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PROFESSOR: So last time we
talked about the zeroth law,

9
00:00:25,900 --> 00:00:29,440
which is the common-sense law,
which says that if you take a

10
00:00:29,440 --> 00:00:32,460
hot object next to a cold
object, heat will flow from

11
00:00:32,460 --> 00:00:36,320
the hot to the cold in a way
that is well defined, and it

12
00:00:36,320 --> 00:00:40,500
allows you to define
temperature.

13
00:00:40,500 --> 00:00:43,390
It allows you to define the
concept of a thermometer.

14
00:00:43,390 --> 00:00:47,890
You have three objects, one of
them could be a thermometer.

15
00:00:47,890 --> 00:00:49,620
You have two of them separated
at a distance.

16
00:00:49,620 --> 00:00:51,920
You take the third one, and you
go from one to the other,

17
00:00:51,920 --> 00:00:55,550
and you see whether heat flows,
when you touch one

18
00:00:55,550 --> 00:00:59,640
object, the middle object,
between those two objects.

19
00:00:59,640 --> 00:01:01,680
Let me talk to you about
temperature scales.

20
00:01:01,680 --> 00:01:04,030
We talked about the
Celsius scale then

21
00:01:04,030 --> 00:01:07,000
the Fahrenheit scale.

22
00:01:07,000 --> 00:01:10,270
The late 1800's were a booming
time for temperature scales.

23
00:01:10,270 --> 00:01:13,380
People didn't really realize
how important it was to

24
00:01:13,380 --> 00:01:16,310
properly define the reference
points: Fahrenheit's

25
00:01:16,310 --> 00:01:24,020
warm-blooded or 96 degrees,
and Romer's 7.5 degrees.

26
00:01:24,020 --> 00:01:26,310
Romer because he didn't want
to go below zero degrees

27
00:01:26,310 --> 00:01:28,870
measuring temperature
outside in Denmark

28
00:01:28,870 --> 00:01:30,540
Those are kind of silly.

29
00:01:30,540 --> 00:01:34,260
But they're the legacy that
we have today, and

30
00:01:34,260 --> 00:01:38,230
that's what we use.

31
00:01:38,230 --> 00:01:42,180
In science, we use somewhat
better temperature scales.

32
00:01:42,180 --> 00:01:46,690
And the temperature scale that
turns out to be well-defined

33
00:01:46,690 --> 00:01:52,410
and ends up giving us the
concept of an absolute zero is

34
00:01:52,410 --> 00:01:55,660
the ideal gas thermometer.

35
00:01:55,660 --> 00:02:03,550
So, let's talk about that
briefly today first.

36
00:02:03,550 --> 00:02:10,880
The ideal gas thermometer.

37
00:02:10,880 --> 00:02:12,250
It's based on Boyle's law.

38
00:02:12,250 --> 00:02:15,330
Boyle's law was an empirical law
that Mr. Boyle discovered

39
00:02:15,330 --> 00:02:18,750
by doing lots of experiments,
and Boyle's law says that the

40
00:02:18,750 --> 00:02:26,640
limit of the quantity pressure
times the molar volume, so

41
00:02:26,640 --> 00:02:30,030
this quantity here, pressure
times the molar volume, as you

42
00:02:30,030 --> 00:02:34,020
let pressure go to zero.

43
00:02:34,020 --> 00:02:37,250
So, you do this measurement, you
measure with the gas, you

44
00:02:37,250 --> 00:02:39,930
measure the pressure and
the molar volume.

45
00:02:39,930 --> 00:02:42,410
Then you change the pressure
again, and you measure the

46
00:02:42,410 --> 00:02:44,140
pressure in the volume, and
you multiply these two

47
00:02:44,140 --> 00:02:46,530
together, and you keep doing
this experiment, getting the

48
00:02:46,530 --> 00:02:50,700
pressure smaller and smaller,
you find that this limit turns

49
00:02:50,700 --> 00:02:53,610
out to be a constant,
independent of the gas.

50
00:02:53,610 --> 00:02:55,610
It doesn't care where
the gas is.

51
00:02:55,610 --> 00:02:58,570
You always get to the
same constant.

52
00:02:58,570 --> 00:03:02,440
And that constant turns out
to be a function of the

53
00:03:02,440 --> 00:03:04,320
temperature.

54
00:03:04,320 --> 00:03:06,910
The only function it is -- it
doesn't care where the gas is.

55
00:03:06,910 --> 00:03:10,000
It only cares where the
temperature is.

56
00:03:10,000 --> 00:03:13,840
All right, so now we have the
makings of a good thermometer

57
00:03:13,840 --> 00:03:15,380
and a good temperature scale.

58
00:03:15,380 --> 00:03:19,010
We have a substance.

59
00:03:19,010 --> 00:03:22,290
The substance could
be any gas.

60
00:03:22,290 --> 00:03:24,620
That's pretty straightforward.

61
00:03:24,620 --> 00:03:32,040
So now we have a substance,
which is

62
00:03:32,040 --> 00:03:36,070
a gas, with a property.

63
00:03:36,070 --> 00:03:41,300
So now the volume of mercury,
or the color of something

64
00:03:41,300 --> 00:03:43,890
which changes with temperature,
or the

65
00:03:43,890 --> 00:03:44,820
resistivity.

66
00:03:44,820 --> 00:03:51,120
In this case here, our property
is the value of the

67
00:03:51,120 --> 00:03:52,980
pressure times the volume,
times the molar volume.

68
00:03:52,980 --> 00:03:56,210
That's the property.

69
00:03:56,210 --> 00:04:02,200
The property is the limit as p
goes to zero of pressure times

70
00:04:02,200 --> 00:04:03,580
molar volume.

71
00:04:03,580 --> 00:04:04,990
It's a number.

72
00:04:04,990 --> 00:04:05,400
Measure it.

73
00:04:05,400 --> 00:04:05,860
It's a number.

74
00:04:05,860 --> 00:04:06,520
It's going to come out.

75
00:04:06,520 --> 00:04:09,500
That's the property that's going
to give us the change in

76
00:04:09,500 --> 00:04:11,040
temperature.

77
00:04:11,040 --> 00:04:12,850
Then we need some reference
points.

78
00:04:12,850 --> 00:04:20,580
And Celsius first used the
boiling point of water, and

79
00:04:20,580 --> 00:04:24,110
called that 100 degrees Celsius,
and the freezing

80
00:04:24,110 --> 00:04:32,410
point of water and called that
zero degrees Celsius.

81
00:04:32,410 --> 00:04:34,870
And then we need an
interpolation scale.

82
00:04:34,870 --> 00:04:37,550
How to go from one reference
point to the

83
00:04:37,550 --> 00:04:40,660
other with this property.

84
00:04:40,660 --> 00:04:43,490
This property, which we're
going to call f(t).

85
00:04:43,490 --> 00:04:50,740
There are many ways you can
connect those two dots.

86
00:04:50,740 --> 00:04:55,560
If I draw a graph, and on one
axis I have this temperature.

87
00:04:55,560 --> 00:04:58,210
The idea of temperature with two
reference points, zero for

88
00:04:58,210 --> 00:05:02,210
the freezing point of water,
100 degrees for the boiling

89
00:05:02,210 --> 00:05:05,170
point of water.

90
00:05:05,170 --> 00:05:10,120
And on the y-axis I've got
the property f(t).

91
00:05:10,120 --> 00:05:16,070
It has some value

92
00:05:16,070 --> 00:05:18,000
corresponding to t equals zero.

93
00:05:18,000 --> 00:05:20,110
So let's get some value
right here.

94
00:05:20,110 --> 00:05:22,070
There's another value connected
to this property

95
00:05:22,070 --> 00:05:26,460
here, when t is equal to 100,
a reference point here.

96
00:05:26,460 --> 00:05:27,740
Now there many ways
I can connect

97
00:05:27,740 --> 00:05:28,930
these two points together.

98
00:05:28,930 --> 00:05:31,940
The simplest way is to
draw a straight line.

99
00:05:31,940 --> 00:05:34,480
It's called the linear
interpolation.

100
00:05:34,480 --> 00:05:39,020
My line is not so straight,
right here.

101
00:05:39,020 --> 00:05:40,470
You could do a different
kind of line.

102
00:05:40,470 --> 00:05:42,760
You could do a quadratic,
let's say.

103
00:05:42,760 --> 00:05:44,040
Something like this.

104
00:05:44,040 --> 00:05:45,860
That would be perfectly
fine interpolation.

105
00:05:45,860 --> 00:05:58,390
All right, we choose to have
a linear interpolation.

106
00:05:58,390 --> 00:06:03,630
That's a choice, and that choice
turns out to be very

107
00:06:03,630 --> 00:06:06,930
interesting and really
important, because if you

108
00:06:06,930 --> 00:06:09,490
connect these two points
together, you get a straight

109
00:06:09,490 --> 00:06:19,520
line that has to intercept
the x-axis at some point.

110
00:06:19,520 --> 00:06:21,290
Now what does it mean to
intercept the x-axis here?

