1 00:00:00,000 --> 00:00:00,900 2 00:00:00,900 --> 00:00:02,770 Let's do some more problems that involve 3 00:00:02,770 --> 00:00:04,450 the ideal gas equation. 4 00:00:04,450 --> 00:00:07,550 Let's say I have a gas in a container and the current 5 00:00:07,549 --> 00:00:12,455 pressure is 3 atmospheres. 6 00:00:12,455 --> 00:00:15,099 7 00:00:15,099 --> 00:00:22,454 And let's say that the volume of the container is 9 liters. 8 00:00:22,454 --> 00:00:27,459 9 00:00:27,460 --> 00:00:31,230 Now, what will the pressure become if my volume goes from 10 00:00:31,230 --> 00:00:35,259 9 liters to 3 liters? 11 00:00:35,259 --> 00:00:39,149 12 00:00:39,149 --> 00:00:42,500 So from the first ideal gas equation video, you can kind 13 00:00:42,500 --> 00:00:45,979 of have the intuition, that you have a bunch of-- and 14 00:00:45,979 --> 00:00:47,609 we're holding-- and this is important. 15 00:00:47,609 --> 00:00:50,670 We're holding the temperature constant, and that's an 16 00:00:50,670 --> 00:00:52,420 important thing to realize. 17 00:00:52,420 --> 00:00:59,840 So in our very original intuition behind the ideal gas 18 00:00:59,840 --> 00:01:02,470 equation we said, look, if we have a certain number of 19 00:01:02,469 --> 00:01:07,340 particles with a certain amount of kinetic energy, and 20 00:01:07,340 --> 00:01:09,480 they're exerting a certain pressure on their container, 21 00:01:09,480 --> 00:01:14,710 and if we were to make the container smaller, we have the 22 00:01:14,709 --> 00:01:16,009 same number of particles. 23 00:01:16,010 --> 00:01:17,300 n doesn't change. 24 00:01:17,299 --> 00:01:20,009 The average kinetic energy doesn't change, so they're 25 00:01:20,010 --> 00:01:21,469 just going to bump into the walls more. 26 00:01:21,469 --> 00:01:27,030 So that when we make the volume smaller, when the 27 00:01:27,030 --> 00:01:30,019 volume goes down, the pressure should go up. 28 00:01:30,019 --> 00:01:32,500 So let's see if we can calculate the exact number. 29 00:01:32,500 --> 00:01:36,250 So we can take our ideal gas equation: pressure times 30 00:01:36,250 --> 00:01:38,859 volume is equal to nRT. 31 00:01:38,859 --> 00:01:41,790 32 00:01:41,790 --> 00:01:45,220 Now, do the number of particles change when I did 33 00:01:45,219 --> 00:01:48,219 this situation when I shrunk the volume? 34 00:01:48,219 --> 00:01:48,640 No! 35 00:01:48,640 --> 00:01:49,920 We have the same number of particles. 36 00:01:49,920 --> 00:01:52,879 I'm just shrinking the container, so n is n, R 37 00:01:52,879 --> 00:01:55,479 doesn't change, that's a constant, and then the 38 00:01:55,480 --> 00:01:57,030 temperature doesn't change. 39 00:01:57,030 --> 00:02:01,560 So my old pressure times volume is going to be equal to 40 00:02:01,560 --> 00:02:04,740 nRT, and my new pressure times volume-- so let me 41 00:02:04,739 --> 00:02:07,280 call this P1 and V1. 42 00:02:07,280 --> 00:02:11,250 43 00:02:11,250 --> 00:02:12,500 That's V2. 44 00:02:12,500 --> 00:02:15,639 45 00:02:15,639 --> 00:02:21,500 V2 is this, and we're trying to figure out P2. 46 00:02:21,500 --> 00:02:23,229 P2 is what? 47 00:02:23,229 --> 00:02:31,609 Well, we know that P1 times V1 is equal to nRT, and we also 48 00:02:31,610 --> 00:02:34,770 know that since temperature and the number of moles of our 49 00:02:34,770 --> 00:02:40,580 gas stay constant, that P2 times V2 is equal to nRT. 50 00:02:40,580 --> 00:02:44,280 And since they both equal the same thing, we can say that 51 00:02:44,280 --> 00:02:47,039 the pressure times the volume, as long as the temperature is 52 00:02:47,039 --> 00:02:49,219 held constant, will be a constant. 