1
00:00:00,030 --> 00:00:02,400
The following content is
provided under a creative

2
00:00:02,400 --> 00:00:03,810
commons license.

3
00:00:03,810 --> 00:00:06,860
Your support will help MIT
OpenCourseWare continue to

4
00:00:06,860 --> 00:00:10,510
offer high-quality educational
resources for free.

5
00:00:10,510 --> 00:00:13,390
To make a donation or view
additional materials from

6
00:00:13,390 --> 00:00:16,320
hundreds of MIT courses, visit
MIT OpenCourseWare at

7
00:00:16,320 --> 00:00:21,210
ocw.mit.edu.

8
00:00:21,210 --> 00:00:25,430
PROFESSOR NELSON: All right,
well last time we finished our

9
00:00:25,430 --> 00:00:28,460
sort of introduction to entropy
which is a difficult

10
00:00:28,460 --> 00:00:32,020
topic, but I hope we got to the
point where we have some

11
00:00:32,020 --> 00:00:34,350
working understanding of
physically what it's

12
00:00:34,350 --> 00:00:36,470
representing for us, and
also an ability to

13
00:00:36,470 --> 00:00:37,310
just calculate it.

14
00:00:37,310 --> 00:00:39,780
So we went through at the end
of the last lecture, a few

15
00:00:39,780 --> 00:00:42,890
examples where we just
calculated changes in entropy

16
00:00:42,890 --> 00:00:45,470
for simple processes like
heating and cooling something

17
00:00:45,470 --> 00:00:49,140
or going through a phase
transition where that process,

18
00:00:49,140 --> 00:00:51,840
that sort of process is
relatively simple because

19
00:00:51,840 --> 00:00:53,200
there's no temperature change.

20
00:00:53,200 --> 00:00:56,240
While the ice is melting, for
example, you're putting heat

21
00:00:56,240 --> 00:00:58,530
into it, but the temperature
is staying

22
00:00:58,530 --> 00:01:01,530
at zero degree Celsius.

23
00:01:01,530 --> 00:01:04,880
And we saw, how to do these
calculations, we need define

24
00:01:04,880 --> 00:01:05,960
reversible paths.

25
00:01:05,960 --> 00:01:09,640
So it was extremely
straightforward to calculate

26
00:01:09,640 --> 00:01:12,920
the entropy of say ice melting
at zero degrees Celsius,

27
00:01:12,920 --> 00:01:16,350
because there the process is
reversible, because that's the

28
00:01:16,350 --> 00:01:18,180
melting temperature.

29
00:01:18,180 --> 00:01:22,700
But if we wanted to calculate
the change in entropy of ice

30
00:01:22,700 --> 00:01:27,520
melting, you know at, once it
had already been cooled to ten

31
00:01:27,520 --> 00:01:31,330
degrees above the melting point,
to ten degrees Celsius,

32
00:01:31,330 --> 00:01:33,420
then in order to find a
reversible path, we had to

33
00:01:33,420 --> 00:01:37,650
say, OK, let's first cool it
down to zero degrees Celsius.

34
00:01:37,650 --> 00:01:40,580
Then let's have it melt, and
then let's warm the liquid

35
00:01:40,580 --> 00:01:43,330
back up to ten degrees Celsius
so we could construct the

36
00:01:43,330 --> 00:01:46,060
sequence of reversible steps
that would get from the same

37
00:01:46,060 --> 00:01:48,590
starting point to the same end
point, and we could calculate

38
00:01:48,590 --> 00:01:52,770
the change in entropy through
that sort of sequence.

39
00:01:52,770 --> 00:01:56,630
So now that we've got at least
some experience doing

40
00:01:56,630 --> 00:01:59,470
calculations of delta S and
we're just thinking a little

41
00:01:59,470 --> 00:02:03,430
bit about entropy, what I'd like
to do is to try to relate

42
00:02:03,430 --> 00:02:06,230
the state variables together
in a useful way.

43
00:02:06,230 --> 00:02:10,920
And and the immediate problem
that I'd like to address is

44
00:02:10,920 --> 00:02:14,930
the fact that right now we
have kind of a cumbersome

45
00:02:14,930 --> 00:02:18,180
expression for energy.

46
00:02:18,180 --> 00:02:29,890
So you know we have u, we look
at du, right it's dq plus dw,

47
00:02:29,890 --> 00:02:34,330
and you know, I don't
like those.

48
00:02:34,330 --> 00:02:38,980
They're path specific, and it
would be nice to be able to do

49
00:02:38,980 --> 00:02:43,390
a calculation of changes
in energy that

50
00:02:43,390 --> 00:02:44,630
didn't depend on path.

51
00:02:44,630 --> 00:02:47,650
You know it's a state
function.

52
00:02:47,650 --> 00:02:51,590
So in principle it seems like it
sure ought to be possible,

53
00:02:51,590 --> 00:02:54,490
and yet so far when we've
actually gone through

54
00:02:54,490 --> 00:02:57,980
calculations of du, we've had
to go and consider the path

55
00:02:57,980 --> 00:03:01,020
and get the heat and get
the work and so forth.

56
00:03:01,020 --> 00:03:04,470
And then we found special cases,
you know an ideal gas

57
00:03:04,470 --> 00:03:07,460
where the temperature, where the
change is only a function

58
00:03:07,460 --> 00:03:09,860
of temperature and so forth,
where we could write this as a

59
00:03:09,860 --> 00:03:12,260
function of state variables,
but nothing general that

60
00:03:12,260 --> 00:03:14,270
really allows us to do the
calculation under all

61
00:03:14,270 --> 00:03:17,360
circumstances.

62
00:03:17,360 --> 00:03:21,130
So let's think about how
to make this better.

63
00:03:21,130 --> 00:03:24,840
So, and what I mean by that is
you've seen examples like, you

64
00:03:24,840 --> 00:03:32,750
know, some special examples
you saw awhile back.

65
00:03:32,750 --> 00:03:39,560
The case where du was Cv dT
minus Cv and this Joule

66
00:03:39,560 --> 00:03:44,350
coefficient d v. But you know
you still need to find those

67
00:03:44,350 --> 00:03:46,600
coefficients for each system.

68
00:03:46,600 --> 00:03:50,980
This isn't a general equation
that tells us how energy

69
00:03:50,980 --> 00:03:54,960
changes in terms of only
functions of state.

70
00:03:54,960 --> 00:03:57,600
Because of things like this and
this -- what I'd really

71
00:03:57,600 --> 00:04:11,800
like is to be able to you know
write du equals something.

72
00:04:11,800 --> 00:04:16,440
And that something, you know,
it can have T and p and

73
00:04:16,440 --> 00:04:17,700
whatever else I need.

74
00:04:17,700 --> 00:04:23,160
It can have S, H, and of course
differentials of any of

75
00:04:23,160 --> 00:04:30,550
those quantities. dT or dp or
dS, dH, you name it, but all

76
00:04:30,550 --> 00:04:32,400
state variables.

77
00:04:32,400 --> 00:04:35,760
That's what I'd much
rather have.

78
00:04:35,760 --> 00:04:39,260
And then, you know, if I want
to, if I've got something like

79
00:04:39,260 --> 00:04:42,230
that and I want to find out how
the energy changes as a

80
00:04:42,230 --> 00:04:45,970
function of volume, so I'll
calculate du/dV With respect

81
00:04:45,970 --> 00:04:51,420
to some selected variable hold
constant, I can do it.

