0:00:01.105,0:00:02.145
Let's do some more problems

0:00:02.145,0:00:04.556
that involve the ideal gas equation.

0:00:04.556,0:00:06.680
Let's say I have a gas in a container

0:00:06.680,0:00:15.323
and the current pressure is 3 atmospheres.

0:00:15.323,0:00:19.756
And let's say that the volume of the container

0:00:19.756,0:00:27.413
is, I don't know, 9 liters.

0:00:27.413,0:00:30.136
Now, what will the pressure become

0:00:30.136,0:00:39.280
if my volume goes from 9 liters to 3 liters?

0:00:39.280,0:00:42.183
So from the first ideal gas equation video

0:00:42.183,0:00:43.348
you can kind of have the intuition

0:00:43.348,0:00:46.935
that you have a bunch of-- and we're holding--

0:00:46.935,0:00:47.901
and this is important.

0:00:47.901,0:00:50.763
We're holding the temperature constant

0:00:50.763,0:00:52.541
and that's an important thing to realize.

0:00:52.541,0:00:58.384
So in our very original intuition

0:00:58.384,0:01:00.354
behind the ideal gas equation we said,

0:01:00.354,0:01:02.990
look, if we have a certain number of particles

0:01:02.990,0:01:06.850
with a certain amount of kinetic energy,

0:01:06.850,0:01:08.826
and they're exerting a certain pressure

0:01:08.826,0:01:09.778
on their container,

0:01:09.778,0:01:14.370
and if we were to make the container smaller,

0:01:14.370,0:01:16.198
we have the same number of particles.

0:01:16.198,0:01:17.434
n doesn't change.

0:01:17.434,0:01:19.882
The average kinetic energy doesn't change,

0:01:19.882,0:01:21.656
so they're just going to bump into the walls more.

0:01:21.656,0:01:24.216
So that when we make the volume smaller,

0:01:24.216,0:01:26.734
when the volume goes up----

0:01:26.734,0:01:27.757
when the volume goes down,

0:01:27.757,0:01:30.068
the pressure should go up.

0:01:30.068,0:01:32.621
So let's see if we can calculate the exact number.

0:01:32.621,0:01:35.429
So we can take our ideal gas equation:

0:01:35.429,0:01:41.872
pressure times volume is equal to nRT.

0:01:41.872,0:01:44.316
Now, do the number of particles change

0:01:44.316,0:01:47.981
when I did this situation when I shrunk the volume?

0:01:47.981,0:01:48.650
No!

0:01:48.650,0:01:49.758
We have the same number of particles.

0:01:49.758,0:01:50.925
I'm just shrinking the container,

0:01:50.925,0:01:55.200
so n is n, R doesn't change, that's a constant,

0:01:55.200,0:01:57.223
and then the temperature doesn't change.

0:01:57.223,0:02:00.319
So my old pressure times volume

0:02:00.319,0:02:02.689
is going to be equal to nRT,

0:02:02.689,0:02:04.311
and my new pressure times volume--

0:02:04.311,0:02:07.946
so let me call this P1 and V1.

0:02:07.946,0:02:11.001
and then P2 is this----

0:02:11.001,0:02:15.595
sorry, that's V2.

0:02:15.595,0:02:21.703
so V2 is this, and we're trying to figure out P2.

0:02:21.703,0:02:23.134
P2 is what?

0:02:23.134,0:02:31.354
Well, we know that P1 times V1 is equal to nRT,

0:02:31.354,0:02:33.396
and we also know that since temperature and

0:02:33.396,0:02:35.984
the number of moles of our gas stay constant,

0:02:35.984,0:02:40.787
that P2 times V2 is equal to nRT.

0:02:40.787,0:02:43.196
And since they both equal the same thing,

0:02:43.196,0:02:45.673
we can say that the pressure times the volume,

0:02:45.673,0:02:47.799
as long as the temperature is held constant,

0:02:47.799,0:02:49.201
will be a constant.

0:02:49.201,0:02:55.764
So P1 times V1 is going to equal P2 times V2.

0:02:55.764,0:02:57.939
So what was P1?

0:02:57.939,0:03:03.233
P1, our initial pressure, was 3 atmospheres.

0:03:06.633,0:03:12.024
So 3 atmospheres times 9 liters is equal to

0:03:12.024,0:03:15.977
our new pressure times 3 liters.

0:03:15.977,0:03:18.992
And if we divide both sides of the equation by 3,

0:03:18.992,0:03:24.701
we get 3 liters cancel out,

0:03:24.701,0:03:33.637
we're left with 9 atmospheres.

0:03:33.637,0:03:34.799
And that should make sense.

0:03:34.799,0:03:39.258
When you decrease the volume by 2/3

0:03:39.258,0:03:40.304
or when you make the volume

0:03:40.304,0:03:42.939
1/3 of your original volume,

0:03:42.939,0:03:46.199
then your pressure increases by a factor of three.