53 00:02:49,219 --> 00:02:56,020 So P1 times V1 is going to equal P2 times V2. 54 00:02:56,020 --> 00:02:58,290 So what was P1? 55 00:02:58,289 --> 00:03:00,724 P1, our initial pressure, was 3 atmospheres. 56 00:03:00,724 --> 00:03:06,699 57 00:03:06,699 --> 00:03:12,579 So 3 atmospheres times 9 liters is equal to our new 58 00:03:12,580 --> 00:03:15,550 pressure times 3 liters. 59 00:03:15,550 --> 00:03:22,760 And if we divide both sides of the equation by 3, we get 3 60 00:03:22,759 --> 00:03:27,965 liters cancel out, we're left with 9 atmospheres. 61 00:03:27,965 --> 00:03:33,879 62 00:03:33,879 --> 00:03:35,009 And that should make sense. 63 00:03:35,009 --> 00:03:39,959 When you decrease the volume by 2/3 or when you make the 64 00:03:39,960 --> 00:03:43,890 volume 1/3 of your original volume, then your pressure 65 00:03:43,889 --> 00:03:46,229 increases by a factor of three. 66 00:03:46,229 --> 00:03:51,389 So this when by times 3, and this went by times 1/3. 67 00:03:51,389 --> 00:03:52,889 That's a useful thing to know in general. 68 00:03:52,889 --> 00:03:57,179 If temperature is held constant, then pressure times 69 00:03:57,180 --> 00:03:59,290 volume are going to be a constant. 70 00:03:59,289 --> 00:04:00,840 Now, you can take that even further. 71 00:04:00,840 --> 00:04:07,920 If we look at PV equals nRT, the two things that we know 72 00:04:07,919 --> 00:04:11,929 don't change in the vast majority of exercises we do is 73 00:04:11,930 --> 00:04:14,640 the number of molecules we're dealing with, and obviously, R 74 00:04:14,639 --> 00:04:15,679 isn't going to change. 75 00:04:15,680 --> 00:04:20,990 So if we divide both sides of this by T, we get PV over T is 76 00:04:20,990 --> 00:04:25,199 equal to nR, or you could say it's equal to a constant. 77 00:04:25,199 --> 00:04:27,959 This is going to be a constant number for any system where 78 00:04:27,959 --> 00:04:30,509 we're not changing the number of molecules in the container. 79 00:04:30,509 --> 00:04:33,389 80 00:04:33,389 --> 00:04:38,310 So if initially we start with pressure one, volume one, and 81 00:04:38,310 --> 00:04:40,360 some temperature one that's going to be 82 00:04:40,360 --> 00:04:41,410 equal to this constant. 83 00:04:41,410 --> 00:04:46,150 And if we change any of them, if we go back to pressure two, 84 00:04:46,149 --> 00:04:49,750 volume two, temperature two, they should still be equal to 85 00:04:49,750 --> 00:04:51,970 this constant, so they equal each other. 86 00:04:51,970 --> 00:04:59,150 So for example, let's say I start off with a pressure of 1 87 00:04:59,149 --> 00:05:01,009 atmosphere. 88 00:05:01,009 --> 00:05:07,589 and I have a volume of-- I'll switch units here just to do 89 00:05:07,589 --> 00:05:10,199 things differently-- 2 meters cubed. 90 00:05:10,199 --> 00:05:20,069 And let's say our temperature is 27 degrees Celsius. 91 00:05:20,069 --> 00:05:22,310 Well, and I just wrote Celsius because I want you to always 92 00:05:22,310 --> 00:05:25,600 remember you have to convert to Kelvin, so 27 degrees plus 93 00:05:25,600 --> 00:05:39,600 273 will get us exactly to 300 Kelvin. 