82
00:04:51,420 --> 00:04:55,530
Right now, in a general sense,
that's cumbersome.

83
00:04:55,530 --> 00:04:58,450
I've got to figure out how to
do that for each particular

84
00:04:58,450 --> 00:05:01,740
case that I want to treat.

85
00:05:01,740 --> 00:05:13,520
So, let's see how we could
construct such a thing.

86
00:05:13,520 --> 00:05:16,580
Let's consider just a reversible
process, at

87
00:05:16,580 --> 00:05:32,120
constant pressure.

88
00:05:32,120 --> 00:05:43,400
So, OK, I've got one and I'm
going to in some path wind up

89
00:05:43,400 --> 00:05:51,100
at state two and I'll write du
is dq, it's reversible in this

90
00:05:51,100 --> 00:05:54,810
case, minus p dV.

91
00:05:58,600 --> 00:06:08,410
And from the second law, we
know that we can write dq

92
00:06:08,410 --> 00:06:12,520
reversible as T dS.

93
00:06:12,520 --> 00:06:18,360
dq over T is dS or entropy.

94
00:06:18,360 --> 00:06:28,150
So, we can write du is
T dS minus p dV.

95
00:06:35,280 --> 00:06:41,660
That's so important, we'll
circle it with colored chalk.

96
00:06:41,660 --> 00:06:43,850
That's how important it is.

97
00:06:43,850 --> 00:06:48,000
You know it's a dramatic
moment.

98
00:06:48,000 --> 00:06:49,640
So let's look at what
we have here.

99
00:06:49,640 --> 00:06:55,930
Here's du and over on this side
we have T, we have S, we

100
00:06:55,930 --> 00:07:00,770
have p and we have V. Suddenly
and simply, it's only

101
00:07:00,770 --> 00:07:02,550
functions of state.

102
00:07:02,550 --> 00:07:05,940
Well that was pretty easy.

103
00:07:05,940 --> 00:07:12,680
So, what that's telling us is
that we can write u this way,

104
00:07:12,680 --> 00:07:15,200
and you know, this is
generally true.

105
00:07:15,200 --> 00:07:18,390
We got to this by considering
a reversible

106
00:07:18,390 --> 00:07:21,670
constant pressure process.

107
00:07:21,670 --> 00:07:26,020
But we know u is a state
variable right.

108
00:07:26,020 --> 00:07:29,130
So this result is going to
be generally applicable.

109
00:07:29,130 --> 00:07:31,090
And it tells us a couple
of things too.

110
00:07:31,090 --> 00:07:36,190
It tells us that in some sense,
the natural variables

111
00:07:36,190 --> 00:07:50,980
for u are these, right, it's a
function of S and V. Those are

112
00:07:50,980 --> 00:07:53,840
natural variables in the sense
that then it written as

113
00:07:53,840 --> 00:08:02,430
functions of those variables, we
only have state quantities

114
00:08:02,430 --> 00:08:06,000
on the right-hand side.

115
00:08:06,000 --> 00:08:09,030
Very, very valuable
expression.

116
00:08:09,030 --> 00:08:12,230
And of course coming out of
that then, we can take

117
00:08:12,230 --> 00:08:17,290
derivatives and at least for
those particular variables, we

118
00:08:17,290 --> 00:08:26,430
can see that du/dS at constant
V is minus p.

119
00:08:26,430 --> 00:08:42,300
And du/dV at constant
S is T. All right,

120
00:08:42,300 --> 00:08:43,930
those fall right out.

121
00:08:43,930 --> 00:08:49,770
Now we can have a similar set
of steps for H, for the

122
00:08:49,770 --> 00:09:10,090
enthalpy, so let's just
look at that.

123
00:09:10,090 --> 00:09:21,150
So H, of course, it's u plus
pV, so dH is just du plus

124
00:09:21,150 --> 00:09:26,220
d(pV), and now there's our
expression for du.

125
00:09:26,220 --> 00:09:29,080
We're going to use
it this way.

126
00:09:29,080 --> 00:09:39,380
So it's T dS minus p dV
plus p dV plus V dp.

127
00:09:39,380 --> 00:09:43,370
And of course these are
going to cancel.

128
00:09:43,370 --> 00:09:51,420
So we can write dH is
T dS plus V dp.

129
00:09:55,210 --> 00:09:57,520
Also important enough
that we'll

130
00:09:57,520 --> 00:10:00,760
highlight it a little bit.

131
00:10:00,760 --> 00:10:03,390
So again let's look at what
we've got on the right-hand

132
00:10:03,390 --> 00:10:08,700
side here, T, S, V, p, only
quantities that are

133
00:10:08,700 --> 00:10:13,220
functions of state.

134
00:10:13,220 --> 00:10:17,290
And of course, we can take the
corresponding derivative, so

135
00:10:17,290 --> 00:10:21,180
let's also be explicit here,
that means that were writing H

136
00:10:21,180 --> 00:10:28,110
as a function of the
variables S and p.

137
00:10:28,110 --> 00:10:53,400
And dH/dS at constant p is T,
and dH/dp at constant S is V.

138
00:10:53,400 --> 00:10:58,900
So now we've got a couple of
really surprisingly simple

139
00:10:58,900 --> 00:11:03,600
expressions that we can use to
describe u and H in terms of

140
00:11:03,600 --> 00:11:05,880
only state variables.

141
00:11:05,880 --> 00:11:10,080
All right?

142
00:11:10,080 --> 00:11:14,320
We also can go from these
expressions, using the chain

143
00:11:14,320 --> 00:11:17,540
rule, to expressions for
particularly useful

144
00:11:17,540 --> 00:11:23,260
expressions for the entropy as
a function of temperature.

145
00:11:23,260 --> 00:11:32,250
So, you know, from du
is T dS minus p dV.

146
00:11:32,250 --> 00:11:38,940
We can rewrite this as
dS is one over T du

147
00:11:38,940 --> 00:11:44,870
plus p over T dV.

148
00:11:48,320 --> 00:11:52,430
And now we can go back, you
know, if we can go back to our

149
00:11:52,430 --> 00:11:58,410
writing of u in terms of, as a
function of T and V, right.

150
00:11:58,410 --> 00:12:09,320
So we can write here du as a
function of T and V, Cv dT

151
00:12:09,320 --> 00:12:17,650
plus du/dV at constant T dV.

152
00:12:17,650 --> 00:12:22,440
The reason we're doing that is
now we can rewrite dS is one

153
00:12:22,440 --> 00:12:28,500
over T times Cv dT, and that's
the only temperature

154
00:12:28,500 --> 00:12:29,990
dependence we're
going to have.

155
00:12:29,990 --> 00:12:33,760
The other part is going to
be a function of volume.

156
00:12:33,760 --> 00:12:45,250
So it's, we've got p over T plus
du/dV at constant T dV,

157
00:12:45,250 --> 00:12:54,170
and what this says then is that
dS/dT at constant V is

158
00:12:54,170 --> 00:13:03,770
just Cv over T. Very useful, not
surprising because of the

159
00:13:03,770 --> 00:13:08,280
relation between heat at
constant volume and Cv, right.

160
00:13:08,280 --> 00:13:15,730
And of course dS is just dq
reversible over T, but this is

161
00:13:15,730 --> 00:13:19,600
telling us, in general, how
the entropy changes with

162
00:13:19,600 --> 00:13:23,640
temperature at constant
volume.