0:03:46.199,0:03:51.571
So this went by times 3, and this went by times 1/3.

0:03:51.571,0:03:52.898
That's a useful thing to know in general.

0:03:52.898,0:03:55.201
If temperature is held constant,

0:03:55.201,0:03:57.478
then pressure times volume

0:03:57.478,0:03:59.111
are going to be a constant.

0:03:59.111,0:04:00.958
Now, you can take that even further.

0:04:00.958,0:04:06.878
If we look at PV equals nRT,

0:04:06.878,0:04:09.162
the two things that we know don't change

0:04:09.162,0:04:11.840
in the vast majority of exercises we do

0:04:11.840,0:04:13.535
is the number of molecules we're dealing with,

0:04:13.535,0:04:15.529
and obviously, R isn't going to change.

0:04:15.529,0:04:18.265
So if we divide both sides of this by T,

0:04:18.265,0:04:23.165
we get PV over T is equal to nR,

0:04:23.165,0:04:24.918
or you could say it's equal to a constant.

0:04:24.918,0:04:27.203
This is going to be a constant number for any system

0:04:27.203,0:04:28.629
where we're not changing

0:04:28.629,0:04:31.524
the number of molecules in the container.

0:04:31.524,0:04:33.373
So, if we are changing the pressure----

0:04:33.373,0:04:35.653
So if initially we start with

0:04:35.653,0:04:40.000
pressure one, volume one, and some temperature one

0:04:40.000,0:04:41.501
that's going to be equal to this constant.

0:04:41.501,0:04:44.192
And if we change any of them,

0:04:44.192,0:04:44.731
we go back to

0:04:44.731,0:04:48.861
pressure two, volume two, temperature two,

0:04:48.861,0:04:50.470
they should still be equal to this constant,

0:04:50.470,0:04:51.467
so they equal each other.

0:04:51.467,0:04:55.350
So for example, let's say I start off with a

0:04:55.350,0:05:01.076
pressure of 1 atmosphere.

0:05:01.076,0:05:05.066
and I have a volume of----

0:05:05.066,0:05:08.613
I'll switch units here just to do things differently

0:05:08.613,0:05:10.639
----2 meters cubed.

0:05:10.639,0:05:20.209
And let's say our temperature is 27 degrees Celsius.

0:05:20.209,0:05:21.742
Well, and I just wrote Celsius

0:05:21.742,0:05:22.697
because I want you to always remember

0:05:22.697,0:05:23.973
you have to convert to Kelvin,

0:05:23.973,0:05:27.830
so 27 degrees plus 273 will get us

0:05:27.830,0:05:33.154
exactly to 300 Kelvin.

0:05:33.154,0:05:39.531
And let's say that our new temperature is

0:05:39.531,0:05:40.631
Actually let's figure out what the new temperature

0:05:40.631,0:05:41.418
is going to be.

0:05:41.418,0:05:46.270
Let's say our new pressure is 2 atmospheres.

0:05:46.270,0:05:47.884
The pressure has increased.

0:05:47.884,0:05:50.014
Let's say we make the container smaller,

0:05:50.014,0:05:52.487
so 1 meter cubed.

0:05:52.487,0:05:55.101
So the container has been decreased by half

0:05:55.101,0:05:56.680
and the pressure is doubled by half.

0:05:56.680,0:05:57.591
So you could guess.

0:05:57.591,0:06:02.154
You know, we have made the pressure higher----

0:06:02.154,0:06:08.179
Let me make the container even smaller.

0:06:08.179,0:06:08.771
Actually, no.

0:06:08.771,0:06:10.709
Let me make the pressure even larger.

0:06:10.709,0:06:14.257
Let me make the pressure into 5 atmospheres.

0:06:14.257,0:06:16.937
Now we want to know what the second temperature is

0:06:16.937,0:06:18.810
and we set up our equation.

0:06:18.810,0:06:19.533
And so we have

0:06:19.533,0:06:28.103
2/300 atmosphere meters cubed per Kelvin

0:06:28.103,0:06:32.687
is equal to 5/T2, our new temperature,

0:06:32.687,0:06:40.148
and then we have 1,500 is equal to 2T2.

0:06:40.148,0:06:41.372
Divide both sides by 2.

0:06:41.372,0:06:46.902
You have T2 is equal to 750 degrees Kelvin,

0:06:46.902,0:06:48.314
which makes sense, right?

0:06:48.314,0:06:50.537
We increased the pressure so much

0:06:50.537,0:06:53.288
and we decreased the volume at the same time

0:06:53.288,0:06:55.638
that the temperature just had to go up.

