1
0:00:02 --> 00:00:08
Liquids are incompressible; 
gases are not incompressible.

2
00:00:09 --> 00:00:15
When you decrease the volume of
a gas by 50%, that's no problem.

3
00:00:14 --> 00:00:20
It's impossible to do that
for a liquid.

4
00:00:18 --> 00:00:24
In liquids, the atoms
and the molecules

5
00:00:20 --> 00:00:26
effectively touch each other,

6
00:00:22 --> 00:00:28
whereas in gases,
they are very far apart,

7
00:00:25 --> 00:00:31
so that's why
you can compress the gases.

8
00:00:27 --> 00:00:33
If you take air
at one atmospheres,

9
00:00:30 --> 00:00:36
the density is a thousand times
less than the density of water.

10
00:00:35 --> 00:00:41
What it tells you is

11
00:00:37 --> 00:00:43
that the molecules are
much further apart.

12
00:00:41 --> 00:00:47
It is an experimental fact
that there is a simple relation

13
00:00:46 --> 00:00:52
between the pressure that you
see there, the volume of a gas,

14
00:00:51 --> 00:00:57
the temperature of a gas
in degrees Kelvin,

15
00:00:54 --> 00:01:00
and the number of molecules
that you have.

16
00:00:57 --> 00:01:03
Now, when you see
the word "molecules,"

17
00:00:59 --> 00:01:05
I may often mean "atoms."

18
00:01:01 --> 00:01:07
I realize that helium and neon
and krypton and argon

19
00:01:06 --> 00:01:12
are atomic gases,

20
00:01:07 --> 00:01:13
and that O2 and H2 and CO2
are molecular gases.

21
00:01:12 --> 00:01:18
So I will use
that word "molecules"

22
00:01:15 --> 00:01:21
even when I mean "atoms,"

23
00:01:16 --> 00:01:22
and maybe vice versa,
just for simplicity.

24
00:01:22 --> 00:01:28
The relation that exists
between these quantities,

25
00:01:29 --> 00:01:35
PV equals nRT: pressure, volume,
n is the number of moles--

26
00:01:40 --> 00:01:46
I'll get back to that--
R is the universal gas constant,

27
00:01:44 --> 00:01:50
which is 8.3 joules
per degree Kelvin,

28
00:01:49 --> 00:01:55
and T must be in degrees Kelvin.

29
00:01:55 --> 00:02:01
So, what is a mole?

30
00:01:58 --> 00:02:04
A mole has always about 6.02
times ten to the 23 molecules,

31
00:02:09 --> 00:02:15
or atoms, in the case
that you have helium,

32
00:02:12 --> 00:02:18
but I will call that molecules.

33
00:02:14 --> 00:02:20
And this number is called
Avogadro's number.

34
00:02:17 --> 00:02:23
 

35
00:02:23 --> 00:02:29
So that's
the definition of a mole.

36
00:02:28 --> 00:02:34
If you take a mole of helium,
or a mole of oxygen,

37
00:02:32 --> 00:02:38
or CO2, or N2,
it doesn't matter,

38
00:02:35 --> 00:02:41
it always has this number
of molecules, approximately.

39
00:02:40 --> 00:02:46
Now, each of these substances
have very different masses.

40
00:02:44 --> 00:02:50
If I take, for instance, carbon,

41
00:02:49 --> 00:02:55
then one mole of carbon would
weigh very close to 12 grams.

42
00:02:53 --> 00:02:59
If I take helium,

43
00:02:55 --> 00:03:01
one mole of helium would weigh
very close to four grams.

44
00:03:00 --> 00:03:06
And if I took oxygen two, O2,

45
00:03:03 --> 00:03:09
then one mole would be
very close to 32 grams.

46
00:03:08 --> 00:03:14
So the masses are
very different in a mole

47
00:03:12 --> 00:03:18
but not the number of molecules
or the number of atoms.

48
00:03:18 --> 00:03:24
When I take a neutral atom,
then we have a nucleus,

49
00:03:25 --> 00:03:31
and the nucleus contains
protons and neutrons.

50
00:03:30 --> 00:03:36
It has Z protons
and it has N neutrons.

51
00:03:34 --> 00:03:40
 

52
00:03:37 --> 00:03:43
The protons
are positively charged,

53
00:03:40 --> 00:03:46
and it has Z electrons
if it is a neutral atom.

54
00:03:45 --> 00:03:51
There is almost no weight
in the electrons;

55
00:03:47 --> 00:03:53
you can almost ignore that.

56
00:03:48 --> 00:03:54
Everything is in the protons
and in the neutrons.

57
00:03:53 --> 00:03:59
N plus Z is called A, and that's
called the atomic mass number.

58
00:04:00 --> 00:04:06
 

59
00:04:02 --> 00:04:08
Let's look at carbon
in a little bit more detail.

60
00:04:08 --> 00:04:14
If we have carbon--

61
00:04:10 --> 00:04:16
and I call it carbon 12 for now,
you'll see shortly why--

62
00:04:14 --> 00:04:20
then carbon has always
six protons in the nucleus;

63
00:04:17 --> 00:04:23
otherwise it isn't carbon.

64
00:04:19 --> 00:04:25
And when it has six neutrons,
then A is 12.

65
00:04:23 --> 00:04:29
That's why we call it carbon 12.

66
00:04:28 --> 00:04:34
So the atomic mass number
of carbon is 12,

67
00:04:31 --> 00:04:37
but if you had,
for instance, carbon 14--

68
00:04:33 --> 00:04:39
which happens
to be radioactive--

69
00:04:35 --> 00:04:41
again, six protons, otherwise
it wouldn't be carbon,

70
00:04:39 --> 00:04:45
you would have
eight neutrons now,

71
00:04:43 --> 00:04:49
and now you would have...
atomic mass number would be 14.

72
00:04:49 --> 00:04:55
A mole is this number in grams,
and so you see carbon...

73
00:04:54 --> 00:05:00
is the atomic mass number
in grams-- you see 12 there.

74
00:04:59 --> 00:05:05
If you go to helium, it has
two protons and two neutrons,

75
00:05:04 --> 00:05:10
so A is four-- that's
why you see your four grams.

76
00:05:08 --> 00:05:14
If you take oxygen, it has eight
protons and eight neutrons,

77
00:05:13 --> 00:05:19
so A is 16, but you have
O2 in gas form,

78
00:05:16 --> 00:05:22
so now your atomic mass number
has to be doubled to 32.

79
00:05:20 --> 00:05:26
And so a mole of O2
is therefore 32 grams.

80
00:05:25 --> 00:05:31
In fact, Avogadro's number
is defined through carbon 12.

81
00:05:30 --> 00:05:36
If you take 12 grams
of carbon 12,

82
00:05:35 --> 00:05:41
and you count the number
of atoms that you have,

83
00:05:37 --> 00:05:43
then you find
exactly Avogadro's number.

84
00:05:40 --> 00:05:46
That's the definition
of that number,

85
00:05:42 --> 00:05:48
and that's very close
to what we have there,

86
00:05:44 --> 00:05:50
6.02 times ten to the 23rd.

87
00:05:47 --> 00:05:53
 

88
00:05:53 --> 00:05:59
The mass of the proton
and the mass of the neutron

89
00:05:57 --> 00:06:03
are nearly equal.

90
00:05:59 --> 00:06:05
 

91
00:06:00 --> 00:06:06
I wrote down m2
for the mass of the neutron;

92
00:06:03 --> 00:06:09
of course,
that should have been m of n.

93
00:06:05 --> 00:06:11
 

94
00:06:06 --> 00:06:12
So the mass of a molecule,

95
00:06:10 --> 00:06:16
or an atom, whatever the case
may be, would be this number A--

96
00:06:15 --> 00:06:21
because that's the sum
of the protons and neutrons--

97
00:06:18 --> 00:06:24
times the mass of the neutrons
and the protons.

98
00:06:21 --> 00:06:27
And so this is A times--

99
00:06:22 --> 00:06:28
approximately,
I should put a wiggle here--

100
00:06:24 --> 00:06:30
1.66 times ten
to the minus 27 kilograms.

101
00:06:30 --> 00:06:36
So that's now
an individual mass

102
00:06:32 --> 00:06:38
of either an atom or a molecule,

103
00:06:34 --> 00:06:40
and all that information,
you have there

104
00:06:36 --> 00:06:42
and that's,
of course, on the Web.

105
00:06:40 --> 00:06:46
So, let's do a trivial example.

106
00:06:44 --> 00:06:50
I take gases, any kind of gas--
you choose whatever you want--

107
00:06:51 --> 00:06:57
and I take one atmosphere.

108
00:06:54 --> 00:07:00
So that means

109
00:06:56 --> 00:07:02
that the pressure is 1.03 times
ten to the fifth pascal.

110
00:07:03 --> 00:07:09
I do it at room temperature,
so T is 293 degrees Kelvin.

111
00:07:12 --> 00:07:18
And I take in all cases
only one mole, so n is one.

112
00:07:19 --> 00:07:25
And I'm asking you now, what
will be the volume of that gas?

113
00:07:23 --> 00:07:29
Well, you take the gas law,

114
00:07:26 --> 00:07:32
and it tells you that V, the
volume, equals nRT divided by P.