163
00:13:23,640 --> 00:13:28,610
We can go through the exact same
procedure with the H to

164
00:13:28,610 --> 00:13:32,110
look at how entropy varies with
temperature at constant

165
00:13:32,110 --> 00:13:36,480
pressure, and we'll get exactly
an analogous set of

166
00:13:36,480 --> 00:13:39,650
steps that will be
Cp over T, right.

167
00:13:39,650 --> 00:14:00,650
So also dS/dT at constant
pressure is Cp over T. OK?

168
00:14:00,650 --> 00:14:05,200
Now I want to carry our
discussion a little bit

169
00:14:05,200 --> 00:14:10,020
further and look at entropy a
little more carefully and in

170
00:14:10,020 --> 00:14:15,380
particular, how it varies
with temperature.

171
00:14:15,380 --> 00:14:18,480
And here's what I really
want to look at.

172
00:14:18,480 --> 00:14:21,470
You know, we've talked about
when we look at changes in u

173
00:14:21,470 --> 00:14:24,770
and changes in H, and we've
done this under lots of

174
00:14:24,770 --> 00:14:26,360
circumstances at this point.

175
00:14:26,360 --> 00:14:29,300
And at various times I've
emphasized, and I'm sure

176
00:14:29,300 --> 00:14:33,000
Professor Bawendi did too, that
when we look at these

177
00:14:33,000 --> 00:14:36,350
quantities we can only define
changes in them.

178
00:14:36,350 --> 00:14:41,580
There's not an absolute scales
for energy or for enthalpy.

179
00:14:41,580 --> 00:14:46,640
We can set the zero in
a particular problem,

180
00:14:46,640 --> 00:14:49,350
arbitrarily.

181
00:14:49,350 --> 00:14:55,050
And so, for example, when we
talked about thermochemistry,

182
00:14:55,050 --> 00:15:00,490
we defined heats of formation,
and then we said, well, the

183
00:15:00,490 --> 00:15:05,310
heat of formation of an element
in its natural state

184
00:15:05,310 --> 00:15:08,390
at room temperature and pressure
we'll call zero.

185
00:15:08,390 --> 00:15:09,790
We called it zero.

186
00:15:09,790 --> 00:15:13,950
If we wanted to put some number
on it, and put energy

187
00:15:13,950 --> 00:15:15,390
in it, we could have
done that.

188
00:15:15,390 --> 00:15:18,300
We defined the zero.

189
00:15:18,300 --> 00:15:22,230
So far, that's, well not just so
far, that's always the way

190
00:15:22,230 --> 00:15:25,970
it will be for quantities like
energy and enthalpy.

191
00:15:25,970 --> 00:15:30,130
Entropy is different.

192
00:15:30,130 --> 00:15:33,520
So let's just see
how that works.

193
00:15:33,520 --> 00:15:52,230
So, let's consider the entropy,
we'll consider as a

194
00:15:52,230 --> 00:16:00,780
function of temperature
and pressure.

195
00:16:00,780 --> 00:16:03,360
First let's just see how it
varies with pressure.

196
00:16:03,360 --> 00:16:05,220
We're going to see -- what
we'll do is consider its

197
00:16:05,220 --> 00:16:08,110
variation to both pressure and
temperature, and the objective

198
00:16:08,110 --> 00:16:10,830
is to say all right, if I've
got some substance at any

199
00:16:10,830 --> 00:16:15,020
arbitrary temperature and
pressure, can I define and

200
00:16:15,020 --> 00:16:19,020
calculate an absolute number
for the entropy?

201
00:16:19,020 --> 00:16:22,190
Not just a change in entropy,
unlike the cases with delta u

202
00:16:22,190 --> 00:16:26,470
and delta H, but an absolute
number that says in absolute

203
00:16:26,470 --> 00:16:29,480
terms the entropy of this
substance at room temperature

204
00:16:29,480 --> 00:16:31,340
and pressure or whatever
temperature and pressure is

205
00:16:31,340 --> 00:16:33,140
this amount.

206
00:16:33,140 --> 00:16:39,160
Something that I can do by
choice of a zero for energy or

207
00:16:39,160 --> 00:16:41,680
for u or H, but here,
I want to look

208
00:16:41,680 --> 00:16:43,920
for an absolute answer.

209
00:16:43,920 --> 00:16:46,510
All right, so let's start by
looking at the pressure

210
00:16:46,510 --> 00:16:48,620
dependence.

211
00:16:48,620 --> 00:16:59,450
So we're going to start with du
is T dS minus p dV, so dS

212
00:16:59,450 --> 00:17:07,130
is du plus p dV over T.

213
00:17:07,130 --> 00:17:14,230
Now let's look, T
being constant.

214
00:17:14,230 --> 00:17:16,910
OK, and now let's specify
a little bit.

215
00:17:16,910 --> 00:17:20,610
I want to make it something
as tractable as possible.

216
00:17:20,610 --> 00:17:27,760
Let's go to an ideal gas.

217
00:17:27,760 --> 00:17:31,660
So then at constant temperature,
that says du is

218
00:17:31,660 --> 00:17:34,650
equal to zero.

219
00:17:34,650 --> 00:17:41,680
So dS at constant temperature
is just p over T dV.

220
00:17:45,450 --> 00:17:57,750
And in the case of an ideal
gas, that's nR dV over V.

221
00:17:57,750 --> 00:18:08,110
And at constant temperature,
that means that d(nRT), which

222
00:18:08,110 --> 00:18:19,470
is the same as d(pV) is equal
to zero, but this

223
00:18:19,470 --> 00:18:23,790
is p dV plus V dp.

224
00:18:27,190 --> 00:18:34,210
So this says that dV over V,
that I've got there, is the

225
00:18:34,210 --> 00:18:39,550
same thing as negative
dp over p.

226
00:18:39,550 --> 00:18:59,900
Right, so I can write that dS
as constant temperature is

227
00:18:59,900 --> 00:19:06,980
minus nR dp over p.

228
00:19:06,980 --> 00:19:07,710
So that's great.

229
00:19:07,710 --> 00:19:10,720
That says now if I know the
entropy at some particular

230
00:19:10,720 --> 00:19:13,350
pressure, I can calculate how
it changes as a function of

231
00:19:13,350 --> 00:19:20,930
pressure, right.

232
00:19:20,930 --> 00:19:33,910
If I know S at some standard
pressure that we can define,

233
00:19:33,910 --> 00:19:42,050
then S at some arbitrary
pressure, is just S of p

234
00:19:42,050 --> 00:19:49,420
naught and T minus the integral
from p naught to p of

235
00:19:49,420 --> 00:19:53,220
nR dp over p.

236
00:19:53,220 --> 00:20:01,940
All right, which is to say it's
S of p naught T minus nR

237
00:20:01,940 --> 00:20:06,800
log of p over p naught.

238
00:20:06,800 --> 00:20:18,400
Right, now normally we'll
define p naught as

239
00:20:18,400 --> 00:20:23,930
equal to one bar.

240
00:20:23,930 --> 00:20:28,750
And often you'll see this simply
written as nR log p.

241
00:20:28,750 --> 00:20:31,060
I don't particularly like to
do that because of course,

242
00:20:31,060 --> 00:20:33,630
then, formally speaking were
looking at something that's

243
00:20:33,630 --> 00:20:36,760
written that has units inside
as the argument of a log.