0:06:55.638,0:06:56.553
Or if you thought of it the other way,

0:06:56.553,0:06:58.176
maybe we increased the temperature

0:06:58.176,0:06:59.500
and that's what drove the pressure

0:06:59.500,0:07:00.691
to be so much higher,

0:07:00.691,0:07:03.874
especially since we decreased the volume.

0:07:03.874,0:07:05.398
I guess the best way to think about is

0:07:05.398,0:07:08.233
this pressure went up so much,

0:07:08.233,0:07:10.196
it went up by factor of five,

0:07:10.196,0:07:12.477
it went from 1 atmosphere to 5 atmospheres,

0:07:12.477,0:07:14.374
because on one level

0:07:14.374,0:07:18.032
we shrunk the volume by a factor of 1/2,

0:07:18.032,0:07:19.685
so that should have doubled the pressure,

0:07:19.685,0:07:21.903
so that should have gotten us to two atmospheres.

0:07:21.903,0:07:23.783
And then we made the temperature a lot higher,

0:07:23.783,0:07:25.407
so we were also bouncing into the container.

0:07:25.407,0:07:27.901
We made the temperature 750 degrees Kelvin,

0:07:27.901,0:07:29.892
so more than double the temperature,

0:07:29.892,0:07:33.879
and then that's what got us to 5 atmospheres.

0:07:33.879,0:07:37.988
Now, one other thing that you'll probably hear about

0:07:37.988,0:07:39.689
is the notion of what happens

0:07:39.689,0:07:42.475
at standard temperature and pressure.

0:07:42.475,0:07:44.038
Let me delete all of the stuff over here.

0:07:44.038,0:07:47.572
Standard temperature and pressure.

0:07:47.572,0:07:51.532
Let me delete all this stuff that I don't need.

0:07:52.886,0:07:56.809
Standard temperature and pressure.

0:07:56.809,0:07:57.466
And I'm bringing it up

0:07:57.466,0:07:58.690
because even though it's called

0:07:58.690,0:07:59.881
standard temperature and pressure,

0:07:59.881,0:08:03.704
and sometimes called STP,

0:08:03.704,0:08:05.740
unfortunately for the world,

0:08:05.740,0:08:07.840
they haven't really standardized

0:08:07.840,0:08:13.742
what the standard pressure and temperature are.

0:08:13.742,0:08:15.916
I went to Wikipedia and I looked it up.

0:08:15.916,0:08:16.882
And the one that you'll probably see

0:08:16.882,0:08:19.986
in most physics classes and most standardized tests

0:08:19.986,0:08:23.935
is standard temperature is 0 degrees celsius,

0:08:23.935,0:08:26.837
which is, of course, 273 degrees Kelvin.

0:08:26.837,0:08:30.302
And standard pressure is 1 atmosphere.

0:08:30.302,0:08:31.241
And here on Wikipedia,

0:08:31.241,0:08:38.533
they wrote it as 101.325 kilopascals,

0:08:38.533,0:08:41.341
or a little more than 101,000 pascals.

0:08:41.341,0:08:44.246
of course, a pascal is a newton per square meter.

0:08:44.246,0:08:45.968
In all of this stuff, the units are really

0:08:45.968,0:08:47.665
the hardest part to get a hold of.

0:08:47.665,0:08:49.741
But let's say that we assume

0:08:49.741,0:08:50.676
these are all different

0:08:50.676,0:08:52.195
standard temperatures and pressures

0:08:52.195,0:08:54.878
based on different standard-making bodies.

0:08:54.878,0:08:55.783
So they can't really agree with each other.

0:08:55.783,0:08:57.259
But let's say we took this as

0:08:57.259,0:09:00.889
the definition of standard temperature and pressure.

0:09:00.889,0:09:04.604
So we're assuming that temperature

0:09:04.604,0:09:07.227
is equal to 0 degrees Celsius,

0:09:07.227,0:09:11.203
which is equal to 273 degrees Kelvin.

0:09:11.203,0:09:15.255
And pressure, we're assuming, is 1 atmosphere,

0:09:15.255,0:09:16.075
which could also be written as

0:09:16.075,0:09:22.440
101.325 or 3/8 kilopascals.

0:09:22.440,0:09:26.349
So my question is if I have an ideal gas

0:09:26.349,0:09:30.021
at standard temperature and pressure,

0:09:30.021,0:09:36.453
how many moles of that do I have in 1 liter?

0:09:36.453,0:09:37.583
No, let me say that the other way.

0:09:37.583,0:09:40.868
How many liters will 1 mole take up?

0:09:40.868,0:09:43.785
So let me say that a little bit more.

0:09:43.785,0:09:46.384
So n is equal to 1 mole.

0:09:46.384,0:09:48.940
So I want to figure out what my volume is.

0:09:48.940,0:09:50.657
So if I have 1 mole of a gas,

0:09:50.657,0:09:55.556
I have 6.02 times 10 to 23 molecules of that gas.

