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PROFESSOR: Last time,
you talked about

9
00:00:22,540 --> 00:00:24,270
the Gibbs free energy.

10
00:00:24,270 --> 00:00:26,570
And the fundamental equations.

11
00:00:26,570 --> 00:00:31,820
And how powerful the fundamental
equations were in

12
00:00:31,820 --> 00:00:35,410
being able to calculate anything
from pressure,

13
00:00:35,410 --> 00:00:40,310
volume, temperature data.

14
00:00:40,310 --> 00:00:44,350
And you saw that the Gibbs free
energy was especially

15
00:00:44,350 --> 00:00:51,730
important for everyday
sort of processes.

16
00:00:51,730 --> 00:00:56,250
Because of the constant
pressure constraint.

17
00:00:56,250 --> 00:00:59,590
And the fact that the intrinsic
variables are

18
00:00:59,590 --> 00:01:02,910
pressure and temperature.

19
00:01:02,910 --> 00:01:05,060
Well, it turns out that the
Gibbs free energy is even more

20
00:01:05,060 --> 00:01:05,850
important than that.

21
00:01:05,850 --> 00:01:08,580
And this is something that it
took me a while to learn.

22
00:01:08,580 --> 00:01:10,690
I had to take thermodynamics
many times to really

23
00:01:10,690 --> 00:01:13,770
appreciate how important
that was.

24
00:01:13,770 --> 00:01:18,480
Even when I was doing research,
at the beginning, I

25
00:01:18,480 --> 00:01:21,770
was a theorist, and I was trying
to calculate different

26
00:01:21,770 --> 00:01:25,440
quantities of liquids and
polymers and in these papers

27
00:01:25,440 --> 00:01:27,225
the first thing they did was
to calculate the Gibbs free

28
00:01:27,225 --> 00:01:29,530
energy and I didn't quite
appreciate why

29
00:01:29,530 --> 00:01:31,050
they were doing that.

30
00:01:31,050 --> 00:01:32,640
And the reason is, is if you've
got the Gibbs free

31
00:01:32,640 --> 00:01:36,320
energy, you got really
everything you need to know.

32
00:01:36,320 --> 00:01:39,000
Because you can get everything
from the Gibbs free energy.

33
00:01:39,000 --> 00:01:42,880
And it really becomes the
fundamental quantity that you

34
00:01:42,880 --> 00:01:44,890
want to have.

35
00:01:44,890 --> 00:01:49,620
So let me give you an example
of how important that is, if

36
00:01:49,620 --> 00:01:52,660
you have an equation that
describes the Gibbs free

37
00:01:52,660 --> 00:01:55,780
energy as a function of pressure
and temperature,

38
00:01:55,780 --> 00:01:59,450
number of moles of
different things.

39
00:01:59,450 --> 00:02:00,570
Different things you
can calculate.

40
00:02:00,570 --> 00:02:03,290
So let's just start from the
fundamental equation for the

41
00:02:03,290 --> 00:02:12,970
Gibbs free energy, dG is
minus S dT plus V dp.

42
00:02:12,970 --> 00:02:19,330
And let's say that you've gotten
some expression, G, as

43
00:02:19,330 --> 00:02:22,670
a function of temperature and
pressure for your system.

44
00:02:22,670 --> 00:02:24,700
It could be, as we're going to
see today that we're going to

45
00:02:24,700 --> 00:02:27,580
increase the number of variables
here, by making the

46
00:02:27,580 --> 00:02:28,670
system more complicated.

47
00:02:28,670 --> 00:02:31,100
So what I'm saying, now what I'm
going to say now, is going

48
00:02:31,100 --> 00:02:35,010
to be more general than just
these two variables here.

49
00:02:35,010 --> 00:02:37,110
So you've gotten this.

50
00:02:37,110 --> 00:02:38,280
So you have this.

51
00:02:38,280 --> 00:02:40,070
You've got the fundamental
equation.

52
00:02:40,070 --> 00:02:41,920
You've got all the other
fundamental equations, and

53
00:02:41,920 --> 00:02:43,700
from there you can calculate
all these quantities.

54
00:02:43,700 --> 00:02:47,270
For instance, you can calculate
an expression for S.

55
00:02:47,270 --> 00:02:50,320
Because you know that S, from
the fundamental equation, is

56
00:02:50,320 --> 00:02:56,920
just the derivative of G, with
respect to T, keeping the

57
00:02:56,920 --> 00:02:58,330
pressure fixed.

58
00:02:58,330 --> 00:03:01,440
So you've got your equation for
G. That translates into an

59
00:03:01,440 --> 00:03:07,360
equation for S. You can get
volume, volume is not one of

60
00:03:07,360 --> 00:03:08,330
the variables.

61
00:03:08,330 --> 00:03:10,340
Temperature and pressure
are the two knobs.

62
00:03:10,340 --> 00:03:13,030
But you can get volume out,
because volume is the

63
00:03:13,030 --> 00:03:17,540
derivative of G, with
respect to pressure.

64
00:03:17,540 --> 00:03:20,420
Keeping the temperature
constant.

65
00:03:20,420 --> 00:03:24,090
And you've got S now,
you got volume.

66
00:03:24,090 --> 00:03:26,410
Do you know where you,
how did you define G

67
00:03:26,410 --> 00:03:27,380
in the first place?

68
00:03:27,380 --> 00:03:32,030
We define G as H minus dS.

69
00:03:32,030 --> 00:03:35,020
One of the many definitions
of g.

70
00:03:35,020 --> 00:03:40,530
Reverse that, you've got now
H as a function of G and

71
00:03:40,530 --> 00:03:44,350
temperature and entropy.

72
00:03:44,350 --> 00:03:48,820
Well, you've got an expression
for G, we just calculated, we

73
00:03:48,820 --> 00:03:51,110
just showed we could get an
expression for S, which is

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00:03:51,110 --> 00:03:52,410
sitting right here.

75
00:03:52,410 --> 00:03:55,170
Temperature is a variable
here, so now we have an

76
00:03:55,170 --> 00:04:00,490
expression for H. And you can
go on like that with every

77
00:04:00,490 --> 00:04:06,230
variable that you've learned
in this class already.

78
00:04:06,230 --> 00:04:12,260
For instance, u is H minus pV.

79
00:04:12,260 --> 00:04:15,200
Well, there's the H here.

80
00:04:15,200 --> 00:04:19,060
We have that equation for H.
We have an equation for V,

81
00:04:19,060 --> 00:04:20,730
coming from here.

82
00:04:20,730 --> 00:04:23,860
So there's nothing
unknown here.

83
00:04:23,860 --> 00:04:29,600
If we have an equation for G
here in terms of temperature

84
00:04:29,600 --> 00:04:31,740
and pressure.

85
00:04:31,740 --> 00:04:33,740
Same thing for the Helmholtz
free energy.

86
00:04:33,740 --> 00:04:36,830
You can even get the heat
capacities out.

87
00:04:36,830 --> 00:04:42,010
Every single one of these
interesting, useful quantities

88
00:04:42,010 --> 00:04:45,590
that one would want to calculate
falls out from the

89
00:04:45,590 --> 00:04:51,380
Gibbs free energy here.