94 00:05:39,600 --> 00:05:41,530 Let's figure out what the new temperature is going to be. 95 00:05:41,529 --> 00:05:46,299 Let's say our new pressure is 2 atmospheres. 96 00:05:46,300 --> 00:05:47,819 The pressure has increased. 97 00:05:47,819 --> 00:05:48,920 Let's say we make the container 98 00:05:48,920 --> 00:05:52,379 smaller, so 1 meter cubed. 99 00:05:52,379 --> 00:05:55,350 So the container has been decreased by half and the 100 00:05:55,350 --> 00:05:56,740 pressure is doubled by half. 101 00:05:56,740 --> 00:06:01,519 102 00:06:01,519 --> 00:06:08,259 Let me make the container even smaller. 103 00:06:08,259 --> 00:06:08,589 Actually, no. 104 00:06:08,589 --> 00:06:10,489 Let me make the pressure even larger. 105 00:06:10,490 --> 00:06:14,470 Let me make the pressure into 5 atmospheres. 106 00:06:14,470 --> 00:06:17,250 Now we want to know what the second temperature is, and we 107 00:06:17,250 --> 00:06:18,779 set up our equation. 108 00:06:18,779 --> 00:06:26,809 And so we have 2/300 atmosphere meters cubed per 109 00:06:26,810 --> 00:06:33,009 Kelvin is equal to 5/T2, our new temperature, and then we 110 00:06:33,009 --> 00:06:39,990 have 1,500 is equal to 2 T2. 111 00:06:39,990 --> 00:06:41,329 Divide both sides by 2. 112 00:06:41,329 --> 00:06:47,439 You have T2 is equal to 750 degrees Kelvin, which makes 113 00:06:47,439 --> 00:06:48,290 sense, right? 114 00:06:48,290 --> 00:06:51,560 We increased the pressure so much and we decreased the 115 00:06:51,560 --> 00:06:54,120 volume at the same time that the temperature 116 00:06:54,120 --> 00:06:55,459 just had to go up. 117 00:06:55,459 --> 00:06:57,409 Or if you thought of it the other way, maybe we increased 118 00:06:57,410 --> 00:06:59,850 the temperature and that's what drove the pressure to be 119 00:06:59,850 --> 00:07:04,420 so much higher, especially since we decreased the volume. 120 00:07:04,420 --> 00:07:07,550 I guess the best way to think about is this pressure went up 121 00:07:07,550 --> 00:07:10,800 so much, it went up by factor of five, it went from 1 122 00:07:10,800 --> 00:07:15,040 atmosphere to 5 atmospheres, because on one level we shrunk 123 00:07:15,040 --> 00:07:18,920 the volume by a factor of 1/2, so that should have doubled 124 00:07:18,920 --> 00:07:20,870 the pressure, so that should have gotten us to two 125 00:07:20,870 --> 00:07:21,840 atmospheres. 126 00:07:21,839 --> 00:07:23,889 And then we made the temperature a lot higher, so 127 00:07:23,889 --> 00:07:25,339 we were also bouncing into the container. 128 00:07:25,339 --> 00:07:28,909 We made the temperature 750 degrees Kelvin, so more than 129 00:07:28,910 --> 00:07:32,280 double the temperature, and then that's what got us to 5 130 00:07:32,279 --> 00:07:33,819 atmospheres. 131 00:07:33,819 --> 00:07:38,310 Now, one other thing that you'll probably hear about is 132 00:07:38,310 --> 00:07:40,500 the notion of what happens at standard 133 00:07:40,500 --> 00:07:41,790 temperature and pressure. 134 00:07:41,790 --> 00:07:43,980 Let me delete all of the stuff over here. 135 00:07:43,980 --> 00:07:47,290 Standard temperature and pressure. 136 00:07:47,290 --> 00:07:52,498 Let me delete all this stuff that I don't need. 137 00:07:52,497 --> 00:07:56,620 Standard temperature and pressure. 