1
00:00:01,105 --> 00:00:02,145
Let's do some more problems

2
00:00:02,145 --> 00:00:04,556
that involve the ideal gas equation.

3
00:00:04,556 --> 00:00:06,680
Let's say I have a gas in a container

4
00:00:06,680 --> 00:00:15,323
and the current pressure is 3 atmospheres.

5
00:00:15,323 --> 00:00:19,756
And let's say that the volume of the container

6
00:00:19,756 --> 00:00:27,413
is, I don't know, 9 liters.

7
00:00:27,413 --> 00:00:30,136
Now, what will the pressure become

8
00:00:30,136 --> 00:00:39,280
if my volume goes from 9 liters to 3 liters?

9
00:00:39,280 --> 00:00:42,183
So from the first ideal gas equation video

10
00:00:42,183 --> 00:00:43,348
you can kind of have the intuition

11
00:00:43,348 --> 00:00:46,935
that you have a bunch of-- and we're holding--

12
00:00:46,935 --> 00:00:47,901
and this is important.

13
00:00:47,901 --> 00:00:50,763
We're holding the temperature constant

14
00:00:50,763 --> 00:00:52,541
and that's an important thing to realize.

15
00:00:52,541 --> 00:00:58,384
So in our very original intuition

16
00:00:58,384 --> 00:01:00,354
behind the ideal gas equation we said,

17
00:01:00,354 --> 00:01:02,990
look, if we have a certain number of particles

18
00:01:02,990 --> 00:01:06,850
with a certain amount of kinetic energy,

19
00:01:06,850 --> 00:01:08,826
and they're exerting a certain pressure

20
00:01:08,826 --> 00:01:09,778
on their container,

21
00:01:09,778 --> 00:01:14,370
and if we were to make the container smaller,

22
00:01:14,370 --> 00:01:16,198
we have the same number of particles.

23
00:01:16,198 --> 00:01:17,434
n doesn't change.

24
00:01:17,434 --> 00:01:19,882
The average kinetic energy doesn't change,

25
00:01:19,882 --> 00:01:21,656
so they're just going to bump into the walls more.

26
00:01:21,656 --> 00:01:24,216
So that when we make the volume smaller,

27
00:01:24,216 --> 00:01:26,734
when the volume goes up----

28
00:01:26,734 --> 00:01:27,757
when the volume goes down,

29
00:01:27,757 --> 00:01:30,068
the pressure should go up.

30
00:01:30,068 --> 00:01:32,621
So let's see if we can calculate the exact number.

31
00:01:32,621 --> 00:01:35,429
So we can take our ideal gas equation:

32
00:01:35,429 --> 00:01:41,872
pressure times volume is equal to nRT.

33
00:01:41,872 --> 00:01:44,316
Now, do the number of particles change

34
00:01:44,316 --> 00:01:47,981
when I did this situation when I shrunk the volume?

35
00:01:47,981 --> 00:01:48,650
No!

36
00:01:48,650 --> 00:01:49,758
We have the same number of particles.

37
00:01:49,758 --> 00:01:50,925
I'm just shrinking the container,

38
00:01:50,925 --> 00:01:55,200
so n is n, R doesn't change, that's a constant,

39
00:01:55,200 --> 00:01:57,223
and then the temperature doesn't change.

40
00:01:57,223 --> 00:02:00,319
So my old pressure times volume

41
00:02:00,319 --> 00:02:02,689
is going to be equal to nRT,

42
00:02:02,689 --> 00:02:04,311
and my new pressure times volume--

43
00:02:04,311 --> 00:02:07,946
so let me call this P1 and V1.

44
00:02:07,946 --> 00:02:11,001
and then P2 is this----

45
00:02:11,001 --> 00:02:15,595
sorry, that's V2.

46
00:02:15,595 --> 00:02:21,703
so V2 is this, and we're trying to figure out P2.

47
00:02:21,703 --> 00:02:23,134
P2 is what?

48
00:02:23,134 --> 00:02:31,354
Well, we know that P1 times V1 is equal to nRT,

49
00:02:31,354 --> 00:02:33,396
and we also know that since temperature and

50
00:02:33,396 --> 00:02:35,984
the number of moles of our gas stay constant,

51
00:02:35,984 --> 00:02:40,787
that P2 times V2 is equal to nRT.

52
00:02:40,787 --> 00:02:43,196
And since they both equal the same thing,

53
00:02:43,196 --> 00:02:45,673
we can say that the pressure times the volume,

54
00:02:45,673 --> 00:02:47,799
as long as the temperature is held constant,

55
00:02:47,799 --> 00:02:49,201
will be a constant.