90
00:04:51,380 --> 00:04:56,730
Any questions on that
important step?

91
00:04:56,730 --> 00:04:59,240
And, really, I can't believe
how clueless I was when I

92
00:04:59,240 --> 00:05:00,650
started doing research.

93
00:05:00,650 --> 00:05:02,970
Because I would go through the
process of calculating G, and

94
00:05:02,970 --> 00:05:05,340
getting G, and et cetera,
et cetera.

95
00:05:05,340 --> 00:05:08,270
I wrote papers, you know,
G equals blah blah blah.

96
00:05:08,270 --> 00:05:11,040
And I didn't realize that that's
why people wanted to

97
00:05:11,040 --> 00:05:15,780
know G. Anyway, I
know better now.

98
00:05:15,780 --> 00:05:21,750
So, there are a few things we
can say about G that are

99
00:05:21,750 --> 00:05:24,430
fairly easy to calculate.

100
00:05:24,430 --> 00:05:29,980
For instance, if I look
at liquids or solids.

101
00:05:29,980 --> 00:05:36,890
And I want to know how G
changes with pressure.

102
00:05:36,890 --> 00:05:49,860
So, I know that the volume here
dG/dp, that dG/dp is the

103
00:05:49,860 --> 00:05:50,400
volume here.

104
00:05:50,400 --> 00:05:54,220
So if I look under constant
temperature, I pick my

105
00:05:54,220 --> 00:05:56,830
fundamental equation under
constant temperature, and I

106
00:05:56,830 --> 00:05:59,270
want to know how G is changing,
I integrate.

107
00:05:59,270 --> 00:06:04,570
So I have dG is equal to V dp.

108
00:06:04,570 --> 00:06:09,620
So if I change my, and I look
at per mole, and if I change

109
00:06:09,620 --> 00:06:14,680
my pressure from p1 to p2, I
integrate from p1 to p2, p1 to

110
00:06:14,680 --> 00:06:23,600
p2 here, final state minus the
initial state is equal to the

111
00:06:23,600 --> 00:06:31,920
integral from p1 to
p2, V dp per mole.

112
00:06:31,920 --> 00:06:35,000
And what can I say, for a liquid
or a solid, the volume

113
00:06:35,000 --> 00:06:38,960
per mole, over a liquid or a
solid, is small and it doesn't

114
00:06:38,960 --> 00:06:41,560
change very much.

115
00:06:41,560 --> 00:06:47,410
So V is small.

116
00:06:47,410 --> 00:06:51,320
And usually these solids and
liquids, you can assume to be

117
00:06:51,320 --> 00:06:53,860
incompressible.

118
00:06:53,860 --> 00:06:55,765
Meaning, as you change
the pressure, the

119
00:06:55,765 --> 00:06:57,700
volume doesn't change.

120
00:06:57,700 --> 00:07:00,670
It's a good approximation.

121
00:07:00,670 --> 00:07:05,310
So when you do your integral
here, you get that G at the

122
00:07:05,310 --> 00:07:11,080
new pressure is G at the old
pressure, then if this isn't

123
00:07:11,080 --> 00:07:13,190
changing very much with
pressure, or not at all, then

124
00:07:13,190 --> 00:07:13,850
you can take it out.

125
00:07:13,850 --> 00:07:16,430
It's just a constant.

126
00:07:16,430 --> 00:07:21,900
Plus V times p2 minus p1.

127
00:07:21,900 --> 00:07:23,940
And so, this is the
incompressible

128
00:07:23,940 --> 00:07:24,800
part, you take it out.

129
00:07:24,800 --> 00:07:28,190
The fact that it's small means
that you can assume that this

130
00:07:28,190 --> 00:07:31,620
is zero, this whole thing is
zero, that it's small enough.

131
00:07:31,620 --> 00:07:35,630
And then you see that G,
approximately doesn't change

132
00:07:35,630 --> 00:07:37,830
with pressure.

133
00:07:37,830 --> 00:07:43,820
Tells you that G, for a liquid
or solid, most of the time you

134
00:07:43,820 --> 00:07:46,230
can assume that it's just a
function of temperature.

135
00:07:46,230 --> 00:07:49,310
Just like we saw for an ideal
gas, that the energy and the

136
00:07:49,310 --> 00:07:51,280
enthalpy were just functions
of temperature.

137
00:07:51,280 --> 00:07:55,430
And that's a useful
approximation.

138
00:07:55,430 --> 00:07:57,640
It's useful, but it's
not completely true.

139
00:07:57,640 --> 00:08:01,330
And if it were true, then
there would not be any

140
00:08:01,330 --> 00:08:03,170
pressure dependents to
phase transitions.

141
00:08:03,170 --> 00:08:04,620
And we know that's
not the case.

142
00:08:04,620 --> 00:08:11,730
We know that if you press on
water when it's close to the

143
00:08:11,730 --> 00:08:21,200
water liquid-solid transition,
that you can lower the melting

144
00:08:21,200 --> 00:08:22,590
point of ice.

145
00:08:22,590 --> 00:08:25,120
You press on ice, and you press
hard enough, and ice

146
00:08:25,120 --> 00:08:29,030
will melt, the temperature is
closer to melting point.

147
00:08:29,030 --> 00:08:30,110
And we'll go through that.

148
00:08:30,110 --> 00:08:33,450
So, that means that there has
to be some sort of pressure

149
00:08:33,450 --> 00:08:34,890
dependence, eventually.

150
00:08:34,890 --> 00:08:36,380
And we'll see that.

151
00:08:36,380 --> 00:08:39,210
This is just an approximation.

152
00:08:39,210 --> 00:08:39,940
What else can we do?

153
00:08:39,940 --> 00:08:44,750
We can calculate, also,
for an ideal gas.

154
00:08:44,750 --> 00:08:50,910
Liquid and solid, we can
do an ideal gas.

155
00:08:50,910 --> 00:08:53,990
So for an ideal gas, again,
starting from the fundamental

156
00:08:53,990 --> 00:08:59,990
equation, we have
dG equals V dp.

157
00:08:59,990 --> 00:09:04,780
We can do it per mole.

158
00:09:04,780 --> 00:09:10,530
So integrate both sides, G(T,
p2) is equal to G at the

159
00:09:10,530 --> 00:09:13,630
initial pressure, plus
the integral from

160
00:09:13,630 --> 00:09:15,890
p1 to p2, the volume.

161
00:09:15,890 --> 00:09:18,080
So instead of putting the
volume, this is an ideal gas

162
00:09:18,080 --> 00:09:20,220
now, we can put the
ideal gas law.

163
00:09:20,220 --> 00:09:22,870
So V is really RT over p.

164
00:09:22,870 --> 00:09:25,890
RT over p dp.

165
00:09:25,890 --> 00:09:27,730
We can integrate this.

166
00:09:27,730 --> 00:09:31,180
Get a log term out.

167
00:09:31,180 --> 00:09:42,080
G(T, p1) plus RT
log p2 over p1.

