WEBVTT 00:01.267 --> 00:07.327 RONALD SMITH: Last time, we talked about how planets retain their 00:07.333 --> 00:08.103 atmospheres. 00:08.100 --> 00:12.230 And I wonder if there are any questions about that 00:12.233 --> 00:16.133 discussion we had last time? 00:16.133 --> 00:20.673 Just to review, we defined the escape velocity, we defined 00:20.667 --> 00:23.967 the molecular speed, and then we talked about the 00:23.967 --> 00:27.267 relationship between those two in regards to how an 00:27.267 --> 00:30.327 atmosphere can be retained by a planet. 00:30.333 --> 00:31.573 Anything on that? 00:34.467 --> 00:37.267 OK, let's get started then with a new subject. 00:37.267 --> 00:41.097 I promised I would spend a few minutes beginning today to 00:41.100 --> 00:44.230 talk about the system of units we're going to be using. 00:44.233 --> 00:46.403 So if you don't object, I'm going to spend about 10 00:46.400 --> 00:49.000 minutes or so on that. 00:49.000 --> 00:50.970 I'm sure it's a review for most of you, but we're going 00:50.967 --> 00:54.497 to talk about this thing called the SI system of units. 00:58.567 --> 01:01.867 We'll be using that throughout the course primarily, although 01:01.867 --> 01:05.197 there are some traditional units that come up in 01:05.200 --> 01:08.730 meteorology and oceanography that don't necessarily fit 01:08.733 --> 01:12.033 into this SI system. 01:12.033 --> 01:13.473 SI is-- 01:13.467 --> 01:14.127 well, it's French. 01:14.133 --> 01:15.933 It's Systeme International. 01:15.933 --> 01:18.503 We just say the International System of Units. 01:18.500 --> 01:20.600 It might also be called the metric system. 01:23.433 --> 01:25.033 And you'll see why in just a minute. 01:25.033 --> 01:27.073 So the three basic-- 01:27.067 --> 01:30.367 or the foundation blocks for the SI system of units are 01:30.367 --> 01:39.667 mass, for which we use the unit kilograms, length, for 01:39.667 --> 01:45.867 which we use the unit meter, and time, for which we use the 01:45.867 --> 01:48.567 unit seconds. 01:48.567 --> 01:51.667 Now, there are some other fundamentals that involve 01:51.667 --> 01:54.197 electric field strength, magnetic field strength. 01:54.200 --> 01:55.270 We're not going to be using those. 01:55.267 --> 01:58.297 So I'm going to just work with these three and then see what 01:58.300 --> 02:01.600 we can build up based on this foundation. 02:01.600 --> 02:06.330 So with these as a foundation, we've got a bunch of really 02:06.333 --> 02:07.803 simple things we can write down. 02:07.800 --> 02:12.700 For example, speed, the rate at which something is moving, 02:12.700 --> 02:17.200 is going to be meters per second, just taking the length 02:17.200 --> 02:19.230 of meters and the time of seconds. 02:19.233 --> 02:24.273 By the way, in current scientific convention, it's 02:24.267 --> 02:29.167 more appropriate to write this m, seconds to the minus one. 02:29.167 --> 02:30.297 Sometimes I'll use this. 02:30.300 --> 02:32.200 Sometimes I'll use that. 02:32.200 --> 02:34.770 Today, this is the more preferred. 02:34.767 --> 02:39.497 In the scientific literature, you'll normally find this 02:39.500 --> 02:46.170 inverse operator used to indicate that it's per second. 02:46.167 --> 02:53.597 Acceleration, of course, is going to be meters per second 02:53.600 --> 02:57.400 per second, so we could write that as meters, second 02:57.400 --> 02:58.930 to the minus two. 02:58.933 --> 03:01.703 It's how fast is the speed changing. 03:04.567 --> 03:10.897 A few other easy ones, area would be meters squared. 03:10.900 --> 03:15.070 Volume would be meters cubed. 03:18.567 --> 03:20.667 And then we'll move into the ones that are 03:20.667 --> 03:21.927 a little bit trickier. 03:21.933 --> 03:25.233 Let's start with force. 03:25.233 --> 03:33.833 Now, the SI unit of force is the Newton, named after Sir 03:33.833 --> 03:36.273 Isaac, of course. 03:36.267 --> 03:38.927 But it's not a fundamental unit. 03:38.933 --> 03:40.573 We can derive it from these. 03:40.567 --> 03:42.667 And the way I do that is just to remember Newton's 03:42.667 --> 03:47.197 Law, F equals ma. 03:47.200 --> 03:51.370 So the unit of force, which is a Newton, can also be written 03:51.367 --> 03:53.797 as the product of-- 03:53.800 --> 03:55.030 well, I can write it this way. 03:55.033 --> 04:01.303 It's going to be kilograms, meters, per second squared. 04:01.300 --> 04:03.100 So that's another way to write a Newton. 04:06.133 --> 04:09.473 We could do a pressure. 04:09.467 --> 04:11.827 Pressure is going to be the subject of 04:11.833 --> 04:14.203 most of today's lecture. 04:14.200 --> 04:22.630 A pressure is a force per unit area, a force per unit area. 04:22.633 --> 04:27.873 The SI name for it is the Pascal, named after the French 04:27.867 --> 04:30.867 scientist who made a fundamental breakthrough in 04:30.867 --> 04:33.367 understanding pressure. 04:33.367 --> 04:37.267 And of course, we can write it in terms of the three building 04:37.267 --> 04:41.067 blocks by realizing, if that's a force per unit area, then a 04:41.067 --> 04:43.767 pascal is going to be a kilogram, 04:43.767 --> 04:46.897 meter per second squared. 04:46.900 --> 04:51.170 And then per area, so it's meters to the minus two. 04:51.167 --> 04:52.267 Let's simplify that. 04:52.267 --> 04:56.397 So it's kilograms, meters to the minus one, seconds 04:56.400 --> 04:58.970 to the minus two. 04:58.967 --> 05:02.497 That is the pressure unit called the Pascal. 05:06.800 --> 05:20.500 Energy, well, the way I think of energy is that it's the 05:20.500 --> 05:25.100 amount of work done as I push something along. 05:25.100 --> 05:27.930 It's usually the product of the force times the distance. 05:31.100 --> 05:33.800 Force pushing over some distance is 05:33.800 --> 05:35.330 an amount of energy. 05:35.333 --> 05:38.033 Well, that makes it easy if you can remember that, because 05:38.033 --> 05:41.433 then I can take the unit of force-- 05:41.433 --> 05:44.103 oh, by the way, the unit of energy in the SI system is 05:44.100 --> 05:50.870 going to be the Joule, J-O-U-L-E. And of course, that 05:50.867 --> 05:55.367 is going to be this times an extra distance, so it's going 05:55.367 --> 05:59.497 to be kilograms, meters squared, seconds 05:59.500 --> 06:01.530 to the minus two. 06:01.533 --> 06:02.773 That will be a Joule. 06:06.467 --> 06:14.767 Power, well, power is the rate of expending energy. 06:14.767 --> 06:18.697 How fast are you using or creating energy? 06:18.700 --> 06:23.770 The SI system unit for power is the familiar watt. 06:23.767 --> 06:28.127 You'll find it stamped on your light bulbs. 