56
00:02:49,201 --> 00:02:55,764
So P1 times V1 is going to equal P2 times V2.

57
00:02:55,764 --> 00:02:57,939
So what was P1?

58
00:02:57,939 --> 00:03:03,233
P1, our initial pressure, was 3 atmospheres.

59
00:03:06,633 --> 00:03:12,024
So 3 atmospheres times 9 liters is equal to

60
00:03:12,024 --> 00:03:15,977
our new pressure times 3 liters.

61
00:03:15,977 --> 00:03:18,992
And if we divide both sides of the equation by 3,

62
00:03:18,992 --> 00:03:24,701
we get 3 liters cancel out,

63
00:03:24,701 --> 00:03:33,637
we're left with 9 atmospheres.

64
00:03:33,637 --> 00:03:34,799
And that should make sense.

65
00:03:34,799 --> 00:03:39,258
When you decrease the volume by 2/3

66
00:03:39,258 --> 00:03:40,304
or when you make the volume

67
00:03:40,304 --> 00:03:42,939
1/3 of your original volume,

68
00:03:42,939 --> 00:03:46,199
then your pressure increases by a factor of three.

69
00:03:46,199 --> 00:03:51,571
So this went by times 3, and this went by times 1/3.

70
00:03:51,571 --> 00:03:52,898
That's a useful thing to know in general.

71
00:03:52,898 --> 00:03:55,201
If temperature is held constant,

72
00:03:55,201 --> 00:03:57,478
then pressure times volume

73
00:03:57,478 --> 00:03:59,111
are going to be a constant.

74
00:03:59,111 --> 00:04:00,958
Now, you can take that even further.

75
00:04:00,958 --> 00:04:06,878
If we look at PV equals nRT,

76
00:04:06,878 --> 00:04:09,162
the two things that we know don't change

77
00:04:09,162 --> 00:04:11,840
in the vast majority of exercises we do

78
00:04:11,840 --> 00:04:13,535
is the number of molecules we're dealing with,

79
00:04:13,535 --> 00:04:15,529
and obviously, R isn't going to change.

80
00:04:15,529 --> 00:04:18,265
So if we divide both sides of this by T,

81
00:04:18,265 --> 00:04:23,165
we get PV over T is equal to nR,

82
00:04:23,165 --> 00:04:24,918
or you could say it's equal to a constant.

83
00:04:24,918 --> 00:04:27,203
This is going to be a constant number for any system

84
00:04:27,203 --> 00:04:28,629
where we're not changing

85
00:04:28,629 --> 00:04:31,524
the number of molecules in the container.

86
00:04:31,524 --> 00:04:33,373
So, if we are changing the pressure----

87
00:04:33,373 --> 00:04:35,653
So if initially we start with

88
00:04:35,653 --> 00:04:40,000
pressure one, volume one, and some temperature one

89
00:04:40,000 --> 00:04:41,501
that's going to be equal to this constant.

90
00:04:41,501 --> 00:04:44,192
And if we change any of them,

91
00:04:44,192 --> 00:04:44,731
we go back to

92
00:04:44,731 --> 00:04:48,861
pressure two, volume two, temperature two,

93
00:04:48,861 --> 00:04:50,470
they should still be equal to this constant,

94
00:04:50,470 --> 00:04:51,467
so they equal each other.

95
00:04:51,467 --> 00:04:55,350
So for example, let's say I start off with a

96
00:04:55,350 --> 00:05:01,076
pressure of 1 atmosphere.

97
00:05:01,076 --> 00:05:05,066
and I have a volume of----

98
00:05:05,066 --> 00:05:08,613
I'll switch units here just to do things differently

99
00:05:08,613 --> 00:05:10,639
----2 meters cubed.

100
00:05:10,639 --> 00:05:20,209
And let's say our temperature is 27 degrees Celsius.

101
00:05:20,209 --> 00:05:21,742
Well, and I just wrote Celsius

102
00:05:21,742 --> 00:05:22,697
because I want you to always remember

103
00:05:22,697 --> 00:05:23,973
you have to convert to Kelvin,

104
00:05:23,973 --> 00:05:27,830
so 27 degrees plus 273 will get us

105
00:05:27,830 --> 00:05:33,154
exactly to 300 Kelvin.

106
00:05:33,154 --> 00:05:39,531
And let's say that our new temperature is

107
00:05:39,531 --> 00:05:40,631
Actually let's figure out what the new temperature

108
00:05:40,631 --> 00:05:41,418
is going to be.