168
00:09:42,080 --> 00:09:46,410
And then it's very useful to
reference everything to the

169
00:09:46,410 --> 00:09:51,040
center state. p1 is equal
to one bar, let's say.

170
00:09:51,040 --> 00:09:58,680
So if you take p1 equals one
bar as our reference point,

171
00:09:58,680 --> 00:10:02,460
and get rid of the little
subscript two here, we can

172
00:10:02,460 --> 00:10:07,170
write G of T at some pressure
p, then is G and the little

173
00:10:07,170 --> 00:10:14,610
naught on top here means
standard state one bar plus RT

174
00:10:14,610 --> 00:10:20,730
log p divided by one bar.

175
00:10:20,730 --> 00:10:26,350
And I put in a little dotted
line here because very often

176
00:10:26,350 --> 00:10:28,550
you just write it without
the one bar and bar.

177
00:10:28,550 --> 00:10:31,640
And you know that there has to
be a one bar, because inside a

178
00:10:31,640 --> 00:10:33,500
log you can't have something
with units.

179
00:10:33,500 --> 00:10:34,360
It has to be unitless.

180
00:10:34,360 --> 00:10:36,420
So you know if you have
something with bar here,

181
00:10:36,420 --> 00:10:38,100
you've got to divide with
something with bar, and there

182
00:10:38,100 --> 00:10:40,940
happens to be one bar here.

183
00:10:40,940 --> 00:10:46,850
So pressure p is G at its
standard state plus RT log p.

184
00:10:46,850 --> 00:10:50,680
And this becomes a
very useful, very

185
00:10:50,680 --> 00:10:56,060
useful, quantity to know.

186
00:10:56,060 --> 00:11:00,190
OK, so G is so important.

187
00:11:00,190 --> 00:11:06,430
And G per mole is so
fundamental, that we're going

188
00:11:06,430 --> 00:11:09,470
to give it a special name.

189
00:11:09,470 --> 00:11:11,610
Not to make your life more
complicated but just because

190
00:11:11,610 --> 00:11:12,410
it's just so important.

191
00:11:12,410 --> 00:11:17,040
We're going to call it the
chemical potential.

192
00:11:17,040 --> 00:11:23,100
So G per mole, we're
going to call mu.

193
00:11:23,100 --> 00:11:24,580
And that's going to be
a chemical potential.

194
00:11:24,580 --> 00:11:25,750
We're going to do a
lot more with the

195
00:11:25,750 --> 00:11:30,390
chemical potential today.

196
00:11:30,390 --> 00:11:33,590
And the reason why we call
potential is because we

197
00:11:33,590 --> 00:11:40,240
already saw that if you've got
something under constant

198
00:11:40,240 --> 00:11:48,640
pressure, temperature, that
you want to use G as the

199
00:11:48,640 --> 00:11:50,290
variable to tell you whether
something is going to be

200
00:11:50,290 --> 00:11:51,950
spontaneous or not.

201
00:11:51,950 --> 00:11:55,190
So you want G to go downhill.

202
00:11:55,190 --> 00:11:57,940
And so, we're going
to be talking

203
00:11:57,940 --> 00:11:59,520
about chemical species.

204
00:11:59,520 --> 00:12:04,410
And instead of having a car up
and down mountains, trying to

205
00:12:04,410 --> 00:12:07,060
go down to the valleys, we're
going to have chemical species

206
00:12:07,060 --> 00:12:09,330
trying to find the valleys.

207
00:12:09,330 --> 00:12:12,850
The potential valleys.

208
00:12:12,850 --> 00:12:14,890
To get to equilibrium.

209
00:12:14,890 --> 00:12:16,870
And so we're going to be looking
at the Gibbs free

210
00:12:16,870 --> 00:12:19,030
energy, or the Gibbs free
energy per mole at that

211
00:12:19,030 --> 00:12:22,220
particular species, and it's
going to want to be as small

212
00:12:22,220 --> 00:12:24,580
as possible.

213
00:12:24,580 --> 00:12:26,410
We're going to want to
minimize the chemical

214
00:12:26,410 --> 00:12:27,600
potentials.

215
00:12:27,600 --> 00:12:29,370
And that's why it's
called potential.

216
00:12:29,370 --> 00:12:34,800
It's like an energy.

217
00:12:34,800 --> 00:12:45,900
So, that's the end of the one
component, thermodynamic

218
00:12:45,900 --> 00:12:48,170
background, before we get
to multi-components.

219
00:12:48,170 --> 00:12:50,950
So it's a good time to
stop again and see

220
00:12:50,950 --> 00:12:52,950
if there's any questions.

221
00:12:52,950 --> 00:13:01,850
Any issues.

222
00:13:01,850 --> 00:13:02,610
OK.

223
00:13:02,610 --> 00:13:06,120
So, so far we've done everything
with one species.

224
00:13:06,120 --> 00:13:09,180
One ideal gas, one liquid,
one solid.

225
00:13:09,180 --> 00:13:12,380
We haven't done anything with
mixtures, except for maybe

226
00:13:12,380 --> 00:13:15,210
looking at the entropy
of mixing.

227
00:13:15,210 --> 00:13:17,990
We saw the entropy of mixing was
really important, because

228
00:13:17,990 --> 00:13:24,470
it drove processes where
energy was constant.

229
00:13:24,470 --> 00:13:27,410
But most of what we care about
in chemistry, at least in

230
00:13:27,410 --> 00:13:31,790
chemical reactions,
species change.

231
00:13:31,790 --> 00:13:32,660
They get destroyed.

232
00:13:32,660 --> 00:13:36,120
New species get created.

233
00:13:36,120 --> 00:13:37,120
There are mixtures.

234
00:13:37,120 --> 00:13:38,440
It's pretty complicated.

235
00:13:38,440 --> 00:13:42,350
For instance, if I take a
reaction of hydrogen gas plus

236
00:13:42,350 --> 00:13:49,630
chlorine gas to form two moles
of HCl gas, I'm destroying

237
00:13:49,630 --> 00:13:52,200
hydrogen, I'm destroying
chlorine, I'm making HCl in

238
00:13:52,200 --> 00:13:54,330
the gas phase.

239
00:13:54,330 --> 00:13:55,630
I get a big mixture
at the end.

240
00:13:55,630 --> 00:13:59,040
I get three different kinds
of species at the end.

241
00:13:59,040 --> 00:14:01,550
So the fundamental equations
that I've been talking about,

242
00:14:01,550 --> 00:14:04,350
that we've been talking
about, they're too

243
00:14:04,350 --> 00:14:06,400
simple for such a system.

244
00:14:06,400 --> 00:14:09,990
Because they all care
about one species.

245
00:14:09,990 --> 00:14:13,480
Even more complicated, for
instance, if I take hydrogen

246
00:14:13,480 --> 00:14:22,310
gas and oxygen gas and I mix
them together to make water,

247
00:14:22,310 --> 00:14:26,540
liquid, for instance, not only
do I have species that are

248
00:14:26,540 --> 00:14:31,240
changing, that are getting
destroyed or created, in this

249
00:14:31,240 --> 00:14:34,250
case here the total number
of moles is changing.

250
00:14:34,250 --> 00:14:37,690
And the phase is changing.