115
00:07:33 --> 00:07:39
You know n is one.

116
00:07:36 --> 00:07:42
You know R, 1.03...
excuse me, you know...

117
00:07:40 --> 00:07:46
(laughs ):
I'm a little bit
ahead of myself.

118
00:07:44 --> 00:07:50
You know R, which is 8.3,

119
00:07:48 --> 00:07:54
you know the temperature,
which is 293,

120
00:07:51 --> 00:07:57
and you know the pressure,

121
00:07:53 --> 00:07:59
which is 1.03 times
ten to the fifth.

122
00:07:56 --> 00:08:02
And when you calculate that,

123
00:07:58 --> 00:08:04
you find something
very close to 24 liters,

124
00:08:01 --> 00:08:07
and a liter is about
a thousand cubic centimeters.

125
00:08:04 --> 00:08:10
And it's independent

126
00:08:05 --> 00:08:11
of whether it's helium
or oxygen or nitrogen or CO2.

127
00:08:08 --> 00:08:14
As long as you have a gas,

128
00:08:10 --> 00:08:16
one mole at one atmosphere
pressure and room temperature

129
00:08:15 --> 00:08:21
always has the same volume
of about 24 liters.

130
00:08:19 --> 00:08:25
If a gas obeys that law exactly,
we call it an ideal gas.

131
00:08:24 --> 00:08:30
That's why we call that
the ideal-gas law.

132
00:08:28 --> 00:08:34
And many gases come
very close to that.

133
00:08:31 --> 00:08:37
In fact,
if you took oxygen, O2,

134
00:08:33 --> 00:08:39
and you take one mole of oxygen

135
00:08:35 --> 00:08:41
at room temperature
and at one atmosphere pressure

136
00:08:39 --> 00:08:45
and you were to calculate
its volume,

137
00:08:41 --> 00:08:47
the actual volume
that you measure

138
00:08:43 --> 00:08:49
is only one-tenth
of a percent smaller

139
00:08:44 --> 00:08:50
than what you would have found
with the ideal-gas law.

140
00:08:47 --> 00:08:53
If you do it at 20 atmospheres,

141
00:08:50 --> 00:08:56
it would still be
only two percent smaller,

142
00:08:53 --> 00:08:59
so it's a very good
approximation in many cases.

143
00:08:56 --> 00:09:02
What is very surprising,
that in this ideal-gas law,

144
00:09:02 --> 00:09:08
the mass of the atoms and the
molecules do not show up at all.

145
00:09:06 --> 00:09:12
And that is very puzzling--
you wouldn't expect that at all.

146
00:09:10 --> 00:09:16
And I'll show you
why you wouldn't expect that.

147
00:09:14 --> 00:09:20
Let's take
two different kinds of gases

148
00:09:18 --> 00:09:24
with very different masses
of the molecules,

149
00:09:21 --> 00:09:27
but we have
the same number of moles,

150
00:09:23 --> 00:09:29
we have the same volume,
we have the same temperature

151
00:09:26 --> 00:09:32
and therefore,
we must have the same pressure,

152
00:09:29 --> 00:09:35
according to the ideal-gas law.

153
00:09:31 --> 00:09:37
But the masses of
the molecules-- very different.

154
00:09:34 --> 00:09:40
So, here we have some
of these molecules

155
00:09:38 --> 00:09:44
and the number of... density
is the same,

156
00:09:41 --> 00:09:47
because the number of atoms
is the same

157
00:09:44 --> 00:09:50
and the volume is the same.

158
00:09:46 --> 00:09:52
Now, these molecules
are flying in all directions

159
00:09:50 --> 00:09:56
with different speeds.

160
00:09:51 --> 00:09:57
I will just now, for simplicity,
take some average speed,

161
00:09:55 --> 00:10:01
and I assume this is going
in this direction.

162
00:09:59 --> 00:10:05
It's heading for the wall
of the container, this area.

163
00:10:05 --> 00:10:11
It hits the wall,
there's an elastic collision,

164
00:10:09 --> 00:10:15
and it comes back
in exactly the same direction.

165
00:10:13 --> 00:10:19
So there is momentum transfer,

166
00:10:15 --> 00:10:21
and the momentum transfer
for one collision is 2mv,

167
00:10:19 --> 00:10:25
because it comes in with mv
in this direction,

168
00:10:23 --> 00:10:29
it comes back with mv
in that direction,

169
00:10:26 --> 00:10:32
so the momentum transfer
is 2mv.

170
00:10:28 --> 00:10:34
But I'm interested in the
momentum transfer per second,

171
00:10:32 --> 00:10:38
not just for one molecule.

172
00:10:34 --> 00:10:40
And now, of course, I have
to multiply by the velocity,

173
00:10:37 --> 00:10:43
because if the velocity is high,

174
00:10:39 --> 00:10:45
you have a lot of bombardments
per second on here.

175
00:10:42 --> 00:10:48
For each bombardment,
this is the momentum transfer,

176
00:10:46 --> 00:10:52
but if there are many,

177
00:10:47 --> 00:10:53
well, you have to multiply that,
of course, then, by the speed.

178
00:10:52 --> 00:10:58
So the momentum transfer
per second is proportional,

179
00:11:00 --> 00:11:06
let's say, to mv squared.

180
00:11:03 --> 00:11:09
mv comes from the momentum,
from one particle,

181
00:11:05 --> 00:11:11
and v comes
from the fact that...

182
00:11:08 --> 00:11:14
the number
that hit it per second.

183
00:11:11 --> 00:11:17
Now, momentum transfer
per second is clearly...

184
00:11:16 --> 00:11:22
It's a force,
proportional to the force,

185
00:11:18 --> 00:11:24
and that is proportional
to the pressure.

186
00:11:21 --> 00:11:27
And yet the pressure is not
affected by the mass, notice?

187
00:11:28 --> 00:11:34
If these are the same, the
pressure must also be the same.

188
00:11:31 --> 00:11:37
And so there's only one
conclusion that you can draw,

189
00:11:33 --> 00:11:39
which is very nonintuitive--

190
00:11:36 --> 00:11:42
that the pressure
can only be the same

191
00:11:39 --> 00:11:45
if, for a given temperature,
this product, mv squared,

192
00:11:44 --> 00:11:50
is independent of the mass
of the molecule.

193
00:11:48 --> 00:11:54
How can mv squared
possibly be independent

194
00:11:51 --> 00:11:57
of the mass of the molecule?

195
00:11:53 --> 00:11:59
There's only one way
that that's possible--

196
00:11:55 --> 00:12:01
that if you take
two different masses,

197
00:11:58 --> 00:12:04
two gases with totally
different mass of the molecules,

198
00:12:01 --> 00:12:07
that this product is always the
same for a given temperature,

199
00:12:04 --> 00:12:10
and that, indeed, is the case.

200
00:12:06 --> 00:12:12
So if you take,
for instance, helium,

201
00:12:09 --> 00:12:15
and you compare that with oxygen

202
00:12:12 --> 00:12:18
and the ratio of these
two masses is four to 32--

203
00:12:17 --> 00:12:23
this is eight times heavier
than this one--

204
00:12:20 --> 00:12:26
and you have them
at a certain temperature,

205
00:12:23 --> 00:12:29
then the mass of the helium

206
00:12:26 --> 00:12:32
times the mean speed
of the helium squared

207
00:12:30 --> 00:12:36
is the mass
of the oxygen molecules

208
00:12:34 --> 00:12:40
times the mean speed squared
of the oxygen.

209
00:12:38 --> 00:12:44
And so what that means is

210
00:12:40 --> 00:12:46
that if the oxygen molecule is
eight times more massive--

211
00:12:47 --> 00:12:53
this one--

212
00:12:48 --> 00:12:54
then the velocity is the square
root of eight times smaller,

213
00:12:52 --> 00:12:58
because the ratio,
32 over four, is eight.

214
00:12:55 --> 00:13:01
So the oxygen molecules
have a lower speed,

215
00:12:58 --> 00:13:04
so that the product, mv squared,
is always the same.

216
00:13:02 --> 00:13:08
Oxygen molecules
at room temperatures

217
00:13:04 --> 00:13:10
have a speed
of about 480 meters per second,

218
00:13:07 --> 00:13:13
and so the helium is the square
root of eight times higher.

219
00:13:11 --> 00:13:17
If you have a mixture
of oxygen and helium,

220
00:13:13 --> 00:13:19
then one gas would have
an average speed

221
00:13:15 --> 00:13:21
of 480 meters per second
for the molecules,

222
00:13:17 --> 00:13:23
and the other would be
1,350 meters per second.

223
00:13:21 --> 00:13:27
That is the only way
that the gas law can hold.

224
00:13:25 --> 00:13:31
It is a consequence
of the gas law.

225
00:13:28 --> 00:13:34
You very often see the gas law
written in a different way,

226
00:13:32 --> 00:13:38
and you see it written

227
00:13:33 --> 00:13:39
as PV equals capital N
times k times T.

228
00:13:38 --> 00:13:44
You see that also there.

229
00:13:41 --> 00:13:47
And this N, now,

230
00:13:43 --> 00:13:49
is the total number
of molecules that you have.