109
00:05:41,418 --> 00:05:46,270
Let's say our new pressure is 2 atmospheres.

110
00:05:46,270 --> 00:05:47,884
The pressure has increased.

111
00:05:47,884 --> 00:05:50,014
Let's say we make the container smaller,

112
00:05:50,014 --> 00:05:52,487
so 1 meter cubed.

113
00:05:52,487 --> 00:05:55,101
So the container has been decreased by half

114
00:05:55,101 --> 00:05:56,680
and the pressure is doubled by half.

115
00:05:56,680 --> 00:05:57,591
So you could guess.

116
00:05:57,591 --> 00:06:02,154
You know, we have made the pressure higher----

117
00:06:02,154 --> 00:06:08,179
Let me make the container even smaller.

118
00:06:08,179 --> 00:06:08,771
Actually, no.

119
00:06:08,771 --> 00:06:10,709
Let me make the pressure even larger.

120
00:06:10,709 --> 00:06:14,257
Let me make the pressure into 5 atmospheres.

121
00:06:14,257 --> 00:06:16,937
Now we want to know what the second temperature is

122
00:06:16,937 --> 00:06:18,810
and we set up our equation.

123
00:06:18,810 --> 00:06:19,533
And so we have

124
00:06:19,533 --> 00:06:28,103
2/300 atmosphere meters cubed per Kelvin

125
00:06:28,103 --> 00:06:32,687
is equal to 5/T2, our new temperature,

126
00:06:32,687 --> 00:06:40,148
and then we have 1,500 is equal to 2T2.

127
00:06:40,148 --> 00:06:41,372
Divide both sides by 2.

128
00:06:41,372 --> 00:06:46,902
You have T2 is equal to 750 degrees Kelvin,

129
00:06:46,902 --> 00:06:48,314
which makes sense, right?

130
00:06:48,314 --> 00:06:50,537
We increased the pressure so much

131
00:06:50,537 --> 00:06:53,288
and we decreased the volume at the same time

132
00:06:53,288 --> 00:06:55,638
that the temperature just had to go up.

133
00:06:55,638 --> 00:06:56,553
Or if you thought of it the other way,

134
00:06:56,553 --> 00:06:58,176
maybe we increased the temperature

135
00:06:58,176 --> 00:06:59,500
and that's what drove the pressure

136
00:06:59,500 --> 00:07:00,691
to be so much higher,

137
00:07:00,691 --> 00:07:03,874
especially since we decreased the volume.

138
00:07:03,874 --> 00:07:05,398
I guess the best way to think about is

139
00:07:05,398 --> 00:07:08,233
this pressure went up so much,

140
00:07:08,233 --> 00:07:10,196
it went up by factor of five,

141
00:07:10,196 --> 00:07:12,477
it went from 1 atmosphere to 5 atmospheres,

142
00:07:12,477 --> 00:07:14,374
because on one level

143
00:07:14,374 --> 00:07:18,032
we shrunk the volume by a factor of 1/2,

144
00:07:18,032 --> 00:07:19,685
so that should have doubled the pressure,

145
00:07:19,685 --> 00:07:21,903
so that should have gotten us to two atmospheres.

146
00:07:21,903 --> 00:07:23,783
And then we made the temperature a lot higher,

147
00:07:23,783 --> 00:07:25,407
so we were also bouncing into the container.

148
00:07:25,407 --> 00:07:27,901
We made the temperature 750 degrees Kelvin,

149
00:07:27,901 --> 00:07:29,892
so more than double the temperature,

150
00:07:29,892 --> 00:07:33,879
and then that's what got us to 5 atmospheres.

151
00:07:33,879 --> 00:07:37,988
Now, one other thing that you'll probably hear about

152
00:07:37,988 --> 00:07:39,689
is the notion of what happens

153
00:07:39,689 --> 00:07:42,475
at standard temperature and pressure.

154
00:07:42,475 --> 00:07:44,038
Let me delete all of the stuff over here.

155
00:07:44,038 --> 00:07:47,572
Standard temperature and pressure.

156
00:07:47,572 --> 00:07:51,532
Let me delete all this stuff that I don't need.

157
00:07:52,886 --> 00:07:56,809
Standard temperature and pressure.