111
00:06:21,290 --> 00:06:24,510
It means that the value
of f(t) for this

112
00:06:24,510 --> 00:06:29,000
temperature is zero.

113
00:06:29,000 --> 00:06:32,120
That means that at this point
right here, f(t)=0.

114
00:06:35,260 --> 00:06:38,120
That means the pressure times
the volume equals

115
00:06:38,120 --> 00:06:43,890
zero, for that gas.

116
00:06:43,890 --> 00:06:47,160
And if you're below this
temperature here, this

117
00:06:47,160 --> 00:06:51,100
quantity, p times v it
would be negative.

118
00:06:51,100 --> 00:06:54,050
Is that possible?

119
00:06:54,050 --> 00:06:55,540
Can we have p v negative?

120
00:06:55,540 --> 00:06:58,470
Yes?

121
00:06:58,470 --> 00:07:01,020
No, it can't be.

122
00:07:01,020 --> 00:07:02,720
Negative pressure doesn't
make any sense, right?

123
00:07:02,720 --> 00:07:04,030
Negative volume doesn't
make any sense.

124
00:07:04,030 --> 00:07:09,860
That means that this part
here, can't happen.

125
00:07:09,860 --> 00:07:14,840
That means that this temperature
right here is the

126
00:07:14,840 --> 00:07:18,610
absolute lowest temperature you
can go to that physically

127
00:07:18,610 --> 00:07:19,920
makes any sense.

128
00:07:19,920 --> 00:07:22,770
That's the absolute zero.

129
00:07:22,770 --> 00:07:27,580
So the concept of an absolute
zero, a temperature below

130
00:07:27,580 --> 00:07:34,130
which you just can't go, that's
directly out of the

131
00:07:34,130 --> 00:07:37,210
scheme here, this linear
interpolation scheme with

132
00:07:37,210 --> 00:07:39,540
these two reference points.

133
00:07:39,540 --> 00:07:42,770
If I had taken as my
interpolation scheme, my white

134
00:07:42,770 --> 00:07:47,750
curve here, I could go to
infinity and have the

135
00:07:47,750 --> 00:07:49,830
equivalent of absolute
zero being at

136
00:07:49,830 --> 00:07:53,530
infinity, minus infinity.

137
00:07:53,530 --> 00:07:59,060
So, this temperature, this
absolute zero here, which is

138
00:07:59,060 --> 00:08:04,300
absolute zero on the
Kelvin scale.

139
00:08:04,300 --> 00:08:05,990
The lowest possible temperature
in the Celsius

140
00:08:05,990 --> 00:08:13,190
scale is minus 273.15
degrees Celsius.

141
00:08:13,190 --> 00:08:19,720
So that begs the notion of
re-referencing our reference

142
00:08:19,720 --> 00:08:23,060
point, of changing our
reference points.

143
00:08:23,060 --> 00:08:26,180
To change a reference point
from this point here being

144
00:08:26,180 --> 00:08:29,290
zero, instead of this point
here being zero.

145
00:08:29,290 --> 00:08:32,180
And so redefining then the
temperature scale to the

146
00:08:32,180 --> 00:08:41,550
Kelvin scale, where t in degrees
Kelvin is equal to t

147
00:08:41,550 --> 00:08:50,040
in degree Celsius,
plus 273.15.

148
00:08:50,040 --> 00:08:55,200
And then you would get
the Kelvin scale.

149
00:08:55,200 --> 00:08:59,760
All right, it turned out that
this thermometer here wasn't

150
00:08:59,760 --> 00:09:02,450
quite perfect either.

151
00:09:02,450 --> 00:09:07,480
Just like Fahrenheit measuring
96 degrees being a

152
00:09:07,480 --> 00:09:11,670
warm-blooded, healthy
man, right,

153
00:09:11,670 --> 00:09:15,540
that's not very accurate.

154
00:09:15,540 --> 00:09:17,590
Our temperature probably
fluctuates during the day a

155
00:09:17,590 --> 00:09:20,720
little bit anyways, it's
not very accurate.

156
00:09:20,720 --> 00:09:23,960
And similarly, the boiling
point, defining that at a 100

157
00:09:23,960 --> 00:09:25,730
degrees Celsius, well that
depends on the pressure.

158
00:09:25,730 --> 00:09:29,330
It depends whether you're in
Denver or you're in Boston.

159
00:09:29,330 --> 00:09:31,790
Water boils at different
temperatures, depending on

160
00:09:31,790 --> 00:09:34,800
what the atmospheric pressure
is; same thing for the

161
00:09:34,800 --> 00:09:37,130
freezing point.

162
00:09:37,130 --> 00:09:38,650
So that means, then, you've
got to define the pressure

163
00:09:38,650 --> 00:09:39,050
pretty well.

164
00:09:39,050 --> 00:09:41,790
You've got to know where
the pressure is.

165
00:09:41,790 --> 00:09:44,190
It would be much better if you
had a reference point that

166
00:09:44,190 --> 00:09:45,560
didn't care where the
pressure was.

167
00:09:45,560 --> 00:09:49,340
Just like our substance doesn't
care where the gas is.

168
00:09:49,340 --> 00:09:52,020
It's kind of universal.

169
00:09:52,020 --> 00:09:55,460
And so now, instead of using
these reference points for the

170
00:09:55,460 --> 00:10:01,490
Kelvin scale, we use the
absolute zero, which isn't

171
00:10:01,490 --> 00:10:03,630
going to care what
the pressure is.

172
00:10:03,630 --> 00:10:05,340
It's the lowest number
you can go to.

173
00:10:05,340 --> 00:10:08,890
And our other reference point
is the triple point of water

174
00:10:08,890 --> 00:10:18,460
-- reference points become zero
Kelvin, absolute zero,

175
00:10:18,460 --> 00:10:22,870
and the triple point.

176
00:10:22,870 --> 00:10:27,100
The triple point of water is
going to be defined as 273.16

177
00:10:27,100 --> 00:10:30,080
degrees Kelvin.

178
00:10:30,080 --> 00:10:35,050
And the triple point of water
is that temperature and

179
00:10:35,050 --> 00:10:38,790
pressure -- there's a unique
temperature and pressure where

180
00:10:38,790 --> 00:10:44,180
water exists in equilibrium
between the liquid phase, the

181
00:10:44,180 --> 00:10:48,230
vapor phase, and the
solid phase.

182
00:10:48,230 --> 00:10:57,010
So the triple point is liquid,
solid, gas, all in

183
00:10:57,010 --> 00:10:58,750
equilibrium.

184
00:10:58,750 --> 00:11:01,740
Now you may think, well
I've seen that before.

185
00:11:01,740 --> 00:11:06,390
You take a glass of ice
water and set it down.

186
00:11:06,390 --> 00:11:09,290
There's the water phase, there's
the ice cube is the

187
00:11:09,290 --> 00:11:15,310
solid phase, and there's
some water, gas, vapor,

188
00:11:15,310 --> 00:11:16,770
and that's one bar.

189
00:11:16,770 --> 00:11:17,830
Where am I going wrong here?

190
00:11:17,830 --> 00:11:25,620
The partial pressure of the
water, of gaseous water, above

191
00:11:25,620 --> 00:11:30,360
that equilibrium of ice
and water is not one

192
00:11:30,360 --> 00:11:33,650
bar, it's much less.

193
00:11:33,650 --> 00:11:38,920
So the partial pressure or the
pressure by which you have

194
00:11:38,920 --> 00:11:46,140
this triple point, happens
to be 6.1 times 10

195
00:11:46,140 --> 00:11:48,040
to the minus 3 bar.

196
00:11:48,040 --> 00:11:49,960
There's hardly any vapor
pressure above

197
00:11:49,960 --> 00:11:55,530
your ice water glass.

198
00:11:55,530 --> 00:12:00,800
So this unique temperature and
unique pressure defines a

199
00:12:00,800 --> 00:12:03,930
triple point everywhere,
and that's a

200
00:12:03,930 --> 00:12:07,110
great reference point.

201
00:12:07,110 --> 00:12:10,970
Any questions?

202
00:12:10,970 --> 00:12:11,300
Great.

203
00:12:11,300 --> 00:12:13,775
So now we have this ideal gas
thermometer, and out of this

204
00:12:13,775 --> 00:12:18,780
ideal gas thermometer, also
comes out the ideal gas law.

205
00:12:18,780 --> 00:12:22,510
Because we can take our
interpolation here, our linear

206
00:12:22,510 --> 00:12:29,110
interpolation, the slope
of this line.

207
00:12:29,110 --> 00:12:31,020
Let's draw it in degrees
Kelvin, instead

208
00:12:31,020 --> 00:12:35,400
of in degrees Celsius.

209
00:12:35,400 --> 00:12:38,670
So we have now temperature
in degrees Kelvin.

210
00:12:38,670 --> 00:12:43,150
We have the quantity
f(t) here.