06:28.133 --> 06:29.273 It's a power unit. 06:29.267 --> 06:32.427 And we know immediately what that's going to be, because we 06:32.433 --> 06:34.433 have energy here. 06:34.433 --> 06:39.903 But this is per unit time, so that's going to be kilograms, 06:39.900 --> 06:43.170 meters squared, second to the minus three. 06:43.167 --> 06:47.167 I've changed that two to a three to get it into a per 06:47.167 --> 06:49.727 unit time system. 06:49.733 --> 06:51.233 Any questions on this yet? 06:54.667 --> 06:58.027 Well, one we'll be using today also is the mass density. 07:04.000 --> 07:07.430 It's how much mass of a fluid or an object is 07:07.433 --> 07:09.033 there per unit volume. 07:09.033 --> 07:11.033 It's a mass per unit volume. 07:11.033 --> 07:13.073 So this is a really trivia one. 07:13.067 --> 07:19.097 There's no name for it, but it's going to be kilograms per 07:19.100 --> 07:22.030 meter cubed. 07:22.033 --> 07:26.403 Or kilograms, meter to the minus three. 07:29.167 --> 07:31.627 So that's not bad once you get the hang of it. 07:31.633 --> 07:34.803 And what you're going to be doing in this course is to be 07:34.800 --> 07:38.100 using various formulas, computing quantities. 07:38.100 --> 07:43.230 And to improve your odds of getting it right, I recommend 07:43.233 --> 07:46.573 that you check the units on every calculation that you do. 07:46.567 --> 07:49.327 If the units don't work out right, then your numerical 07:49.333 --> 07:51.673 answer's going to be wrong as well. 07:51.667 --> 07:53.067 So let me give you an example of that. 07:53.067 --> 07:54.627 The one we're going to be working on today is the 07:54.633 --> 07:56.033 perfect gas law. 07:56.033 --> 07:59.903 One way to write it is P equals rho RT. 08:02.600 --> 08:09.300 P is the pressure, rho is the density, the mass density, R 08:09.300 --> 08:11.630 is the gas constant, and T is the temperature. 08:11.633 --> 08:13.673 Now, let me write out the units for this. 08:13.667 --> 08:15.897 Pressure, we already know-- 08:15.900 --> 08:16.930 where did I put pressure? 08:16.933 --> 08:18.203 There it is. 08:18.200 --> 08:24.670 Pressure is kilograms, meters to the minus one, seconds to 08:24.667 --> 08:27.097 the minus two. 08:27.100 --> 08:30.500 Now, that should be equal to the product of the units of 08:30.500 --> 08:31.530 all these other things. 08:31.533 --> 08:34.233 The units of mass density we've already said are 08:34.233 --> 08:40.003 kilograms per meter cubed. 08:40.000 --> 08:42.370 The units of the gas constant I'll give you. 08:42.367 --> 08:48.397 It's joules per kilogram per degree Kelvin. 08:48.400 --> 08:53.000 And the temperature will be in Kelvins as well. 08:53.000 --> 08:54.870 So is that going to cancel out? 08:54.867 --> 08:57.867 Well, it's not completely clear yet, because we haven't 08:57.867 --> 09:01.097 taken apart this joule yet to see what's inside that. 09:01.100 --> 09:03.700 But it looks like we're going to get rid of the kilograms 09:03.700 --> 09:07.000 OK, and it looks like the Kelvins are 09:07.000 --> 09:08.300 going to cancel out. 09:08.300 --> 09:10.430 But what is in that joule? 09:10.433 --> 09:12.903 Well, that joule is here. 09:12.900 --> 09:21.230 It is a kilogram, meter squared per second squared. 09:21.233 --> 09:24.303 And it looks like that's going to work, because we've got a 09:24.300 --> 09:26.500 meters cubed downstairs. 09:26.500 --> 09:28.670 That's going to take that meters squared and make it 09:28.667 --> 09:32.527 into a meters to the minus one, and then that's going to 09:32.533 --> 09:36.573 exactly balance with the left-hand side. 09:36.567 --> 09:39.197 You see how that works? 09:39.200 --> 09:42.470 So this is a calculation that should be going on in the 09:42.467 --> 09:47.927 background whenever you are working on a numerical problem 09:47.933 --> 09:50.033 to be sure you've got the units right. 09:52.700 --> 09:53.970 No questions on that? 09:57.267 --> 10:01.327 Well, the focus today is talking about pressure and the 10:01.333 --> 10:02.633 perfect gas law. 10:13.200 --> 10:14.470 How many of you have seen the perfect gas 10:14.467 --> 10:17.927 law before in courses? 10:17.933 --> 10:20.803 Most of you. 10:20.800 --> 10:27.970 Well, let's first imagine a box full of gas molecules, but 10:27.967 --> 10:29.897 they're moving around. 10:29.900 --> 10:32.570 We know what the typical molecular speed is. 10:32.567 --> 10:35.097 They're colliding off each other, but they're also 10:35.100 --> 10:39.770 occasionally bouncing off the wall of that chamber. 10:39.767 --> 10:42.997 And every time they do that, they impart a little bit of 10:43.000 --> 10:46.370 force to the wall that they bounce off of, and that's 10:46.367 --> 10:47.067 called pressure. 10:47.067 --> 10:51.567 It's the repeated bouncing of molecules off the side of a 10:51.567 --> 10:55.067 box that gives rise to this quantity we call pressure. 10:55.067 --> 10:57.227 It's going to depend, of course, on the number of 10:57.233 --> 11:14.833 molecules, their speed, and their mass, in principle. 11:14.833 --> 11:18.173 At least, it might depend on these things. 11:18.167 --> 11:22.167 Now, if you've taken a course in chemistry, you probably 11:22.167 --> 11:25.667 have seen the perfect gas law written this way: pressure 11:25.667 --> 11:27.697 times volume equals mRT. 11:31.500 --> 11:33.770 That's probably the most familiar way to write the 11:33.767 --> 11:37.427 perfect gas law in a chemistry course. 11:37.433 --> 11:43.473 Here, P is the pressure, V is the volume of the container 11:43.467 --> 11:47.897 that you have it in, m is the number of moles-- 11:47.900 --> 11:49.770 let me write this out, number of moles. 11:56.900 --> 11:58.370 This is the gas constant. 12:02.967 --> 12:05.127 That's the temperature, of course, in Kelvins. 12:10.533 --> 12:14.703 That's the volume of the container. 12:14.700 --> 12:15.970 And that's the pressure. 12:23.200 --> 12:26.470 Do you remember what a mole is? 12:26.467 --> 12:30.527 A mole is a certain number of molecules. 12:30.533 --> 12:31.773 Avogadro's number-- 12:41.533 --> 12:55.403 which is, if I remember, it's 6.02 times 10 to the 23-- 12:55.400 --> 12:58.700 is a number of molecules of any gas in a mole. 12:58.700 --> 13:04.630 So this would be the number of moles you have in that box. 13:04.633 --> 13:08.003 The interesting thing about this formula is that it seems 13:08.000 --> 13:12.570 to be independent of the mass of the molecule. 13:12.567 --> 13:15.727 While I speculated that this might depend on the mass, it 13:15.733 --> 13:20.773 turns out that it doesn't depend on the mass. 13:20.767 --> 13:22.797 You would think that a molecule that has a heavier 13:22.800 --> 13:26.530 mass would impart more force as it bounces off the wall. 13:26.533 --> 13:29.873 But you may remember from last time that at a given 13:29.867 --> 13:35.297 temperature, heavier molecules move more slowly. 