158
00:07:56,809 --> 00:07:57,466
And I'm bringing it up

159
00:07:57,466 --> 00:07:58,690
because even though it's called

160
00:07:58,690 --> 00:07:59,881
standard temperature and pressure,

161
00:07:59,881 --> 00:08:03,704
and sometimes called STP,

162
00:08:03,704 --> 00:08:05,740
unfortunately for the world,

163
00:08:05,740 --> 00:08:07,840
they haven't really standardized

164
00:08:07,840 --> 00:08:13,742
what the standard pressure and temperature are.

165
00:08:13,742 --> 00:08:15,916
I went to Wikipedia and I looked it up.

166
00:08:15,916 --> 00:08:16,882
And the one that you'll probably see

167
00:08:16,882 --> 00:08:19,986
in most physics classes and most standardized tests

168
00:08:19,986 --> 00:08:23,935
is standard temperature is 0 degrees celsius,

169
00:08:23,935 --> 00:08:26,837
which is, of course, 273 degrees Kelvin.

170
00:08:26,837 --> 00:08:30,302
And standard pressure is 1 atmosphere.

171
00:08:30,302 --> 00:08:31,241
And here on Wikipedia,

172
00:08:31,241 --> 00:08:38,533
they wrote it as 101.325 kilopascals,

173
00:08:38,533 --> 00:08:41,341
or a little more than 101,000 pascals.

174
00:08:41,341 --> 00:08:44,246
of course, a pascal is a newton per square meter.

175
00:08:44,246 --> 00:08:45,968
In all of this stuff, the units are really

176
00:08:45,968 --> 00:08:47,665
the hardest part to get a hold of.

177
00:08:47,665 --> 00:08:49,741
But let's say that we assume

178
00:08:49,741 --> 00:08:50,676
these are all different

179
00:08:50,676 --> 00:08:52,195
standard temperatures and pressures

180
00:08:52,195 --> 00:08:54,878
based on different standard-making bodies.

181
00:08:54,878 --> 00:08:55,783
So they can't really agree with each other.

182
00:08:55,783 --> 00:08:57,259
But let's say we took this as

183
00:08:57,259 --> 00:09:00,889
the definition of standard temperature and pressure.

184
00:09:00,889 --> 00:09:04,604
So we're assuming that temperature

185
00:09:04,604 --> 00:09:07,227
is equal to 0 degrees Celsius,

186
00:09:07,227 --> 00:09:11,203
which is equal to 273 degrees Kelvin.

187
00:09:11,203 --> 00:09:15,255
And pressure, we're assuming, is 1 atmosphere,

188
00:09:15,255 --> 00:09:16,075
which could also be written as

189
00:09:16,075 --> 00:09:22,440
101.325 or 3/8 kilopascals.

190
00:09:22,440 --> 00:09:26,349
So my question is if I have an ideal gas

191
00:09:26,349 --> 00:09:30,021
at standard temperature and pressure,

192
00:09:30,021 --> 00:09:36,453
how many moles of that do I have in 1 liter?

193
00:09:36,453 --> 00:09:37,583
No, let me say that the other way.

194
00:09:37,583 --> 00:09:40,868
How many liters will 1 mole take up?

195
00:09:40,868 --> 00:09:43,785
So let me say that a little bit more.

196
00:09:43,785 --> 00:09:46,384
So n is equal to 1 mole.

197
00:09:46,384 --> 00:09:48,940
So I want to figure out what my volume is.

198
00:09:48,940 --> 00:09:50,657
So if I have 1 mole of a gas,

199
00:09:50,657 --> 00:09:55,556
I have 6.02 times 10 to 23 molecules of that gas.

200
00:09:55,556 --> 00:09:58,456
It's at standard pressure, 1 atmosphere,

201
00:09:58,456 --> 00:10:01,002
and at standard temperature, 273 degrees,

202
00:10:01,002 --> 00:10:03,455
what is the volume of that gas?

203
00:10:03,455 --> 00:10:07,745
So let's apply PV is equal to nRT.

204
00:10:07,745 --> 00:10:10,096
Pressure is 1 atmosphere,

205
00:10:10,096 --> 00:10:11,748
but remember we're dealing with atmospheres.

206
00:10:11,748 --> 00:10:15,362
1 atmosphere times volume

207
00:10:15,362 --> 00:10:16,656
that's what we're solving for.

208
00:10:16,656 --> 00:10:18,043
I'll do that in purple

209
00:10:18,043 --> 00:10:22,007
is equal to 1 mole, we have 1 mole of the gas,

210
00:10:22,007 --> 00:10:29,312
times R, times temperature, times 273.

211
00:10:29,312 --> 00:10:31,786
Now this is in Kelvin; this is in moles.

212
00:10:31,786 --> 00:10:39,508
We want our volume in liters.