211
00:12:43,150 --> 00:12:50,480
We have an interpolation scheme
between zero and 273.16

212
00:12:50,480 --> 00:12:53,710
with two values for this
quantity, and we have a linear

213
00:12:53,710 --> 00:12:57,760
interpolation that defines our
temperature scale, our Kelvin

214
00:12:57,760 --> 00:12:59,350
temperature scale.

215
00:12:59,350 --> 00:13:07,150
And so the slope of this thing
is f(t) at the triple point,

216
00:13:07,150 --> 00:13:10,120
which is this point here, this
is the temperature of the

217
00:13:10,120 --> 00:13:16,650
triple point of water,
divided by 273.16.

218
00:13:16,650 --> 00:13:18,710
That's the slope of that line.

219
00:13:18,710 --> 00:13:25,660
The quantity here, which is
f (t of the triple point),

220
00:13:25,660 --> 00:13:33,480
divided by the value
of the x-axis here.

221
00:13:33,480 --> 00:13:37,080
So that's the slope, and the
intercept is zero, so the

222
00:13:37,080 --> 00:13:45,730
function f(t), you just
multiply by t here.

223
00:13:45,730 --> 00:13:50,250
This is the slope. f(t)
is just the limit.

224
00:13:50,250 --> 00:13:56,910
As p goes to zero of
p times v bar.

225
00:13:56,910 --> 00:14:00,080
And so now we have this
quantity, p times v bar, and

226
00:14:00,080 --> 00:14:04,350
the limit of p goes to zero is
equal to a constant times the

227
00:14:04,350 --> 00:14:07,500
temperature.

228
00:14:07,500 --> 00:14:09,200
That's a universal statement.

229
00:14:09,200 --> 00:14:11,050
It's true of every gas.

230
00:14:11,050 --> 00:14:14,410
I didn't say this is only true
of hydrogen or nitrogen, This

231
00:14:14,410 --> 00:14:20,890
is any gas because I'm taking
this limit p equals to zero.

232
00:14:20,890 --> 00:14:22,370
Now this constant is
just a constant.

233
00:14:22,370 --> 00:14:23,250
I'm going to call it r.

234
00:14:23,250 --> 00:14:25,760
I'm going to call it r.

235
00:14:25,760 --> 00:14:31,520
It's going to be the gas
constant, and now I have r

236
00:14:31,520 --> 00:14:37,950
times t is equal to the limit,
p goes to zero of p r.

237
00:14:37,950 --> 00:14:44,700
It's true for any gas, and if I
remove this limit here, r t

238
00:14:44,700 --> 00:14:52,600
is equal to p v bar, I'm going
to call that an ideal gas.

239
00:14:52,600 --> 00:14:55,020
See, this is the property
of an ideal gas.

240
00:14:55,020 --> 00:14:55,760
What does it mean, ideal gas?

241
00:14:55,760 --> 00:14:58,200
It means that the molecules or
the atoms and the gas don't

242
00:14:58,200 --> 00:15:01,110
know about each other.

243
00:15:01,110 --> 00:15:02,670
They effectively
have no volume.

244
00:15:02,670 --> 00:15:04,170
They have no interactions
with each other.

245
00:15:04,170 --> 00:15:06,290
They occupy the same
volume in space.

246
00:15:06,290 --> 00:15:08,230
They don't care that there
are other atoms

247
00:15:08,230 --> 00:15:09,900
and molecules around.

248
00:15:09,900 --> 00:15:11,280
So that's basically what
you do when you

249
00:15:11,280 --> 00:15:12,890
take p goes to zero.

250
00:15:12,890 --> 00:15:16,360
You make the volume infinitely
large, the density of the gas

251
00:15:16,360 --> 00:15:17,820
infinitely small.

252
00:15:17,820 --> 00:15:19,990
The atoms or molecules in the
gas don't know that there are

253
00:15:19,990 --> 00:15:22,880
other atoms and molecules in the
gas, and then you end up

254
00:15:22,880 --> 00:15:26,360
with this universal property.

255
00:15:26,360 --> 00:15:29,040
All right, so gases that have
this universal property, even

256
00:15:29,040 --> 00:15:32,810
when the pressure is not zero,
those are the ideal gases.

257
00:15:32,810 --> 00:15:35,450
And for the sake of this class,
we're going to consider

258
00:15:35,450 --> 00:15:40,860
most gases to be ideal gases.

259
00:15:40,860 --> 00:15:44,690
Questions?

260
00:15:44,690 --> 00:15:50,530
So now, this equation here
relates three state functions

261
00:15:50,530 --> 00:15:53,010
together: the pressure, the
volume, and the temperature.

262
00:15:53,010 --> 00:15:56,510
Now, if you remember, we said
that if you had a substance,

263
00:15:56,510 --> 00:16:00,100
if you knew the number of moles
and two properties, you

264
00:16:00,100 --> 00:16:02,180
knew everything about the gas.

265
00:16:02,180 --> 00:16:11,040
Which means that you can
re-write this in the form,

266
00:16:11,040 --> 00:16:16,250
volume, for instance, is equal
to the function of n, p, t..

267
00:16:16,250 --> 00:16:26,660
In this case, V = (nRT)/P. Have
two quantities and the

268
00:16:26,660 --> 00:16:29,930
number of moles gives you
another property.

269
00:16:29,930 --> 00:16:31,280
You don't need to
know the volume.

270
00:16:31,280 --> 00:16:32,690
All you need to know is the
pressure and temperature and

271
00:16:32,690 --> 00:16:34,200
the number of moles
to get the volume.

272
00:16:34,200 --> 00:16:36,340
This is called an equation
of state.

273
00:16:36,340 --> 00:16:43,530
It relate state properties
to each other.

274
00:16:43,530 --> 00:16:46,270
In this case it relates the
volume to the pressure and the

275
00:16:46,270 --> 00:16:49,120
temperature.

276
00:16:49,120 --> 00:16:55,220
Now, if you're an engineer, and
you use the ideal gas law

277
00:16:55,220 --> 00:17:01,160
to design a chemical plant or
a boiler or an electrical

278
00:17:01,160 --> 00:17:04,420
plant, you know, a steam plant,
you're going to be in

279
00:17:04,420 --> 00:17:08,740
big trouble.

280
00:17:08,740 --> 00:17:14,080
Your plant is going to blow up,
because the ideal gas law

281
00:17:14,080 --> 00:17:16,890
works only in very small range
of pressures and temperatures

282
00:17:16,890 --> 00:17:18,480
for most gases.

283
00:17:18,480 --> 00:17:25,110
So, we have other equations
of states for real gases.

284
00:17:25,110 --> 00:17:27,440
This is an equation of state
for an ideal gases.

285
00:17:27,440 --> 00:17:30,280
For real gases, there's a whole
bunch of equation the

286
00:17:30,280 --> 00:17:33,805
states that you can find in
textbooks, and I'm just going

287
00:17:33,805 --> 00:17:37,620
to go through a few of them.

288
00:17:37,620 --> 00:17:39,610
The first one uses something
called a

289
00:17:39,610 --> 00:17:45,020
compressibility factor, z.

290
00:17:45,020 --> 00:17:46,280
Compressibility factor, z.

291
00:17:46,280 --> 00:17:51,620
And instead of writing PV = RT,
which would be the ideal

292
00:17:51,620 --> 00:17:55,580
gas law, we put a fudge
factor in there.

293
00:17:55,580 --> 00:17:59,620
And the fudge factor
is called z.

294
00:17:59,620 --> 00:18:04,460
Now we can put real instead of
ideal for our volume. z is the

295
00:18:04,460 --> 00:18:11,050
compressibility factor, and z is
the ratio of the volume of

296
00:18:11,050 --> 00:18:14,310
the real gas divided
by what it would be

297
00:18:14,310 --> 00:18:21,240
were it an ideal gas.

298
00:18:21,240 --> 00:18:27,120
So, if z is less than 1, then
the real gas is more compact

299
00:18:27,120 --> 00:18:28,920
then the ideal gas.

300
00:18:28,920 --> 00:18:30,320
It's a smaller volume.

301
00:18:30,320 --> 00:18:33,230
If z is greater than 1, then
the real gas means that the

302
00:18:33,230 --> 00:18:36,090
atoms and molecules in the real
gas are repelling each

303
00:18:36,090 --> 00:18:40,270
other and wants to have
a bigger volume.

304
00:18:40,270 --> 00:18:41,490
And you can find these

305
00:18:41,490 --> 00:18:43,190
compressibility factors in tables.

306
00:18:43,190 --> 00:18:46,560
If you want to know the
compressibility factors for

307
00:18:46,560 --> 00:18:49,180
water, for steam, at a certain
pressure and temperature, you

308
00:18:49,180 --> 00:18:52,420
go to a table and you find it.

309
00:18:52,420 --> 00:18:58,800
So that's one example of a
real equation of state.

310
00:18:58,800 --> 00:19:02,920
Not a very useful one for our
purposes in this class here.

311
00:19:02,920 --> 00:19:06,370
Another one is the
virial expansion.

312
00:19:06,370 --> 00:19:08,920
It's a little bit more useful.