13:35.300 --> 13:38.800 So in fact, those two factors cancel out in 13:38.800 --> 13:41.330 the perfect gas law. 13:41.333 --> 13:44.573 So the pressure you get depends only on the number of 13:44.567 --> 13:46.997 molecules that you have, not on the 13:47.000 --> 13:48.670 mass of those molecules. 13:48.667 --> 13:52.397 That's a bit of a surprise, so be aware of that. 13:55.733 --> 13:57.503 What gets a little bit confusing is that in 13:57.500 --> 13:59.970 atmospheric science, we don't use the perfect 13:59.967 --> 14:02.727 gas law in this form. 14:02.733 --> 14:05.103 So I'm going to give you the form in which we will be using 14:05.100 --> 14:06.800 it in this class. 14:12.133 --> 14:13.633 Stop me if you have questions. 14:25.733 --> 14:27.973 We're going to write the perfect gas law as P equals 14:27.967 --> 14:41.397 rho RT, where this is the pressure again, this is the 14:41.400 --> 14:48.470 mass density, that is the gas constant 14:48.467 --> 14:50.327 for the gas in question-- 14:50.333 --> 14:56.333 we call it the specific gas constant, not the universal 14:56.333 --> 14:57.273 gas constant-- 14:57.267 --> 15:00.727 and that again is the temperature in degrees Kelvin. 15:00.733 --> 15:02.603 What we've done here-- 15:02.600 --> 15:05.000 I think you can see it if you compare the two-- 15:05.000 --> 15:08.500 basically we've said that the air density, the mass density 15:08.500 --> 15:09.770 is going to be-- 15:12.100 --> 15:17.230 it's going to be the number of moles per unit volume times 15:17.233 --> 15:18.473 the molecular weight. 15:20.833 --> 15:24.703 So the more molecules you have and the heavier each molecule 15:24.700 --> 15:27.300 is in a given volume, that's going to 15:27.300 --> 15:28.930 determine the mass density. 15:28.933 --> 15:34.103 So I've used this formula, if you like, to rewrite that so 15:34.100 --> 15:37.070 that it looks like this form that we use 15:37.067 --> 15:38.267 in atmospheric science. 15:38.267 --> 15:38.727 Question? 15:38.733 --> 15:40.133 STUDENT: I have a question. 15:40.133 --> 15:42.673 For Avogadro's number, is it 10 to the negative 3 or 15:42.667 --> 15:42.967 negative 23? 15:42.967 --> 15:46.467 PROFESSOR: 10-- what did I write there? 15:46.467 --> 15:48.297 Oh, thank you. 15:48.300 --> 15:50.600 10 to the 23. 15:50.600 --> 15:51.600 10 to the plus 23. 15:51.600 --> 15:53.530 That's the number of molecules in a-- 15:53.533 --> 15:54.903 thanks very much. 15:54.900 --> 15:57.430 Yeah, that's your job out there, to keep 15:57.433 --> 15:58.703 me honest on this. 16:01.600 --> 16:05.130 So what is this specific gas constant, then? 16:05.133 --> 16:07.873 If you follow the math through, you can see that 16:07.867 --> 16:14.097 we've defined the specific gas constant as being the 16:14.100 --> 16:19.300 universal gas constant divided by the molecular weight. 16:19.300 --> 16:21.800 So when you're using air, that'll be one number. 16:21.800 --> 16:23.730 When you're using hydrogen, that'll be a different 16:23.733 --> 16:25.203 number, and so on. 16:25.200 --> 16:27.900 So what's the advantage of this? 16:27.900 --> 16:29.970 We seem like we've made things more complicated, because we 16:29.967 --> 16:32.427 no longer have a universal gas constant. 16:32.433 --> 16:35.673 It's because we want to get at this mass density. 16:35.667 --> 16:37.327 That's important in atmospheric science. 16:37.333 --> 16:40.633 We want to know, how dense is the air? 16:40.633 --> 16:41.673 And that's why we want it. 16:41.667 --> 16:44.227 We don't want to work in terms of number of molecules. 16:44.233 --> 16:48.573 We want to work in terms of the mass of the air. 16:48.567 --> 16:49.567 So let's do an example. 16:49.567 --> 16:54.127 First of all, for air, then, let me put it 16:54.133 --> 16:56.903 subscript air there. 16:56.900 --> 17:01.500 The average molecular weight for air is 29. 17:01.500 --> 17:04.870 This is 8,314. 17:04.867 --> 17:08.067 And so that turns out to be roughly 287. 17:08.067 --> 17:10.667 And the units on that are joules per 17:10.667 --> 17:14.997 kilogram per Kelvin. 17:15.000 --> 17:19.900 So that's the specific gas constant for air, which is the 17:19.900 --> 17:24.230 gas we have most abundantly, of course, in our atmosphere. 17:24.233 --> 17:26.173 Let's work out a quick example of that. 17:29.633 --> 17:30.633 Let's say-- 17:30.633 --> 17:32.703 and I'll try to make it somewhat similar to this 17:32.700 --> 17:35.470 room-- let's say the temperature is 15 degrees 17:35.467 --> 17:44.397 Celsius and the air density, I somehow know that's 1.2 17:44.400 --> 17:48.500 kilograms per cubic meter. 17:48.500 --> 17:50.830 What will be the pressure? 17:50.833 --> 17:54.633 Well, first of all, we have to convert this to Kelvins, so 17:54.633 --> 18:01.033 it's going to be 15 plus 273.1. 18:01.033 --> 18:05.233 That's going to be about 288.1. 18:05.233 --> 18:08.473 And then we're ready to plug it into the formula that's 18:08.467 --> 18:22.697 going to be 1.2 times 287 times 288.1. 18:22.700 --> 18:29.300 And that comes out to be 99,221.7 pascals. 18:29.300 --> 18:30.770 The unit on that is going to be pascals. 18:33.467 --> 18:39.027 1.2, which is the air density, the specific gas constant, 18:39.033 --> 18:45.973 287, and the temperature expressed in Kelvins, 288.1. 18:45.967 --> 18:47.197 Questions on that? 18:50.733 --> 18:53.303 Now, what good is this? 18:53.300 --> 18:57.100 This is a very useful formula, but it's not as useful as one 18:57.100 --> 18:59.070 might think in every application. 18:59.067 --> 19:04.767 First of all, for air, we can take that as known. 19:04.767 --> 19:07.497 But in general, as you move around the atmosphere, the 19:07.500 --> 19:10.870 other three things will be changing. 19:10.867 --> 19:16.027 And so if I know one of these, like temperature, well, that 19:16.033 --> 19:18.733 formula's pretty useless, because I don't know either of 19:18.733 --> 19:20.733 the other two. 19:20.733 --> 19:26.433 So this formula is best used when you know two of those 19:26.433 --> 19:28.833 quantities and need to get the third. 19:28.833 --> 19:31.733 For example, if you knew density and temperature, that 19:31.733 --> 19:33.133 would give you pressure. 19:33.133 --> 19:35.333 If you knew pressure and density, you could solve that 19:35.333 --> 19:36.903 for temperature, and so on. 19:36.900 --> 19:40.600 So it's useful, but it's not everything we would like. 19:40.600 --> 19:41.100 Question? 19:41.100 --> 19:44.200 STUDENT: Just going back to the pascals, do you want us to 19:44.200 --> 19:47.500 express it in pascals or kilograms per meter cubed? 