1 00:00:00,320 --> 00:00:01,880 - [Voiceover] So let's think a little bit about the 2 00:00:01,880 --> 00:00:04,130 Maxwell-Boltzmann distribution. 3 00:00:04,130 --> 00:00:05,360 And this right over here, this is 4 00:00:05,360 --> 00:00:07,540 a picture of James Clerk Maxwell. 5 00:00:07,540 --> 00:00:09,140 And I really like this picture, it's with his 6 00:00:09,140 --> 00:00:12,430 wife Katherine Maxwell and I guess this is their dog. 7 00:00:12,430 --> 00:00:15,800 And James Maxwell, he is a titan of physics 8 00:00:15,800 --> 00:00:17,550 famous for Maxwell's equations. 9 00:00:17,550 --> 00:00:19,550 He also did some of the foundational work 10 00:00:19,550 --> 00:00:22,030 on color photography and he was involved in 11 00:00:22,030 --> 00:00:23,720 thinking about, "Well, what's the distribution 12 00:00:23,720 --> 00:00:26,520 of speeds of air particles 13 00:00:26,520 --> 00:00:29,020 of idealized gas particles?" 14 00:00:29,020 --> 00:00:32,420 And this gentleman over here, this is Ludwig Boltzmann. 15 00:00:32,420 --> 00:00:34,960 And he's considered the father or one 16 00:00:34,960 --> 00:00:37,880 of the founding fathers of statistical mechanics. 17 00:00:37,880 --> 00:00:41,360 And together, through the Maxwell-Boltzman distribution 18 00:00:41,360 --> 00:00:42,480 they didn't collaborate, but they 19 00:00:42,480 --> 00:00:44,530 independently came to the same distribution. 20 00:00:44,530 --> 00:00:46,490 They were able to describe, "Well, what's the 21 00:00:46,490 --> 00:00:50,737 distribution of the speeds of air particles?" 22 00:00:50,737 --> 00:00:52,290 So let's back up a little bit or let's just 23 00:00:52,290 --> 00:00:54,100 do a little bit of a thought experiment. 24 00:00:54,100 --> 00:00:56,900 So let's say that I have a container here. 25 00:00:57,420 --> 00:00:59,240 Let's say that I have a container here. 26 00:00:59,240 --> 00:01:00,880 And let's say it has air. 27 00:01:01,290 --> 00:01:03,720 And air is actually made up mostly of nitrogen. 28 00:01:03,720 --> 00:01:05,349 Let's just say it just has only nitrogen in it 29 00:01:05,349 --> 00:01:06,778 just to simplify things. 30 00:01:06,778 --> 00:01:10,844 So let me just draw some nitrogen molecules in there. 31 00:01:10,844 --> 00:01:13,558 And let's say that I have a thermometer. 32 00:01:13,558 --> 00:01:15,590 I put a thermometer in there. 33 00:01:15,590 --> 00:01:19,936 And the thermometer 34 00:01:20,320 --> 00:01:23,810 reads a temperature of 300 Kelvin. 35 00:01:24,190 --> 00:01:27,320 What does this temperature of 300 Kelvin mean? 36 00:01:27,660 --> 00:01:29,510 Well, in our everyday life, we have 37 00:01:29,510 --> 00:01:31,730 kind of a visceral sense of temperature. 38 00:01:31,730 --> 00:01:33,200 Hey, I don't wanna touch something that's hot. 39 00:01:33,200 --> 00:01:34,630 It's going to burn me. 40 00:01:34,630 --> 00:01:38,650 Or that cold thing, it's gonna make me shiver. 41 00:01:38,650 --> 00:01:40,080 And that's how our brain 42 00:01:40,080 --> 00:01:42,300 processes this thing called temperature. 43 00:01:42,300 --> 00:01:45,760 But what's actually going on at a molecular scale? 44 00:01:45,760 --> 00:01:47,630 Well, temperature, one way to think 45 00:01:47,630 --> 00:01:49,420 about temperature, this would be a very 46 00:01:49,420 --> 00:01:51,530 accurate way to think about temperature 47 00:01:51,530 --> 00:01:52,750 is that tempera- 48 00:01:52,750 --> 00:01:54,140 I'm spelling it wrong. 