313
00:19:08,920 --> 00:19:13,060
What you do is you take that
fudge factor, and you expand

314
00:19:13,060 --> 00:19:15,490
it out into a Taylor series.

315
00:19:15,490 --> 00:19:23,306
So, we have the p v real
over r t is equal to z.

316
00:19:23,306 --> 00:19:28,200
Now, we're going to take z and
say all right, under most

317
00:19:28,200 --> 00:19:32,610
conditions, it's pretty close to
1, when it's an ideal gas.

318
00:19:32,610 --> 00:19:37,620
And then we have to add
corrections to that, and the

319
00:19:37,620 --> 00:19:41,320
corrections are going to
be more important, the

320
00:19:41,320 --> 00:19:44,190
larger the volume is.

321
00:19:44,190 --> 00:19:47,686
Remember, it's the limit of p
times v goes to zero, so if

322
00:19:47,686 --> 00:19:50,800
you have a large volume with a
large pressure, then you're

323
00:19:50,800 --> 00:19:52,240
out of the ideal gas regime.

324
00:19:52,240 --> 00:20:00,100
So let's take Taylor series in
one over the volume, it's

325
00:20:00,100 --> 00:20:04,070
going to be one over the volume
squared, etcetera.

326
00:20:04,070 --> 00:20:07,370
And these factors on top, which
are going to depend on

327
00:20:07,370 --> 00:20:11,900
the temperature, are the virial
coefficients, and those

328
00:20:11,900 --> 00:20:14,070
depend on the substance.

329
00:20:14,070 --> 00:20:17,660
So you have this p B(t) here.

330
00:20:17,660 --> 00:20:25,040
This is called a second
virial coefficient.

331
00:20:25,040 --> 00:20:29,700
And then, so you can get,
you can actually find a

332
00:20:29,700 --> 00:20:31,080
graph of this B(t).

333
00:20:31,080 --> 00:20:32,240
It's going to look something
like this.

334
00:20:32,240 --> 00:20:39,130
It's the function of
temperature, as B(t).

335
00:20:39,130 --> 00:20:41,690
There's going to be some
temperature where B(t) is

336
00:20:41,690 --> 00:20:42,060
equal to zero.

337
00:20:42,060 --> 00:20:45,970
In that case, your gas is
going to look awfully

338
00:20:45,970 --> 00:20:46,570
like an ideal gas.

339
00:20:46,570 --> 00:20:50,960
Above some temperature is going
to be positive, below

340
00:20:50,960 --> 00:20:53,790
some temperature is going
to be negative.

341
00:20:53,790 --> 00:20:57,630
Generally, we ignore the
high order terms here.

342
00:20:57,630 --> 00:21:00,320
So again, if you do a
calculation where you're close

343
00:21:00,320 --> 00:21:04,567
enough to the ideal gas, and
you need to design your, if

344
00:21:04,567 --> 00:21:07,050
you have an engineer designing
something that's got a bunch

345
00:21:07,050 --> 00:21:13,590
of gases around, this is
a useful thing to use.

346
00:21:13,590 --> 00:21:18,480
Now, the most interesting one
for our class, the equation of

347
00:21:18,480 --> 00:21:20,700
state that's the most
interesting, is the Van der

348
00:21:20,700 --> 00:21:24,976
Waals equation of state,
developed by Mr. Van

349
00:21:24,976 --> 00:21:29,230
der Waals in 1873.

350
00:21:29,230 --> 00:21:31,320
And the beauty of that equation
of state is that it

351
00:21:31,320 --> 00:21:37,460
only relies on two parameters.

352
00:21:37,460 --> 00:21:38,700
So let's build it up.

353
00:21:38,700 --> 00:21:43,090
Let's see where it comes from.

354
00:21:43,090 --> 00:21:44,400
Let me just first write it
down, the Van der Waals

355
00:21:44,400 --> 00:21:53,480
equation of state. p plus a over
v bar squared times v bar

356
00:21:53,480 --> 00:21:57,390
minus b equals r t.

357
00:21:57,390 --> 00:22:02,030
All right, if you take a equal
to zero, these are the two

358
00:22:02,030 --> 00:22:03,250
parameters, a and b.

359
00:22:03,250 --> 00:22:05,250
If you take those two equal
to zero, you have p v

360
00:22:05,250 --> 00:22:05,960
is equal to r t.

361
00:22:05,960 --> 00:22:08,950
That's the ideal gas law.

362
00:22:08,950 --> 00:22:09,780
Let's build this up.

363
00:22:09,780 --> 00:22:11,745
Let's see where this comes from,
where these parameters a

364
00:22:11,745 --> 00:22:14,110
and b comes from.

365
00:22:14,110 --> 00:22:15,560
So, the first thing we're going
to do is we're going to

366
00:22:15,560 --> 00:22:20,360
take our gas in our box,
let's build a box

367
00:22:20,360 --> 00:22:21,770
full of gases here.

368
00:22:21,770 --> 00:22:24,760
We've got a bunch of gas
molecules or atoms.

369
00:22:24,760 --> 00:22:30,230
OK, there's the volume
of a box here.

370
00:22:30,230 --> 00:22:34,440
While these gas molecules
or atoms through first

371
00:22:34,440 --> 00:22:37,770
approximation, are like
hard spheres.

372
00:22:37,770 --> 00:22:39,390
They occupy a certain volume.

373
00:22:39,390 --> 00:22:45,020
Each atom or molecule occupies
a particular volume.

374
00:22:45,020 --> 00:22:58,370
And so, we can call b is the
volume per mole of the hard

375
00:22:58,370 --> 00:23:06,390
spheres, volume per mole that is
the little sphere that the

376
00:23:06,390 --> 00:23:07,760
molecules are.

377
00:23:07,760 --> 00:23:11,790
So that the volume that is
available to any one of those

378
00:23:11,790 --> 00:23:16,880
spheres is actually smaller than
v. Because you've got all

379
00:23:16,880 --> 00:23:19,570
these other little spheres
around, so the actual volume

380
00:23:19,570 --> 00:23:23,180
seen by any one of those spheres
is smaller than v. So

381
00:23:23,180 --> 00:23:29,250
when we take our ideal gas law,
p v bar is equal to r t

382
00:23:29,250 --> 00:23:32,790
we have to replace v bar by the
actual volume available to

383
00:23:32,790 --> 00:23:34,030
this hard sphere.

384
00:23:34,030 --> 00:23:42,110
So instead of v bar, we write
p v bar minus b, equal r t.

385
00:23:42,110 --> 00:23:47,890
OK, that's the hard sphere
volume of the spheres.

386
00:23:47,890 --> 00:23:52,770
Now, those molecules or atoms
that are in here, also feel

387
00:23:52,770 --> 00:23:53,640
each other.

388
00:23:53,640 --> 00:23:55,680
There are a whole bunch of
forces that you learn in

389
00:23:55,680 --> 00:23:57,390
5.112, 5.111 like with Van
der Waals' attractions

390
00:23:57,390 --> 00:23:58,280
and things like this.

391
00:23:58,280 --> 00:24:04,320
So there are attractive forces,
or repulsive forces

392
00:24:04,320 --> 00:24:13,090
that these molecules feel, and
that's going to change the

393
00:24:13,090 --> 00:24:16,280
pressure that the
molecules feel.

394
00:24:16,280 --> 00:24:19,480
For instance, if I have,
what is pressure?

395
00:24:19,480 --> 00:24:21,330
Pressure is when you have one
of these hard spheres

396
00:24:21,330 --> 00:24:23,410
colliding against the wall.

397
00:24:23,410 --> 00:24:24,950
There's the hard sphere.

398
00:24:24,950 --> 00:24:27,300
It wants to collide against the
wall to create a force on

399
00:24:27,300 --> 00:24:30,150
the wall, and I have a couple
of the hard spheres that are

400
00:24:30,150 --> 00:24:34,090
nearby, right, and in the
absence of any interactions, I

401
00:24:34,090 --> 00:24:34,900
get a certain pressure.

402
00:24:34,900 --> 00:24:37,210
This thing would but careen
into the wall, kaboom!

403
00:24:37,210 --> 00:24:41,750
You'd have this little force,
but in the presence of these

404
00:24:41,750 --> 00:24:45,190
interactions, you've got these
other molecules here that are

405
00:24:45,190 --> 00:24:51,630
watching this, you know, their
partner sort of wants to do

406
00:24:51,630 --> 00:24:54,112
damage to themselves, like
hitting that wall,

407
00:24:54,112 --> 00:24:54,830
and they say, no!

408
00:24:54,830 --> 00:24:56,600
Come back, come back, right?

409
00:24:56,600 --> 00:24:59,470
There is an attractive force.

410
00:24:59,470 --> 00:25:01,130
There are no other molecules
on that side of the wall.

411
00:25:01,130 --> 00:25:04,980
So there's an attractive force
that makes the velocity within

412
00:25:04,980 --> 00:25:05,960
not quite as fast.