19:47.500 --> 19:51.570 PROFESSOR: For pressure, you should express it in pascals. 19:51.567 --> 19:59.827 Or what is sometimes a more frequent unit in meteorology 19:59.833 --> 20:01.103 is a millibar. 20:03.033 --> 20:10.203 Millibar, which is sometimes written as a hectopascal, 20:10.200 --> 20:12.100 which is one one-hundredth of a pascal [correction: one 20:12.100 --> 20:13.170 hundred Pascals]. 20:13.167 --> 20:19.927 So this would be 992.217 hectopascals. 20:22.833 --> 20:26.073 In the meteorological literature, you'll often find 20:26.067 --> 20:29.727 hectopascals used instead of pascals. 20:29.733 --> 20:30.833 It's easy to do the conversion. 20:30.833 --> 20:34.073 Just divide by 100 if you're going that way, or multiply by 20:34.067 --> 20:39.967 100 if you're going that way. 20:39.967 --> 20:44.867 Now, there is an application, a direct application, for the 20:44.867 --> 20:48.097 perfect gas law that I'm going to show you now that is really 20:48.100 --> 20:52.330 of fundamental importance for how the atmosphere works. 20:52.333 --> 20:55.003 And so I'm going to go through this a little carefully, 20:55.000 --> 20:59.600 because it is something we'll be meeting over and over again 20:59.600 --> 21:01.730 in the course. 21:01.733 --> 21:06.773 And there's a very simple idea that I'm sure you are aware 21:06.767 --> 21:11.667 of, and that is warm air rises, and cold air sinks. 21:14.533 --> 21:17.273 I'd like to actually prove that to you. 21:17.267 --> 21:19.497 It seems like a trivial thing, but I'd like to 21:19.500 --> 21:21.400 prove that to you. 21:21.400 --> 21:24.130 And to do that, I'm going to have to define something 21:24.133 --> 21:26.303 called the buoyancy force. 21:34.933 --> 21:49.903 The buoyancy force is a pressure force on an object 21:49.900 --> 22:00.300 immersed in a liquid or a fluid in a gravity field. 22:05.900 --> 22:09.000 Now, this is very easy to imagine, because if you've 22:09.000 --> 22:14.600 ever taken a basketball or a beach ball into a pool and 22:14.600 --> 22:18.800 tried to push it down in the water, you know there's a 22:18.800 --> 22:24.900 rather large force resisting that trying to make that ball 22:24.900 --> 22:26.770 quickly lift back up to the top. 22:26.767 --> 22:29.597 That's the buoyancy force. 22:29.600 --> 22:34.200 And for example, here's the top of the water. 22:34.200 --> 22:38.100 There's your basketball or your beach ball. 22:38.100 --> 22:40.400 You're trying to hold it down there with your hand. 22:40.400 --> 22:43.230 There's something very strong pushing it up. 22:43.233 --> 22:45.703 What's pushing that ball up? 22:45.700 --> 22:49.400 What's the physics of that? 22:49.400 --> 22:49.870 Anybody? 22:49.867 --> 22:52.867 What's pushing that ball up? 22:52.867 --> 22:53.327 Yeah. 22:53.333 --> 22:54.833 STUDENT: The displaced water? 22:54.833 --> 22:55.603 PROFESSOR: Yes. 22:55.600 --> 22:56.870 But how does it work? 22:59.800 --> 23:00.630 In the back? 23:00.633 --> 23:02.203 STUDENT: Is it because the ball is less dense 23:02.200 --> 23:03.070 than the water is? 23:03.067 --> 23:05.967 PROFESSOR: The ball is less dense than the water, but I'm 23:05.967 --> 23:07.967 looking for a more detailed mechanism. 23:07.967 --> 23:08.397 Yes. 23:08.400 --> 23:14.000 STUDENT: The same amount of force is the weight of the 23:14.000 --> 23:14.470 baseball bat? 23:14.467 --> 23:15.367 PROFESSOR: Yes. 23:15.367 --> 23:17.267 But actually how does it act? 23:17.267 --> 23:17.997 What's the physics? 23:18.000 --> 23:21.030 How is acting on that ball? 23:21.033 --> 23:21.673 So you're right. 23:21.667 --> 23:23.827 It depends on the water displaced. 23:23.833 --> 23:25.333 That's going to be Archimedes' Law. 23:25.333 --> 23:27.203 I'm going to put that on the board in just a moment. 23:27.200 --> 23:27.630 Yes. 23:27.633 --> 23:30.433 STUDENT: [INAUDIBLE] 23:30.433 --> 23:32.833 outside pressure pushing down on the water? 23:32.833 --> 23:35.933 PROFESSOR: Well, it'll have a bit to do with that, but 23:35.933 --> 23:37.603 that's not going to be having to do with the 23:37.600 --> 23:38.700 pressure coming down here. 23:38.700 --> 23:41.330 It's going to have to do with variations in pressure within 23:41.333 --> 23:42.633 that liquid. 23:42.633 --> 23:44.103 Anybody else? 23:44.100 --> 23:44.530 Yes. 23:44.533 --> 23:45.673 STUDENT: [INAUDIBLE] 23:45.667 --> 23:47.027 PROFESSOR: Yes. 23:47.033 --> 23:49.973 So as you go down in this fluid, the pressure's getting 23:49.967 --> 23:51.167 greater and greater. 23:51.167 --> 23:56.627 That means the pressure acting up on the bottom is greater 23:56.633 --> 24:01.303 than the pressure acting down on the top. 24:01.300 --> 24:04.900 So that's why I said it's got to be a liquid with some mass 24:04.900 --> 24:05.930 in a gravity field. 24:05.933 --> 24:09.033 Because only in a gravity field will there be that 24:09.033 --> 24:11.573 increase in pressure as you go down. 24:11.567 --> 24:13.897 So when you push that beach ball down there, realize the 24:13.900 --> 24:16.600 pressure at the bottom of the ball is greater than the 24:16.600 --> 24:18.830 pressure at the top of the ball. 24:18.833 --> 24:22.903 And that is what's causing this buoyancy force. 24:22.900 --> 24:24.330 So that's step one. 24:24.333 --> 24:29.973 By the way, let's quantify that using the comment that 24:29.967 --> 24:30.997 was made earlier. 24:31.000 --> 24:32.670 What is Archimedes' Law? 24:38.167 --> 24:47.367 Archimedes' Law said that that buoyancy force is equal to the 24:47.367 --> 24:51.827 weight of the water displaced, or the weight of the-- let's 24:51.833 --> 24:54.333 call it water-- the weight of the fluid displaced. 24:59.500 --> 25:03.470 So in order to compute that force, we just have to know 25:03.467 --> 25:09.067 how much water would be there if the object were not there. 25:09.067 --> 25:12.567 In this case, it would be the volume of the object 25:12.567 --> 25:15.927 multiplied times the density of the fluid. 25:15.933 --> 25:20.773 But it's the weight, not the mass, so this has to be 25:20.767 --> 25:24.967 multiplied by little g, the acceleration of gravity. 25:24.967 --> 25:29.497 If you have something of mass m, its weight is the product 25:29.500 --> 25:34.600 of mass and the acceleration of gravity. 25:34.600 --> 25:36.030 So that's Archimedes' Law. 25:36.033 --> 25:37.303 That's the buoyancy force. 25:37.300 --> 25:40.370 Now, it's acting in the atmosphere all the time 25:40.367 --> 25:44.427 whenever you have a little parcel of air that's at a 25:44.433 --> 25:48.003 different temperature than its surroundings. 25:48.000 --> 25:50.270 And that's what I want to work out, and that's where the 25:50.267 --> 26:00.367 perfect gas law is going to be a very nice thing to have. 26:00.367 --> 26:01.897 So I've worked out an example. 