49 00:01:55,270 --> 00:01:59,480 Temperature is proportional to average kinetic energy 50 00:01:59,480 --> 00:02:02,919 of the molecules in that system. 51 00:02:02,919 --> 00:02:03,900 So let me write it this way. 52 00:02:03,900 --> 00:02:08,636 Temperature is proportional to average kinetic energy. 53 00:02:08,636 --> 00:02:09,850 Average 54 00:02:10,430 --> 00:02:13,110 kinetic 55 00:02:13,110 --> 00:02:15,740 energy 56 00:02:15,740 --> 00:02:16,720 in the system. 57 00:02:18,010 --> 00:02:19,600 I'll just write average kinetic energy. 58 00:02:19,600 --> 00:02:21,952 So let's make that a little bit more concrete. 59 00:02:21,952 --> 00:02:25,810 So let's say that I have two containers. 60 00:02:25,810 --> 00:02:27,297 So it's one container. 61 00:02:27,297 --> 00:02:28,364 Whoops. 62 00:02:28,364 --> 00:02:31,480 And two containers right over here. 63 00:02:31,480 --> 00:02:33,240 And let's say they have the same 64 00:02:33,240 --> 00:02:36,060 number of molecules of nitrogen gas 65 00:02:36,060 --> 00:02:37,440 And I'm just gonna draw 10 here. 66 00:02:37,440 --> 00:02:38,830 This obviously is not realistic 67 00:02:38,830 --> 00:02:40,350 you'd have many, many more molecules. 68 00:02:40,350 --> 00:02:45,350 One, two, three, four, five, six, seven, eight, nine, ten. 69 00:02:45,648 --> 00:02:50,640 One, two, three, four, five, six, seven, eight, nine, ten. 70 00:02:51,030 --> 00:02:52,760 And let's say we know that the 71 00:02:52,760 --> 00:02:55,210 temperature here is 300 Kelvin. 72 00:02:56,440 --> 00:02:58,940 So the temperature of this system is 300 Kelvin. 73 00:02:58,940 --> 00:03:01,500 And the temperature of this system is 200 Kelvin. 74 00:03:02,040 --> 00:03:05,900 So if I wanted to visualize what these molecules are doing 75 00:03:05,900 --> 00:03:07,450 they're all moving around, they're bumping 76 00:03:07,450 --> 00:03:10,200 they don't all move together in unison. 77 00:03:10,540 --> 00:03:12,810 The average kinetic energy of the molecules 78 00:03:12,810 --> 00:03:15,010 in this system is going to be higher. 79 00:03:15,010 --> 00:03:16,440 And so maybe you have 80 00:03:16,440 --> 00:03:18,965 this molecule is moving in that direction. 81 00:03:18,965 --> 00:03:20,830 So that's its velocity. 82 00:03:20,830 --> 00:03:22,610 This one has this velocity. 83 00:03:22,610 --> 00:03:23,770 This one's going there. 84 00:03:23,770 --> 00:03:25,800 This one might not be moving much at all. 85 00:03:25,800 --> 00:03:28,100 This one might be going really fast that way. 86 00:03:28,100 --> 00:03:30,340 This one might be going super fast that way. 87 00:03:30,340 --> 00:03:32,040 This is doing that. 88 00:03:32,270 --> 00:03:33,416 This is doing that. 89 00:03:33,712 --> 00:03:35,130 This is doing that. 90 00:03:35,132 --> 00:03:37,200 So if you were to now compare it to this system 91 00:03:37,200 --> 00:03:40,750 this system, you could still have a molecule 92 00:03:40,750 --> 00:03:41,800 that is going really fast. 93 00:03:41,800 --> 00:03:43,060 Maybe this molecule is going faster 94 00:03:43,060 --> 00:03:44,800 than any of the molecules over here. 95 00:03:44,800 --> 00:03:46,840 But on average, the molecules here 96 00:03:46,840 --> 00:03:48,730 have a lower kinetic energy. 