231
00:13:46 --> 00:13:52
Don't confuse that
with little n,

232
00:13:48 --> 00:13:54
which is the number
of moles that you have.

233
00:13:51 --> 00:13:57
So this is the total number
of molecules,

234
00:13:54 --> 00:14:00
and k is called
Boltzmann's constant.

235
00:13:56 --> 00:14:02
And since Nk must be
the same as nR,

236
00:14:02 --> 00:14:08
if you compare the two gas laws,

237
00:14:04 --> 00:14:10
the one up there
and the one here,

238
00:14:07 --> 00:14:13
you can see that the way
that k is defined,

239
00:14:10 --> 00:14:16
it is nothing but R
divided by N of A.

240
00:14:13 --> 00:14:19
k is R divided
by Avogadro's number,

241
00:14:16 --> 00:14:22
because little n, remember,

242
00:14:18 --> 00:14:24
is the total number
of molecules that you have

243
00:14:22 --> 00:14:28
divided by N of A.

244
00:14:23 --> 00:14:29
So if you substitute that,
you get this as a result,

245
00:14:27 --> 00:14:33
and so this is 8.3 divided
by 6.02 times ten to the 23rd,

246
00:14:32 --> 00:14:38
and that is

247
00:14:34 --> 00:14:40
about 1.38 times ten to the
minus 23rd joules per Kelvin.

248
00:14:38 --> 00:14:44
And you can use that number.

249
00:14:40 --> 00:14:46
If you want to, you can use
either this relationship

250
00:14:43 --> 00:14:49
or you can use that one,
whichever is convenient.

251
00:14:46 --> 00:14:52
They are identical.

252
00:14:49 --> 00:14:55
So, let's now bring
the ideal-gas law to a test.

253
00:14:54 --> 00:15:00
And the way I'm going to bring
it to a test is as follows:

254
00:15:00 --> 00:15:06
I have here a volume, copper.

255
00:15:05 --> 00:15:11
There's air inside.

256
00:15:08 --> 00:15:14
And I'm going to heat that up.

257
00:15:11 --> 00:15:17
I'll make you a drawing
on the blackboard.

258
00:15:16 --> 00:15:22
Here is that volume,

259
00:15:18 --> 00:15:24
and there's
an extremely thin tube,

260
00:15:20 --> 00:15:26
which has almost no volume,

261
00:15:22 --> 00:15:28
and at the end is
a pressure gauge.

262
00:15:25 --> 00:15:31
So there is
a pressure gauge here

263
00:15:27 --> 00:15:33
which gives us the pressure,

264
00:15:29 --> 00:15:35
and the pressure is given
in pounds per square inch,

265
00:15:33 --> 00:15:39
much to my regret,
but that's the way it is.

266
00:15:36 --> 00:15:42
For your recollection,

267
00:15:38 --> 00:15:44
one atmosphere is approximately
15 pounds per square inch.

268
00:15:44 --> 00:15:50
It is also a gauge
that measures overpressure.

269
00:15:47 --> 00:15:53
In other words, if you expose it
to one atmosphere,

270
00:15:51 --> 00:15:57
it will read zero.

271
00:15:52 --> 00:15:58
Just when you go
to the gas station

272
00:15:54 --> 00:16:00
and you measure
the pressure of your tires,

273
00:15:56 --> 00:16:02
that is also a gauge
that measures overpressure,

274
00:15:58 --> 00:16:04
the difference
between inside and outside.

275
00:16:03 --> 00:16:09
This has a valve here,

276
00:16:05 --> 00:16:11
and I can connect it
with the outside world.

277
00:16:08 --> 00:16:14
And when I do that,

278
00:16:09 --> 00:16:15
then the pressure is very close
to one atmosphere,

279
00:16:12 --> 00:16:18
regardless of
what the temperature is here,

280
00:16:15 --> 00:16:21
because it's connected
to the universe,

281
00:16:17 --> 00:16:23
so the pressure in here
will then be one atmosphere.

282
00:16:20 --> 00:16:26
And that's the situation
that we have now

283
00:16:23 --> 00:16:29
when this valve is open,
and it is in melting ice,

284
00:16:26 --> 00:16:32
And so melting ice--
T1 is 273 degrees Kelvin.

285
00:16:35 --> 00:16:41
P1 is one atmosphere, and V1 is
some value that I don't know.

286
00:16:43 --> 00:16:49
I close this valve.

287
00:16:44 --> 00:16:50
Thereby the number of moles
of air in there is fixed;

288
00:16:48 --> 00:16:54
that's not going to change.

289
00:16:50 --> 00:16:56
So we have one atmosphere
inside now,

290
00:16:53 --> 00:16:59
and whatever n is,
it's of no importance,

291
00:16:56 --> 00:17:02
but it's not going to change.

292
00:16:58 --> 00:17:04
Now I'm going to put it
in boiling water,

293
00:17:01 --> 00:17:07
and so I know now that T2
will become 373 degrees Kelvin.

294
00:17:08 --> 00:17:14
I want to know now what P2 is,

295
00:17:10 --> 00:17:16
and the volume is hardly
going to change at all.

296
00:17:15 --> 00:17:21
It changes because of the
expansion coefficient of copper.

297
00:17:18 --> 00:17:24
And I made a small calculation
based on this size,

298
00:17:22 --> 00:17:28
and it turns out

299
00:17:23 --> 00:17:29
that the extra volume
that you get because of the heat

300
00:17:27 --> 00:17:33
is only .5 percent
of the original volume,

301
00:17:30 --> 00:17:36
so we can forget the fact

302
00:17:32 --> 00:17:38
that V2 is just
a hair larger than V1.

303
00:17:34 --> 00:17:40
It's close enough to say
that these are the same.

304
00:17:38 --> 00:17:44
And of course we have
the same number of moles.

305
00:17:41 --> 00:17:47
So now we can predict
with the gas law

306
00:17:43 --> 00:17:49
what the pressure is going to be

307
00:17:45 --> 00:17:51
when we stick it
in the boiling water.

308
00:17:47 --> 00:17:53
So we're going to get
that P1 V1 equals n R T1;

309
00:17:58 --> 00:18:04
P2 V2 equals n R T2.

310
00:18:02 --> 00:18:08
I divide these two equations,
lose my n and my R--

311
00:18:06 --> 00:18:12
we agreed that
the volume was the same--

312
00:18:10 --> 00:18:16
and so we find that P1 divided
by P2, T1 divided by T2,

313
00:18:14 --> 00:18:20
or the pressure P2,
which is our goal,

314
00:18:18 --> 00:18:24
equals the one atmosphere
pressure, which is P1,

315
00:18:21 --> 00:18:27
times T2 divided by T1.

316
00:18:23 --> 00:18:29
And that is...
T2 is 373, T1 is 273,

317
00:18:29 --> 00:18:35
so it is one atmosphere
times 373 divided by 273.

318
00:18:35 --> 00:18:41
That's what
the gas law predicts.

319
00:18:37 --> 00:18:43
And that ratio is 1.366,

320
00:18:41 --> 00:18:47
so P2 is then 1.366
times one atmosphere.

321
00:18:47 --> 00:18:53
Now, this gauge is
an overpressure gauge,

322
00:18:49 --> 00:18:55
so therefore you're not going
to see this number,

323
00:18:52 --> 00:18:58
but you're going to see the
difference with one atmosphere.

324
00:18:55 --> 00:19:01
So what the gauge will show us
is 0.366 times one atmosphere,

325
00:19:03 --> 00:19:09
and since it is calibrated
in pounds per square inch,

326
00:19:08 --> 00:19:14
it is 0.366 times
15 pounds per square inch,

327
00:19:14 --> 00:19:20
and that is about...

328
00:19:15 --> 00:19:21
something like 5.4 pounds
per square inch.

329
00:19:20 --> 00:19:26
So that's what I predict,
and we'll see how close we get.

330
00:19:25 --> 00:19:31
You're going to see this,
if all works well, on...

331
00:19:29 --> 00:19:35
 

332
00:19:32 --> 00:19:38
Yeah, there it is.

333
00:19:35 --> 00:19:41
I'm going to set you...

334
00:19:36 --> 00:19:42
the light situation
a little better for you.

335
00:19:39 --> 00:19:45
 

336
00:19:41 --> 00:19:47
I can even make it
a little darker.

337
00:19:45 --> 00:19:51
So, you see here... this gauge.

338
00:19:54 --> 00:20:00
Note that it is zero,
even though it is in ice water.

339
00:19:58 --> 00:20:04
This valve is open,

340
00:20:00 --> 00:20:06
and so it is at zero because
it measures only overpressure.

341
00:20:04 --> 00:20:10
 

342
00:20:06 --> 00:20:12
And now we're going
to close this valve.

343
00:20:08 --> 00:20:14
 

344
00:20:10 --> 00:20:16
The valve is now closed.

345
00:20:13 --> 00:20:19
And we're going to stick it
here as this object;

346
00:20:19 --> 00:20:25
that is the volume.

347
00:20:21 --> 00:20:27
It looks like a bathroom floater
to me, in the toilet flush.

348
00:20:25 --> 00:20:31
That's what it is, probably.

349
00:20:27 --> 00:20:33
And now it goes
into boiling water.

350
00:20:30 --> 00:20:36
And now look at the pressure.