26:01.900 --> 26:07.800 I've imagined a little piece of air-- 26:07.800 --> 26:10.500 maybe it's about this big-- 26:10.500 --> 26:11.830 that's got a certain pressure. 26:11.833 --> 26:14.803 I'm going to use the subscript p, because I'm calling this a 26:14.800 --> 26:20.070 parcel, a little parcel of air. 26:20.067 --> 26:24.127 It's got a density, and it's got a temperature. 26:24.133 --> 26:27.133 And then surrounding it is the environment. 26:27.133 --> 26:30.133 That'll be the pressure in the environment, the density of 26:30.133 --> 26:32.333 the environment, and the temperature of the 26:32.333 --> 26:33.873 environment. 26:33.867 --> 26:36.327 And my goal is to find out the buoyancy force 26:36.333 --> 26:38.433 acting on that parcel. 26:38.433 --> 26:41.033 I want to know, if it's warm, is it going to rise, or is it 26:41.033 --> 26:41.633 going to sink? 26:41.633 --> 26:44.733 And so on and so forth. 26:44.733 --> 26:47.373 Now, we're going to have to make some assumptions, but 26:47.367 --> 26:50.427 they're going to be very good assumptions. 26:50.433 --> 26:52.733 The first assumption is we're going to assume that the 26:52.733 --> 26:56.273 pressure in the parcel is equal to the pressure of the 26:56.267 --> 26:58.427 environment. 26:58.433 --> 27:00.473 So the pressure here is equal to the pressure there. 27:00.467 --> 27:01.697 Why would that be? 27:04.133 --> 27:06.133 If you had air that was at different pressure than its 27:06.133 --> 27:08.473 environment, let's say at greater pressure, it would 27:08.467 --> 27:13.327 immediately expand until the pressure matched. 27:13.333 --> 27:17.403 If you don't believe that, blow up a balloon so you got 27:17.400 --> 27:19.530 the pressure in that thing a little bit higher than the 27:19.533 --> 27:22.373 environment, and then pop it. 27:22.367 --> 27:26.127 Well, the instant you pop it, now the rubber is gone. 27:26.133 --> 27:28.633 You've got that high-pressure air, and what does it do? 27:28.633 --> 27:31.673 It immediately expands in order to 27:31.667 --> 27:33.827 equalize the pressure. 27:33.833 --> 27:37.133 So this idea of equalizing pressure happens very, very 27:37.133 --> 27:41.303 rapidly, and that's why I can assume that these two 27:41.300 --> 27:44.500 pressures are equal. 27:44.500 --> 27:46.900 Let me put in some numbers to this. 27:46.900 --> 27:55.900 Let's say that these pressures are equal to 80,000 pascals. 27:55.900 --> 27:57.530 The temperature of the environment 27:57.533 --> 28:01.373 let's say is 275 Kelvin. 28:01.367 --> 28:05.897 The temperature of the parcel let's say is 277 Kelvin, so 28:05.900 --> 28:09.730 just a two-degree difference between the two. 28:09.733 --> 28:14.133 And I'm going to compute the density for both. 28:14.133 --> 28:16.773 Rho for the environment is going to be P for the 28:16.767 --> 28:22.127 environment over R and TE. 28:22.133 --> 28:31.173 So it'll be 80,000 divided by 287 divided by 275, and that's 28:31.167 --> 28:33.067 going to be 1.0136. 28:33.067 --> 28:38.067 The units will be kilograms per cubic meter. 28:38.067 --> 28:41.827 That's the density of air in the environment. 28:41.833 --> 28:45.203 The density of air in the parcel is going to be the same 28:45.200 --> 28:52.130 pressure, 80,000, the same gas constant, 287, but the 28:52.133 --> 28:53.233 temperature's a little bit different. 28:53.233 --> 28:55.673 It's 277. 28:55.667 --> 29:03.427 So that's going to be 1.0063 kilograms per cubic meter. 29:03.433 --> 29:07.703 So what I've shown you here is that the density of the 29:07.700 --> 29:13.300 environment is a little bit greater than the density of 29:13.300 --> 29:14.230 the parcel itself. 29:14.233 --> 29:15.703 Now, what does that mean in terms of buoyancy? 29:18.800 --> 29:20.230 I think I'll move back over here. 29:34.400 --> 29:35.670 Here's my parcel. 29:37.800 --> 29:48.170 The gravity force pulling down on that is going to be the 29:48.167 --> 29:50.767 mass of the parcel times gravity. 29:50.767 --> 29:53.827 We already talked about that, so it's going to be the volume 29:53.833 --> 30:01.273 times the density of the parcel, rho sub p times g. 30:01.267 --> 30:10.897 The buoyancy force acting up is going to be the volume-- 30:10.900 --> 30:15.300 well, we're using Archimedes' Law now, so it's going to be 30:15.300 --> 30:16.230 this quantity here. 30:16.233 --> 30:20.673 It's going to be the volume again times the density of the 30:20.667 --> 30:22.127 environment-- 30:22.133 --> 30:23.803 that's the fluid that's been displaced-- 30:26.933 --> 30:28.173 times g. 30:30.467 --> 30:32.397 Well, now you can see immediately what's going to 30:32.400 --> 30:33.700 happen here. 30:33.700 --> 30:36.870 The V's are the same for both, g is the same for both, but 30:36.867 --> 30:39.797 the densities appear differently. 30:39.800 --> 30:42.970 The down force is related to the density of the parcel. 30:42.967 --> 30:46.227 The up force is related to the density of the environment. 30:46.233 --> 30:50.873 In our case, the density of the environment is less, so-- 30:54.033 --> 30:54.973 is that right? 30:54.967 --> 30:56.127 Greater. 30:56.133 --> 31:02.073 So this one is going to be a little bit less. 31:02.067 --> 31:03.297 That's going to be a little bit greater. 31:03.300 --> 31:06.670 And the net buoyancy force is up. 31:06.667 --> 31:11.867 So what I've proven here is that a parcel of air, if it's 31:11.867 --> 31:15.967 equilibrated its pressure with the environment, is going to 31:15.967 --> 31:17.897 be less dense. 31:17.900 --> 31:20.630 Therefore, it's going to have a buoyancy force that's going 31:20.633 --> 31:22.533 to make it rise. 31:22.533 --> 31:27.833 Well, this is probably the basic physics of what happens 31:27.833 --> 31:31.703 in the atmosphere to generate all the wind circulations, to 31:31.700 --> 31:34.400 generate clouds, sea breezes. 31:34.400 --> 31:37.300 Almost everything you can think of in the atmosphere, 31:37.300 --> 31:41.470 any air motion, probably can be tracked back to this simple 31:41.467 --> 31:45.867 little idea, that temperature differences, if the pressure 31:45.867 --> 31:48.867 is equilibrated, will generate buoyancy forces, 31:48.867 --> 31:49.897 either up or down. 31:49.900 --> 31:54.100 Now, if I had chosen a cooler temperature for the parcel, 31:54.100 --> 32:00.430 let's say 273, of course, then everything would be reversed. 32:00.433 --> 32:03.703 The parcel would be denser than air, this vector would be 32:03.700 --> 32:07.400 larger than that one, and the parcel of air would sink. 32:07.400 --> 32:10.400 So it works both ways. 32:10.400 --> 32:13.330 Now, this is a tricky argument, a number of steps. 