244
00:20:36,760 --> 00:20:39,660
Of course it's understood when
you see that, and you're

245
00:20:39,660 --> 00:20:42,780
likely to see it in various
places, it's understood when

246
00:20:42,780 --> 00:20:46,830
you see that the quantity p is
always supposed to be divided

247
00:20:46,830 --> 00:20:53,050
by one bar, and the units
then are taken care of.

248
00:20:53,050 --> 00:21:01,600
For one mole, we can write the
molar quantities S of p and T,

249
00:21:01,600 --> 00:21:16,400
is S, S naught of T minus
R log p over p naught.

250
00:21:16,400 --> 00:21:24,330
All right, so that's our
pressure dependence.

251
00:21:24,330 --> 00:21:25,820
What about that?

252
00:21:25,820 --> 00:21:29,630
We still don't really have a
formulation for calculating

253
00:21:29,630 --> 00:21:32,620
this, or you know, defining it
or whatever we're going to do

254
00:21:32,620 --> 00:21:57,000
to allow us to know it.

255
00:21:57,000 --> 00:22:02,670
Well, let's just consider the
entropy as a function of

256
00:22:02,670 --> 00:22:06,770
temperature, starting all the
way down at zero degrees

257
00:22:06,770 --> 00:22:09,600
Kelvin, and going up to whatever
temperature we want

258
00:22:09,600 --> 00:22:23,460
to consider.

259
00:22:23,460 --> 00:22:28,140
Now, we certainly do know how
to calculate delta S for all

260
00:22:28,140 --> 00:22:30,750
that because we've seen how to
calculate delta S if you just

261
00:22:30,750 --> 00:22:34,760
heat something up, and we've
seen how to calculate delta S

262
00:22:34,760 --> 00:22:37,640
when something under goes a
phase transition, right.

263
00:22:37,640 --> 00:22:39,840
Presumably, if we're starting at
zero Kelvin, we're starting

264
00:22:39,840 --> 00:22:42,780
in a solid state.

265
00:22:42,780 --> 00:22:46,490
As we heat it up, depending on
the material, it may melt at

266
00:22:46,490 --> 00:22:47,090
some temperature.

267
00:22:47,090 --> 00:22:49,510
If we keep keep heating it
up, it'll boil at some

268
00:22:49,510 --> 00:22:54,110
temperature, but we know how to
treat all of that, right.

269
00:22:54,110 --> 00:22:56,670
So let's just consider something
that undergoes that

270
00:22:56,670 --> 00:22:58,130
set of changes.

271
00:22:58,130 --> 00:23:05,180
So, we've got some substance
A, solid, zero

272
00:23:05,180 --> 00:23:13,480
degrees Kelvin, one bar.

273
00:23:13,480 --> 00:23:16,200
Here's process one.

274
00:23:16,200 --> 00:23:20,340
It goes to A, it's a solid
at the melting

275
00:23:20,340 --> 00:23:29,500
temperature and one bar.

276
00:23:29,500 --> 00:23:34,570
Process two is it turns into
a liquid at the melting

277
00:23:34,570 --> 00:23:41,430
temperature and one bar.

278
00:23:41,430 --> 00:23:50,750
Process three is we heat it up
some more, up to the boiling

279
00:23:50,750 --> 00:23:56,190
temperature at one bar.

280
00:23:56,190 --> 00:24:03,010
Process four is it evaporates,
so now it's a gas at the

281
00:24:03,010 --> 00:24:07,620
boiling temperature
and one bar.

282
00:24:07,620 --> 00:24:13,350
Finally, we heat it up some
more, so now it's a gas at

283
00:24:13,350 --> 00:24:19,730
temperature T and one bar.

284
00:24:19,730 --> 00:24:21,510
And if we wanted to,
we can go further.

285
00:24:21,510 --> 00:24:26,200
We can make it a gas at
temperature and whatever

286
00:24:26,200 --> 00:24:26,900
pressure we want.

287
00:24:26,900 --> 00:24:36,420
That part we already know how
to take care of, right.

288
00:24:36,420 --> 00:24:41,410
Well, let's look at what happens
to S, all right.

289
00:24:41,410 --> 00:24:46,110
S, a molar enthalpy at T and
p, where we're going to

290
00:24:46,110 --> 00:24:56,000
finally end up, is, well it's s
zero at zero Kelvin and one

291
00:24:56,000 --> 00:25:02,470
bar or one bar is implied by
the superscripts here.

292
00:25:02,470 --> 00:25:07,080
And then we have delta S for
step one, and delta S for step

293
00:25:07,080 --> 00:25:09,730
two and so on.

294
00:25:09,730 --> 00:25:13,470
So all right, let's, we can
label this six so we to all

295
00:25:13,470 --> 00:25:37,890
the way to delta
S for step six.

296
00:25:37,890 --> 00:25:40,400
Well, so we can do that.

297
00:25:40,400 --> 00:25:51,300
It's S of the material at T
and p is S naught at zero

298
00:25:51,300 --> 00:25:55,620
Kelvin, plus, here's
for process one.

299
00:25:55,620 --> 00:26:01,310
We heat it up from zero Kelvin
up to the melting point.

300
00:26:01,310 --> 00:26:08,690
Cp of the solid more heat
capacity, divided by T dT.

301
00:26:08,690 --> 00:26:15,050
We can calculate delta S for
heating something up, right.

302
00:26:15,050 --> 00:26:21,230
Plus, now we've got the heat of
fusion to melt the stuff,

303
00:26:21,230 --> 00:26:28,310
so it's just delta H naught of
fusion, divided by Tm right.

304
00:26:28,310 --> 00:26:29,250
We saw that last time.

305
00:26:29,250 --> 00:26:32,890
In other words, remember,
we're just looking at q

306
00:26:32,890 --> 00:26:38,160
reversible over T to get delta
S, and it's just given by the

307
00:26:38,160 --> 00:26:40,640
heat of fusion.

308
00:26:40,640 --> 00:26:44,910
All right, then let's go from
the melting point to the

309
00:26:44,910 --> 00:26:45,830
boiling point.

310
00:26:45,830 --> 00:26:50,200
So it's Cp now it's the heat
capacity, the molar heat

311
00:26:50,200 --> 00:26:54,920
capacity of the liquid,
divided by T dT.

312
00:26:54,920 --> 00:26:59,540
We're heating up the liquid.

313
00:26:59,540 --> 00:27:01,550
And then there's vaporization.

314
00:27:01,550 --> 00:27:11,200
Delta H of vaporization over
T at the boiling point.

315
00:27:11,200 --> 00:27:14,050
Then we can go from the boiling
point to our final

316
00:27:14,050 --> 00:27:19,880
temperature T. Now it's the
molar heat capacity of the gas

317
00:27:19,880 --> 00:27:29,440
over T dT minus R log
p over p naught.

318
00:27:29,440 --> 00:27:34,200
OK, so that's everything, and
these are all things that we

319
00:27:34,200 --> 00:27:37,160
know how to do.

320
00:27:37,160 --> 00:27:38,790
Just about.

321
00:27:38,790 --> 00:27:42,300
OK, this one we're going to have
to think about, but all

322
00:27:42,300 --> 00:27:44,530
the changes we know how
to calculate, right.

323
00:27:44,530 --> 00:27:56,000
So if we plot this, S, and
let's just do this as a

324
00:27:56,000 --> 00:27:58,000
function of temperature.