413
00:25:05,960 --> 00:25:08,710
The force is not quite as strong
as it was without this

414
00:25:08,710 --> 00:25:09,570
attractive force.

415
00:25:09,570 --> 00:25:13,040
So the real pressure is not
quite the same because of this

416
00:25:13,040 --> 00:25:16,020
attractive force as it was,
as it would be without the

417
00:25:16,020 --> 00:25:17,170
attractive forces.

418
00:25:17,170 --> 00:25:20,840
The pressure is a little bit
less in this case here.

419
00:25:20,840 --> 00:25:25,270
So instead of this p here.

420
00:25:25,270 --> 00:25:30,370
Now if I re-write this equation
here as p is equal to

421
00:25:30,370 --> 00:25:35,980
r t divided by v bar minus
b, just re-writing this

422
00:25:35,980 --> 00:25:39,260
equation as it is.

423
00:25:39,260 --> 00:25:41,540
So the pressure is going to
depend on how strong this

424
00:25:41,540 --> 00:25:46,680
attractive force is.

425
00:25:46,680 --> 00:25:49,880
So the pressure is going to be
less if there's a strong

426
00:25:49,880 --> 00:25:51,150
attractive force.

427
00:25:51,150 --> 00:25:55,680
And the 1 over v squared is a
statistical, is basically a

428
00:25:55,680 --> 00:26:00,800
probability of having another
molecule, a second molecule in

429
00:26:00,800 --> 00:26:02,240
the volume of space.

430
00:26:02,240 --> 00:26:06,970
So, if the molar volume is
small, then one over v bar is

431
00:26:06,970 --> 00:26:10,190
large, there's a large
probability of having two

432
00:26:10,190 --> 00:26:13,180
spheres together in
the same volume.

433
00:26:13,180 --> 00:26:16,480
If the molar volume is large,
that means that there's a lot

434
00:26:16,480 --> 00:26:19,630
of room for the molecules, and
they're now going to be close

435
00:26:19,630 --> 00:26:20,760
to each other, and
so this isn't

436
00:26:20,760 --> 00:26:22,570
going to be as important.

437
00:26:22,570 --> 00:26:26,620
So, a is the strength of the
interaction, v bar is how

438
00:26:26,620 --> 00:26:29,710
likely they are to be
close to each other.

439
00:26:29,710 --> 00:26:30,840
And that's going to
affect the actual

440
00:26:30,840 --> 00:26:35,220
pressure seen by the gas.

441
00:26:35,220 --> 00:26:43,020
And a is greater than zero when
you have the attraction.

442
00:26:43,020 --> 00:26:47,980
And that gives use the Van der
Waals' equation of state, with

443
00:26:47,980 --> 00:26:50,650
two parameters, the hard
sphere volume and the

444
00:26:50,650 --> 00:26:52,970
attraction.

445
00:26:52,970 --> 00:26:54,930
You don't have to go look
up in tables or books.

446
00:26:54,930 --> 00:26:57,830
You don't have to have all the
values of the second virial

447
00:26:57,830 --> 00:27:01,490
coefficient, or the fudge
factor, just two variables

448
00:27:01,490 --> 00:27:04,750
that make physical sense, and
you get an equation of state

449
00:27:04,750 --> 00:27:07,060
which is a reasonable equation
of state, and that's the power

450
00:27:07,060 --> 00:27:09,500
of the Van der Waals' equation
of state, and that's the one

451
00:27:09,500 --> 00:27:12,080
we're going to be using
later on this class to

452
00:27:12,080 --> 00:27:14,100
describe real gases.

453
00:27:14,100 --> 00:27:19,530
Question?

454
00:27:19,530 --> 00:27:22,740
OK, so we've done
the zeroth law.

455
00:27:22,740 --> 00:27:25,540
We've done temperature,
equations of state.

456
00:27:25,540 --> 00:27:27,040
We're ready for the first law.

457
00:27:27,040 --> 00:27:28,410
We're just going to go to
through these laws pretty

458
00:27:28,410 --> 00:27:31,560
quickly here.

459
00:27:31,560 --> 00:27:34,330
Remember, the first law
is the upbeat law.

460
00:27:34,330 --> 00:27:35,910
It's the one that says,
hey, you know,

461
00:27:35,910 --> 00:27:37,790
life is all rosy here.

462
00:27:37,790 --> 00:27:42,275
We can take energy from fossil
fuels and burn it up and make

463
00:27:42,275 --> 00:27:46,540
it heat, and change that
energy into work.

464
00:27:46,540 --> 00:27:49,460
And it's the same energy, and
we probably can do that with

465
00:27:49,460 --> 00:27:51,310
100% efficiency.

466
00:27:51,310 --> 00:27:54,560
We can take heat from the air
surrounding us and run our car

467
00:27:54,560 --> 00:27:56,350
on it with 100% efficiency.

468
00:27:56,350 --> 00:27:59,570
Is this possible?

469
00:27:59,570 --> 00:28:01,620
That's what the first law says,
it's possible; work is

470
00:28:01,620 --> 00:28:05,570
heat, and heat is work, and
they're the same thing.

471
00:28:05,570 --> 00:28:08,650
You can break even, maybe.

472
00:28:08,650 --> 00:28:10,760
So let's go back and
see what work is.

473
00:28:10,760 --> 00:28:17,640
Let's go back to our
freshman physics.

474
00:28:17,640 --> 00:28:24,220
Work, work is if you take a
force, and you push something

475
00:28:24,220 --> 00:28:27,770
a certain distance,
you do work on it.

476
00:28:27,770 --> 00:28:32,670
So if I take my chalk here and
I push on it, I'm doing work

477
00:28:32,670 --> 00:28:33,890
to push that chalk.

478
00:28:33,890 --> 00:28:37,980
Force times distance is work.

479
00:28:37,980 --> 00:28:40,080
The applied force times
the distance.

480
00:28:40,080 --> 00:28:42,370
There are many kinds of work.

481
00:28:42,370 --> 00:28:45,700
There's electrical work, take
the motor, you plug it into

482
00:28:45,700 --> 00:28:48,140
the wall, electricity makes
the fan go around, that's

483
00:28:48,140 --> 00:28:49,170
electrical work.

484
00:28:49,170 --> 00:28:50,830
There's magnetic work.

485
00:28:50,830 --> 00:28:55,920
There is work due to gravity.

486
00:28:55,920 --> 00:28:58,210
In this class here, we're going
to stick to one kind of

487
00:28:58,210 --> 00:29:03,690
work which is expansion work.

488
00:29:03,690 --> 00:29:06,360
So expansion work, for instance,
or compression work,

489
00:29:06,360 --> 00:29:11,870
is if you have a piston
with a gas in it.

490
00:29:11,870 --> 00:29:15,040
All right, you put a pressure
on this piston here, and you

491
00:29:15,040 --> 00:29:18,470
compress the gas down.

492
00:29:18,470 --> 00:29:20,470
This is compression work.

493
00:29:20,470 --> 00:29:26,150
Now the volume gets smaller.
p external here.

494
00:29:26,150 --> 00:29:30,180
Pressure, the piston goes
down by some volume l.

495
00:29:30,180 --> 00:29:35,740
The piston has a cross-sectional
area, a, and

496
00:29:35,740 --> 00:29:41,990
the force -- pressure is
force per volume area.

497
00:29:41,990 --> 00:29:48,210
So the force that you're pushing
down on here is the

498
00:29:48,210 --> 00:29:51,190
external pressure
times the area.

499
00:29:51,190 --> 00:29:55,120
Pressure is force
per volume area.

500
00:29:55,120 --> 00:29:57,870
That's the force you're
using to push down.

501
00:29:57,870 --> 00:30:03,880
Now the work that's it is
calculated when you push down

502
00:30:03,880 --> 00:30:07,560
with the pressure on this piston
here, that work is

503
00:30:07,560 --> 00:30:19,500
force times distance, f times
I. f is p external times a,

504
00:30:19,500 --> 00:30:20,820
times the distance l.

505
00:30:20,820 --> 00:30:27,130
So that's p external times
the change in the volume.

506
00:30:27,130 --> 00:30:32,230
The area times this distance is
a volume, and that is the

507
00:30:32,230 --> 00:30:34,120
change in volume from going
to the initial state

508
00:30:34,120 --> 00:30:37,240
to the final state.

509
00:30:37,240 --> 00:30:39,540
Now we need to have
a convention.

510
00:30:39,540 --> 00:30:40,650
We've got force.

511
00:30:40,650 --> 00:30:44,690
Work is force times distance,
it's p external times delta v,

512
00:30:44,690 --> 00:30:47,000
and I'm going to be stressing
a lot that this is the

513
00:30:47,000 --> 00:30:47,860
external pressure.

514
00:30:47,860 --> 00:30:50,710
This is the pressure that you're
applying against the

515
00:30:50,710 --> 00:30:54,310
piston, not the pressure
of the gas.

516
00:30:54,310 --> 00:30:58,270
It's the pressure the external
world is applying on this poor

517
00:30:58,270 --> 00:31:01,290
system here.