351
00:20:31 --> 00:20:37
There it goes--
three, 31/2, four...

352
00:20:35 --> 00:20:41
That's overpressure
in pounds per square inch.

353
00:20:37 --> 00:20:43
Four and a half.

354
00:20:39 --> 00:20:45
 

355
00:20:42 --> 00:20:48
I'll just give it
a little bit more time.

356
00:20:44 --> 00:20:50
It would take a while,
of course,

357
00:20:46 --> 00:20:52
because gas is
a very good insulator,

358
00:20:49 --> 00:20:55
so it may take a while for that
gas in the copper ball to...

359
00:20:54 --> 00:21:00
It's already close to five.

360
00:20:57 --> 00:21:03
We don't have to wait,
of course,

361
00:20:59 --> 00:21:05
for the thing to become
all the way 5.5, 5.4,

362
00:21:04 --> 00:21:10
but it will probably go up
if we wait.

363
00:21:08 --> 00:21:14
It's now five.

364
00:21:10 --> 00:21:16
Not much sense in waiting.

365
00:21:13 --> 00:21:19
What I want you to realize,
what happens...

366
00:21:16 --> 00:21:22
or I want to ask you, actually,

367
00:21:18 --> 00:21:24
what happens
if I open the valve now?

368
00:21:20 --> 00:21:26
 

369
00:21:22 --> 00:21:28
Yeah?

370
00:21:24 --> 00:21:30
What happens?

371
00:21:26 --> 00:21:32
Yeah.

372
00:21:27 --> 00:21:33
So if I open the valve,
then one atmosphere,

373
00:21:30 --> 00:21:36
some of the high-pressure stuff
gets out,

374
00:21:32 --> 00:21:38
one atmosphere settles inside,

375
00:21:35 --> 00:21:41
and so, since this one is
an overpressure gauge,

376
00:21:38 --> 00:21:44
it will be zero.

377
00:21:40 --> 00:21:46
So right now, it's... it's five.

378
00:21:45 --> 00:21:51
That's fine with me--
that's very close.

379
00:21:47 --> 00:21:53
That's within eight percent.

380
00:21:49 --> 00:21:55
Previous class, we had 5.3,

381
00:21:51 --> 00:21:57
but we were
a little bit more patient.

382
00:21:54 --> 00:22:00
I will open the valve now,
and watch what happens.

383
00:21:58 --> 00:22:04
It goes back to zero.

384
00:21:59 --> 00:22:05
So some air was let out,

385
00:22:01 --> 00:22:07
because the pressure
of the air inside was higher,

386
00:22:06 --> 00:22:12
was higher than one atmosphere.

387
00:22:09 --> 00:22:15
So, we've seen a test,
a reasonable test,

388
00:22:14 --> 00:22:20
first application
of the ideal-gas law.

389
00:22:19 --> 00:22:25
 

390
00:22:24 --> 00:22:30
A gas can turn into a liquid,
and a liquid can become a solid,

391
00:22:30 --> 00:22:36
and that depends entirely
on the kind of substance,

392
00:22:34 --> 00:22:40
the temperature, and
the pressure that we have.

393
00:22:38 --> 00:22:44
And this brings us to the field
of what we call phase diagrams.

394
00:22:44 --> 00:22:50
I will show you your first
schematic of a phase diagram,

395
00:22:48 --> 00:22:54
which is also on the Web.

396
00:22:50 --> 00:22:56
Here you see a phase diagram--
very intuitive.

397
00:22:53 --> 00:22:59
Here is pressure,
here is temperature.

398
00:22:57 --> 00:23:03
Imagine that we have a cylinder
and we put gas in that cylinder

399
00:23:02 --> 00:23:08
and we put a piston on top

400
00:23:04 --> 00:23:10
and we push it down,
slowly pushing it down.

401
00:23:07 --> 00:23:13
So we start here with gas
at a particular temperature,

402
00:23:11 --> 00:23:17
which we're not going to change,

403
00:23:13 --> 00:23:19
and we slowly push
the piston down.

404
00:23:15 --> 00:23:21
In this trajectory,
the ideal-gas law would hold.

405
00:23:18 --> 00:23:24
Temperature remains constant,
so if you look at the gas law,

406
00:23:22 --> 00:23:28
PV remains constant.

407
00:23:24 --> 00:23:30
That's called Boyle's Law,
by the way,

408
00:23:26 --> 00:23:32
that the product of pressure
and volume remains constant.

409
00:23:28 --> 00:23:34
So the pressure in the gas
goes up, the volume goes down,

410
00:23:31 --> 00:23:37
pressure goes up,
volume goes down,

411
00:23:33 --> 00:23:39
pressure goes up,
until I hit this point.

412
00:23:36 --> 00:23:42
And now liquid
is going to be formed,

413
00:23:39 --> 00:23:45
so the pressure is now
high enough at this temperature

414
00:23:44 --> 00:23:50
to form liquid.

415
00:23:46 --> 00:23:52
If I push further,
the pressure will not go up.

416
00:23:49 --> 00:23:55
All the gas will first
turn into liquid,

417
00:23:52 --> 00:23:58
all of it,
until the last molecule;

418
00:23:55 --> 00:24:01
and not until
everything has become liquid

419
00:23:58 --> 00:24:04
can I push even further
onto the liquid,

420
00:24:01 --> 00:24:07
to increase the pressure
on the liquid.

421
00:24:03 --> 00:24:09
It may be a silly thing to do,
but I could do that.

422
00:24:06 --> 00:24:12
You're not going to compress it
very much, but you can try.

423
00:24:09 --> 00:24:15
And in some cases, if you put
a tremendous pressure on it,

424
00:24:13 --> 00:24:19
you may turn the liquid
into a solid.

425
00:24:15 --> 00:24:21
And then you reach this domain,
where you have a solid.

426
00:24:19 --> 00:24:25
What is much less intuitive--

427
00:24:21 --> 00:24:27
that if you did it
at a lower temperature

428
00:24:24 --> 00:24:30
and you squeezed the volume so
that the pressure would go up,

429
00:24:27 --> 00:24:33
that you may now reach the point
here in this phase diagram

430
00:24:30 --> 00:24:36
whereby no liquid is formed--
no condensation of liquid--

431
00:24:34 --> 00:24:40
but you get immediately
the formation of crystals.

432
00:24:37 --> 00:24:43
So you go from the gas phase

433
00:24:38 --> 00:24:44
immediately
into the solid phase.

434
00:24:40 --> 00:24:46
If you push further down
the piston,

435
00:24:43 --> 00:24:49
the pressure will not go up

436
00:24:46 --> 00:24:52
until all the gas
has become solid,

437
00:24:49 --> 00:24:55
and then it will continue
to go further.

438
00:24:52 --> 00:24:58
Suppose this were one atmosphere
here, and I took some ice--

439
00:24:57 --> 00:25:03
or you can take a piece of iron
at one atmosphere--

440
00:25:01 --> 00:25:07
and it's a very low temperature,
it's a solid.

441
00:25:03 --> 00:25:09
And I start heating it up,

442
00:25:05 --> 00:25:11
but I keep the pressure
one atmosphere.

443
00:25:07 --> 00:25:13
It's a solid,
it's still a solid.

444
00:25:09 --> 00:25:15
At this point,
it begins to melt.

445
00:25:12 --> 00:25:18
This will be the melting point.

446
00:25:14 --> 00:25:20
And when I keep heating it,
the temperature will not go up

447
00:25:18 --> 00:25:24
until all the solid
has been melted into liquid.

448
00:25:21 --> 00:25:27
Then I can increase
the temperature.

449
00:25:23 --> 00:25:29
Then the liquid will get hotter
until you reach this line.

450
00:25:27 --> 00:25:33
And when you reach this line,

451
00:25:29 --> 00:25:35
some of the liquid
will turn into gas.

452
00:25:31 --> 00:25:37
It will boil at one atmosphere.

453
00:25:33 --> 00:25:39
You will see it boil.

454
00:25:35 --> 00:25:41
You cannot increase
the temperature.

455
00:25:37 --> 00:25:43
If it is water, it will stay
at 100 degrees centigrade.

456
00:25:40 --> 00:25:46
There's nothing you can do until
all the liquid has become gas.

457
00:25:44 --> 00:25:50
We call that water vapor.

458
00:25:46 --> 00:25:52
Then after that, the temperature
can be further increased.

459
00:25:50 --> 00:25:56
So this point would be a
melting point for ice and water,

460
00:25:54 --> 00:26:00
and this would be the boiling
point at one atmosphere.

461
00:25:57 --> 00:26:03
So that's the idea
behind a phase diagram,

462
00:26:00 --> 00:26:06
and we are going
to use them today

463
00:26:02 --> 00:26:08
for some of our experiments.

464
00:26:05 --> 00:26:11
I have here a fire extinguisher,

465
00:26:09 --> 00:26:15
and a fire extinguisher
is filled with CO2--

466
00:26:13 --> 00:26:19
that's a given.

467
00:26:15 --> 00:26:21
And I ask myself the question--

468
00:26:18 --> 00:26:24
seriously, this is really
a question that I ask myself;

469
00:26:21 --> 00:26:27
it's not something
I made up for you--

470
00:26:23 --> 00:26:29
I ask myself the question:

471
00:26:26 --> 00:26:32
Is there liquid inside,
liquid carbon dioxide,

472
00:26:31 --> 00:26:37
or is there gas inside?