138 00:07:56,620 --> 00:07:58,610 And I'm bringing it up because even though it's called 139 00:07:58,610 --> 00:08:00,930 standard temperature and pressure, and sometimes called 140 00:08:00,930 --> 00:08:06,800 STP, unfortunately for the world, they haven't really 141 00:08:06,800 --> 00:08:09,990 standardized what the standard pressure and temperature are. 142 00:08:09,990 --> 00:08:14,150 143 00:08:14,149 --> 00:08:15,810 I went to Wikipedia and I looked it up. 144 00:08:15,810 --> 00:08:17,720 And the one that you'll probably see in most physics 145 00:08:17,720 --> 00:08:21,910 classes and most standardized tests is standard temperature 146 00:08:21,910 --> 00:08:24,250 is 0 degrees celsius, which is, of 147 00:08:24,250 --> 00:08:26,740 course, 273 degrees Kelvin. 148 00:08:26,740 --> 00:08:30,030 And standard pressure is 1 atmosphere. 149 00:08:30,029 --> 00:08:36,918 And here on Wikipedia, they wrote it as 101.325 150 00:08:36,918 --> 00:08:40,904 kilopascals, or a little more than 101,000 pascals. 151 00:08:40,904 --> 00:08:44,129 And of course, a pascal is a newton per square meter. 152 00:08:44,129 --> 00:08:46,470 In all of this stuff, the units are really the hardest 153 00:08:46,470 --> 00:08:47,620 part to get a hold of. 154 00:08:47,620 --> 00:08:50,779 But let's say that we assume that these are all different 155 00:08:50,779 --> 00:08:53,549 standard temperatures and pressures based on different 156 00:08:53,549 --> 00:08:54,839 standard-making bodies. 157 00:08:54,840 --> 00:08:56,149 So they can't really agree with each other. 158 00:08:56,149 --> 00:08:59,059 But let's say we took this as the definition of standard 159 00:08:59,059 --> 00:09:01,359 temperature and pressure. 160 00:09:01,360 --> 00:09:05,629 So we're assuming that temperature is equal to 0 161 00:09:05,629 --> 00:09:11,029 degrees Celsius, which is equal to 273 degrees Kelvin. 162 00:09:11,029 --> 00:09:15,429 And pressure, we're assuming, is 1 atmosphere, which could 163 00:09:15,429 --> 00:09:22,429 also be written as 101.325 or 3/8 kilopascals. 164 00:09:22,429 --> 00:09:27,059 So my question is if I have an ideal gas at standard 165 00:09:27,059 --> 00:09:33,939 temperature and pressure, how many moles of that do I have 166 00:09:33,940 --> 00:09:36,330 in 1 liter? 167 00:09:36,330 --> 00:09:37,629 No, let me say that the other way. 168 00:09:37,629 --> 00:09:40,500 How many liters will 1 mole take up? 169 00:09:40,500 --> 00:09:43,690 So let me say that a little bit more. 170 00:09:43,690 --> 00:09:46,290 So n is equal to 1 mole. 171 00:09:46,289 --> 00:09:48,899 So I want to figure out what my volume is. 172 00:09:48,899 --> 00:09:53,809 So if I have 1 mole of a gas, I have 6.02 times 10 to 23 173 00:09:53,809 --> 00:09:55,579 molecules of that gas. 174 00:09:55,580 --> 00:09:59,150 It's at standard pressure, 1 atmosphere, and at standard 175 00:09:59,149 --> 00:10:03,389 temperature, 273 degrees, what is the volume of that gas? 176 00:10:03,389 --> 00:10:05,590 So let's apply PV is equal to nRT. 177 00:10:05,590 --> 00:10:08,320 178 00:10:08,320 --> 00:10:10,970 Pressure is 1 atmosphere, but remember we're dealing with 179 00:10:10,970 --> 00:10:11,660 atmospheres. 180 00:10:11,659 --> 00:10:16,500 1 atmosphere times volume-- that's what we're solving for. 181 00:10:16,500 --> 00:10:24,500 I'll do that and purple-- is equal to 1 mole times R times 182 00:10:24,500 --> 00:10:29,100 temperature, times 273. 183 00:10:29,100 --> 00:10:31,080 Now this is in Kelvin; this is in moles. 184 00:10:31,080 --> 00:10:33,590 185 00:10:33,590 --> 00:10:34,840 We want our volume in liters. 