00:23:40,550 And when we use that definition the change in our 525 00:23:40,550 --> 00:23:43,910 s-value from this position to this position, for a 526 00:23:43,910 --> 00:23:48,290 quasi-static process, ended up being this. 527 00:23:48,290 --> 00:23:50,870 Nk natural log of 2. 528 00:23:50,870 --> 00:23:53,100 Now, this is a state function. 529 00:23:53,100 --> 00:23:54,050 State variable. 530 00:23:54,050 --> 00:23:56,000 It's not dependent on the path. 531 00:23:56,000 --> 00:24:00,720 So any process that gets from here, that gets from this 532 00:24:00,720 --> 00:24:05,320 point to that point, has to have the same change in s. 533 00:24:05,320 --> 00:24:09,190 So the delta s for any process is going to be equal to that 534 00:24:09,190 --> 00:24:13,720 same value, which was N, in this case, k, times the 535 00:24:13,720 --> 00:24:15,090 natural log of 2. 536 00:24:15,090 --> 00:24:17,610 Any system, by our definition, right? 537 00:24:17,610 --> 00:24:18,380 It's the state variable. 538 00:24:18,380 --> 00:24:21,120 I don't care whether it disappeared, or the path was 539 00:24:21,120 --> 00:24:22,390 some crazy path. 540 00:24:22,390 --> 00:24:22,715 It's a state. 541 00:24:22,715 --> 00:24:26,260 It's only a function of that and of that, our change in s. 542 00:24:26,260 --> 00:24:29,410 So given that, even this system-- we said that this 543 00:24:29,410 --> 00:24:32,520 system that we started the video out with, it started off 544 00:24:32,520 --> 00:24:35,920 at this same V1, and it got to the same V2. 545 00:24:35,920 --> 00:24:39,670 So by the definition of the previous video, by this 546 00:24:39,670 --> 00:24:45,550 definition, its change in s is going to be the number of 547 00:24:45,550 --> 00:24:48,880 molecules times some constant times the natural log of 2. 548 00:24:48,880 --> 00:24:53,590 Now, that's the same exact result we got when we thought 549 00:24:53,590 --> 00:24:56,050 about it from a statistical point of view, when we were 550 00:24:56,050 --> 00:24:59,170 saying, how many more different states can the 551 00:24:59,170 --> 00:25:01,330 system take on? 552 00:25:01,330 --> 00:25:04,980 And what's mind-blowing here is that what we started off 553 00:25:04,980 --> 00:25:08,560 with was just kind of a nice, you know, macrostate in our 554 00:25:08,560 --> 00:25:10,850 little Carnot engine world, that we didn't really know 555 00:25:10,850 --> 00:25:11,760 what it meant. 556 00:25:11,760 --> 00:25:15,000 But we got the same exact result that when we try to do 557 00:25:15,000 --> 00:25:17,110 it from a measuring the number of states the 558 00:25:17,110 --> 00:25:18,670 system could take on. 559 00:25:18,670 --> 00:25:23,590 So all of this has been a long, two-video-winded version 560 00:25:23,590 --> 00:25:25,210 of an introduction to entropy. 561 00:25:25,210 --> 00:25:29,530 562 00:25:29,530 --> 00:25:33,660 And in thermodynamics, a change in entropy-- entropy is 563 00:25:33,660 --> 00:25:38,180 s, or I think of it, s for states-- the thermodynamic, or 564 00:25:38,180 --> 00:25:42,840 Carnot cycle, or Carnot engine world, is defined as the 565 00:25:42,840 --> 00:25:44,740 change in entropy is defined as the heat added to the 566 00:25:44,740 --> 00:25:46,230 system divided by the temperature at 567 00:25:46,230 --> 00:25:47,510 which it was added. 568 00:25:47,510 --> 00:25:52,410 Now, in our statistical mechanics world, we can define 569 00:25:52,410 --> 00:25:59,350 entropy as some constant-- and it's especially convenient, 570 00:25:59,350 --> 00:26:04,540 this is Boltzmann's constant-- some constant times the 571 00:26:04,540 --> 00:26:09,080 natural log of the number of states we have. Sometimes it's 572 00:26:09,080 --> 00:26:12,110 written as omega, sometimes other things. 