32:13.333 --> 32:16.333 So I'd be pleased to stop for a minute or two and take 32:16.333 --> 32:18.903 questions on this. 32:18.900 --> 32:19.400 Yes. 32:19.400 --> 32:22.800 STUDENT: So it equalizes pressure, but at the expense 32:22.800 --> 32:23.800 of equalizing temperature? 32:23.800 --> 32:24.700 PROFESSOR: That's right. 32:24.700 --> 32:25.830 So that's a very good question. 32:25.833 --> 32:28.003 The question is, why does it equalize pressure? 32:28.000 --> 32:29.900 Why doesn't it equalize temperature or density? 32:29.900 --> 32:31.970 Well, they're different quantities. 32:31.967 --> 32:35.297 Pressure is a force per unit area, and that's the thing 32:35.300 --> 32:37.500 that wants to equalize. 32:37.500 --> 32:39.700 There's no quick process-- 32:39.700 --> 32:42.800 temperature might equalize over an hour or two, because 32:42.800 --> 32:45.900 they're in contact with each other, but not that 32:45.900 --> 32:48.900 instantaneous equilibration like you get 32:48.900 --> 32:50.930 with a balloon popping. 32:50.933 --> 32:53.103 That's pressure equilibration, and it's 32:53.100 --> 32:54.970 fast and it's physical. 32:54.967 --> 32:57.827 And the other two either are slower, or just there's no 32:57.833 --> 32:59.503 tendency for that at all. 32:59.500 --> 33:00.170 But that's right. 33:00.167 --> 33:02.567 That's the key part of the argument, isn't it, that of 33:02.567 --> 33:05.297 these three quantities we're talking about, the pressure 33:05.300 --> 33:09.130 wants to equilibrate, but the other two do not. 33:09.133 --> 33:12.573 And that's what leads rise to the whole concept of buoyancy, 33:12.567 --> 33:14.967 warm air rising, cold air sinking. 33:14.967 --> 33:18.967 It all comes from the way the pressure equilibrates. 33:18.967 --> 33:20.497 That's key. 33:20.500 --> 33:21.730 Other questions on this? 33:25.733 --> 33:27.003 Anything? 33:29.067 --> 33:31.767 Well, that went through pretty quickly. 33:31.767 --> 33:33.767 I wanted to-- 33:33.767 --> 33:34.127 oh, wait. 33:34.133 --> 33:35.973 Let's do another example. 33:35.967 --> 33:39.327 I want to do another example, because-- 33:51.400 --> 33:58.670 let's say that I've got my parcel, and everything's 33:58.667 --> 33:59.897 defined as before. 34:02.567 --> 34:05.227 But now I've got-- 34:05.233 --> 34:06.903 let's say-- what did I use? 34:06.900 --> 34:11.400 I've got helium in here, helium. 34:11.400 --> 34:15.200 And I've got air out here. 34:15.200 --> 34:16.500 The pressures are equal. 34:19.733 --> 34:21.703 In this case, I'm going to say the temperature 34:21.700 --> 34:22.930 are equal as well. 34:25.633 --> 34:27.903 But are the densities equal? 34:27.900 --> 34:30.230 Do you think the densities are going to be equal 34:30.233 --> 34:31.933 in these two cases? 34:31.933 --> 34:34.003 No. 34:34.000 --> 34:37.230 And let's see how that's going to work out. 34:37.233 --> 34:43.273 The density of the environment is going to be the pressure of 34:43.267 --> 34:53.327 the environment with the gas constant for air, 287, and 34:53.333 --> 34:55.773 then the temperature of the environment. 34:55.767 --> 34:58.967 The density of the parcel is going to be pressure of the 34:58.967 --> 35:00.927 parcel over-- 35:00.933 --> 35:01.333 now, let's see. 35:01.333 --> 35:04.433 What's going to be the gas constant for helium? 35:04.433 --> 35:11.003 8,314 divided by the molecular weight of helium, which you 35:11.000 --> 35:13.030 recall is four. 35:13.033 --> 35:25.633 The gas constant for helium's about 2,079. 35:25.633 --> 35:28.333 So look, even if the pressures are the same and the 35:28.333 --> 35:30.473 temperatures are the same, because they're different 35:30.467 --> 35:35.297 gases, the densities are going to be very, very different. 35:35.300 --> 35:39.770 And so the helium balloon is going to have a smaller mass, 35:39.767 --> 35:42.367 smaller density than the air that it's displaced. 35:42.367 --> 35:43.267 It's going to rise. 35:43.267 --> 35:46.067 It's going to have a buoyancy force that rises. 35:46.067 --> 35:49.197 And in this case, it comes in through the different gas 35:49.200 --> 35:53.630 constant, which in turn arises because of the different 35:53.633 --> 35:56.473 molecular weights. 35:56.467 --> 35:58.927 So later on in the course, in the lab, we'll be launching 35:58.933 --> 36:03.333 helium-filled balloons, and you can think back at that 36:03.333 --> 36:06.233 moment and realize, ah, that's what's going on. 36:06.233 --> 36:07.903 That's why there's a buoyancy force. 36:07.900 --> 36:12.800 That's why that balloon wants to rise is because it has a 36:12.800 --> 36:14.600 different gas constant, because it has a different 36:14.600 --> 36:15.300 molecular weight. 36:15.300 --> 36:17.130 It's a lighter gas. 36:17.133 --> 36:21.033 Each molecule has a smaller mass than the 36:21.033 --> 36:22.703 air molecules do. 36:28.533 --> 36:29.803 Any questions there? 36:46.433 --> 36:47.973 I can't leave this subject without 36:47.967 --> 36:49.567 mentioning mixtures of gases. 36:56.833 --> 37:02.033 So we imagine this same box, and it's got some A molecules, 37:02.033 --> 37:04.673 and it's got some B molecules. 37:04.667 --> 37:06.527 And they're all bouncing around off the 37:06.533 --> 37:08.373 walls and so on. 37:08.367 --> 37:14.127 There's a mixture of gas A and gas B. What is the deal there? 37:14.133 --> 37:18.333 When you mix two gasses together, what relationship do 37:18.333 --> 37:21.933 they have to each other? 37:21.933 --> 37:27.333 I can tell you pretty clearly what's going to happen. 37:27.333 --> 37:34.573 The temperatures are going to quickly equilibrate. 37:34.567 --> 37:37.727 Even if the masses are different, because they're 37:37.733 --> 37:42.503 bouncing into each other frequently, thousands of times 37:42.500 --> 37:47.170 per second, the temperature of the A and B molecules will 37:47.167 --> 37:51.297 quickly come to the same value. 37:51.300 --> 37:56.270 The pressure is additive. 37:59.667 --> 38:03.067 In other words, we can define the pressure that the A 38:03.067 --> 38:08.027 molecules are making, we can call that PA, and the pressure 38:08.033 --> 38:11.833 that the B molecules are making, that's PB. 38:11.833 --> 38:16.573 And the total pressure, P Total, is just 38:16.567 --> 38:17.927 the sum of the two. 38:17.933 --> 38:23.033 So we use this term partial pressure-- 38:23.033 --> 38:26.303 partial pressure of A, partial pressure of B-- and they add 38:26.300 --> 38:29.170 up to give the total pressure. 38:29.167 --> 38:31.797 So for example, if the pressure in this room-- 38:31.800 --> 38:34.000 let's call it P Total for the moment-- 38:34.000 --> 38:37.970 is about 1,013 millibars-- 38:37.967 --> 38:41.367 or that is to say 1,013 with two more 38:41.367 --> 38:44.567 decimal places pascals-- 38:44.567 --> 38:48.767 part of that is due to the nitrogen molecules. 