473
00:26:33 --> 00:26:39
And if so,
what could the pressure be?

474
00:26:36 --> 00:26:42
 

475
00:26:38 --> 00:26:44
So I measured
the volume of that tank.

476
00:26:41 --> 00:26:47
It's about 40 centimeters high

477
00:26:43 --> 00:26:49
and it has a diameter
of about 15 centimeters.

478
00:26:46 --> 00:26:52
So it's a cylinder.

479
00:26:47 --> 00:26:53
 

480
00:26:50 --> 00:26:56
This is about 40 centimeters and
this is about 15 centimeters.

481
00:26:57 --> 00:27:03
So the volume--
easy to calculate:

482
00:27:00 --> 00:27:06
2.3 times ten to the minus three
cubic meters.

483
00:27:04 --> 00:27:10
It's clear that it is
at room temperature--

484
00:27:08 --> 00:27:14
that's nonnegotiable-- so
that's about 293 degrees Kelvin,

485
00:27:13 --> 00:27:19
the same temperature
that we are at.

486
00:27:16 --> 00:27:22
So now I want to know how many
moles I have, the little n.

487
00:27:23 --> 00:27:29
And what I read on the label

488
00:27:25 --> 00:27:31
that this tank, when it is full,
weighs 31 pounds,

489
00:27:29 --> 00:27:35
but when it's empty, no CO2
inside, it weighs 21 pounds...

490
00:27:33 --> 00:27:39
So I have the mass of the gas--

491
00:27:36 --> 00:27:42
of the CO2 gas, whatever it is,
maybe it's liquid--

492
00:27:39 --> 00:27:45
is ten pounds.

493
00:27:43 --> 00:27:49
That's a given.

494
00:27:45 --> 00:27:51
And that is 4,500 grams--
a pound is 450 grams--

495
00:27:50 --> 00:27:56
so I know what n is,

496
00:27:52 --> 00:27:58
because the atomic mass number
of CO2 is 12 plus 32 is 44.

497
00:27:58 --> 00:28:04
And so that is 4,500 divided by
44; that's close enough to 100.

498
00:28:04 --> 00:28:10
So I have 100 moles.

499
00:28:06 --> 00:28:12
So now I can ask myself,
what is the pressure?

500
00:28:09 --> 00:28:15
If this were a gas,
what would be the pressure?

501
00:28:12 --> 00:28:18
Well, if this were a gas,
then the pressure P would be

502
00:28:18 --> 00:28:24
n times R times T
divided by the volume.

503
00:28:22 --> 00:28:28
I stuck in the numbers,
and out comes ten to the eighth.

504
00:28:28 --> 00:28:34
Ten to the eighth pascal--
anenormous number.

505
00:28:33 --> 00:28:39
Ten to the eighth pascal
is about 1,000 atmospheres,

506
00:28:39 --> 00:28:45
so I doubted very much
whether there is gas inside,

507
00:28:43 --> 00:28:49
because I said to myself,

508
00:28:45 --> 00:28:51
at that high pressure, CO2
probably becomes a liquid.

509
00:28:50 --> 00:28:56
And so I looked up on the Web--

510
00:28:52 --> 00:28:58
in fact, Dave Pooley did that
for me, my graduate student--

511
00:28:57 --> 00:29:03
looked on the Web,

512
00:28:59 --> 00:29:05
and we found the phase diagram
for carbon dioxide.

513
00:29:03 --> 00:29:09
And what do you see?

514
00:29:05 --> 00:29:11
This is pressure in atmospheres.

515
00:29:07 --> 00:29:13
It's a strange scale,
because it goes five, ten, 15,

516
00:29:10 --> 00:29:16
then there is an interruption
and it goes to 73.

517
00:29:13 --> 00:29:19
And this is the temperature,
zero degrees and 20 degrees.

518
00:29:16 --> 00:29:22
And so at 20 degrees...

519
00:29:18 --> 00:29:24
Since we know, if there
is liquid inside there...

520
00:29:20 --> 00:29:26
If it were a liquid,

521
00:29:21 --> 00:29:27
then the liquid has to be
in equilibrium with the gas.

522
00:29:24 --> 00:29:30
So you go up here,

523
00:29:25 --> 00:29:31
and you see there's no way
the 1,000 atmospheres,

524
00:29:29 --> 00:29:35
the 1,000 atmospheres somewhere
there in the corridor.

525
00:29:32 --> 00:29:38
So already at a temperature...

526
00:29:34 --> 00:29:40
Already at a pressure of
something like 60 atmospheres,

527
00:29:38 --> 00:29:44
it begins to be liquid.

528
00:29:39 --> 00:29:45
You can't see
that 60 here on this scale

529
00:29:42 --> 00:29:48
because it jumps there.

530
00:29:44 --> 00:29:50
But I called
the fire department,

531
00:29:46 --> 00:29:52
and they said it's about
900 pounds per square inch,

532
00:29:49 --> 00:29:55
which is 60 atmospheres.

533
00:29:50 --> 00:29:56
And so this... this canister
contains, then, liquid and gas

534
00:29:54 --> 00:30:00
and stays exactly at that line.

535
00:29:56 --> 00:30:02
It cannot be higher, it cannot
be lower-- think about that--

536
00:30:00 --> 00:30:06
because gas and liquid at that
line exist in coexistence.

537
00:30:03 --> 00:30:09
And the only way they can do
that is exactly at that pressure

538
00:30:07 --> 00:30:13
if the temperature is
20 degrees.

539
00:30:09 --> 00:30:15
And when you release,
when you open the valve,

540
00:30:12 --> 00:30:18
then the liquid CO2
will turn into gas,

541
00:30:15 --> 00:30:21
but the pressure will
always remain 60 atmospheres

542
00:30:18 --> 00:30:24
until you have used up all
the liquid, and not until then

543
00:30:22 --> 00:30:28
will the pressure come
below 60 atmospheres.

544
00:30:26 --> 00:30:32
 

545
00:30:32 --> 00:30:38
We dealt earlier with
the hydrostatic equilibrium,

546
00:30:38 --> 00:30:44
hydrostatic equilibrium
of fluids in general,

547
00:30:42 --> 00:30:48
but we used it only for liquids,

548
00:30:45 --> 00:30:51
to calculate
hydrostatic pressure.

549
00:30:48 --> 00:30:54
And the equation
for hydrostatic equilibrium--

550
00:30:53 --> 00:30:59
you will see that,
undoubtedly, on the final--

551
00:30:57 --> 00:31:03
equals minus rho times g.

552
00:31:00 --> 00:31:06
And this is very easy to use
for a liquid,

553
00:31:03 --> 00:31:09
because a liquid
is incompressible,

554
00:31:07 --> 00:31:13
so rho is not
a function of pressure.

555
00:31:10 --> 00:31:16
So you can integrate this out
very easily, as we did.

556
00:31:14 --> 00:31:20
You get a linear relation
between P and y.

557
00:31:17 --> 00:31:23
So if I have here y

558
00:31:21 --> 00:31:27
and I have here
hydrostatic pressure

559
00:31:25 --> 00:31:31
and let this be y zero--

560
00:31:27 --> 00:31:33
that is, the sea level,
I call that zero--

561
00:31:31 --> 00:31:37
and let this be minus 4
meters, 4,000 meters lower,

562
00:31:37 --> 00:31:43
then the pressure
just goes like this.

563
00:31:40 --> 00:31:46
Here it will be
around 400 atmospheres,

564
00:31:44 --> 00:31:50
and it drops linearly.

565
00:31:46 --> 00:31:52
When you go up to the surface,
it drops linearly.

566
00:31:50 --> 00:31:56
Well, it may be
one atmosphere here

567
00:31:52 --> 00:31:58
because that's the barometric
pressure, but that's a detail.

568
00:31:55 --> 00:32:01
I really want this to be
the hydrostatic pressure,

569
00:31:58 --> 00:32:04
so this is rho...
minus rho times g times y.

570
00:32:03 --> 00:32:09
For gases, this would be
very different, though,

571
00:32:06 --> 00:32:12
because with a gas, the density
does depend on pressure.

572
00:32:10 --> 00:32:16
And now I will calculate,
I will derive for you

573
00:32:14 --> 00:32:20
how the pressure changes
with altitude in our atmosphere,

574
00:32:18 --> 00:32:24
and it's going to be
very different from this.

575
00:32:21 --> 00:32:27
And I will do that
under the following assumption--

576
00:32:24 --> 00:32:30
which is not
an ideal assumption,

577
00:32:27 --> 00:32:33
but it's not very bad--

578
00:32:28 --> 00:32:34
namely, that the temperature
in our atmosphere

579
00:32:31 --> 00:32:37
is roughly constant everywhere.

580
00:32:33 --> 00:32:39
And we'll take
zero degrees centigrade.

581
00:32:35 --> 00:32:41
Here it's a little warmer;

582
00:32:37 --> 00:32:43
when you go up,
it's a little colder.

583
00:32:41 --> 00:32:47
We call that
an isothermal atmosphere.