325
00:27:58,000 --> 00:28:02,360
I don't have pressure
in here explicitly.

326
00:28:02,360 --> 00:28:08,770
Well, it's going to change as
I warm up the solid, soon

327
00:28:08,770 --> 00:28:14,750
we're really starting
at zero Kelvin.

328
00:28:14,750 --> 00:28:17,690
This stuff is all positive,
right, so the change in

329
00:28:17,690 --> 00:28:19,810
entropy is going
to be positive.

330
00:28:19,810 --> 00:28:23,070
Entropy is going to increase
as this happens, and then

331
00:28:23,070 --> 00:28:26,450
there's a change right at some
fixed temperature as the

332
00:28:26,450 --> 00:28:30,210
material melts.

333
00:28:30,210 --> 00:28:33,570
So here is step one.

334
00:28:33,570 --> 00:28:37,220
Here is step two, right, this
must be the melting

335
00:28:37,220 --> 00:28:40,170
temperature.

336
00:28:40,170 --> 00:28:45,130
And then there's another
heating step.

337
00:28:45,130 --> 00:28:48,590
Well, strictly speaking, I'm
going to run out of space here

338
00:28:48,590 --> 00:28:52,430
if I'm not careful, so I'm
going to be a little more

339
00:28:52,430 --> 00:28:55,470
careful here.

340
00:28:55,470 --> 00:28:59,930
One, two, three.

341
00:28:59,930 --> 00:29:01,690
I'm heating it up
a little more.

342
00:29:01,690 --> 00:29:03,650
Entropy is still increasing,
right.

343
00:29:03,650 --> 00:29:05,520
So I've done this.

344
00:29:05,520 --> 00:29:06,670
I've done this.

345
00:29:06,670 --> 00:29:09,870
Now I've heated up the liquid.

346
00:29:09,870 --> 00:29:13,760
Now, I'm going to boil the
liquid, so it's going to have

347
00:29:13,760 --> 00:29:17,100
some change in entropy.

348
00:29:17,100 --> 00:29:21,770
This must be my boiling point,
and now there's some further

349
00:29:21,770 --> 00:29:25,550
change in the gas, and that gets
me to whatever my final

350
00:29:25,550 --> 00:29:32,040
temperature is, right, that
I'm going to reach.

351
00:29:32,040 --> 00:29:36,700
Four and five, great.

352
00:29:36,700 --> 00:29:42,020
So there is monotonic increase
in the entropy.

353
00:29:42,020 --> 00:29:52,380
OK, so we're there, except
for this value.

354
00:29:52,380 --> 00:29:59,760
That one stinking
little number --

355
00:29:59,760 --> 00:30:03,530
S naught at zero Kelvin.

356
00:30:03,530 --> 00:30:07,320
That's the only thing we
don't know so far.

357
00:30:07,320 --> 00:30:13,470
So, for this we need some
additional input.

358
00:30:13,470 --> 00:30:17,610
We got some input of the
sort that we need

359
00:30:17,610 --> 00:30:18,820
in 1905 from Nernst.

360
00:30:18,820 --> 00:30:26,810
Nernst deduced that as you go
down from zero Kelvin for any

361
00:30:26,810 --> 00:30:31,130
process, the change in entropy
gets smaller and smaller.

362
00:30:31,130 --> 00:30:35,690
It approaches zero.

363
00:30:35,690 --> 00:30:38,490
Now, that actually was certainly
an important

364
00:30:38,490 --> 00:30:43,360
advance, but it was superseded
by such an important advance

365
00:30:43,360 --> 00:30:47,280
that I'm not even going to
reward it by placing it on the

366
00:30:47,280 --> 00:30:49,380
blackboard.

367
00:30:49,380 --> 00:30:52,430
Forget highlight, color,
forget it.

368
00:30:52,430 --> 00:30:59,550
Because Planck, six years later,
in 1911, deduced a

369
00:30:59,550 --> 00:31:04,480
stronger statement which
is extremely useful,

370
00:31:04,480 --> 00:31:06,200
and it's the following.

371
00:31:06,200 --> 00:31:11,120
What he showed is that as
temperature approaches zero

372
00:31:11,120 --> 00:31:15,730
Kelvin, for a pure
substance in it's

373
00:31:15,730 --> 00:31:20,620
crystalline state, S is zero.

374
00:31:20,620 --> 00:31:22,990
A much stronger statement,
right.

375
00:31:22,990 --> 00:31:27,350
A stronger statement than the
idea that changes in S get

376
00:31:27,350 --> 00:31:30,180
very small as you approach
zero Kelvin.

377
00:31:30,180 --> 00:31:34,100
No, he's saying we can make a
statement about that absolute

378
00:31:34,100 --> 00:31:39,240
number S goes to zero as
temperature goes to zero.

379
00:31:39,240 --> 00:31:43,380
For a pure substance in it's
crystalline state.

380
00:31:43,380 --> 00:31:48,310
So that is monumentally
important.

381
00:31:48,310 --> 00:32:14,680
So as T goes to zero Kelvin, S
goes to zero, for every pure

382
00:32:14,680 --> 00:32:21,600
substance in its, and I'll
sort of interject here,

383
00:32:21,600 --> 00:32:30,640
perfect crystalline state.

384
00:32:30,640 --> 00:32:32,690
That's really an
amazing result.

385
00:32:32,690 --> 00:32:36,420
So what it's saying is I'm
down at zero Kelvin.

386
00:32:36,420 --> 00:32:40,310
Minimally, I've somehow cooled
it as much as I possibly

387
00:32:40,310 --> 00:32:44,930
could, and I've got my perfect
crystal lattice.

388
00:32:44,930 --> 00:32:47,380
It could be an atomic crystal
like this, or, you know, it

389
00:32:47,380 --> 00:32:50,560
could be molecules.

390
00:32:50,560 --> 00:32:54,060
But they're all exactly where
they belong in their locations

391
00:32:54,060 --> 00:32:57,520
in the crystal, and the absolute
entropy is something

392
00:32:57,520 --> 00:33:02,960
I can define and it's zero.

393
00:33:02,960 --> 00:33:07,030
So S equals zero.

394
00:33:07,030 --> 00:33:19,160
Perfect, pure crystal,
all right.

395
00:33:19,160 --> 00:33:26,860
OK, well this came out of a
microscopic description of

396
00:33:26,860 --> 00:33:32,180
entropy that I briefly alluded
to last lecture, and again

397
00:33:32,180 --> 00:33:36,620
we'll go into in more detail
in a few lectures hence.

398
00:33:36,620 --> 00:33:42,400
But the result that I mentioned,
the general result,

399
00:33:42,400 --> 00:33:48,120
was that S was R over Na
Avogadro's number, times the

400
00:33:48,120 --> 00:34:07,800
log of this omega number of
microscopic states available

401
00:34:07,800 --> 00:34:10,210
to the system that
I'm considering.

402
00:34:10,210 --> 00:34:14,000
Now normally for a macroscopic
system, I've got just an

403
00:34:14,000 --> 00:34:19,530
astronomical number of
microscopic states.

404
00:34:19,530 --> 00:34:22,680
You know, that could mean in
a liquid, different little

405
00:34:22,680 --> 00:34:25,930
configurations of the molecules
around each other.

406
00:34:25,930 --> 00:34:27,940
They're all different states.