38:48.767 --> 38:50.867 That's the partial pressure of the nitrogen. 38:50.867 --> 38:53.267 Part of it's due to the oxygen molecules. 38:53.267 --> 38:54.867 Part of it's due to the argon. 38:54.867 --> 38:57.567 There's also some water vapor in this room. 38:57.567 --> 38:59.797 Water vapor is contributing something 38:59.800 --> 39:00.930 to that total pressure. 39:00.933 --> 39:03.933 So when you're measuring pressure in a gas, you're 39:03.933 --> 39:08.673 measuring the sum of all the pressures of the components 39:08.667 --> 39:11.467 within that gas. 39:11.467 --> 39:14.497 Usually, we don't need to know that, but occasionally, that's 39:14.500 --> 39:18.530 the way we keep track of how much of these other gases you 39:18.533 --> 39:21.203 have. Someone might say, well, the partial pressure of water 39:21.200 --> 39:24.470 vapor is three millibars today or something like that. 39:24.467 --> 39:27.827 That's the contribution water vapor is making to the total 39:27.833 --> 39:29.673 pressure on this particular day. 39:29.667 --> 39:31.227 So it's a useful quantity. 39:35.333 --> 39:40.603 Let me remind you what the atmospheric composition is for 39:40.600 --> 39:43.470 our atmosphere. 39:49.667 --> 39:54.327 For air on Earth, it's primarily 39:54.333 --> 39:57.233 nitrogen, oxygen, and argon. 40:02.133 --> 40:08.703 I'm going to give you the number two ways: by volume, 40:08.700 --> 40:10.130 which is what the chemists say-- 40:12.933 --> 40:17.303 I prefer to remember that that is by molecule, by the number 40:17.300 --> 40:18.530 of molecules-- 40:23.600 --> 40:25.870 and I'm going to also give it to you by mass. 40:31.467 --> 40:42.567 For nitrogen, it's 78.1% by volume and 75.5% by mass. 40:42.567 --> 40:47.627 In other words, 78% of the molecules in this room are 40:47.633 --> 40:54.733 nitrogen, but 75.5% of the mass of the gas in this room 40:54.733 --> 40:57.033 is the nitrogen. 40:57.033 --> 41:02.003 Oxygen, 21.0% and 23.2%. 41:02.000 --> 41:05.730 Argon, 0.9% and 1.3%. 41:05.733 --> 41:07.903 Just remember, there's this difference because the 41:07.900 --> 41:10.300 molecules have different masses. 41:10.300 --> 41:11.170 Some are heavier. 41:11.167 --> 41:11.827 Some are lighter. 41:11.833 --> 41:15.073 So whether you've counted up the molecules and are 41:15.067 --> 41:18.497 representing the fraction that way, or whether you're 41:18.500 --> 41:20.630 counting up the masses, you're going to get slightly 41:20.633 --> 41:23.703 different numbers for the two. 41:23.700 --> 41:29.500 Now I've chosen, and the convention is to define that 41:29.500 --> 41:34.700 part as being the air, because these proportions are constant 41:34.700 --> 41:37.030 everywhere you go in the atmosphere. 41:37.033 --> 41:40.003 If I go to the North Pole, the Equator, the South Pole, if I 41:40.000 --> 41:43.830 go high in the atmosphere, winter or summer, these 41:43.833 --> 41:46.433 proportions are unchanging. 41:46.433 --> 41:49.303 So we call that air. 41:49.300 --> 41:51.930 But then there are other gases as well. 41:51.933 --> 41:54.303 And sometimes they're called trace gases. 41:57.867 --> 42:00.267 Sometimes they're just called variable gases. 42:04.567 --> 42:06.667 They're found in varying proportions 42:06.667 --> 42:09.027 depending where you are. 42:09.033 --> 42:10.133 Let me give you an example. 42:10.133 --> 42:12.403 Probably the most important one is water, 42:12.400 --> 42:15.630 water vapor, H2O. 42:15.633 --> 42:20.233 And it's found anywhere from-- 42:20.233 --> 42:29.903 well, from let's say one part per 100, 10 to the minus two, 42:29.900 --> 42:32.670 to really as small as you want to go, maybe 10 to the minus 42:32.667 --> 42:37.897 five, by volume. 42:42.433 --> 42:48.803 CO2, another very important gas, is more thoroughly mixed, 42:48.800 --> 42:52.170 but not perfectly mixed. 42:52.167 --> 42:58.097 A typical value these days might be about 395 parts per 42:58.100 --> 43:04.570 million by volume, ppmv. So we're using this method, we're 43:04.567 --> 43:05.997 counting molecules. 43:06.000 --> 43:10.270 I could write that as 395 times 10 to the 43:10.267 --> 43:13.327 minus six by volume. 43:19.867 --> 43:25.967 That varies only up and down by about 5%, plus or minus 5%. 43:25.967 --> 43:31.197 So that's nearly well mixed, but not quite as thoroughly 43:31.200 --> 43:35.630 mixed as these gases within the atmosphere-- 43:35.633 --> 43:38.033 within the air. 43:38.033 --> 43:41.433 Some other molecules I mentioned last time on the 43:41.433 --> 43:45.003 slide you should be aware of are methane, nitrous 43:45.000 --> 43:47.630 oxide--N2O-- and ozone. 43:47.633 --> 43:51.303 And just be aware that those and a few other gases will pop 43:51.300 --> 43:54.400 up from time to time in this course, and we'll be wanting 43:54.400 --> 44:01.430 to know what their partial pressure is, what their mass 44:01.433 --> 44:05.273 ratio is, what their ratio by molecules is. 44:05.267 --> 44:08.167 We can convert back and forth between these different 44:08.167 --> 44:15.367 measures using the formulas that I've given you today. 44:15.367 --> 44:17.967 Any questions here? 44:17.967 --> 44:22.027 We've actually covered a lot of I think somewhat confusing 44:22.033 --> 44:24.573 material, so I want to be sure we take a few 44:24.567 --> 44:25.867 more minutes for questions. 44:25.867 --> 44:26.167 Yes. 44:26.167 --> 44:30.367 STUDENT: For the water vapor, the numbers there, it's 10 to 44:30.367 --> 44:32.427 the negative 2 - there is 10 to the negative 5. 44:32.433 --> 44:33.633 PROFESSOR: Negative five. 44:33.633 --> 44:34.903 Thank you. 44:38.133 --> 44:41.533 For example, I don't have an instrument with me to measure 44:41.533 --> 44:44.433 this-- we'll be doing it in lab-- but yesterday and today 44:44.433 --> 44:46.633 have been rather humid days. 44:46.633 --> 44:50.003 So this means that this number is going to be a little larger 44:50.000 --> 44:53.030 than it would have been last week, when we had a drier 44:53.033 --> 44:53.803 atmosphere. 44:53.800 --> 44:56.900 So that's an example of how that number fluctuates. 44:56.900 --> 44:59.730 This fraction isn't changed between last week and this 44:59.733 --> 45:01.273 week, but this one has. 45:01.267 --> 45:04.597 So these are variable ones, and these are constant 45:04.600 --> 45:05.200 proportions. 45:05.200 --> 45:05.630 Yes. 45:05.633 --> 45:09.403 STUDENT: When it's 100% humidity, about 45:09.400 --> 45:10.370 where is that range? 45:10.367 --> 45:12.667 PROFESSOR: Well, so that depends on the temperature. 