584
00:32:44 --> 00:32:50
 

585
00:32:49 --> 00:32:55
What is the density of a gas?

586
00:32:52 --> 00:32:58
Well, it is the mass of a gas
divided by its volume.

587
00:32:56 --> 00:33:02
I take a certain volume and
I have N molecules in there.

588
00:33:01 --> 00:33:07
And each molecule has mass m

589
00:33:04 --> 00:33:10
and this is the volume,
so this is the density.

590
00:33:09 --> 00:33:15
But now I go to
my gas law there, and I say,

591
00:33:13 --> 00:33:19
"Aha! Capital N,

592
00:33:15 --> 00:33:21
"which is the number of
molecules divided by the volume,

593
00:33:19 --> 00:33:25
is also P divided by kT."

594
00:33:20 --> 00:33:26
So this is
P divided by kT times m.

595
00:33:25 --> 00:33:31
So now I take this equation,
and I say,

596
00:33:30 --> 00:33:36
"Aha! dP/dy equals minus
Pm divided by kT times g."

597
00:33:44 --> 00:33:50
I bring the P under here
and I bring the dy there,

598
00:33:48 --> 00:33:54
so we get dP divided by P
equals minus mg divided by kT--

599
00:33:58 --> 00:34:04
which is some kind
of a constant-- times dy.

600
00:34:02 --> 00:34:08
Let's first talk
about that constant.

601
00:34:05 --> 00:34:11
That constant must have
a dimension,

602
00:34:08 --> 00:34:14
one divided by meters,
because this is dimensionless--

603
00:34:12 --> 00:34:18
pressure divided by pressure
has no dimension.

604
00:34:14 --> 00:34:20
dy has the dimension of length,

605
00:34:17 --> 00:34:23
so this must have the dimension
of one over length.

606
00:34:21 --> 00:34:27
In fact, I can calculate
what kT over mg is.

607
00:34:25 --> 00:34:31
That should, then, have
a dimension of length.

608
00:34:29 --> 00:34:35
I know k; I know T; I take zero
degrees centigrade, so T is 273;

609
00:34:35 --> 00:34:41
I know what g is.

610
00:34:36 --> 00:34:42
What do I take for the
molecule... a molecule of air?

611
00:34:41 --> 00:34:47
What is an air molecule?

612
00:34:43 --> 00:34:49
Well, we have 20% oxygen,
we have 80% nitrogen.

613
00:34:48 --> 00:34:54
The atomic mass number
of oxygen is 32, nitrogen is 28.

614
00:34:53 --> 00:34:59
But really, there is
more nitrogen than oxygen.

615
00:34:58 --> 00:35:04
So take 29 as
a reasonable atomic mass number

616
00:35:02 --> 00:35:08
for a mean mass
of an air molecule,

617
00:35:05 --> 00:35:11
and so you will get, then,
that it is roughly

618
00:35:07 --> 00:35:13
29 times the 1.66 times
ten to the minus 27 kilograms.

619
00:35:13 --> 00:35:19
And you stick that
in that equation,

620
00:35:16 --> 00:35:22
you can't be too far off.

621
00:35:18 --> 00:35:24
And what you find,
that this is 8,000 meters.

622
00:35:21 --> 00:35:27
It has the unit length,

623
00:35:23 --> 00:35:29
or it is eight kilometers,
and we call this H zero.

624
00:35:30 --> 00:35:36
So I will rewrite this a little.

625
00:35:32 --> 00:35:38
 

626
00:35:36 --> 00:35:42
We're almost done
with our integration.

627
00:35:39 --> 00:35:45
So I will rewrite the equation

628
00:35:42 --> 00:35:48
and introduce for that constant
one over H zero,

629
00:35:48 --> 00:35:54
because, remember, I turned it
upside down there to get length.

630
00:35:52 --> 00:35:58
So we have dP over P equals
minus one over H zero times dy.

631
00:36:00 --> 00:36:06
I integrate this between P zero,
which is sea level,

632
00:36:05 --> 00:36:11
and P at some altitude h,

633
00:36:07 --> 00:36:13
and so dy between zero,
sea level, and altitude h.

634
00:36:12 --> 00:36:18
And that's an easy integral,

635
00:36:14 --> 00:36:20
so I get ln P at altitude h
divided by P zero

636
00:36:26 --> 00:36:32
equals minus h divided
by H zero,

637
00:36:29 --> 00:36:35
because an integral of dy
from zero to h is simply h.

638
00:36:35 --> 00:36:41
And so what do I find now?

639
00:36:38 --> 00:36:44
That the pressure at altitude h

640
00:36:41 --> 00:36:47
equals P zero times e to
the minus h divided by H zero.

641
00:36:50 --> 00:36:56
This is the altitude
in the atmosphere,

642
00:36:53 --> 00:36:59
and if you take this H,

643
00:36:54 --> 00:37:00
then this is the altitude
in kilometers.

644
00:36:56 --> 00:37:02
H zero would then be
eight kilometers.

645
00:37:00 --> 00:37:06
And so if you use this equation,

646
00:37:02 --> 00:37:08
you can calculate
what the pressure is

647
00:37:04 --> 00:37:10
at the various altitudes
in our atmosphere,

648
00:37:06 --> 00:37:12
and that's not
a bad approximation.

649
00:37:08 --> 00:37:14
Everest is 8.9 kilometers high.

650
00:37:10 --> 00:37:16
If you use this equation,
you will find

651
00:37:12 --> 00:37:18
that the atmospheric
pressure is there

652
00:37:15 --> 00:37:21
only one-third
of what we have here.

653
00:37:18 --> 00:37:24
Not enough oxygen to live.

654
00:37:20 --> 00:37:26
I did quite a bit of observing

655
00:37:23 --> 00:37:29
at an optical observatory
in Chile

656
00:37:26 --> 00:37:32
which was at an altitude
of 2,400 meters.

657
00:37:29 --> 00:37:35
At 2,400 meters, the pressure

658
00:37:31 --> 00:37:37
is 3/4 of an atmosphere
according to this equation,

659
00:37:35 --> 00:37:41
and water doesn't boil there
at 100 degrees centigrade.

660
00:37:39 --> 00:37:45
Here at sea level, it does,

661
00:37:41 --> 00:37:47
but at 3/4 of an atmosphere, it
boils at 92 degrees centigrade,

662
00:37:45 --> 00:37:51
so you can never time to get
a soft-boiled egg.

663
00:37:48 --> 00:37:54
You can never time
that properly,

664
00:37:51 --> 00:37:57
because you're used
to the 100 degrees centigrade.

665
00:37:53 --> 00:37:59
In fact, in the kitchen,
there were tables

666
00:37:55 --> 00:38:01
which indicated how long
you would have to boil potatoes

667
00:37:58 --> 00:38:04
to get them to what you want to
at 90 degrees centigrade.

668
00:38:01 --> 00:38:07
That's all you can get--
you can't go any higher.

669
00:38:03 --> 00:38:09
On Mount Everest, water will
boil at 72 degrees centigrade,

670
00:38:07 --> 00:38:13
so there's no way
you can get yourself there

671
00:38:10 --> 00:38:16
some real hot food.

672
00:38:11 --> 00:38:17
So you need a pressure cooker
there, of course.

673
00:38:14 --> 00:38:20
If you go
to 30 kilometers altitude,

674
00:38:16 --> 00:38:22
and you ask this equation
what the pressure is,

675
00:38:19 --> 00:38:25
it's 1/45 of an atmosphere,

676
00:38:20 --> 00:38:26
it's only
17 millimeters mercury.

677
00:38:23 --> 00:38:29
Water at 20 degrees centigrade
would boil at that altitude.

678
00:38:29 --> 00:38:35
I want to show you
the phase diagram of water.

679
00:38:33 --> 00:38:39
 

680
00:38:39 --> 00:38:45
This is the phase diagram
of water.

681
00:38:41 --> 00:38:47
 

682
00:38:45 --> 00:38:51
And what I want to do is,
I want to take some water

683
00:38:51 --> 00:38:57
and bring it to an altitude
of about 30 kilometers.

684
00:38:56 --> 00:39:02
At 30 kilometers, the water
at 20 degrees centigrade

685
00:39:01 --> 00:39:07
should start to boil.

686
00:39:04 --> 00:39:10
And how do I know that?

687
00:39:06 --> 00:39:12
This is the phase diagram
for water.

688
00:39:09 --> 00:39:15
Zero degrees centigrade,
100 degrees centigrade.

689
00:39:13 --> 00:39:19
This is pressure
in millimeters mercury,

690
00:39:16 --> 00:39:22
and the scales are
not very clear.

691
00:39:20 --> 00:39:26
This is all we have.

692
00:39:22 --> 00:39:28
This is again what Dave Pooley
got from the Web for me.

693
00:39:27 --> 00:39:33
Now, if you take
20 degrees somewhere here,

694
00:39:31 --> 00:39:37
and you have water at one
atmospheres and 20 degrees--

695
00:39:35 --> 00:39:41
which is
what we have in this room--

696
00:39:38 --> 00:39:44
and I'm going to lower
the pressure on it...