407
00:34:27,940 --> 00:34:33,660
Huge amounts of possible states,
and the gas even more.

408
00:34:33,660 --> 00:34:36,060
But if I go to zero Kelvin,
and I've got a perfect

409
00:34:36,060 --> 00:34:40,080
crystal, every atom,
every molecule is

410
00:34:40,080 --> 00:34:42,680
exactly in its place.

411
00:34:42,680 --> 00:34:46,900
How many possible different
states is that?

412
00:34:46,900 --> 00:34:48,600
It's one.

413
00:34:48,600 --> 00:34:51,230
There aren't any more
possible states.

414
00:34:51,230 --> 00:34:55,880
I've localized every identical
lateral molecule in its

415
00:34:55,880 --> 00:34:59,850
particular place,
and it's done.

416
00:34:59,850 --> 00:35:03,490
And you know, if I start
worrying about the various

417
00:35:03,490 --> 00:35:06,780
things that would matter under
ordinary conditions, right,

418
00:35:06,780 --> 00:35:09,080
you know, maybe at higher
temperature, I'd have some

419
00:35:09,080 --> 00:35:13,100
molecules in excited
vibration levels or

420
00:35:13,100 --> 00:35:14,740
maybe electronic levels.

421
00:35:14,740 --> 00:35:17,970
Maybe if it's hydrogen,
maybe everything isn't

422
00:35:17,970 --> 00:35:18,860
in the ground state.

423
00:35:18,860 --> 00:35:21,770
It's not all in the 1s orbital
but in higher levels.

424
00:35:21,770 --> 00:35:24,790
Then there's be lots of states
available, right, even of only

425
00:35:24,790 --> 00:35:28,350
one atom in the whole crystal
is excited, well there's one

426
00:35:28,350 --> 00:35:30,690
state that would have
it be this atom.

427
00:35:30,690 --> 00:35:33,020
A different one would have it
be this atom, and so forth.

428
00:35:33,020 --> 00:35:36,960
Already, there'd be an enormous
number of states, but

429
00:35:36,960 --> 00:35:41,320
at zero Kelvin, there's no
thermal lexitation of any of

430
00:35:41,320 --> 00:35:42,960
that stuff.

431
00:35:42,960 --> 00:35:46,230
Things are in the lowest states,
and they're in their

432
00:35:46,230 --> 00:35:48,450
proper positions.

433
00:35:48,450 --> 00:35:52,100
There's only one state for the
whole system, so that's why

434
00:35:52,100 --> 00:35:56,030
the entropy is zero in
a perfect crystal.

435
00:35:56,030 --> 00:36:01,610
At zero degrees Kelvin.

436
00:36:01,610 --> 00:36:05,310
Now, there are things that may
appear to violate that.

437
00:36:05,310 --> 00:36:07,480
Now of course you can make
measurements of entropy,

438
00:36:07,480 --> 00:36:14,800
right, so it can be verified
that this is the case.

439
00:36:14,800 --> 00:36:18,440
There are some things
that would appear to

440
00:36:18,440 --> 00:36:21,750
violate that result.

441
00:36:21,750 --> 00:36:25,670
You know, you can make
measurements of entropies, and

442
00:36:25,670 --> 00:36:29,860
for example let's take carbon
monoxide crystal, CO.

443
00:36:29,860 --> 00:36:32,200
So let's say this is
a crystal lattice,

444
00:36:32,200 --> 00:36:33,800
it's diatomic molecules.

445
00:36:33,800 --> 00:36:38,500
It's carbon oxygen carbon
oxygen carbon oxygen.

446
00:36:38,500 --> 00:36:42,370
All there in perfect place
in the crystal lattice.

447
00:36:42,370 --> 00:36:45,710
Well it turns out when you form
the crystal every now and

448
00:36:45,710 --> 00:36:51,880
then -- you know, let's
put it in color.

449
00:36:51,880 --> 00:36:56,690
Let's put it in the color that
we'll use to signify something

450
00:36:56,690 --> 00:37:01,250
in some way evil.

451
00:37:01,250 --> 00:37:03,140
No insult to people who
like that color.

452
00:37:03,140 --> 00:37:05,100
I kind of like it in fact.

453
00:37:05,100 --> 00:37:12,020
OK, so you know, you're making
the crystal, cooling it,

454
00:37:12,020 --> 00:37:14,500
started out maybe in the gas
phase, start cooling it,

455
00:37:14,500 --> 00:37:15,140
starts crystallizing.

456
00:37:15,140 --> 00:37:22,870
Gets colder and colder, but you
know, carbon monoxide is

457
00:37:22,870 --> 00:37:25,820
pretty easy for those things
to flip sides.

458
00:37:25,820 --> 00:37:28,030
And even in the crystalline
state, even though it's a

459
00:37:28,030 --> 00:37:30,990
crystal, so the molecules center
of mass are all where

460
00:37:30,990 --> 00:37:35,150
they belong, still at ordinary
temperatures they will be able

461
00:37:35,150 --> 00:37:36,890
to rotate a bit.

462
00:37:36,890 --> 00:37:40,280
So even in the crystalline
state, when it's originally

463
00:37:40,280 --> 00:37:43,380
formed, not at zero Kelvin,
there's thermal energy around.

464
00:37:43,380 --> 00:37:45,360
These things can to get knocked
around, and the

465
00:37:45,360 --> 00:37:47,020
orientations can change.

466
00:37:47,020 --> 00:37:50,350
Now you start cooling it, and
you know, by and large they'll

467
00:37:50,350 --> 00:37:54,010
all go into the proper
orientation.

468
00:37:54,010 --> 00:37:57,500
Right, that's the lowest energy
state, but you know,

469
00:37:57,500 --> 00:37:59,640
there are all sorts of kinetic
things involved.

470
00:37:59,640 --> 00:38:01,710
There it takes time for the
flipping, depending on how

471
00:38:01,710 --> 00:38:05,310
long, how slowly it was
cooled, and so forth.

472
00:38:05,310 --> 00:38:08,390
May never happen, and then it's
cooled, and then anything

473
00:38:08,390 --> 00:38:11,230
that's left in the
other orientation

474
00:38:11,230 --> 00:38:12,370
is frozen in there.

475
00:38:12,370 --> 00:38:13,680
There's no thermal
energy anymore.

476
00:38:13,680 --> 00:38:16,470
It can't find a way
to reorient.

477
00:38:16,470 --> 00:38:18,210
That's it.

478
00:38:18,210 --> 00:38:21,010
All right, let's say we're down
to zero Kelvin, and out

479
00:38:21,010 --> 00:38:24,040
of the whole crystal, we've
got a mole of molecules.

480
00:38:24,040 --> 00:38:27,560
One of them is in the
wrong orientation.

481
00:38:27,560 --> 00:38:30,080
Now how many states do
we have like that

482
00:38:30,080 --> 00:38:32,130
that would be possible?

483
00:38:32,130 --> 00:38:34,020
We'd have a mole of
states, right.

484
00:38:34,020 --> 00:38:34,690
It could be this one.

485
00:38:34,690 --> 00:38:36,470
It could be that one, right.

486
00:38:36,470 --> 00:38:38,480
Or in general, of course
really there's a whole

487
00:38:38,480 --> 00:38:41,160
distribution of them, and they
could be anywhere, and pairs

488
00:38:41,160 --> 00:38:43,790
of them could be, and it doesn't
take long to get to

489
00:38:43,790 --> 00:38:47,340
really large numbers.