518
00:31:01,290 --> 00:31:02,950
OK, but we need a
convention here.

519
00:31:02,950 --> 00:31:06,180
The convention, and then
we need to stick to it.

520
00:31:06,180 --> 00:31:07,890
And this convention,
unfortunately, has changed

521
00:31:07,890 --> 00:31:08,700
over the ages.

522
00:31:08,700 --> 00:31:12,280
But we're going to pick one,
and we're going to stick to

523
00:31:12,280 --> 00:31:17,740
it, which is that if the
environment does work on the

524
00:31:17,740 --> 00:31:22,020
system, if we push down on this
thing and do work on it,

525
00:31:22,020 --> 00:31:32,790
to compress it, then we call
that work negative work.

526
00:31:32,790 --> 00:31:38,130
No, we call that work
positive work.

527
00:31:38,130 --> 00:31:40,230
All right, so that means we need
to put a negative sign

528
00:31:40,230 --> 00:31:48,610
right here, by convention.

529
00:31:48,610 --> 00:31:56,090
So if delta v is negative, in
this case delta v is negative,

530
00:31:56,090 --> 00:31:58,540
OK, delta v is negative,
pressure is a positive number,

531
00:31:58,540 --> 00:32:02,520
negative times negative
is positive, work is

532
00:32:02,520 --> 00:32:04,070
greater than zero.

533
00:32:04,070 --> 00:32:09,520
We're doing work on the
system, to the system.

534
00:32:09,520 --> 00:32:13,480
In this case here,
work is positive.

535
00:32:13,480 --> 00:32:16,920
If you have expansion on the
other side, if the system is

536
00:32:16,920 --> 00:32:18,830
expanding in the other
direction, if you're going

537
00:32:18,830 --> 00:32:27,890
this way, right, you're going to
do work to the environment.

538
00:32:27,890 --> 00:32:29,710
There might be a mass here.

539
00:32:29,710 --> 00:32:30,650
This could be a car.

540
00:32:30,650 --> 00:32:33,360
Pistons in the car, right,
so the piston goes up.

541
00:32:33,360 --> 00:32:34,980
That's going to drive
the wheels.

542
00:32:34,980 --> 00:32:36,210
The car is going
to go forward.

543
00:32:36,210 --> 00:32:38,780
You're doing work on
the environment.

544
00:32:38,780 --> 00:32:41,700
Delta v is going to
be negative. w

545
00:32:41,700 --> 00:32:43,320
is going to be negative.

546
00:32:43,320 --> 00:32:45,650
Sorry, I got it backwards
again.

547
00:32:45,650 --> 00:32:46,710
Delta v is positive
in this direction

548
00:32:46,710 --> 00:32:50,100
here, the work is negative.

549
00:32:50,100 --> 00:32:55,060
So work on the system
is positive.

550
00:32:55,060 --> 00:32:57,570
Work done by the system
is negative.

551
00:32:57,570 --> 00:33:00,850
Convention, OK, this negative
sign is just a pure

552
00:33:00,850 --> 00:33:01,760
convention.

553
00:33:01,760 --> 00:33:02,760
You just got to use
it all the time.

554
00:33:02,760 --> 00:33:07,150
If you use an old textbook,
written when I was taking

555
00:33:07,150 --> 00:33:10,430
thermodynamics, they have the
opposite convention, and it's

556
00:33:10,430 --> 00:33:11,410
very confusing.

557
00:33:11,410 --> 00:33:15,360
But now we've all agreed on this
convention, and work is

558
00:33:15,360 --> 00:33:20,450
going to be with the
negative sign here.

559
00:33:20,450 --> 00:33:25,910
OK, any questions?

560
00:33:25,910 --> 00:33:28,990
This is an example where the
external pressure here is kept

561
00:33:28,990 --> 00:33:31,610
fixed as the volume changes,
but it doesn't

562
00:33:31,610 --> 00:33:33,140
have to be kept fixed.

563
00:33:33,140 --> 00:33:36,060
I could change my external
pressure through the whole

564
00:33:36,060 --> 00:33:37,560
process, and that's the path.

565
00:33:37,560 --> 00:33:40,190
We talked about the path last
time being very important.

566
00:33:40,190 --> 00:33:41,670
Defining the path.

567
00:33:41,670 --> 00:33:46,850
So if I have a path where my
pressure is changing, then I

568
00:33:46,850 --> 00:33:49,920
can't go directly from
this large volume

569
00:33:49,920 --> 00:33:50,630
to this small volume.

570
00:33:50,630 --> 00:33:55,730
I have to go in little steps,
infinitely small steps.

571
00:33:55,730 --> 00:34:00,260
So, instead of writing work is
the negative of p external

572
00:34:00,260 --> 00:34:07,810
times delta v, I'm going to
write a differential. dw is

573
00:34:07,810 --> 00:34:14,040
minus p external dv, where this
depends on the path, it

574
00:34:14,040 --> 00:34:18,800
depends on path and is changing
as v and p change.

575
00:34:18,800 --> 00:34:22,620
Now I'm going to add a
little thing here.

576
00:34:22,620 --> 00:34:26,790
I'm going to put a little
bar right here.

577
00:34:26,790 --> 00:34:31,010
And the little bar here means
that this dw that I'm putting

578
00:34:31,010 --> 00:34:37,440
here is not an exact
differential.

579
00:34:37,440 --> 00:34:42,070
What do I mean by that?

580
00:34:42,070 --> 00:34:45,890
I mean that if I take the
integral of this to find out

581
00:34:45,890 --> 00:34:50,070
how much work I've done
on the system, I

582
00:34:50,070 --> 00:34:52,060
need to know the path.

583
00:34:52,060 --> 00:34:53,550
That's what this means here.

584
00:34:53,550 --> 00:34:58,450
It's not enough to know the
initial state and the final

585
00:34:58,450 --> 00:35:00,630
state to find what w is.

586
00:35:00,630 --> 00:35:04,880
You also need to know
how you got there.

587
00:35:04,880 --> 00:35:07,990
This is very different from the
functions of state, like

588
00:35:07,990 --> 00:35:10,360
pressure and temperature.

589
00:35:10,360 --> 00:35:11,690
There's a volume, there's
a temperature, than

590
00:35:11,690 --> 00:35:13,150
the pressure here.

591
00:35:13,150 --> 00:35:14,810
There's other volume,
temperature and pressure here,

592
00:35:14,810 --> 00:35:17,660
corresponding to this
system here.

593
00:35:17,660 --> 00:35:20,600
And this volume, temperature and
pressure doesn't care how

594
00:35:20,600 --> 00:35:21,890
you got there.

595
00:35:21,890 --> 00:35:23,540
It is what it is.

596
00:35:23,540 --> 00:35:26,720
It defines the state
of the system.

597
00:35:26,720 --> 00:35:29,090
The amount of work you've
put in to get here

598
00:35:29,090 --> 00:35:30,260
depends on the path.

599
00:35:30,260 --> 00:35:32,800
It's not a function of state.

600
00:35:32,800 --> 00:35:34,260
It's not an exact
differential.

601
00:35:34,260 --> 00:35:37,510
So the delta v here is an
exact differential,

602
00:35:37,510 --> 00:35:40,390
but this dw is not.

603
00:35:40,390 --> 00:35:42,910
That's going to be
really important.

604
00:35:42,910 --> 00:35:45,130
So if you want to find out how
much work you've done, you

605
00:35:45,130 --> 00:35:49,050
take the integral from the
initial state to the final

606
00:35:49,050 --> 00:36:01,350
state of dw minus from one to
two p external dv, and you've

607
00:36:01,350 --> 00:36:05,630
got to know what the path is.

608
00:36:05,630 --> 00:36:15,220
So let's look at this path
dependence briefly here.

609
00:36:15,220 --> 00:36:20,280
We're going to do two different
paths, and see how

610
00:36:20,280 --> 00:36:24,420
they're different in terms of
the work that comes out.

611
00:36:24,420 --> 00:36:28,220
So we're going to take an ideal
gas, we can assume that

612
00:36:28,220 --> 00:36:28,750
it's ideal.

613
00:36:28,750 --> 00:36:31,990
Let's take argon, for
instance, a nice,

614
00:36:31,990 --> 00:36:33,200
non-interacting gas.

615
00:36:33,200 --> 00:36:35,530
We're going to do
a compression.

616
00:36:35,530 --> 00:36:41,640
We're going to take argon, with
a certain gas, certain

617
00:36:41,640 --> 00:36:45,640
pressure p1, volume V1, and
we're going to a final state

618
00:36:45,640 --> 00:36:50,520
argon, gas, p2, V2.

619
00:36:50,520 --> 00:36:58,730
Where V1 is greater than V2,
and p1 is less than p2.

620
00:36:58,730 --> 00:37:07,650
So if I draw this on a p
v diagram, so there is

621
00:37:07,650 --> 00:37:09,090
volume on this axis.

622
00:37:09,090 --> 00:37:11,510
There's pressure on this axis.

623
00:37:11,510 --> 00:37:13,170
There is V1 here.