45:12.667 --> 45:14.267 So the relative humidity-- we'll talk about 45:14.267 --> 45:15.667 this in great detail-- 45:15.667 --> 45:17.827 the relative humidity is a measure of how much water 45:17.833 --> 45:21.533 vapor you have to the maximum that can be held 45:21.533 --> 45:22.873 in the vapor state. 45:22.867 --> 45:25.367 And because that second number is so strongly 45:25.367 --> 45:27.427 temperature-dependent, I can't give you a 45:27.433 --> 45:28.473 fixed number for this. 45:28.467 --> 45:29.697 It'll depend on the temperature. 45:29.700 --> 45:31.670 But we'll talk about that later on, because that's so 45:31.667 --> 45:34.367 important for how clouds form and so on. 45:37.133 --> 45:42.073 I think I may have time to do one other thing before we quit 45:42.067 --> 45:49.797 today, and that's to talk about how density and pressure 45:49.800 --> 45:51.030 change with altitude. 46:00.333 --> 46:03.533 First of all, just some background information. 46:03.533 --> 46:08.403 The typical sea-level density, of course it varies 46:08.400 --> 46:09.170 from place to place. 46:09.167 --> 46:12.597 But if you want to work out a problem and you're not given 46:12.600 --> 46:16.070 enough information, you should know, for example, in this 46:16.067 --> 46:22.697 room, the density is about 1.2 kilograms per cubic meter. 46:22.700 --> 46:30.070 And a typical sea-level pressure is about 1,013 46:30.067 --> 46:35.097 millibars or 1,013 two more zeros pascals. 46:35.100 --> 46:39.970 So let's take that as just basic climatological 46:39.967 --> 46:40.667 information. 46:40.667 --> 46:41.997 But now I'm interested in how those 46:42.000 --> 46:44.630 numbers change with height. 46:44.633 --> 46:52.233 Typically, if I plot pressure on the x-axis and height using 46:52.233 --> 46:55.773 the letter z on the y-axis, it looks like this. 46:55.767 --> 47:01.627 It decreases rapidly at first, then less rapidly, and so on, 47:01.633 --> 47:07.633 asymptotically, but never actually reaching zero. 47:07.633 --> 47:10.803 And if I plot air density, it looks very much the same. 47:17.300 --> 47:21.430 It's such a simple relationship that we'd like to 47:21.433 --> 47:24.033 be able to have a formula for it. 47:24.033 --> 47:26.873 And there is a nice one, but we have to make an 47:26.867 --> 47:28.397 approximation now. 47:28.400 --> 47:32.430 We have to assume that the temperature is approximately 47:32.433 --> 47:39.603 constant with height, which is not a very good approximation, 47:39.600 --> 47:43.000 but we're looking here just to get a rough-- maybe an 47:43.000 --> 47:46.970 estimate at the 10% level or the 20% level, something in 47:46.967 --> 47:47.697 that range. 47:47.700 --> 47:51.330 But if this approximation is used, then I can write down a 47:51.333 --> 47:54.233 formula for the pressure as a function of height. 47:54.233 --> 47:58.973 It's the pressure at sea level times e to the minus z over H 47:58.967 --> 47:59.667 sub S. 47:59.667 --> 48:03.997 And the density follows exactly the same formula. 48:04.000 --> 48:08.830 The density at sea level, rho sub SL-- 48:08.833 --> 48:12.403 I'm using Greek letter rho for density-- 48:12.400 --> 48:17.400 e to the minus z over H sub S. Now, if you're familiar with 48:17.400 --> 48:19.830 this exponential function, you would have already 48:19.833 --> 48:20.933 recognized it here. 48:20.933 --> 48:25.433 This is the behavior of the exponential function. 48:25.433 --> 48:30.473 It drops rapidly at first and then more slowly as you go on. 48:33.167 --> 48:35.797 This thing is called the density for the scale height, 48:35.800 --> 48:37.570 the scale height for the atmosphere. 48:40.367 --> 48:45.067 It is a measure of how fast the pressure and density 48:45.067 --> 48:47.597 decrease as you go up. 48:47.600 --> 48:50.230 And there's a very simple formula for it. 48:50.233 --> 48:58.573 It's RT over g, the gas constant times the temperature 48:58.567 --> 49:01.697 divided by the surface gravity. 49:01.700 --> 49:03.770 Let's work it out for Earth. 49:03.767 --> 49:07.127 Air has a gas constant of 287. 49:07.133 --> 49:11.933 Let's take 288 for the temperature of the air and 49:11.933 --> 49:15.673 9.81 for the acceleration of gravity. 49:15.667 --> 49:23.597 That turns out to be approximately 8,400 meters. 49:23.600 --> 49:28.600 Every time you go up 8,400, meters, you tick off another 49:28.600 --> 49:32.270 fractional decrease in atmospheric 49:32.267 --> 49:34.897 pressure and density. 49:34.900 --> 49:38.400 We just have a minute left, so I can do a 49:38.400 --> 49:39.630 quick example of this. 49:44.133 --> 49:45.303 Let's say we've got an aircraft 49:45.300 --> 49:48.730 flying at 37,000 feet. 49:48.733 --> 49:52.433 That's typically what an airliner would fly at. 49:52.433 --> 49:55.833 And you'd like to know what is the pressure and density of 49:55.833 --> 49:57.773 the air just outside the cabin? 49:57.767 --> 50:01.267 Of course, the cabin itself is pressurized so you can breathe 50:01.267 --> 50:05.397 and maintain consciousness, but what is the air 50:05.400 --> 50:06.830 temperature and pressure just outside? 50:06.833 --> 50:10.403 Well, first of all, we have to convert this to meters. 50:10.400 --> 50:15.800 That's going to be 11,278 meters. 50:15.800 --> 50:17.530 And then I'm going to put it into this formula. 50:17.533 --> 50:22.073 So the pressure at that height z is going to be the pressure 50:22.067 --> 50:29.597 at sea level, 101,300, times e to the minus 50:29.600 --> 50:35.500 11,278 divided by 8,400. 50:35.500 --> 50:38.600 I hope you know how to do that in your scientific calculator 50:38.600 --> 50:39.800 with the minus sign in there. 50:39.800 --> 50:41.200 Practice that. 50:41.200 --> 50:44.530 That comes out to be-- 50:44.533 --> 50:45.233 let's see-- 50:45.233 --> 50:49.803 26,524 pascals. 50:49.800 --> 50:55.900 Well, that's about a quarter of the pressure at sea level. 50:55.900 --> 50:58.730 And density would be something very similar. 50:58.733 --> 51:07.533 It'll be 1.2 times e to the minus 11,278 over 8,400. 51:07.533 --> 51:12.433 And that's going to come out to be 0.31. 51:12.433 --> 51:16.233 Units are kilograms per cubic meter. 51:16.233 --> 51:19.573 So that also is about a quarter of 51:19.567 --> 51:20.797 what you started with. 51:20.800 --> 51:22.830 So that's not much. 51:22.833 --> 51:27.103 In other words, at typical airliner flight level, the 51:27.100 --> 51:29.200 density and pressure that you're flying through is only 51:29.200 --> 51:33.270 about one quarter that you have here at sea level, and 51:33.267 --> 51:35.897 that's why the cabin has to be pressurized. 51:35.900 --> 51:38.500 We're really out of time now, so we'll move on to some new 51:38.500 --> 51:40.400 material on Wednesday.