490
00:38:47,340 --> 00:38:49,390
So the entropy won't be zero.

491
00:38:49,390 --> 00:38:51,310
Entropy of a perfect
crystal would mean

492
00:38:51,310 --> 00:38:54,460
it's perfectly ordered.

493
00:38:54,460 --> 00:38:56,970
So that's zero.

494
00:38:56,970 --> 00:38:58,980
But things like that not
withstanding, and of course

495
00:38:58,980 --> 00:39:00,520
it's the same if you have
a mixed crystal.

496
00:39:00,520 --> 00:39:03,920
In a sense I've described a
mixed crystal where the

497
00:39:03,920 --> 00:39:06,590
mixture is a mixture of carbon
monoxide pointing this way,

498
00:39:06,590 --> 00:39:08,680
and carbon monoxide
pointing this way.

499
00:39:08,680 --> 00:39:12,580
But a real mixed crystal with
two different constituents,

500
00:39:12,580 --> 00:39:14,970
well of course, then you
have all the possible

501
00:39:14,970 --> 00:39:17,790
configurations where, you know,
they could be here and

502
00:39:17,790 --> 00:39:19,800
here and here, and then you
could move one of them around

503
00:39:19,800 --> 00:39:21,480
and so forth, there
are zillions

504
00:39:21,480 --> 00:39:25,880
and zillions of states.

505
00:39:25,880 --> 00:39:30,920
But for a pure crystal, in
perfect form, you really have

506
00:39:30,920 --> 00:39:32,895
only one configuration,
and your entropy

507
00:39:32,895 --> 00:39:34,600
is therefore zero.

508
00:39:34,600 --> 00:39:44,490
OK, so now, we can go back
and we can do this.

509
00:39:44,490 --> 00:39:47,100
And at least in principle, even
for things that don't

510
00:39:47,100 --> 00:39:49,720
form perfect crystals, we could
calculate the change in

511
00:39:49,720 --> 00:39:54,190
entropy going from the perfect
ordered crystal to something

512
00:39:54,190 --> 00:39:56,930
else with some degree of
disorder and keep going and

513
00:39:56,930 --> 00:40:00,130
change the temperature and
do all these things.

514
00:40:00,130 --> 00:40:03,970
So the real point is that this
is extremely powerful because

515
00:40:03,970 --> 00:40:07,400
given this, we really can
calculate absolute numbers for

516
00:40:07,400 --> 00:40:10,640
the entropies of substances, at
ordinary, not just at zero

517
00:40:10,640 --> 00:40:13,280
Kelvin, but using this which,
you know, this is a really

518
00:40:13,280 --> 00:40:15,390
very straightforward
procedure.

519
00:40:15,390 --> 00:40:18,790
And in fact these things are
really quite easy to measure.

520
00:40:18,790 --> 00:40:21,530
You know, you do calorimetry,
you can measure those delta H

521
00:40:21,530 --> 00:40:25,240
of fusions, right, delta
H of vaporization.

522
00:40:25,240 --> 00:40:27,400
You can measure the heat
capacities, the things in the

523
00:40:27,400 --> 00:40:28,160
calorimeter.

524
00:40:28,160 --> 00:40:30,370
You'll see how much heat
is needed to raise the

525
00:40:30,370 --> 00:40:31,230
temperature a degree.

526
00:40:31,230 --> 00:40:33,405
That give you your heat capacity
for the gas or the

527
00:40:33,405 --> 00:40:34,630
liquid or the solid.

528
00:40:34,630 --> 00:40:37,930
So in fact, it's extremely
practical to make all those

529
00:40:37,930 --> 00:40:40,680
measurements, and you can easily
find those values of

530
00:40:40,680 --> 00:40:44,710
the heat capacities and the
delta H's tabulated for a huge

531
00:40:44,710 --> 00:40:47,190
number of substances.

532
00:40:47,190 --> 00:40:50,620
So this is, in fact, the
practical procedure then of

533
00:40:50,620 --> 00:40:54,140
protocol for calculating
absolute entropies of all

534
00:40:54,140 --> 00:41:00,960
sorts of materials at ordinary
temperatures and pressures.

535
00:41:00,960 --> 00:41:03,690
Very important.

536
00:41:03,690 --> 00:41:12,000
Now, one of those corollaries
to this law is that in fact

537
00:41:12,000 --> 00:41:16,010
it's impossible to reduce the
temperature of any substance,

538
00:41:16,010 --> 00:41:20,680
any system, all the way
to absolute, exact, no

539
00:41:20,680 --> 00:41:23,000
approximations, zero Kelvin.

540
00:41:23,000 --> 00:41:29,000
Because you can't quite get
down to zero Kelvin.

541
00:41:29,000 --> 00:41:31,600
And there are various ways that
you can see that this

542
00:41:31,600 --> 00:41:43,280
must be the case.

543
00:41:43,280 --> 00:41:53,250
But here's one way to
think about it.

544
00:41:53,250 --> 00:42:08,300
So, let's just write
that first.

545
00:42:08,300 --> 00:42:16,420
All right, can't get quite
down to zero Kelvin.

546
00:42:16,420 --> 00:42:29,130
All right, let's consider
a mole of an ideal gas.

547
00:42:29,130 --> 00:42:41,160
So p is RT over V. And let's
start at T1 and V1, and now

548
00:42:41,160 --> 00:42:45,900
let's bring it down to some
lower temperature, T2 in some

549
00:42:45,900 --> 00:42:48,640
spontaneous process.

550
00:42:48,640 --> 00:42:50,490
We'll make it adiabatic
so it's like an

551
00:42:50,490 --> 00:42:53,450
irreversible expansion.

552
00:42:53,450 --> 00:42:57,440
I just want to calculate what
delta S would be there in

553
00:42:57,440 --> 00:42:59,310
terms of T and V.

554
00:42:59,310 --> 00:43:21,400
So, well, du is T
dS minus p dV.

555
00:43:25,630 --> 00:43:33,700
dS is du over T plus
p over T dV.

556
00:43:33,700 --> 00:43:42,370
But we can also write du is
Cv dT in this case, right?

557
00:43:42,370 --> 00:43:55,080
So that says that p over T,
that's R over V, and we can

558
00:43:55,080 --> 00:44:31,106
write, dS is Cv dT over T plus
R dV over V. It's Cv, I'll

559
00:44:31,106 --> 00:44:49,300
write it as d(log T) plus R
d(log V), and so delta S, Cv

560
00:44:49,300 --> 00:44:56,180
log of T2 minus log of T1, if
T2 is my final temperature,

561
00:44:56,180 --> 00:45:07,350
plus R log of V2 minus
log of V1, okay.

562
00:45:07,350 --> 00:45:16,940
Or Cv log of T2 over T1 plus
R log of V2 over V1.

563
00:45:16,940 --> 00:45:35,290
Well, what does it mean
when T2 is zero?

564
00:45:35,290 --> 00:45:41,510
Well, I don't know
what it means.

565
00:45:41,510 --> 00:45:51,880
This turns into negative
infinity.

566
00:45:51,880 --> 00:45:57,010
So we're going to write it again
as our either evil or at

567
00:45:57,010 --> 00:46:00,030
least unattainable color.

568
00:46:00,030 --> 00:46:02,590
What's going to happen?