697
00:39:41 --> 00:39:47
So I'm going to put it
in the bell jar there,

698
00:39:44 --> 00:39:50
and I'm going to take
all the air out

699
00:39:46 --> 00:39:52
so that the pressure goes
down and down and down,

700
00:39:48 --> 00:39:54
but the temperature
is not changing,

701
00:39:50 --> 00:39:56
you go down in the liquid phase,
stays liquid... stays liquid...

702
00:39:56 --> 00:40:02
stays liquid... stays liquid
until you reach this point,

703
00:40:00 --> 00:40:06
and then you have coexistence
between the gas and the liquid--

704
00:40:04 --> 00:40:10
we call that vapor,
in the case of water--

705
00:40:06 --> 00:40:12
and that means
it will start to boil.

706
00:40:11 --> 00:40:17
And that happens--
and I looked that up--

707
00:40:14 --> 00:40:20
at a pressure of
about 17 millimeters mercury,

708
00:40:17 --> 00:40:23
which is equivalent
to 30 kilometers altitude.

709
00:40:21 --> 00:40:27
And so we have here some water,
room temperature.

710
00:40:26 --> 00:40:32
Put it here,
put it in a bell jar.

711
00:40:31 --> 00:40:37
 

712
00:40:35 --> 00:40:41
This is a 19th-century
vintage of a bell jar.

713
00:40:42 --> 00:40:48
It's very slow.

714
00:40:43 --> 00:40:49
It will take
at least five minutes

715
00:40:45 --> 00:40:51
before we reach
that low pressure.

716
00:40:49 --> 00:40:55
But you will see here
the... the wineglass.

717
00:40:54 --> 00:41:00
 

718
00:40:57 --> 00:41:03
There it is.

719
00:40:58 --> 00:41:04
And we will keep an eye on it,
on and off,

720
00:41:00 --> 00:41:06
and when it starts to boil, I
can even read the pressure here,

721
00:41:04 --> 00:41:10
but that's not so important.

722
00:41:05 --> 00:41:11
I want you to appreciate

723
00:41:07 --> 00:41:13
the fact that if you keep
pumping long enough

724
00:41:09 --> 00:41:15
that you will hit the line

725
00:41:12 --> 00:41:18
where gas and liquid are
in coexistence with each other,

726
00:41:16 --> 00:41:22
and that is our definition
of boiling.

727
00:41:20 --> 00:41:26
So, let me start the pumping

728
00:41:22 --> 00:41:28
and in the meantime,
we will do something else,

729
00:41:26 --> 00:41:32
because it will take
quite some time

730
00:41:29 --> 00:41:35
before that vintage pump
reaches a decent low pressure,

731
00:41:33 --> 00:41:39
which is what we want.

732
00:41:35 --> 00:41:41
We have to go down to about
15, 20 millimeters mercury.

733
00:41:42 --> 00:41:48
In the meantime, we'll work on
something that is quite similar.

734
00:41:49 --> 00:41:55
I have a paint can--
you see the paint can there.

735
00:41:53 --> 00:41:59
You've seen
that paint can before.

736
00:41:55 --> 00:42:01
You remember it--
we evacuated it and it imploded,

737
00:42:00 --> 00:42:06
and we understood
why it imploded.

738
00:42:04 --> 00:42:10
We filled it
with one atmosphere air.

739
00:42:07 --> 00:42:13
This was 25 centimeters,
and this was 15 centimeters.

740
00:42:12 --> 00:42:18
And we took the air out,

741
00:42:14 --> 00:42:20
and then you get an overpressure
of one atmosphere,

742
00:42:18 --> 00:42:24
which is one kilogram
per square centimeter.

743
00:42:21 --> 00:42:27
And this front cover alone is
about 375 square centimeters,

744
00:42:26 --> 00:42:32
and so the force ishuge!

745
00:42:27 --> 00:42:33
And it imploded.

746
00:42:28 --> 00:42:34
You've seen it happen
in front of your eyes

747
00:42:31 --> 00:42:37
when I pumped it out.

748
00:42:32 --> 00:42:38
Today I'm going to do
something more subtle

749
00:42:36 --> 00:42:42
but with the same effect.

750
00:42:38 --> 00:42:44
I'm going to put in here
a little bit of water.

751
00:42:41 --> 00:42:47
Here is a little bit of water--
there it goes.

752
00:42:45 --> 00:42:51
And I'm going to bring
this water to a boil.

753
00:42:50 --> 00:42:56
At this moment, there is
one atmosphere air in that can.

754
00:42:57 --> 00:43:03
But as the water starts to boil,

755
00:43:00 --> 00:43:06
the vapor pressure of the water
at 100 degrees centigrade

756
00:43:04 --> 00:43:10
becomes one atmosphere.

757
00:43:06 --> 00:43:12
And so the can fills exclusively
with water vapor

758
00:43:09 --> 00:43:15
and drives out all the air.

759
00:43:10 --> 00:43:16
So the air is gone-- the moment
that we see steam coming out,

760
00:43:14 --> 00:43:20
the air is gone.

761
00:43:16 --> 00:43:22
Then I will tighten it,
I will close it,

762
00:43:20 --> 00:43:26
and I will put the can here
and let it cool.

763
00:43:24 --> 00:43:30
What would be

764
00:43:25 --> 00:43:31
the water vapor pressure
at 20 degrees centigrade?

765
00:43:27 --> 00:43:33
I just told you that shortly.

766
00:43:30 --> 00:43:36
That is about
17 millimeters mercury.

767
00:43:33 --> 00:43:39
That is 1/45 of an atmosphere.

768
00:43:35 --> 00:43:41
In other words, if this can kept
its volume and didn't implode,

769
00:43:40 --> 00:43:46
by the time it reaches
20 degrees centigrade,

770
00:43:43 --> 00:43:49
the pressure in here would be

771
00:43:44 --> 00:43:50
only a few percent
of one atmospheric pressure,

772
00:43:47 --> 00:43:53
so it's like having
a vacuum in there.

773
00:43:49 --> 00:43:55
And so clearly,
the can will implode.

774
00:43:53 --> 00:43:59
So we'll try to get the water
out, to get it to boil.

775
00:43:58 --> 00:44:04
 

776
00:44:04 --> 00:44:10
I think it is boiling.

777
00:44:07 --> 00:44:13
I have to make sure
that all the air is out.

778
00:44:10 --> 00:44:16
I really want
pure water vapor in there.

779
00:44:12 --> 00:44:18
 

780
00:44:17 --> 00:44:23
Yeah, looks good.

781
00:44:21 --> 00:44:27
Looks fine.

782
00:44:22 --> 00:44:28
 

783
00:44:34 --> 00:44:40
There it goes.

784
00:44:35 --> 00:44:41
So now, the vapor pressure
goes down,

785
00:44:42 --> 00:44:48
the gas-- if you want
to call it the gas,

786
00:44:45 --> 00:44:51
which is what it is--
condenses into liquid,

787
00:44:48 --> 00:44:54
because at lower temperature,
it will start to condense.

788
00:44:52 --> 00:44:58
And what it does, it's going
to walk down this line.

789
00:44:56 --> 00:45:02
 

790
00:45:01 --> 00:45:07
Ah! The water is boiling!

791
00:45:03 --> 00:45:09
You see that?

792
00:45:05 --> 00:45:11
Water is boiling.

793
00:45:07 --> 00:45:13
20 degrees centigrade.

794
00:45:10 --> 00:45:16
Water is boiling--
okay, so we've seen that.

795
00:45:14 --> 00:45:20
Let's go back to the...

796
00:45:16 --> 00:45:22
 

797
00:45:18 --> 00:45:24
Shall we go back to water?

798
00:45:20 --> 00:45:26
We started with boiling water
here at one atmosphere,

799
00:45:24 --> 00:45:30
100 degrees centigrade.

800
00:45:26 --> 00:45:32
As the temperature goes down,
it must stay on this line,

801
00:45:30 --> 00:45:36
because water and liquid...
liquid and the vapor are

802
00:45:34 --> 00:45:40
in thermal equilibrium
with each other,

803
00:45:36 --> 00:45:42
and as it comes down this line,

804
00:45:38 --> 00:45:44
you see the pressure goes
down and down and down.

805
00:45:41 --> 00:45:47
And by the time that it is
20 degrees centigrade,

806
00:45:44 --> 00:45:50
we would be back
at that 17 millimeters mercury.

807
00:45:47 --> 00:45:53
Now, if this can is leaking,
which I think it is,

808
00:45:50 --> 00:45:56
because it should
already have collapsed...

809
00:45:52 --> 00:45:58
If the can is leaking,
of course,

810
00:45:54 --> 00:46:00
then that would be
a different story.

811
00:45:57 --> 00:46:03
Then it will not do
what we want it to do.

812
00:46:01 --> 00:46:07
So I may have to try this again.

813
00:46:03 --> 00:46:09
 

814
00:46:07 --> 00:46:13
So, we do it again
with another can.

815
00:46:10 --> 00:46:16
See, that already
should have imploded.

816
00:46:13 --> 00:46:19
 

817
00:46:17 --> 00:46:23
So we'll boil this one.

818
00:46:20 --> 00:46:26
Be a little patient,
and we will try it again.

819
00:46:23 --> 00:46:29
Yeah.