251
00:14:37,690 --> 00:14:40,450
Got all sorts of changes
going on here.

252
00:14:40,450 --> 00:14:43,390
And so if I want to understand
equilibrium, if I want to

253
00:14:43,390 --> 00:14:46,790
understand the direction
of time for these more

254
00:14:46,790 --> 00:14:49,330
complicated processes, I have
to be able to take into

255
00:14:49,330 --> 00:14:55,590
account, in an easy way, these
mixing processes, these phase

256
00:14:55,590 --> 00:15:03,730
changes, these changes in
the number of moles.

257
00:15:03,730 --> 00:15:07,000
And that's what we're going
to talk about today.

258
00:15:07,000 --> 00:15:10,010
We're going to try to change our
fundamental equations to

259
00:15:10,010 --> 00:15:12,140
make them a little bit more
complicated so that we can

260
00:15:12,140 --> 00:15:13,500
deal with these sorts
of problems.

261
00:15:13,500 --> 00:15:15,070
Because those are the real
problems we need

262
00:15:15,070 --> 00:15:19,950
to keep track of.

263
00:15:19,950 --> 00:15:23,540
And the ultimate goal, then,
of changing our fundamental

264
00:15:23,540 --> 00:15:27,320
questions is to derive
equilibrium from first

265
00:15:27,320 --> 00:15:28,320
principles.

266
00:15:28,320 --> 00:15:31,190
To really understand chemical
equilibrium, which you've all

267
00:15:31,190 --> 00:15:31,680
seen before.

268
00:15:31,680 --> 00:15:35,410
You've all used the chemical
equilibrium constant K, you've

269
00:15:35,410 --> 00:15:36,770
done problems.

270
00:15:36,770 --> 00:15:40,400
But you've been given,
basically, the equilibrium

271
00:15:40,400 --> 00:15:43,870
constant, and not really derived
it, understood where

272
00:15:43,870 --> 00:15:48,790
it came from.

273
00:15:48,790 --> 00:15:53,800
OK, another simple example here,
which is actually the

274
00:15:53,800 --> 00:15:57,820
one that we're going to be
looking at in the first case.

275
00:15:57,820 --> 00:16:02,520
Where there's a change
going on, is just to

276
00:16:02,520 --> 00:16:04,360
look at a phase change.

277
00:16:04,360 --> 00:16:07,840
H2O liquid going to H2O solid.

278
00:16:07,840 --> 00:16:10,040
There's a phase change, you
can think of it as one

279
00:16:10,040 --> 00:16:12,590
species, the H2O liquid
sort of changing

280
00:16:12,590 --> 00:16:14,710
into an H2O a solid.

281
00:16:14,710 --> 00:16:17,340
It's the same chemical in this
case here, there's no change

282
00:16:17,340 --> 00:16:20,890
in the molecules.

283
00:16:20,890 --> 00:16:27,640
But it's still a change that
we have to account for.

284
00:16:27,640 --> 00:16:32,910
Another example that's also
simple like this, that you

285
00:16:32,910 --> 00:16:41,990
all, I'm sure, have seen before,
suppose I take a cell.

286
00:16:41,990 --> 00:16:44,780
My cell here, full of water.

287
00:16:44,780 --> 00:16:50,930
And then I put my cell, let's
say I take a human cell.

288
00:16:50,930 --> 00:16:53,000
My skin or something.

289
00:16:53,000 --> 00:17:03,350
And I take it and I put
it in distilled water.

290
00:17:03,350 --> 00:17:07,600
What's going to happen
to the cell?

291
00:17:07,600 --> 00:17:10,520
Is it going to be happy?

292
00:17:10,520 --> 00:17:15,580
What's going to happen to it?

293
00:17:15,580 --> 00:17:17,330
It's going to burst, right?

294
00:17:17,330 --> 00:17:18,470
Why is it going to burst?

295
00:17:18,470 --> 00:17:26,060
Anybody have an idea why
it's going to burst?

296
00:17:26,060 --> 00:17:26,460
Yes.

297
00:17:26,460 --> 00:17:38,260
STUDENT: [INAUDIBLE]

298
00:17:38,260 --> 00:17:38,780
PROFESSOR: That's right.

299
00:17:38,780 --> 00:17:42,960
So the water wants to go from,
you're completely right.

300
00:17:42,960 --> 00:17:46,330
But let me rephrase it in a
thermodynamic language here.

301
00:17:46,330 --> 00:17:48,770
The water is going to go from
a place of high chemical

302
00:17:48,770 --> 00:17:52,720
potential to low chemical
potential.

303
00:17:52,720 --> 00:17:56,650
And the cell can't take all
that water in there.

304
00:17:56,650 --> 00:17:59,290
The membrane's going
to try to swell.

305
00:17:59,290 --> 00:18:00,700
And eventually burst, right?

306
00:18:00,700 --> 00:18:05,680
Same thing if you if you take
a, go fishing, go to the

307
00:18:05,680 --> 00:18:09,040
Atlantic Ocean and then get
a nice cod or something.

308
00:18:09,040 --> 00:18:13,950
Bring it back and on your
sailboat, you dump it in a tub

309
00:18:13,950 --> 00:18:15,770
of fresh water.

310
00:18:15,770 --> 00:18:18,380
Is that cod going to be happy?

311
00:18:18,380 --> 00:18:22,390
It's not going to be happy at
all, right, because its

312
00:18:22,390 --> 00:18:26,910
biology is geared towards
living in salt water.

313
00:18:26,910 --> 00:18:31,170
And turns out that the chemical
potential of water,

314
00:18:31,170 --> 00:18:32,860
in salt water, is lower
than the chemical

315
00:18:32,860 --> 00:18:34,830
potential of pure water.

316
00:18:34,830 --> 00:18:37,590
And so when you put the cod in
there, the chemical potential

317
00:18:37,590 --> 00:18:40,340
of the water and the cod, is
lower than the chemical

318
00:18:40,340 --> 00:18:42,480
potential of the fresh water
you have on the outside.

319
00:18:42,480 --> 00:18:44,180
And the fresh water
wants to be at a

320
00:18:44,180 --> 00:18:44,965
lower chemical potential.

321
00:18:44,965 --> 00:18:48,410
It rushes into the cod, and
well, the cod does what the

322
00:18:48,410 --> 00:18:50,960
cell does, when you put
it in distilled water.

323
00:18:50,960 --> 00:18:53,980
It sort of bloats.

324
00:18:53,980 --> 00:18:57,920
It isn't very happy.

325
00:18:57,920 --> 00:19:01,150
OK, so but all these things
are basically

326
00:19:01,150 --> 00:19:02,250
the same idea here.

327
00:19:02,250 --> 00:19:05,220
Where you have a complicated
change, where species are

328
00:19:05,220 --> 00:19:07,320
mixing, and things like this.

329
00:19:07,320 --> 00:19:09,500
And it turns out the chemical
potential is going to tell us

330
00:19:09,500 --> 00:19:12,430
all about how to think
about that.

331
00:19:12,430 --> 00:19:16,650
That's why the chemical
potential is so important.