569
00:46:02,590 --> 00:46:06,850
I mean you could say, well, we
can counteract it by having

570
00:46:06,850 --> 00:46:08,890
this go to plus infinity,
right.

571
00:46:08,890 --> 00:46:11,930
Make the volume infinite.

572
00:46:11,930 --> 00:46:13,945
In other words, have the
expansion be in through an

573
00:46:13,945 --> 00:46:14,970
infinite volume.

574
00:46:14,970 --> 00:46:18,180
Of course that's impossible.

575
00:46:18,180 --> 00:46:20,620
That would be the only
way to counteract the

576
00:46:20,620 --> 00:46:24,280
divergence of this term.

577
00:46:24,280 --> 00:46:29,370
In practice, you really cannot
get to absolute zero, but it

578
00:46:29,370 --> 00:46:32,660
is possible to get
extremely close.

579
00:46:32,660 --> 00:46:34,110
That's doable experimentally.

580
00:46:34,110 --> 00:46:38,040
It's possible to get down to
some micro or nano Kelvin

581
00:46:38,040 --> 00:46:41,560
temperatures, right.

582
00:46:41,560 --> 00:46:49,420
In fact, our MIT physicist,
Wolfgang Ketterle, by bringing

583
00:46:49,420 --> 00:46:53,370
atoms and molecules down to
extremely low temperatures,

584
00:46:53,370 --> 00:46:57,290
was able to see them all reach
the very lowest possible

585
00:46:57,290 --> 00:46:59,100
quantum state available
to them.

586
00:46:59,100 --> 00:47:02,500
All sorts of unusual properties
emerge in that kind

587
00:47:02,500 --> 00:47:06,510
of state, where the atoms and
molecules behave coherently.

588
00:47:06,510 --> 00:47:10,290
You can make matter waves,
right, and see interferences

589
00:47:10,290 --> 00:47:12,740
among them because of the fact
that they're all in this

590
00:47:12,740 --> 00:47:14,730
lowest quantum state.

591
00:47:14,730 --> 00:47:17,780
So it is possible to get
extremely low temperatures,

592
00:47:17,780 --> 00:47:23,130
but never absolute zero.

593
00:47:23,130 --> 00:47:34,240
Here's another way to
think about it.

594
00:47:34,240 --> 00:47:37,320
You could consider what happens
to the absolute

595
00:47:37,320 --> 00:47:56,790
entropy, starting at T
equals zero, right.

596
00:47:56,790 --> 00:48:00,220
So we already saw that, you
know, that we can go, the

597
00:48:00,220 --> 00:48:04,310
first step will go from zero to
some temperature of Cp over

598
00:48:04,310 --> 00:48:11,880
T for the solid dT if we start
at zero Kelvin and warm up.

599
00:48:11,880 --> 00:48:16,710
Now, already we've
got a problem.

600
00:48:16,710 --> 00:48:23,390
If this initial temperature
really is zero, what happens?

601
00:48:23,390 --> 00:48:26,530
This diverge, right.

602
00:48:26,530 --> 00:48:31,820
Well, in fact, what that
suggests is as you approach

603
00:48:31,820 --> 00:48:40,380
zero Kelvin, the heat capacity
also approaches zero.

604
00:48:40,380 --> 00:48:54,340
So, does this go to infinity as
T approaches zero Kelvin?

605
00:48:54,340 --> 00:49:05,450
Well, not if Cp of the solid
approaches zero as T

606
00:49:05,450 --> 00:49:09,320
approaches zero Kelvin.

607
00:49:09,320 --> 00:49:15,910
And in fact, we can measure Cp,
heat capacities at very

608
00:49:15,910 --> 00:49:20,050
low temperatures, and what
we find is, they do.

609
00:49:20,050 --> 00:49:27,860
They do go to zero as you
approach zero Kelvin.

610
00:49:27,860 --> 00:49:35,530
So in fact, Cp of T approaches
zero as T

611
00:49:35,530 --> 00:49:38,700
approaches zero Kelvin.

612
00:49:38,700 --> 00:49:41,320
Good!

613
00:49:41,320 --> 00:49:49,020
But one thing that this says is,
remember, that dT is dq at

614
00:49:49,020 --> 00:49:51,700
the heat in divided by Cp.

615
00:49:55,990 --> 00:49:58,200
And this is getting really,
really small.

616
00:49:58,200 --> 00:49:59,720
It's going to zero
as temperature

617
00:49:59,720 --> 00:50:01,370
goes to zero, right.

618
00:50:01,370 --> 00:50:03,440
Which means that this
is enormous.

619
00:50:03,440 --> 00:50:09,770
What it says is even the tiniest
amount the heat input

620
00:50:09,770 --> 00:50:15,130
leads to a significant change
increase in the temperature.

621
00:50:15,130 --> 00:50:17,540
So, this is another way of
understanding why it just

622
00:50:17,540 --> 00:50:21,380
becomes impossible to lower the
temperature of a system to

623
00:50:21,380 --> 00:50:25,700
absolute zero, because any kind
of contact, and I mean

624
00:50:25,700 --> 00:50:27,470
any kind, like let's
say you've got the

625
00:50:27,470 --> 00:50:29,000
system in some cryostat.

626
00:50:29,000 --> 00:50:31,790
Of course the walls of the
cryostat aren't at zero

627
00:50:31,790 --> 00:50:33,650
Kelvin, but somewhere in here
it's at zero Kelvin.

628
00:50:33,650 --> 00:50:36,300
You can say, okay I won't
won't make it in

629
00:50:36,300 --> 00:50:37,870
contact with the walls.

630
00:50:37,870 --> 00:50:44,540
You can try, but actually light,
photons that emit from

631
00:50:44,540 --> 00:50:47,450
the walls go into the sample.

632
00:50:47,450 --> 00:50:50,800
You wouldn't think that heats
something up very much, and it

633
00:50:50,800 --> 00:50:53,160
doesn't heat it up very
much, but we're not

634
00:50:53,160 --> 00:50:55,170
talking about very much.

635
00:50:55,170 --> 00:50:57,400
It does heat it up by enough
that you can't get the

636
00:50:57,400 --> 00:50:58,850
absolute zero.

637
00:50:58,850 --> 00:51:02,660
In other words, somehow it will
be in contact with stuff

638
00:51:02,660 --> 00:51:06,880
around that is not at absolute
zero Kelvin.

639
00:51:06,880 --> 00:51:10,910
And even that kind of radiative
contact is enough,

640
00:51:10,910 --> 00:51:16,730
in fact, to make it not
reach absolute zero.

641
00:51:16,730 --> 00:51:19,610
So the point is, one way or
another, you can easily see

642
00:51:19,610 --> 00:51:23,090
that it becomes impossible to
keep pulling heat out of

643
00:51:23,090 --> 00:51:26,770
something and keep it down at
the temperature that's right

644
00:51:26,770 --> 00:51:30,310
there at zero, but again it's
possible to get very close.

645
00:51:30,310 --> 00:51:33,620
All right, what we're going to
be able to do next time is

646
00:51:33,620 --> 00:51:37,180
take what we've seen so far, and
develop the conditions for

647
00:51:37,180 --> 00:51:38,200
reaching equilibrium.

648
00:51:38,200 --> 00:51:41,640
So in a general sense, we'll be
able to tell which way the

649
00:51:41,640 --> 00:51:45,480
processes go left unto
themselves to move toward

650
00:51:45,480 --> 00:51:46,730
equilibrium.