213
00:10:39,508 --> 00:10:41,562
So which version of R should we use?

214
00:10:41,562 --> 00:10:44,414
Well, we're dealing with atmospheres.

215
00:10:44,414 --> 00:10:46,609
We want our volume in liters,

216
00:10:46,609 --> 00:10:48,029
and of course, we have moles in Kelvin,

217
00:10:48,029 --> 00:10:50,531
so we'll use this version, 0.082.

218
00:10:50,531 --> 00:10:52,210
So this is 1,

219
00:10:52,210 --> 00:10:54,866
so we can ignore the 1 there, the 1 there.

220
00:10:54,866 --> 00:10:56,388
So the volume is equal to

221
00:10:56,388 --> 00:11:02,204
0.082 times 273 degrees Kelvin,

222
00:11:02,204 --> 00:11:19,229
and that is 0.082 times 273 is equal to 22.4 liters.

223
00:11:19,229 --> 00:11:21,429
So if I have any ideal gas,

224
00:11:21,429 --> 00:11:24,079
and all gases don't behave ideally ideal,

225
00:11:24,079 --> 00:11:25,475
but if I have an ideal gas

226
00:11:25,475 --> 00:11:26,930
and it's at standard temperature,

227
00:11:26,930 --> 00:11:29,099
which is at 0 degrees Celsius,

228
00:11:29,099 --> 00:11:30,423
or the freezing point of water,

229
00:11:30,423 --> 00:11:32,423
which is also 273 degrees Kelvin,

230
00:11:32,423 --> 00:11:33,713
and I have a mole of it,

231
00:11:33,713 --> 00:11:37,559
and it's at standard pressure, 1 atmosphere,

232
00:11:37,559 --> 00:11:42,479
that gas should take up exactly 22.4 liters.

233
00:11:42,479 --> 00:11:44,796
And if you wanted to know how many meters cubed

234
00:11:44,796 --> 00:11:46,385
it's going to take up.

235
00:11:46,385 --> 00:11:50,987
well, you could just say 22.4 liters times----

236
00:11:50,987 --> 00:11:53,236
now, how many meters cubed are there----

237
00:11:53,236 --> 00:11:57,501
so for every 1 meter cubed, you have 1,000 liters.

238
00:11:57,501 --> 00:11:59,627
I know that seems like a lot, but it's true.

239
00:11:59,627 --> 00:12:02,482
Just think about how big a meter cubed is.

240
00:12:02,482 --> 00:12:09,365
So this would be equal to 0.0224 meters cubed.

241
00:12:09,365 --> 00:12:12,450
If you have something at 1 atmosphere, a mole of it,

242
00:12:12,450 --> 00:12:14,748
and at 0 degrees Celsius.

243
00:12:14,748 --> 00:12:16,083
Anyway, this is actually

244
00:12:16,083 --> 00:12:17,712
a useful number to know sometimes.

245
00:12:17,712 --> 00:12:22,248
They'll often say you have 2 moles

246
00:12:22,248 --> 00:12:25,292
at standard temperature and pressure.

247
00:12:25,292 --> 00:12:26,966
How many liters is it going to take up?

248
00:12:26,966 --> 00:12:29,614
Well, 1 mole will take up this many,

249
00:12:29,614 --> 00:12:31,780
and so 2 moles at standard temperature and pressure

250
00:12:31,780 --> 00:12:33,436
will take up twice as much,

251
00:12:33,436 --> 00:12:34,805
because you're just taking PV equals nRT

252
00:12:34,805 --> 00:12:36,272
and just doubling.

253
00:12:36,272 --> 00:12:38,790
Everything else is being held constant.

254
00:12:38,790 --> 00:12:40,992
The pressure, everything else is being held constant,

255
00:12:40,992 --> 00:12:43,043
so if you double the number of moles,

256
00:12:43,043 --> 00:12:44,206
you're going to double the volume it takes up.

257
00:12:44,206 --> 00:12:46,107
Or if you half the number of moles,

258
00:12:46,107 --> 00:12:47,674
you're going to half the volume it takes up.

259
00:12:47,674 --> 00:12:49,656
So it's a useful thing to know that in liters

260
00:12:49,656 --> 00:12:52,015
at standard temperature and pressure,

261
00:12:52,015 --> 00:12:52,911
where standard temperature and pressure

262
00:12:52,911 --> 00:12:56,516
is defined as 1 atmosphere and 273 degrees Kelvin,

263
00:12:56,516 --> 00:13:00,159
an idea gas will take up 22.4 liters of volume