332
00:19:16,650 --> 00:19:19,070
So we're going to go
back to these two

333
00:19:19,070 --> 00:19:23,940
examples here many times.

334
00:19:23,940 --> 00:19:26,000
So let's take the simplest
example here.

335
00:19:26,000 --> 00:19:32,900
Let's go back and derive
some equations.

336
00:19:32,900 --> 00:19:37,280
Let's take our simplest example
that's not a one

337
00:19:37,280 --> 00:19:41,000
species system, but
has two species.

338
00:19:41,000 --> 00:19:45,820
Species 1 and 2.

339
00:19:45,820 --> 00:19:50,450
And n1 and n2 are the number of
moles of species 1 and 2.

340
00:19:50,450 --> 00:19:52,490
And then we're going
to see if I make a

341
00:19:52,490 --> 00:19:55,150
perturbation in my system.

342
00:19:55,150 --> 00:19:57,940
I change the number of
moles of 1, or the

343
00:19:57,940 --> 00:19:58,850
number of moles of 2.

344
00:19:58,850 --> 00:20:03,030
How does this affect the
Gibbs free energy?

345
00:20:03,030 --> 00:20:05,730
That's the question we're
going to post.

346
00:20:05,730 --> 00:20:08,880
And our goal is to find a new
fundamental equation for G

347
00:20:08,880 --> 00:20:12,670
that includes the number of
moles of the different species

348
00:20:12,670 --> 00:20:13,730
as they change.

349
00:20:13,730 --> 00:20:16,730
Because in chemistry they're
going to be changing.

350
00:20:16,730 --> 00:20:19,330
They're not going to be fixed.

351
00:20:19,330 --> 00:20:23,560
So what we want is just purely
mathematically formally, take

352
00:20:23,560 --> 00:20:27,200
the differential of the Gibbs
free energy, which we know is

353
00:20:27,200 --> 00:20:33,060
dG/dT, keeping pressure, the
number of moles of 1, the

354
00:20:33,060 --> 00:20:35,910
number of 2 constant, dT.

355
00:20:35,910 --> 00:20:43,820
That is, dG/dp constant
temperature, n1 and n2 dp.

356
00:20:43,820 --> 00:20:48,920
And then we have our two more
variables now, dG/dn1,

357
00:20:48,920 --> 00:20:53,570
remember this is just a formal
statement keeping temperature

358
00:20:53,570 --> 00:21:03,000
and pressure and n2 constant.
dn1 plus dG/dn2, dn2 keeping

359
00:21:03,000 --> 00:21:07,090
temperature and pressure
and n1 constant here.

360
00:21:07,090 --> 00:21:08,910
I'm not writing anything new
here, I'm just telling you

361
00:21:08,910 --> 00:21:13,890
what the definition of the
differential is here, for G.

362
00:21:13,890 --> 00:21:16,950
We already know what some
of these quantities are.

363
00:21:16,950 --> 00:21:23,410
We know that this is the
entropy, minus the entropy.

364
00:21:23,410 --> 00:21:28,930
This here is the volume.

365
00:21:28,930 --> 00:21:31,440
And I know the answer already.

366
00:21:31,440 --> 00:21:34,180
But I'm going to define
it anyways.

367
00:21:34,180 --> 00:21:37,710
And we're going to prove it.

368
00:21:37,710 --> 00:21:41,610
I'm going to define this
as the chemical

369
00:21:41,610 --> 00:21:46,210
potential for species 1.

370
00:21:46,210 --> 00:21:51,640
I'm going to define this, I'm
going to give it a symbol,

371
00:21:51,640 --> 00:21:56,960
chemical potential mu,
for species 2.

372
00:21:56,960 --> 00:22:02,490
So that I can write my new
fundamental equation as dG as

373
00:22:02,490 --> 00:22:04,890
minus S dT.

374
00:22:04,890 --> 00:22:11,570
Plus V dp plus, and if I have
more than two species present,

375
00:22:11,570 --> 00:22:17,060
the sum of all species in my
mixture times the chemical

376
00:22:17,060 --> 00:22:21,380
potential of that
species, dni.

377
00:22:21,380 --> 00:22:27,510
The change in the number of
moles of that species.

378
00:22:27,510 --> 00:22:36,500
So, this quantity mu, that I've
just defined, dG/dni,

379
00:22:36,500 --> 00:22:39,700
keeping the temperature, the
pressure and all the n's,

380
00:22:39,700 --> 00:22:44,220
except for the i'th one
constant, that is

381
00:22:44,220 --> 00:22:45,180
an intensive quantity.

382
00:22:45,180 --> 00:22:53,250
Because G scales with size,
scales, with size of system.

383
00:22:53,250 --> 00:22:56,470
G is intensive. n, obviously,
scales with

384
00:22:56,470 --> 00:22:59,800
the size of the system.

385
00:22:59,800 --> 00:23:04,000
Also intensive, and you take
the ratio of two extensive

386
00:23:04,000 --> 00:23:05,640
variables, you get an intensive
variable which

387
00:23:05,640 --> 00:23:08,610
doesn't care about the
size of the system.

388
00:23:08,610 --> 00:23:09,890
Which is a good thing.

389
00:23:09,890 --> 00:23:12,730
For what we've been
talking about.

390
00:23:12,730 --> 00:23:15,670
Intensive.

391
00:23:15,670 --> 00:23:19,500
If I'm talking about putting a
freshwater fish and dumping it

392
00:23:19,500 --> 00:23:22,870
in the, putting it in the
Atlantic Ocean, the chemical

393
00:23:22,870 --> 00:23:25,750
potential of the water in the
Atlantic ocean better not care

394
00:23:25,750 --> 00:23:29,440
whether the Atlantic Ocean
is huge or even huger.

395
00:23:29,440 --> 00:23:31,760
It just cares about the
local environment.

396
00:23:31,760 --> 00:23:39,150
Just cares that it that wants to
be in that freshwater fish.

397
00:23:39,150 --> 00:23:42,720
So the chemical potential
is intensive.

398
00:23:42,720 --> 00:23:47,690
Just as we've written
it down here for a

399
00:23:47,690 --> 00:23:50,150
single species system.

400
00:23:50,150 --> 00:23:51,710
It's the Gibbs energy,
free energy per mole.

401
00:23:51,710 --> 00:23:53,330
Which we haven't proven yet.

402
00:23:53,330 --> 00:23:54,880
We haven't proven yet here.

403
00:23:54,880 --> 00:23:57,490
We've just defined it this way
and we're going to prove that

404
00:23:57,490 --> 00:24:01,940
in fact the mu's are the Gibbs
free energies per mole for

405
00:24:01,940 --> 00:24:07,820
each of the species
in our system.

406
00:24:07,820 --> 00:24:14,240
OK, so this is now our first
new fundamental equation.

407
00:24:14,240 --> 00:24:17,840
All we did was to add
the sum here.

408
00:24:17,840 --> 00:24:19,970
Now, we started the lecture by
saying that if you have the

409
00:24:19,970 --> 00:24:24,170
Gibbs free energy, you've
got everything.