97 00:03:48,730 --> 00:03:50,780 So this one maybe is doing this. 98 00:03:50,780 --> 00:03:52,750 I'm going to see if I can draw... 99 00:03:52,750 --> 00:03:56,520 On average, they're going to have a lower kinetic energy. 100 00:03:56,520 --> 00:03:58,110 That doesn't mean all of these molecules 101 00:03:58,110 --> 00:04:00,110 are necessarily slower than all of these molecules 102 00:04:00,110 --> 00:04:02,160 or have lower kinetic energy than all of these molecules. 103 00:04:02,160 --> 00:04:06,901 But on average they're going to have less kinetic energy. 104 00:04:06,901 --> 00:04:08,930 And we can actually draw a distribution. 105 00:04:08,930 --> 00:04:10,600 And this distribution, that is 106 00:04:10,600 --> 00:04:13,382 the Maxwell-Boltzmann distribution. 107 00:04:13,382 --> 00:04:15,110 So if we... 108 00:04:15,110 --> 00:04:18,350 Let me draw a little coordinate plane here. 109 00:04:18,850 --> 00:04:23,850 So let me draw a coordinate plane. 110 00:04:24,260 --> 00:04:28,160 So, if on this axis, I were to put speed. 111 00:04:28,670 --> 00:04:30,140 If I were to put speed. 112 00:04:30,140 --> 00:04:33,532 And on this axis, I would put number of molecules. 113 00:04:33,532 --> 00:04:38,340 Number of molecules. 114 00:04:38,690 --> 00:04:39,931 Right over here. 115 00:04:39,931 --> 00:04:43,310 For this system, the system that is at 300 Kelvin 116 00:04:43,310 --> 00:04:46,092 the distribution might look like this. 117 00:04:46,092 --> 00:04:47,831 So it might look 118 00:04:47,831 --> 00:04:48,910 the distribution... 119 00:04:48,910 --> 00:04:50,340 Let me do this in a new color. 120 00:04:50,690 --> 00:04:53,120 So, the distribution 121 00:04:53,120 --> 00:04:54,840 this is gonna be all of the molecules. 122 00:04:54,840 --> 00:04:59,330 The distribution might look like this. 123 00:04:59,700 --> 00:05:00,890 Might look like this. 124 00:05:00,890 --> 00:05:02,920 And this would actually be the Maxwell-Boltzmann 125 00:05:02,920 --> 00:05:05,172 distribution for this system 126 00:05:05,172 --> 00:05:07,928 For system, let's call this system A. 127 00:05:07,928 --> 00:05:10,428 System A, right over here. 128 00:05:10,428 --> 00:05:14,710 And this system, that has a lower temperature 129 00:05:14,710 --> 00:05:17,238 which means it also has a lower kinetic energy. 130 00:05:17,238 --> 00:05:19,676 The distribution of its particles... 131 00:05:19,676 --> 00:05:22,615 So the most likely, the most probable... 132 00:05:22,615 --> 00:05:24,710 You're going to have the highest number of molecules 133 00:05:24,710 --> 00:05:25,900 at a slower speed. 134 00:05:25,900 --> 00:05:27,330 Let's say you're gonna have it at this speed 135 00:05:27,330 --> 00:05:28,603 right over here. 136 00:05:28,603 --> 00:05:33,600 So its distribution might look something like this. 137 00:05:35,000 --> 00:05:37,010 So it might look something like that. 138 00:05:37,740 --> 00:05:38,980 Now why is this one... 139 00:05:38,980 --> 00:05:40,310 It might make sense to you that 140 00:05:40,310 --> 00:05:42,020 okay, the most probable 141 00:05:42,020 --> 00:05:45,500 the speed at which I have the most molecules 142 00:05:45,500 --> 00:05:48,460 I get that that's going to be lower than the speed 143 00:05:48,460 --> 00:05:51,310 at which I have the most molecules in system A 144 00:05:51,310 --> 00:05:54,280 because I have, because on average 145 00:05:54,280 --> 00:05:56,050 these things have less kinetic energy. 