0:09:55.556,0:09:58.456
It's at standard pressure, 1 atmosphere,

0:09:58.456,0:10:01.002
and at standard temperature, 273 degrees,

0:10:01.002,0:10:03.455
what is the volume of that gas?

0:10:03.455,0:10:07.745
So let's apply PV is equal to nRT.

0:10:07.745,0:10:10.096
Pressure is 1 atmosphere,

0:10:10.096,0:10:11.748
but remember we're dealing with atmospheres.

0:10:11.748,0:10:15.362
1 atmosphere times volume

0:10:15.362,0:10:16.656
that's what we're solving for.

0:10:16.656,0:10:18.043
I'll do that in purple

0:10:18.043,0:10:22.007
is equal to 1 mole, we have 1 mole of the gas,

0:10:22.007,0:10:29.312
times R, times temperature, times 273.

0:10:29.312,0:10:31.786
Now this is in Kelvin; this is in moles.

0:10:31.786,0:10:39.508
We want our volume in liters.

0:10:39.508,0:10:41.562
So which version of R should we use?

0:10:41.562,0:10:44.414
Well, we're dealing with atmospheres.

0:10:44.414,0:10:46.609
We want our volume in liters,

0:10:46.609,0:10:48.029
and of course, we have moles in Kelvin,

0:10:48.029,0:10:50.531
so we'll use this version, 0.082.

0:10:50.531,0:10:52.210
So this is 1,

0:10:52.210,0:10:54.866
so we can ignore the 1 there, the 1 there.

0:10:54.866,0:10:56.388
So the volume is equal to

0:10:56.388,0:11:02.204
0.082 times 273 degrees Kelvin,

0:11:02.204,0:11:19.229
and that is 0.082 times 273 is equal to 22.4 liters.

0:11:19.229,0:11:21.429
So if I have any ideal gas,

0:11:21.429,0:11:24.079
and all gases don't behave ideally ideal,

0:11:24.079,0:11:25.475
but if I have an ideal gas

0:11:25.475,0:11:26.930
and it's at standard temperature,

0:11:26.930,0:11:29.099
which is at 0 degrees Celsius,

0:11:29.099,0:11:30.423
or the freezing point of water,

0:11:30.423,0:11:32.423
which is also 273 degrees Kelvin,

0:11:32.423,0:11:33.713
and I have a mole of it,

0:11:33.713,0:11:37.559
and it's at standard pressure, 1 atmosphere,

0:11:37.559,0:11:42.479
that gas should take up exactly 22.4 liters.

0:11:42.479,0:11:44.796
And if you wanted to know how many meters cubed

0:11:44.796,0:11:46.385
it's going to take up.

0:11:46.385,0:11:50.987
well, you could just say 22.4 liters times----

0:11:50.987,0:11:53.236
now, how many meters cubed are there----

0:11:53.236,0:11:57.501
so for every 1 meter cubed, you have 1,000 liters.

0:11:57.501,0:11:59.627
I know that seems like a lot, but it's true.

0:11:59.627,0:12:02.482
Just think about how big a meter cubed is.

0:12:02.482,0:12:09.365
So this would be equal to 0.0224 meters cubed.

0:12:09.365,0:12:12.450
If you have something at 1 atmosphere, a mole of it,

0:12:12.450,0:12:14.748
and at 0 degrees Celsius.

0:12:14.748,0:12:16.083
Anyway, this is actually

0:12:16.083,0:12:17.712
a useful number to know sometimes.

0:12:17.712,0:12:22.248
They'll often say you have 2 moles

0:12:22.248,0:12:25.292
at standard temperature and pressure.

0:12:25.292,0:12:26.966
How many liters is it going to take up?

0:12:26.966,0:12:29.614
Well, 1 mole will take up this many,

0:12:29.614,0:12:31.780
and so 2 moles at standard temperature and pressure

0:12:31.780,0:12:33.436
will take up twice as much,

0:12:33.436,0:12:34.805
because you're just taking PV equals nRT

0:12:34.805,0:12:36.272
and just doubling.

0:12:36.272,0:12:38.790
Everything else is being held constant.

0:12:38.790,0:12:40.992
The pressure, everything else is being held constant,

0:12:40.992,0:12:43.043
so if you double the number of moles,

0:12:43.043,0:12:44.206
you're going to double the volume it takes up.

0:12:44.206,0:12:46.107
Or if you half the number of moles,

0:12:46.107,0:12:47.674
you're going to half the volume it takes up.

0:12:47.674,0:12:49.656
So it's a useful thing to know that in liters

0:12:49.656,0:12:52.015
at standard temperature and pressure,

0:12:52.015,0:12:52.911
where standard temperature and pressure

0:12:52.911,0:12:56.516
is defined as 1 atmosphere and 273 degrees Kelvin,

0:12:56.516,0:13:00.159
an idea gas will take up 22.4 liters of volume