410
00:24:24,170 --> 00:24:30,220
And we wrote equations that are
covered, were we can get

411
00:24:30,220 --> 00:24:35,160
S, we can get V, we can get H,
we can get u, we can get A. So

412
00:24:35,160 --> 00:24:37,380
now that we have a fundamental
question for G, we've got our

413
00:24:37,380 --> 00:24:40,470
new fundamental equations
for everything else.

414
00:24:40,470 --> 00:24:45,290
Without really thinking
too much.

415
00:24:45,290 --> 00:24:50,260
We go back to our definitions.

416
00:24:50,260 --> 00:24:54,840
Enthalpy is G minus TS.

417
00:24:54,840 --> 00:25:06,550
So dH is dG minus d of TS. dG,
we've got our new fundamental

418
00:25:06,550 --> 00:25:09,450
equation for G. We
plug it in here.

419
00:25:09,450 --> 00:25:12,440
Expand things out with a T dS
here, we can write immediately

420
00:25:12,440 --> 00:25:19,840
a fundamental equation
for H. dH is T dS.

421
00:25:19,840 --> 00:25:21,210
The beginning is going
to look just like

422
00:25:21,210 --> 00:25:22,310
what you've seen before.

423
00:25:22,310 --> 00:25:23,540
Plus V dp.

424
00:25:23,540 --> 00:25:28,630
Plus an extra term, which is
exactly the same extra term

425
00:25:28,630 --> 00:25:30,100
that we had in the fundamental
equation for

426
00:25:30,100 --> 00:25:33,550
G. Exactly the same.

427
00:25:33,550 --> 00:25:37,090
And every one of the other
fundamental questions can be

428
00:25:37,090 --> 00:25:40,600
derived in the similar way from
G. And they're going to

429
00:25:40,600 --> 00:25:50,280
be what you had before minus S
dT plus minus p dV plus the

430
00:25:50,280 --> 00:25:59,800
sum of the mu i's, dni,
and du is T dS minus p

431
00:25:59,800 --> 00:26:05,420
dV plus i mu i dni.

432
00:26:08,940 --> 00:26:13,320
So immediately we can see that
this mu, this quantity mu that

433
00:26:13,320 --> 00:26:16,930
we've defined as the derivative
of G with respect

434
00:26:16,930 --> 00:26:22,080
to the n's, we can write many
other equations for mu.

435
00:26:22,080 --> 00:26:24,880
There are many other
ways to derive it.

436
00:26:24,880 --> 00:26:29,020
Because this is the differential
for H. This is

437
00:26:29,020 --> 00:26:31,820
the first derivative of H
with respect to entropy.

438
00:26:31,820 --> 00:26:34,290
This is the derivative of H
with respect to pressure.

439
00:26:34,290 --> 00:26:36,840
And this is the derivative
of H with respect to n.

440
00:26:36,840 --> 00:26:38,320
Just formally, that's
what it is.

441
00:26:38,320 --> 00:26:41,020
When you write a differential.

442
00:26:41,020 --> 00:26:52,430
So formally, this is also mu i,
is dH/dni, keeping, now, we

443
00:26:52,430 --> 00:26:56,590
have to be very careful, keeping
the entropy and the

444
00:26:56,590 --> 00:26:57,410
pressure constant.

445
00:26:57,410 --> 00:26:59,900
Because those are
the variables.

446
00:26:59,900 --> 00:27:03,430
Keeping the entropy, and the
pressure, and all the other

447
00:27:03,430 --> 00:27:08,890
n's, constant.

448
00:27:08,890 --> 00:27:16,190
We can also write it as
dA/dni, keeping the

449
00:27:16,190 --> 00:27:18,520
temperature and the
volume constant.

450
00:27:18,520 --> 00:27:27,570
And all the other n's, or we
can write it as du/dni,

451
00:27:27,570 --> 00:27:34,400
keeping that this entropy and
the volume, and all the other

452
00:27:34,400 --> 00:27:35,420
n's, constant.

453
00:27:35,420 --> 00:27:42,060
So we have many ways to write
the chemical potential.

454
00:27:42,060 --> 00:27:51,460
They give you all
the same result.

455
00:27:51,460 --> 00:27:53,900
So this is the formal.

456
00:27:53,900 --> 00:27:59,640
Sort of the formal part of
the chemical potential.

457
00:27:59,640 --> 00:28:01,530
Now, what we really want to
show is that the chemical

458
00:28:01,530 --> 00:28:03,240
potential really is connected
to the Gibbs

459
00:28:03,240 --> 00:28:05,060
free energy per mole.

460
00:28:05,060 --> 00:28:08,490
That's going to be
the useful part.

461
00:28:08,490 --> 00:28:27,740
Let me get rid of this here.

462
00:28:27,740 --> 00:28:32,610
So I said earlier, at the
beginning of the lecture, that

463
00:28:32,610 --> 00:28:34,880
the Gibbs free energy per mole
was so important, we were

464
00:28:34,880 --> 00:28:37,010
going to call it the
chemical potential.

465
00:28:37,010 --> 00:28:38,510
And I said that here.

466
00:28:38,510 --> 00:28:39,950
And then I said, well,
we're going to define

467
00:28:39,950 --> 00:28:40,690
this here, the chemical.

468
00:28:40,690 --> 00:28:44,450
But I haven't equated
the two yet.

469
00:28:44,450 --> 00:28:46,710
I haven't proven to you that
in fact this quantity here,

470
00:28:46,710 --> 00:28:49,610
which we've formally defined
as the derivative of G with

471
00:28:49,610 --> 00:28:53,045
respect to n, is the Gibbs
free energy per

472
00:28:53,045 --> 00:28:54,930
mole, for this species.

473
00:28:54,930 --> 00:29:03,010
In fact, what we want to show
is that if I take the sum of

474
00:29:03,010 --> 00:29:08,260
all the chemical potentials,
times the number of moles per

475
00:29:08,260 --> 00:29:15,410
species, that that is the
total Gibbs free energy.

476
00:29:15,410 --> 00:29:19,480
In other words, that the
chemical potential for one

477
00:29:19,480 --> 00:29:25,390
species in the mixture is the
Gibbs free energy per mole for

478
00:29:25,390 --> 00:29:29,960
that species.

479
00:29:29,960 --> 00:29:34,280
Once we have that idea, then
we'll be able to talk about

480
00:29:34,280 --> 00:29:38,450
the concept of chemical
potential as this thing that

481
00:29:38,450 --> 00:29:40,360
we can use to look
at equilibrium.

482
00:29:40,360 --> 00:29:42,430
To look at going downhill
for the species.

483
00:29:42,430 --> 00:29:45,340
To see why the cell bursts
and all these things.

484
00:29:45,340 --> 00:29:47,420
Because now we understand that
Gibbs free energy is so

485
00:29:47,420 --> 00:29:49,170
important for equilibrium.

486
00:29:49,170 --> 00:29:50,680
We don't understand that
quite yet, with

487
00:29:50,680 --> 00:29:51,510
the chemical potential.

488
00:29:51,510 --> 00:29:54,690
So we got to make that
relation here.