624
00:37:13,170 --> 00:37:15,160
There's V2 here.

625
00:37:15,160 --> 00:37:18,610
There's p1 here, and p2 here.

626
00:37:18,610 --> 00:37:20,410
So I'm starting at p1, V1.

627
00:37:20,410 --> 00:37:25,960
I'm starting right here.

628
00:37:25,960 --> 00:37:30,870
And I'm going to
end right here.

629
00:37:30,870 --> 00:37:34,610
Initial find -- there are many
ways I can get from one state

630
00:37:34,610 --> 00:37:35,860
to the other.

631
00:37:35,860 --> 00:37:40,660
Draw any sort of line
to go here, right?

632
00:37:40,660 --> 00:37:43,400
There are a couple obvious ones,
which we're going to --

633
00:37:43,400 --> 00:37:45,730
we can calculate, which
we're going to do.

634
00:37:45,730 --> 00:37:52,450
So, the first obvious one
is to take V1 to V2

635
00:37:52,450 --> 00:38:01,240
first with p constant.

636
00:38:01,240 --> 00:38:03,190
So take this path here.

637
00:38:03,190 --> 00:38:08,000
I take V1 to V2 first, keeping
the pressure constant at p1,

638
00:38:08,000 --> 00:38:12,440
then I take p1 to p2 keeping
the volume constant at V2.

639
00:38:12,440 --> 00:38:14,770
Let's call this path 1.

640
00:38:14,770 --> 00:38:21,690
Then you take p1 to p2
with V constant.

641
00:38:21,690 --> 00:38:31,440
An isobaric process followed by
a constant volume process.

642
00:38:31,440 --> 00:38:33,070
You could also do a
different path.

643
00:38:33,070 --> 00:38:42,580
You could do, let me draw p v,
there's my initial state.

644
00:38:42,580 --> 00:38:47,860
My final state here, I could
take, first, I could change

645
00:38:47,860 --> 00:38:51,920
the pressure, and then
change the volume.

646
00:38:51,920 --> 00:38:59,390
So the second process, if you
take p1 to p2, V constant, and

647
00:38:59,390 --> 00:39:04,140
then you take V1 to V2
with p constant.

648
00:39:04,140 --> 00:39:10,200
This is path number two.

649
00:39:10,200 --> 00:39:12,635
Both are perfectly fine paths,
and I'm going to assume that

650
00:39:12,635 --> 00:39:15,330
these paths are also
reversible.

651
00:39:15,330 --> 00:39:19,970
Let's assume that both are
reversible, meaning that I'm

652
00:39:19,970 --> 00:39:25,230
doing this pretty slowly, so
as I change, let's say I'm

653
00:39:25,230 --> 00:39:30,250
changing my volumes here, V1
to V2, it's happening, I'm

654
00:39:30,250 --> 00:39:33,070
compressing it slowly, slowly,
slowly so that at any point I

655
00:39:33,070 --> 00:39:39,340
could reverse the process
without losing energy, right?

656
00:39:39,340 --> 00:39:47,070
It's always an equilibrium.

657
00:39:47,070 --> 00:39:51,770
All right, let's calculate the
work that's involved with

658
00:39:51,770 --> 00:39:55,100
these two processes.

659
00:39:55,100 --> 00:39:58,870
Remember it's the external
pressure that's important.

660
00:39:58,870 --> 00:40:02,420
In this case, because it's
a reversible process, the

661
00:40:02,420 --> 00:40:06,050
external pressure turns out to
be always the same as the

662
00:40:06,050 --> 00:40:07,230
internal pressure.

663
00:40:07,230 --> 00:40:13,860
It's reversible, that means
that p external, equals p.

664
00:40:13,860 --> 00:40:16,330
I'm doing it very slowly so that
I'm always in equilibrium

665
00:40:16,330 --> 00:40:18,740
between the external pressure
and the internal pressure so I

666
00:40:18,740 --> 00:40:23,240
can go back and forth.

667
00:40:23,240 --> 00:40:26,110
So, let's calculate w1.

668
00:40:26,110 --> 00:40:28,180
The work for path one.

669
00:40:28,180 --> 00:40:31,360
First thing is I change the
volume from V1 to V2 The

670
00:40:31,360 --> 00:40:34,430
external pressure is kept
constant, p1, so it's minus

671
00:40:34,430 --> 00:40:39,750
the integral from 1,
V1 to V2, p1, dv.

672
00:40:39,750 --> 00:40:49,490
And then the next step here is
I'm going from -- the pressure

673
00:40:49,490 --> 00:40:50,270
is changing.

674
00:40:50,270 --> 00:40:58,370
I'm going from V2 to V2 dv
-- what do you think this

675
00:40:58,370 --> 00:40:58,870
integral is?

676
00:40:58,870 --> 00:41:03,460
Right, so this is easy
part, zero here.

677
00:41:03,460 --> 00:41:04,440
This one is also pretty easy.

678
00:41:04,440 --> 00:41:12,470
That's minus p1 times V2 minus
V1. p1 times V2 minus V1.

679
00:41:12,470 --> 00:41:18,650
What that turns out to be,
this area right here.

680
00:41:18,650 --> 00:41:19,890
It's V1 minus V2 times p1.

681
00:41:19,890 --> 00:41:23,120
This is w1 here.

682
00:41:23,120 --> 00:41:34,170
OK, I can re-write this as p1
time V1 minus V2 and get rid

683
00:41:34,170 --> 00:41:36,640
of this negative sign here.

684
00:41:36,640 --> 00:41:44,230
Now V1 is bigger than V2,
so this is positive.

685
00:41:44,230 --> 00:41:50,060
So I am compressing, I'm doing
work to the system, positive

686
00:41:50,060 --> 00:41:54,100
work everything follows
our convention.

687
00:41:54,100 --> 00:42:01,430
Number two here, OK, the first
thing I do is I change the

688
00:42:01,430 --> 00:42:06,600
pressure under constant volume,
V1, V1 minus p dv, and

689
00:42:06,600 --> 00:42:11,870
then I change the volume
from V1 to V2 and

690
00:42:11,870 --> 00:42:14,460
then this is p2, dv.

691
00:42:14,460 --> 00:42:19,050
This first integral is zero V1
to V1, then I get minus p2

692
00:42:19,050 --> 00:42:25,000
times V2 minus V1 or p2
times V1 minus V2.

693
00:42:25,000 --> 00:42:27,090
Again, a positive number.

694
00:42:27,090 --> 00:42:29,760
I'm doing work to the system to
go from the initial state

695
00:42:29,760 --> 00:42:33,680
to the final state.

696
00:42:33,680 --> 00:42:37,480
But it's not the same as w1.

697
00:42:37,480 --> 00:42:41,520
In this case, I have p1 times
delta V. In this case here, I

698
00:42:41,520 --> 00:42:51,400
have p2 times delta V. And p2
is bigger than p1. w2 is

699
00:42:51,400 --> 00:42:56,710
bigger than w1.

700
00:42:56,710 --> 00:43:00,450
The amount of work that you're
doing on the system depends on

701
00:43:00,450 --> 00:43:04,720
the path that you take.

702
00:43:04,720 --> 00:43:08,370
All right, how do
I, practically

703
00:43:08,370 --> 00:43:10,600
speaking, how do I do this?

704
00:43:10,600 --> 00:43:12,960
Anybody have an idea?

705
00:43:12,960 --> 00:43:23,090
How do I keep p1 constant while
I'm lowering the volume?

706
00:43:23,090 --> 00:43:24,000
STUDENT: Change the
temperature?

707
00:43:24,000 --> 00:43:24,420
PROFESSOR: Change the
temperature, right.

708
00:43:24,420 --> 00:43:31,960
So what I'm doing here is I'm
cooling, and then when I'm

709
00:43:31,960 --> 00:43:34,870
sitting at a fixed volume and
I'm increasing the pressure,

710
00:43:34,870 --> 00:43:37,890
what am I doing?

711
00:43:37,890 --> 00:43:38,560
I'm heating, right?

712
00:43:38,560 --> 00:43:40,120
So I'm doing cooling
and heating cycles.

713
00:43:40,120 --> 00:43:45,750
So in this case here, I
cool and then I heat.

714
00:43:45,750 --> 00:43:47,710
In this case here, I heat
and then I cool.

715
00:43:47,710 --> 00:43:52,730
All right, so I'm burning some
energy, I'm burning some fuel

716
00:43:52,730 --> 00:43:59,300
to do this somehow, to get
that work to happen.

717
00:43:59,300 --> 00:44:05,060
All right, now suppose that I
took these two paths, and

718
00:44:05,060 --> 00:44:11,050
coupled them together.

719
00:44:11,050 --> 00:44:12,880
So in this case, it's the amount
of work is the area

720
00:44:12,880 --> 00:44:14,630
under that curve.

721
00:44:14,630 --> 00:44:17,370
And in this case here, the
amount of work is bigger, w2

722
00:44:17,370 --> 00:44:21,010
is bigger, and it's the
area under this curve.