146 00:05:56,050 --> 00:05:58,235 They're going to have less speed. 147 00:05:58,235 --> 00:06:00,184 But why is this peak higher? 148 00:06:00,184 --> 00:06:01,550 Well, you gotta remember we're talking about 149 00:06:01,550 --> 00:06:03,080 the same number of molecules. 150 00:06:03,080 --> 00:06:05,350 So if we have the same number of molecules that means 151 00:06:05,350 --> 00:06:07,910 that the areas under these curves need to be the same. 152 00:06:07,910 --> 00:06:10,900 So if this one is narrower, it's going to be taller. 153 00:06:10,900 --> 00:06:12,760 And if I were gonna, if I were to somehow 154 00:06:12,760 --> 00:06:15,380 raise the temperature of this system even more. 155 00:06:15,380 --> 00:06:17,780 Let's say I create a third system or I get this 156 00:06:17,780 --> 00:06:20,020 or let's say I were to heat it up to 400 Kelvin. 157 00:06:20,020 --> 00:06:24,120 Well then my distribution would look 158 00:06:24,120 --> 00:06:27,470 something like this. 159 00:06:27,470 --> 00:06:29,940 So this is if I heated it up. 160 00:06:30,270 --> 00:06:32,800 Heated up. 161 00:06:32,800 --> 00:06:35,620 And so this is all the Maxwell-Boltzmann distribution is. 162 00:06:35,620 --> 00:06:39,800 I'm not giving you the more involved, hairy equation for it 163 00:06:39,800 --> 00:06:42,128 but really the idea of what it is. 164 00:06:42,128 --> 00:06:43,530 It's a pretty neat idea. 165 00:06:43,530 --> 00:06:46,010 And actually when you actually think about the actual 166 00:06:46,010 --> 00:06:49,690 speeds of some of these particles, even the air around you 167 00:06:49,690 --> 00:06:52,060 I'm gonna say, "Oh, it looks pretty stationary to me." 168 00:06:52,060 --> 00:06:54,700 But it turns out in the air around you is mostly nitrogen. 169 00:06:54,700 --> 00:06:59,000 That the most probable speed of 170 00:06:59,000 --> 00:07:00,410 if you picked a random nitrogen 171 00:07:00,410 --> 00:07:02,618 molecule around you right now. 172 00:07:02,618 --> 00:07:04,340 So the most probable speed. 173 00:07:04,340 --> 00:07:05,160 I'm gonna write this down 174 00:07:05,160 --> 00:07:07,039 'cause this is pretty mindblowing. 175 00:07:07,039 --> 00:07:10,780 Most probable speed at room temperature. 176 00:07:10,784 --> 00:07:15,780 Probable speed 177 00:07:16,197 --> 00:07:20,154 of N2 at room temperature. 178 00:07:20,154 --> 00:07:23,960 Room temperature. 179 00:07:25,230 --> 00:07:27,400 So let's say this that this was the Maxwell-Boltzmann 180 00:07:27,400 --> 00:07:30,960 distribution for nitrogen at room temperature. 181 00:07:30,960 --> 00:07:32,620 Let's say that that's, let's say we make 182 00:07:32,620 --> 00:07:35,450 we call room temperature 300 Kelvin. 183 00:07:35,450 --> 00:07:37,870 This most probable speed right over here 184 00:07:37,870 --> 00:07:40,030 the one where we have the most molecules 185 00:07:40,030 --> 00:07:41,400 the one where we're gonna have the most 186 00:07:41,400 --> 00:07:43,070 molecules at that speed. 187 00:07:43,630 --> 00:07:45,990 In fact, guess what that is going to be before I tell you 188 00:07:45,990 --> 00:07:47,635 'cause it's actually mind boggling. 