489
00:29:54,690 --> 00:29:58,120
We need to go from the formal
definition to a relation that

490
00:29:58,120 --> 00:30:00,170
we can understand better,
because it includes the Gibbs

491
00:30:00,170 --> 00:30:02,810
free energy.

492
00:30:02,810 --> 00:30:07,670
OK, so that's our goal now.

493
00:30:07,670 --> 00:30:08,790
So let's see.

494
00:30:08,790 --> 00:30:09,920
Let's formally do this now.

495
00:30:09,920 --> 00:30:15,360
Let's define, let's
derive this.

496
00:30:15,360 --> 00:30:15,670
OK.

497
00:30:15,670 --> 00:30:21,890
So remember, our goal in this
derivation is to show that

498
00:30:21,890 --> 00:30:24,620
this is true.

499
00:30:24,620 --> 00:30:27,150
Or that this is true, here.

500
00:30:27,150 --> 00:30:28,770
And again, we're going
to start with the

501
00:30:28,770 --> 00:30:30,010
simplest system possible.

502
00:30:30,010 --> 00:30:33,890
We're going to start with
a two component system.

503
00:30:33,890 --> 00:30:46,300
And we can easily generalize
to multi-component.

504
00:30:46,300 --> 00:30:50,310
And in our derivation, what
we're going to be after is,

505
00:30:50,310 --> 00:30:51,750
we're going to start with the
Gibbs free energy, because

506
00:30:51,750 --> 00:30:53,640
that's where we always
start with.

507
00:30:53,640 --> 00:30:57,360
And we're going to remember that
by definition, mu i is

508
00:30:57,360 --> 00:31:04,640
dG/dni, So if somehow in our
derivation dG/dni falls out,

509
00:31:04,640 --> 00:31:05,640
that would be great.

510
00:31:05,640 --> 00:31:08,120
Because we'll be able to replace
this derivative with

511
00:31:08,120 --> 00:31:09,460
the chemical potential.

512
00:31:09,460 --> 00:31:11,650
So the goal was to find
something where this falls

513
00:31:11,650 --> 00:31:17,360
out, so we can replace it with
the chemical potential.

514
00:31:17,360 --> 00:31:18,700
We're going to start with
the fact that G is

515
00:31:18,700 --> 00:31:21,050
an extensive variable.

516
00:31:21,050 --> 00:31:24,380
So if I take G at a temperature
and pressure times

517
00:31:24,380 --> 00:31:28,800
some scaling factor for the
size of my system, lambda,

518
00:31:28,800 --> 00:31:31,770
number of moles of n1, lambda
times the number of moles of

519
00:31:31,770 --> 00:31:35,250
n2, if I double the size
of the system,

520
00:31:35,250 --> 00:31:36,290
lambda is equal to 2.

521
00:31:36,290 --> 00:31:38,820
If I half it, lambda
is equal to 1/2.

522
00:31:38,820 --> 00:31:41,290
Because it's extensive, this
is the same thing as lambda

523
00:31:41,290 --> 00:31:46,600
times G of temperature
and pressure, n1, n2.

524
00:31:46,600 --> 00:31:50,190
Just rewriting the fact that
Gibbs free energy is an

525
00:31:50,190 --> 00:31:52,390
extensive property.

526
00:31:52,390 --> 00:31:54,470
And lambda is an arbitrary
number here.

527
00:31:54,470 --> 00:31:56,480
Arbitrary variable.

528
00:31:56,480 --> 00:31:58,420
Now I'm going to take the
derivative of both sides with

529
00:31:58,420 --> 00:32:00,420
respect to lambda.

530
00:32:00,420 --> 00:32:04,550
So I'm going to take d d lambda
of this side here.

531
00:32:04,550 --> 00:32:12,010
And d d lambda of
that side here.

532
00:32:12,010 --> 00:32:14,070
Now, lambda here is inside
the variable here.

533
00:32:14,070 --> 00:32:16,900
So I'm going to have to
use the chain rule.

534
00:32:16,900 --> 00:32:18,760
To do this properly.

535
00:32:18,760 --> 00:32:30,290
So this is going to be dG/d
lambda n1, there's lambda

536
00:32:30,290 --> 00:32:32,510
sitting in the variable
lambda n1 here, times

537
00:32:32,510 --> 00:32:37,620
d lambda n1 d lambda.

538
00:32:37,620 --> 00:32:47,950
Plus dG/d lambda n2 times
d lambda n2 d lambda.

539
00:32:47,950 --> 00:32:49,470
Using the chain rule.

540
00:32:49,470 --> 00:32:50,930
And on this side here,
lambda's sitting

541
00:32:50,930 --> 00:32:51,820
straight out here.

542
00:32:51,820 --> 00:32:53,370
So this is very easy.

543
00:32:53,370 --> 00:32:55,020
This is G(T, p, n1, n2).

544
00:33:06,390 --> 00:33:07,130
Now, this is good.

545
00:33:07,130 --> 00:33:09,340
Because this is what
I'm looking for.

546
00:33:09,340 --> 00:33:11,850
I'm looking for the derivative
of g with respect to the

547
00:33:11,850 --> 00:33:12,850
number of moles.

548
00:33:12,850 --> 00:33:14,290
Because that's the chemical
potential.

549
00:33:14,290 --> 00:33:17,470
That was my goal up here, to
make sure in the derivation,

550
00:33:17,470 --> 00:33:19,880
somehow, this came out.

551
00:33:19,880 --> 00:33:24,820
And so it's coming
out right there.

552
00:33:24,820 --> 00:33:28,460
Right here and right here.

553
00:33:28,460 --> 00:33:31,800
Since the number of moles
is lambda n1, that first

554
00:33:31,800 --> 00:33:33,650
derivative here is just
the chemical potential

555
00:33:33,650 --> 00:33:35,450
of species 1 there.

556
00:33:35,450 --> 00:33:39,670
So mu 1, then we have d
lambda n1 d lambda.

557
00:33:39,670 --> 00:33:43,170
Well, d lambda n1 d lambda,
that's just n1.

558
00:33:43,170 --> 00:33:47,230
It's lambda times the number n1
that doesn't have anything

559
00:33:47,230 --> 00:33:48,070
to do with lambda.

560
00:33:48,070 --> 00:33:50,680
So this is n1.

561
00:33:50,680 --> 00:33:53,900
This is the chemical potential
of species 2.

562
00:33:53,900 --> 00:33:56,080
Again, the derivative of lambda
n2 with respect to

563
00:33:56,080 --> 00:33:58,400
lambda is just n2.

564
00:33:58,400 --> 00:34:04,830
And there is G here.

565
00:34:04,830 --> 00:34:07,260
It's a fairly simple derivation,
but it gets us

566
00:34:07,260 --> 00:34:08,750
exactly what we want.

567
00:34:08,750 --> 00:34:13,280
An association between this
formal definition of mu, up

568
00:34:13,280 --> 00:34:17,340
here, directly from taking
the differential.

569
00:34:17,340 --> 00:34:20,480
How much more formal can you
be, mathematically, here?

570
00:34:20,480 --> 00:34:24,120
To associating this formal
definition to

571
00:34:24,120 --> 00:34:27,540
the Gibbs free energy.