723
00:44:21,010 --> 00:44:26,670
Now, suppose I took this two
paths, and I took -- couple

724
00:44:26,670 --> 00:44:28,790
them together with one the
reverse of the other.

725
00:44:28,790 --> 00:44:32,970
So I have my initial state, my
final state, my initial state,

726
00:44:32,970 --> 00:44:35,250
my final state here.

727
00:44:35,250 --> 00:44:40,310
And I start by taking
my first path here.

728
00:44:40,310 --> 00:44:44,770
I cool, I heat.

729
00:44:44,770 --> 00:44:46,190
So there's w1.

730
00:44:46,190 --> 00:44:52,580
So the w total that I'm going
to get, is w1, and then

731
00:44:52,580 --> 00:44:56,920
instead of the path from V1
to, from 1 to 2 going like

732
00:44:56,920 --> 00:45:08,150
this as we had before, I'm going
to take it backwards.

733
00:45:08,150 --> 00:45:10,260
If I go backwards, to work --
everything is symmetric, the

734
00:45:10,260 --> 00:45:13,580
work becomes the negative from
what I had calculated before,

735
00:45:13,580 --> 00:45:21,200
so this becomes minus what I
calculated before for w2.

736
00:45:21,200 --> 00:45:29,070
The total work, in this case
here, is p1 times V1 minus p2

737
00:45:29,070 --> 00:45:38,460
times V1 minus V2, it's p1 minus
p2 times V1 minus V2.

738
00:45:38,460 --> 00:45:41,710
This is a positive number,
p1 is smaller than p2.

739
00:45:41,710 --> 00:45:43,280
This is a negative number.

740
00:45:43,280 --> 00:45:48,240
The total work is
less than zero.

741
00:45:48,240 --> 00:45:50,890
That's the work that the
system is doing to the

742
00:45:50,890 --> 00:45:53,210
environment.

743
00:45:53,210 --> 00:45:54,550
I'm doing work to
the environment.

744
00:45:54,550 --> 00:45:56,830
The work is negative, which
means that work is being done

745
00:45:56,830 --> 00:45:57,780
to the environment.

746
00:45:57,780 --> 00:46:07,450
And that work is the area
inside the rectangle.

747
00:46:07,450 --> 00:46:10,720
What you've built
is an engine.

748
00:46:10,720 --> 00:46:17,700
You cool, you heat, you heat,
you cool, you get back to the

749
00:46:17,700 --> 00:46:20,760
same place, but you've just done
work to the environment.

750
00:46:20,760 --> 00:46:23,520
You've just built
a heat engine.

751
00:46:23,520 --> 00:46:28,770
You take fuel, rather you take
something that's warm, and you

752
00:46:28,770 --> 00:46:31,380
put it in contact with the
atmosphere, it cools down.

753
00:46:31,380 --> 00:46:34,040
You take your fuel, you
heat it up again.

754
00:46:34,040 --> 00:46:35,390
It expands.

755
00:46:35,390 --> 00:46:39,060
You change your constraints on
your system, you heat it up

756
00:46:39,060 --> 00:46:43,520
some more, then you take the
heat source away, and you put

757
00:46:43,520 --> 00:46:46,250
it back in contact with
the atmosphere.

758
00:46:46,250 --> 00:46:48,726
And you cool it a little bit,
change the constraints, cool

759
00:46:48,726 --> 00:46:51,050
it a little bit more, and heat,
and you've got a closed

760
00:46:51,050 --> 00:46:53,700
cycle engine.

761
00:46:53,700 --> 00:46:54,860
We're going to work
with some more

762
00:46:54,860 --> 00:46:57,520
complicated engines before.

763
00:46:57,520 --> 00:47:00,350
But the important part here is
that the work is not zero.

764
00:47:00,350 --> 00:47:01,440
You're starting at one point.

765
00:47:01,440 --> 00:47:04,800
You're going around a cycle and
you're going back to the

766
00:47:04,800 --> 00:47:05,420
same point.

767
00:47:05,420 --> 00:47:07,410
The pressure, temperature, and
volume are exactly the same

768
00:47:07,410 --> 00:47:08,710
here as when you started out.

769
00:47:08,710 --> 00:47:10,080
But the w is not zero.

770
00:47:10,080 --> 00:47:15,280
The w, for the closed path, and
when I put a circle there

771
00:47:15,280 --> 00:47:18,280
on my integral that means a
closed path, when you start

772
00:47:18,280 --> 00:47:26,600
and end at the same point,
right, this is not zero.

773
00:47:26,600 --> 00:47:29,320
If you had an exact
differential, the exact

774
00:47:29,320 --> 00:47:31,710
differential around a closed
path, you would get zero.

775
00:47:31,710 --> 00:47:36,800
It wouldn't care where
the path is.

776
00:47:36,800 --> 00:47:37,960
Here this cares where
the path is.

777
00:47:37,960 --> 00:47:44,030
So, work is not a function
of state.

778
00:47:44,030 --> 00:47:53,620
Any questions on work before we
move on to heat, briefly?

779
00:47:53,620 --> 00:48:05,930
So heat is a quantity that
flows into a substance,

780
00:48:05,930 --> 00:48:08,570
something that flows into a
substance that changes it's

781
00:48:08,570 --> 00:48:13,880
temperature, very
broadly defined.

782
00:48:13,880 --> 00:48:16,250
And, again, we have a sign
convention for heat.

783
00:48:16,250 --> 00:48:20,960
So heat, we're going
to call that q.

784
00:48:20,960 --> 00:48:23,710
And our sign convention is
that if we change our

785
00:48:23,710 --> 00:48:31,930
temperature from T1 to T2, where
T2 it's greater than T1

786
00:48:31,930 --> 00:48:36,990
then heat is going
to be positive.

787
00:48:36,990 --> 00:48:39,900
Heat needs to go into the
system to change the

788
00:48:39,900 --> 00:48:43,100
temperature and make it go up.

789
00:48:43,100 --> 00:48:45,970
If the temperature of the system
goes down, heat flows

790
00:48:45,970 --> 00:48:48,200
down heat flows out of the
system, and we call that

791
00:48:48,200 --> 00:48:50,550
negative q.

792
00:48:50,550 --> 00:48:54,300
Same convention is
for w, basically.

793
00:48:54,300 --> 00:48:57,000
Now, you can have a change of
temperature without any heat

794
00:48:57,000 --> 00:48:58,470
being involved.

795
00:48:58,470 --> 00:49:05,010
I can take an insulated box,
and I can have a chemical

796
00:49:05,010 --> 00:49:08,540
reaction in that
insulated box.

797
00:49:08,540 --> 00:49:10,170
I can take a heat pack,
like the kind

798
00:49:10,170 --> 00:49:11,850
you buy at a pharmacy.

799
00:49:11,850 --> 00:49:15,110
Break it up.

800
00:49:15,110 --> 00:49:17,570
It gets hot.

801
00:49:17,570 --> 00:49:20,470
There's no heat flowing from the
environment to the system.

802
00:49:20,470 --> 00:49:22,510
I have to define my terms.

803
00:49:22,510 --> 00:49:24,860
My system is whatever's
inside the box.

804
00:49:24,860 --> 00:49:26,380
It's insulated.

805
00:49:26,380 --> 00:49:28,220
It's a closed system.

806
00:49:28,220 --> 00:49:30,090
In fact, it's an isolated
system.

807
00:49:30,090 --> 00:49:32,700
There's no energy or
matter that can go

808
00:49:32,700 --> 00:49:33,990
through that boundary.

809
00:49:33,990 --> 00:49:37,180
Yet, the temperature goes up.

810
00:49:37,180 --> 00:49:44,160
So, I can have a temperature
change which is an adiabatic

811
00:49:44,160 --> 00:49:45,040
temperature change.

812
00:49:45,040 --> 00:49:52,460
Adiabatic means without heat.

813
00:49:52,460 --> 00:49:55,170
Or I could have a non-adiabatic,
I could take

814
00:49:55,170 --> 00:49:59,420
the same temperature change, by
taking a flame, or a heat

815
00:49:59,420 --> 00:50:04,110
source and heating
up my substance.

816
00:50:04,110 --> 00:50:09,980
So, clearly q is going to
depend on the path.

817
00:50:09,980 --> 00:50:13,640
I'm going from T1 to T2, and
I have two ways to go here.

818
00:50:13,640 --> 00:50:15,030
One is non-adiabatic.

819
00:50:15,030 --> 00:50:21,630
One is adiabatic.

820
00:50:21,630 --> 00:50:24,450
All right, now what we're going
to learn next time, and

821
00:50:24,450 --> 00:50:28,870
Bob Field is going to teach the
lecture next time, is how

822
00:50:28,870 --> 00:50:31,680
heat and work are related, and
how they're really the same

823
00:50:31,680 --> 00:50:36,540
thing, and how they're related
through the first law, through

824
00:50:36,540 --> 00:50:38,560
energy conservation.

825
00:50:38,560 --> 00:50:42,430
OK, I'll see you on
Wednesday then.