820
00:46:25 --> 00:46:31
 

821
00:46:29 --> 00:46:35
In the meantime, I want
to challenge you a little bit

822
00:46:33 --> 00:46:39
and expose you
to a demonstration

823
00:46:35 --> 00:46:41
which is kind of bizarre,
and I want you to tell me

824
00:46:39 --> 00:46:45
why it doesn't behave
the way I want it to behave.

825
00:46:43 --> 00:46:49
I have balloons, small balloons
here, which are filled with air,

826
00:46:49 --> 00:46:55
and I'm going to put them
in liquid nitrogen.

827
00:46:54 --> 00:47:00
So they have a certain volume.

828
00:46:56 --> 00:47:02
I don't care
what that volume is.

829
00:46:59 --> 00:47:05
They are at room temperature,
which is 293 degrees,

830
00:47:02 --> 00:47:08
and the pressure inside is
very close to one atmosphere.

831
00:47:06 --> 00:47:12
There's almost no overpressure
in those balloons.

832
00:47:10 --> 00:47:16
I'm going to make the
temperature 77 degrees Kelvin,

833
00:47:14 --> 00:47:20
which is liquid nitrogen.

834
00:47:15 --> 00:47:21
The pressure will remain
very closely one atmosphere.

835
00:47:18 --> 00:47:24
I think balloons don't have
very much overpressure,

836
00:47:21 --> 00:47:27
no matter what you do with them.

837
00:47:23 --> 00:47:29
And so I want to know
what the volume is,

838
00:47:27 --> 00:47:33
how much
they're going to shrink.

839
00:47:29 --> 00:47:35
Well, I apply the ideal-gas law

840
00:47:31 --> 00:47:37
and the number of molecules
is not going to change,

841
00:47:35 --> 00:47:41
R is not going to change,

842
00:47:38 --> 00:47:44
so the new volume
is going to be the old volume

843
00:47:41 --> 00:47:47
times the new temperature
divided by the old temperature.

844
00:47:46 --> 00:47:52
And so that is the old volume
times 77 divided by 293,

845
00:47:53 --> 00:47:59
if the ideal-gas law holds.

846
00:47:56 --> 00:48:02
And that is one-quarter of V1.

847
00:47:59 --> 00:48:05
So the volume becomes
four times less.

848
00:48:04 --> 00:48:10
If the volume is
four times less,

849
00:48:06 --> 00:48:12
then the radius becomes 60%
of what it originally was,

850
00:48:11 --> 00:48:17
because R cubed goes
with the volume.

851
00:48:15 --> 00:48:21
So R2 is about 60% of R1,
so it should be very noticeable.

852
00:48:21 --> 00:48:27
A balloon this big
should become this big.

853
00:48:24 --> 00:48:30
What you will see, however,
is something very different.

854
00:48:29 --> 00:48:35
Okay, I think we are okay
on this one now.

855
00:48:32 --> 00:48:38
We'll try to put the cap back
on, if I can find the cap.

856
00:48:36 --> 00:48:42
Oh, yeah, there it is.

857
00:48:38 --> 00:48:44
 

858
00:48:41 --> 00:48:47
Now, let me tighten it

859
00:48:43 --> 00:48:49
a little better
than I did the first time.

860
00:48:45 --> 00:48:51
 

861
00:48:57 --> 00:49:03
And let's see now if it cools...
whether it does better.

862
00:49:02 --> 00:49:08
 

863
00:49:04 --> 00:49:10
Clearly, the other one
was never properly sealed.

864
00:49:07 --> 00:49:13
 

865
00:49:09 --> 00:49:15
So, here we have the balloons,

866
00:49:12 --> 00:49:18
and I'm going to dip them
in liquid nitrogen.

867
00:49:16 --> 00:49:22
 

868
00:49:22 --> 00:49:28
Come on, can.

869
00:49:25 --> 00:49:31
It would be quite a coincidence
if that one is also leaking.

870
00:49:31 --> 00:49:37
Normally,
they collapse in seconds.

871
00:49:34 --> 00:49:40
This one doesn't want
to collapse.

872
00:49:36 --> 00:49:42
(can clanks )

873
00:49:38 --> 00:49:44
There it goes.
There it goes.

874
00:49:42 --> 00:49:48
It's making obscene noises--
there it goes, there it goes.

875
00:49:46 --> 00:49:52
Good!

876
00:49:47 --> 00:49:53
(can clanking )

877
00:49:49 --> 00:49:55
You will see that--
look at the balloons.

878
00:49:52 --> 00:49:58
Whew!

879
00:49:53 --> 00:49:59
Here's a balloon.

880
00:49:56 --> 00:50:02
And here is a balloon.

881
00:49:58 --> 00:50:04
Anyone's birthday today here?

882
00:50:01 --> 00:50:07
Must be someone.

883
00:50:02 --> 00:50:08
200 kids-- not kids,
200 grown-ups.

884
00:50:05 --> 00:50:11
Someone must be... birthday.

885
00:50:07 --> 00:50:13
50% chance-- no one's birthday?

886
00:50:10 --> 00:50:16
Hard to believe.

887
00:50:11 --> 00:50:17
Okay, so I predict that
if I put it in liquid nitrogen,

888
00:50:14 --> 00:50:20
which I have here,

889
00:50:15 --> 00:50:21
that the radius will become
60% of what it was.

890
00:50:20 --> 00:50:26
So it shrinks a little--
there we go.

891
00:50:23 --> 00:50:29
And what you see is
something very, very different,

892
00:50:26 --> 00:50:32
and that I want you to explain.

893
00:50:29 --> 00:50:35
And you have
all the tools available.

894
00:50:32 --> 00:50:38
Keep in mind, I put it
in liquid nitrogen--

895
00:50:34 --> 00:50:40
remember that when you're
looking for a solution.

896
00:50:38 --> 00:50:44
Okay, there's almost
no volume left.

897
00:50:40 --> 00:50:46
It's like a flat pancake--
it's nothing.

898
00:50:43 --> 00:50:49
Now it comes up, of course,

899
00:50:45 --> 00:50:51
because now it goes back
to room temperature.

900
00:50:49 --> 00:50:55
(can clanks,
clatters onto floor )

901
00:50:51 --> 00:50:57
My goodness,
it's having a hard time there.

902
00:50:56 --> 00:51:02
Why is it not...

903
00:50:58 --> 00:51:04
(balloon pops )

904
00:50:59 --> 00:51:05
(class laughs )

905
00:51:02 --> 00:51:08
Why is it not one-quarter?

906
00:51:06 --> 00:51:12
Why is it so much smaller
than what you expect?

907
00:51:11 --> 00:51:17
I'll do one more.

908
00:51:13 --> 00:51:19
If you come here--
why don't you come here?--

909
00:51:16 --> 00:51:22
you will see that it is nothing.

910
00:51:17 --> 00:51:23
The volume is effectively zero.

911
00:51:21 --> 00:51:27
You see that?

912
00:51:22 --> 00:51:28
Don't be worried.

913
00:51:23 --> 00:51:29
(class laughs )

914
00:51:24 --> 00:51:30
 

915
00:51:27 --> 00:51:33
And now it comes up.

916
00:51:29 --> 00:51:35
Last question for you
to think about this weekend.

917
00:51:33 --> 00:51:39
I have here a can
with tennis balls,

918
00:51:36 --> 00:51:42
and when you open this can,

919
00:51:37 --> 00:51:43
as everyone knows
who plays tennis, you hear...

920
00:51:40 --> 00:51:46
(makes whooshing sound )

921
00:51:41 --> 00:51:47
When you go to Europe
and you buy coffee,

922
00:51:44 --> 00:51:50
and you open the coffee can,
you hear...

923
00:51:46 --> 00:51:52
(makes whooshing sound )

924
00:51:47 --> 00:51:53
People like that;
they think that's good.

925
00:51:50 --> 00:51:56
A little bit of vacuum in there,
or something like that,

926
00:51:53 --> 00:51:59
makes the coffee stay longer,
better, fresher.

927
00:51:56 --> 00:52:02
Baloney, but it doesn't matter.

928
00:51:58 --> 00:52:04
In any case, these tennis balls,
in the same tradition,

929
00:52:01 --> 00:52:07
you open it up and you hear...
(makes whooshing sound )

930
00:52:04 --> 00:52:10
Now comes the question for you:

931
00:52:06 --> 00:52:12
Is the pressure inside the can
higher than one atmosphere?

932
00:52:09 --> 00:52:15
Or is it lower?

933
00:52:10 --> 00:52:16
It cannot be the same, because
then you wouldn't hear...

934
00:52:13 --> 00:52:19
(makes whooshing sound )

935
00:52:14 --> 00:52:20
And if so, why would the
pressure inside be different?

936
00:52:17 --> 00:52:23
And I'll give you one clue,
and the clue is crucial.

937
00:52:20 --> 00:52:26
You open the can,
you don't play with the balls,

938
00:52:23 --> 00:52:29
you wait two days,
and the balls are useless.

939
00:52:25 --> 00:52:31
You can't play
with them anymore.

940
00:52:27 --> 00:52:33
That should give you a clue.

941
00:52:29 --> 00:52:35
Think about it-- think about
the liquid nitrogen balloons.

942
00:52:33 --> 00:52:39
Have a good weekend--
see you Monday.

943
00:52:35 --> 00:52:41