186 00:10:34,840 --> 00:10:39,840 187 00:10:39,840 --> 00:10:42,210 So which version of R should we use? 188 00:10:42,210 --> 00:10:44,860 Well, we're dealing with atmospheres. 189 00:10:44,860 --> 00:10:47,409 We want our volume in liters, and of course, we have moles 190 00:10:47,409 --> 00:10:50,549 in Kelvin, so we'll use this version, 0.082. 191 00:10:50,549 --> 00:10:54,789 So this is 1, so we can ignore the 1 there, the 1 there. 192 00:10:54,789 --> 00:11:00,819 So the volume is equal to 0.082 times 273 degrees 193 00:11:00,820 --> 00:11:10,790 Kelvin, and that is 0.082 times 273 is 194 00:11:10,789 --> 00:11:19,209 equal to 22.4 liters. 195 00:11:19,210 --> 00:11:22,690 So if I have any ideal gas, and all gases don't behave 196 00:11:22,690 --> 00:11:25,800 ideally ideal, but if I have an ideal gas and it's at 197 00:11:25,799 --> 00:11:29,149 standard temperature, which is at 0 degrees Celsius, or the 198 00:11:29,149 --> 00:11:32,105 freezing point of water, which is also 273 degrees Kelvin, 199 00:11:32,105 --> 00:11:36,700 and I have a mole of it, and it's at standard pressure, 1 200 00:11:36,700 --> 00:11:42,590 atmosphere, that gas should take up exactly 22.4 liters. 201 00:11:42,590 --> 00:11:45,370 And if you wanted to know how many meters cubed it's going 202 00:11:45,370 --> 00:11:50,779 to take up., well, you could just say 22.4 liters times-- 203 00:11:50,779 --> 00:11:54,569 now, how many meters cubed are there-- so for every 1 meter 204 00:11:54,570 --> 00:11:57,060 cubed, you have 1,000 liters. 205 00:11:57,059 --> 00:11:59,739 I know that seems like a lot, but it's true. 206 00:11:59,740 --> 00:12:02,460 Just think about how big a meter cubed is. 207 00:12:02,460 --> 00:12:09,250 So this would be equal to 0.0224 meters cubed. 208 00:12:09,250 --> 00:12:12,929 If you have something at 1 atmosphere, a mole of it, and 209 00:12:12,929 --> 00:12:14,579 at 0 degrees Celsius. 210 00:12:14,580 --> 00:12:16,580 Anyway, this is actually a useful 211 00:12:16,580 --> 00:12:17,680 number to know sometimes. 212 00:12:17,679 --> 00:12:24,279 They'll often say, you have 2 moles at standard temperature 213 00:12:24,279 --> 00:12:25,069 and pressure. 214 00:12:25,070 --> 00:12:26,950 How many liters is it going to take up? 215 00:12:26,950 --> 00:12:30,920 Well, 1 mole will take up this many, and so 2 moles at 216 00:12:30,919 --> 00:12:32,750 standard temperature and pressure will take up twice as 217 00:12:32,750 --> 00:12:34,340 much, because you're just taking PV equals 218 00:12:34,340 --> 00:12:36,750 nRT and just doubling. 219 00:12:36,750 --> 00:12:38,264 Everything else is being held constant. 220 00:12:38,264 --> 00:12:41,039 The pressure, everything else is being held constant, 221 00:12:41,039 --> 00:12:43,480 so if you double the number of moles, you're going to double 222 00:12:43,480 --> 00:12:44,570 the volume it takes up. 223 00:12:44,570 --> 00:12:46,780 Or if you half the number of moles, you're going to half 224 00:12:46,779 --> 00:12:47,639 the volume it takes up. 225 00:12:47,639 --> 00:12:50,669 So it's a useful thing to know that in liters at standard 226 00:12:50,669 --> 00:12:52,979 temperature and pressure, where standard temperature and 227 00:12:52,980 --> 00:12:55,850 pressure is defined as 1 atmosphere and 273 degrees 228 00:12:55,850 --> 00:13:01,040 Kelvin, an ideal gas will take up 22.4 liters of volume.