573 00:26:12,110 --> 00:26:14,450 But this time, it's the number of states we have. 574 00:26:14,450 --> 00:26:17,150 And what just showed in this video is, these are equivalent 575 00:26:17,150 --> 00:26:17,910 definitions. 576 00:26:17,910 --> 00:26:18,030 or. 577 00:26:18,030 --> 00:26:21,020 At least for that one case I just showed you. 578 00:26:21,020 --> 00:26:22,490 These are equivalent definitions. 579 00:26:22,490 --> 00:26:25,580 When we used the number of states for this, how much did 580 00:26:25,580 --> 00:26:29,630 it increase, we got this result. 581 00:26:29,630 --> 00:26:32,400 And then when we used the thermodynamic definition of 582 00:26:32,400 --> 00:26:34,290 it, we got that same result. 583 00:26:34,290 --> 00:26:36,940 And if we assume that this constant is the same as that 584 00:26:36,940 --> 00:26:40,610 constant, if they're both Boltzmann's constant, both 1.3 585 00:26:40,610 --> 00:26:43,610 times 10 to the minus 23, then our definitions are 586 00:26:43,610 --> 00:26:44,540 equivalent. 587 00:26:44,540 --> 00:26:47,990 And so the intuition of entropy-- in the last one, we 588 00:26:47,990 --> 00:26:48,940 were kind of struggling with that. 589 00:26:48,940 --> 00:26:50,730 We just defined it this way, but we were like, what does 590 00:26:50,730 --> 00:26:51,990 that really mean? 591 00:26:51,990 --> 00:26:55,630 What change in entropy means, is just how many more states 592 00:26:55,630 --> 00:26:57,640 can the system take on? 593 00:26:57,640 --> 00:27:00,310 You know, sometimes when you learn it in your high school 594 00:27:00,310 --> 00:27:02,910 chemistry class, they'll call it disorder. 595 00:27:02,910 --> 00:27:04,400 And it is disorder. 596 00:27:04,400 --> 00:27:08,600 But I don't want you to think that somehow, you know, a 597 00:27:08,600 --> 00:27:11,950 messy room has higher entropy than a clean room, which some 598 00:27:11,950 --> 00:27:13,800 people sometimes use as an example. 599 00:27:13,800 --> 00:27:15,070 That's not the case. 600 00:27:15,070 --> 00:27:19,100 What you should say is, is that a stadium full of people 601 00:27:19,100 --> 00:27:23,010 has more states than a stadium without people in it. 602 00:27:23,010 --> 00:27:24,690 That has more entropy. 603 00:27:24,690 --> 00:27:26,640 Or actually, I should even be careful there. 604 00:27:26,640 --> 00:27:32,590 Let me say, a stadium at a high temperature has more 605 00:27:32,590 --> 00:27:36,180 entropy than the inside of my refrigerator. 606 00:27:36,180 --> 00:27:38,630 That the particles in that stadium have more potential 607 00:27:38,630 --> 00:27:41,910 states than the particles in my refrigerator. 608 00:27:41,910 --> 00:27:44,610 Now I'm going to leave you there, and we're going to take 609 00:27:44,610 --> 00:27:47,200 our definitions here, which I think are pretty profound-- 610 00:27:47,200 --> 00:27:50,650 this and this is the same definition-- and we're going 611 00:27:50,650 --> 00:27:54,360 to apply that to talk about the second law of 612 00:27:54,360 --> 00:27:54,960 thermodynamics. 613 00:27:54,960 --> 00:27:56,100 And actually, just a little aside. 614 00:27:56,100 --> 00:28:02,940 I wrote omega here, but in our example, this was 2 to the N. 615 00:28:02,940 --> 00:28:04,200 And so that's why it's simplified. 616 00:28:04,200 --> 00:28:13,450 This was x the first time, and then the second time, when we 617 00:28:13,450 --> 00:28:17,340 double the size of our room, or our volume, it was x to the 618 00:28:17,340 --> 00:28:18,350 N times 2 to N. 619 00:28:18,350 --> 00:28:22,420 I just want to make sure you realize what omega relates to, 620 00:28:22,420 --> 00:28:24,350 relative to what I just went through. 621 00:28:24,350 --> 00:00:00,000 Anyway, see you in the next video.