189 00:07:47,635 --> 00:07:49,930 Well, it turns out that it is approximately 190 00:07:49,930 --> 00:07:53,690 400, 400 and actually at 300 Kelvin 191 00:07:53,690 --> 00:07:57,846 it's gonna be 422 meters per second. 192 00:07:57,846 --> 00:08:00,510 422 meters per second. 193 00:08:00,510 --> 00:08:04,220 Imagine something traveling 422 meters in a second. 194 00:08:04,220 --> 00:08:06,700 And if you're used to thinking in terms of miles per hour 195 00:08:06,700 --> 00:08:09,860 this is approximately 944 196 00:08:09,860 --> 00:08:12,730 miles per hour. 197 00:08:12,730 --> 00:08:14,927 So right now, around you 198 00:08:14,927 --> 00:08:16,850 you have, actually 199 00:08:16,850 --> 00:08:20,340 the most probable, the highest number 200 00:08:20,340 --> 00:08:22,300 of the nitrogen molecules around you 201 00:08:22,300 --> 00:08:24,900 are traveling at roughly this speed 202 00:08:24,900 --> 00:08:26,280 and they're bumping into you. 203 00:08:26,280 --> 00:08:27,860 That's actually what's giving you air pressure. 204 00:08:27,860 --> 00:08:29,480 And not just that speed, there are actually ones 205 00:08:29,480 --> 00:08:31,190 that are travelling even faster than that. 206 00:08:31,190 --> 00:08:34,570 Even faster than 422 meters per second. 207 00:08:34,570 --> 00:08:35,210 Even faster. 208 00:08:35,210 --> 00:08:37,730 There's particles around you traveling faster 209 00:08:37,730 --> 00:08:39,210 than a thousand miles per hour 210 00:08:39,210 --> 00:08:41,980 and they are bumping into your body as we speak. 211 00:08:41,980 --> 00:08:43,650 And you might say, "Well, why doesn't that hurt?" 212 00:08:43,650 --> 00:08:47,040 Well, that gives you a sense of how small the mass 213 00:08:47,040 --> 00:08:48,870 of a nitrogen molecule is, that it can 214 00:08:48,870 --> 00:08:51,640 bump into you at a thousand miles per hour 215 00:08:51,640 --> 00:08:52,640 and you really don't feel it. 216 00:08:52,640 --> 00:08:55,554 It feels just like the ambient air pressure. 217 00:08:55,554 --> 00:08:57,030 Now, when you first look at this, you're like 218 00:08:57,030 --> 00:08:59,300 wait, 422 meters per second? 219 00:08:59,300 --> 00:09:01,510 That's faster than the speed of sound. 220 00:09:01,510 --> 00:09:04,770 The speed of sound is around 340 meters per second. 221 00:09:04,770 --> 00:09:06,000 Well, how can this be? 222 00:09:06,000 --> 00:09:06,860 Well, just think about it. 223 00:09:06,860 --> 00:09:09,220 Sound is transmitted through the air 224 00:09:09,220 --> 00:09:10,970 through collisions of particles. 225 00:09:10,970 --> 00:09:13,420 So the particles themselves have to be moving 226 00:09:13,420 --> 00:09:15,300 or at least some of them, have to be moving 227 00:09:15,300 --> 00:09:16,780 faster than the speed of sound. 228 00:09:16,780 --> 00:09:18,960 So, not all of the things around you 229 00:09:18,960 --> 00:09:19,910 are moving this fast and they're 230 00:09:19,910 --> 00:09:20,810 moving in all different directions. 231 00:09:20,810 --> 00:09:23,298 Some of them might not be moving much at all. 232 00:09:23,298 --> 00:09:26,520 But some of them are moving quite incredibly fast. 233 00:09:26,520 --> 00:00:00,000 So, I don't know, I find that a little bit mindblowing.