572
00:34:27,540 --> 00:34:34,880
Number of moles times mu 1,
number of moles times mu 2,

573
00:34:34,880 --> 00:34:40,630
this is the Gibbs free energy
per mole of species 1.

574
00:34:40,630 --> 00:34:45,410
Gibbs free energy per
mole of species 2.

575
00:34:45,410 --> 00:34:49,040
The sum of all the species of
the Gibbs free energy per mole

576
00:34:49,040 --> 00:34:52,200
of species i times the number
of moles of species i

577
00:34:52,200 --> 00:34:57,850
is G. Or, mu i.

578
00:34:57,850 --> 00:35:00,600
Voila, we've done it.

579
00:35:00,600 --> 00:35:02,500
This is what we wanted.

580
00:35:02,500 --> 00:35:06,360
The chemical potential is the
Gibbs free energy per mole.

581
00:35:06,360 --> 00:35:09,120
And in the mixture, it's the
Gibbs free energy per mole of

582
00:35:09,120 --> 00:35:11,800
the individual species
in that mixture.

583
00:35:11,800 --> 00:35:13,730
And if you want to know what
the total Gibbs free energy

584
00:35:13,730 --> 00:35:16,320
is, because if you have an
equilibrium, what you care

585
00:35:16,320 --> 00:35:18,240
about is the total Gibbs
free energy.

586
00:35:18,240 --> 00:35:21,750
It's not the Gibbs free energy
for one particular species.

587
00:35:21,750 --> 00:35:24,020
What's going to tell you whether
you have a minimum or

588
00:35:24,020 --> 00:35:29,220
not in your system, whether
you're at equilibrium, where

589
00:35:29,220 --> 00:35:32,500
you're at the lowest state
possible, is the total Gibbs

590
00:35:32,500 --> 00:35:35,350
free energy.

591
00:35:35,350 --> 00:35:37,010
Now we'll be able to
manipulate chemical

592
00:35:37,010 --> 00:35:40,120
potentials, of the individual
species, to get

593
00:35:40,120 --> 00:35:45,810
at this number here.

594
00:35:45,810 --> 00:35:46,820
Any questions?

595
00:35:46,820 --> 00:35:50,810
This is really, we're going
to see this over

596
00:35:50,810 --> 00:35:51,500
and over again now.

597
00:35:51,500 --> 00:35:52,030
This chemical potential.

598
00:35:52,030 --> 00:35:53,100
This idea.

599
00:35:53,100 --> 00:36:04,460
And it's not an easy concept.

600
00:36:04,460 --> 00:36:06,560
OK, let me give you
an example, then,

601
00:36:06,560 --> 00:36:13,140
of the water melting.

602
00:36:13,140 --> 00:36:17,350
And how the chemical potential
comes in, now, instead of

603
00:36:17,350 --> 00:36:18,470
using the chemical potential.

604
00:36:18,470 --> 00:36:30,160
Instead of the Gibbs
free energy.

605
00:36:30,160 --> 00:36:31,290
This is the phase transition.

606
00:36:31,290 --> 00:36:35,190
But it's not very different than
the cell bursting when

607
00:36:35,190 --> 00:36:39,410
you put it in distilled water.

608
00:36:39,410 --> 00:36:44,570
So, we take a glass of water
with an ice cube in it.

609
00:36:44,570 --> 00:36:47,580
H2O liquid.

610
00:36:47,580 --> 00:36:50,640
This is H2O solid.

611
00:36:50,640 --> 00:36:53,920
And I'm looking at the
melting process.

612
00:36:53,920 --> 00:36:58,470
I'm looking at a process where
I take a small number of

613
00:36:58,470 --> 00:37:01,950
molecules of water from
the solid phase.

614
00:37:01,950 --> 00:37:04,550
And I bring them to
the liquid phase.

615
00:37:04,550 --> 00:37:10,570
And I want to know, is this
process spontaneous or in

616
00:37:10,570 --> 00:37:17,970
equilibrium, or not possible?

617
00:37:17,970 --> 00:37:19,100
Is this process going
to go on?

618
00:37:19,100 --> 00:37:21,010
Is the direction of time
that this is melting.

619
00:37:21,010 --> 00:37:24,890
And I want to do this formally
thermodynamically.

620
00:37:24,890 --> 00:37:27,370
In terms of the chemical
potentials.

621
00:37:27,370 --> 00:37:30,750
That's going to be what we're
going to be talking about.

622
00:37:30,750 --> 00:37:35,020
So, formally then, what's going
on is, I'm taking nl

623
00:37:35,020 --> 00:37:41,960
moles of liquid water, H2O
liquid, which is in here.

624
00:37:41,960 --> 00:37:47,520
Plus ns moles of solid water.

625
00:37:47,520 --> 00:37:50,670
And I'm, this is my
initial state.

626
00:37:50,670 --> 00:37:59,440
My final state is nl, plus a
small number of moles, dn, of

627
00:37:59,440 --> 00:38:06,900
H2O liquids, H2O liquid, plus
ns minus dn, there's a

628
00:38:06,900 --> 00:38:09,150
conservation of the number
of molecules here.

629
00:38:09,150 --> 00:38:10,880
Whatever I add to the
liquid has to come

630
00:38:10,880 --> 00:38:13,040
from the solid here.

631
00:38:13,040 --> 00:38:19,440
Of H2O solid, of ice.

632
00:38:19,440 --> 00:38:23,210
And to know if this is
spontaneous or not, if this is

633
00:38:23,210 --> 00:38:26,540
done under constant temperature
and pressure, what

634
00:38:26,540 --> 00:38:31,070
variable should we look at?

635
00:38:31,070 --> 00:38:31,900
G, right.

636
00:38:31,900 --> 00:38:33,640
We want to look at the
Gibbs free energy.

637
00:38:33,640 --> 00:38:38,260
So what is G doing during
this process here?

638
00:38:38,260 --> 00:38:42,530
What is delta G here?

639
00:38:42,530 --> 00:38:45,010
Well, we have a way of doing
it now, in terms of the

640
00:38:45,010 --> 00:38:45,630
chemical potentials.

641
00:38:45,630 --> 00:38:51,720
Because we've just shown that
this is the case here.

642
00:38:51,720 --> 00:38:56,140
So G is the sum of the chemical
potentials times the

643
00:38:56,140 --> 00:38:58,020
number of moles in
the species.

644
00:38:58,020 --> 00:39:07,290
Therefore, delta G is going to
be equal to mu for the number

645
00:39:07,290 --> 00:39:09,240
of moles of l.

646
00:39:09,240 --> 00:39:20,320
Liquid, dn, number of moles
of liquid, plus mu s dns.

647
00:39:20,320 --> 00:39:22,840
So the change in G is going to
be equal to the chemical

648
00:39:22,840 --> 00:39:25,700
potential times the change in
the species, which is in a

649
00:39:25,700 --> 00:39:28,780
liquid form, plus the chemical
potential of the solid.

650
00:39:28,780 --> 00:39:32,520
Times the change in the species
in the solid form.

651
00:39:32,520 --> 00:39:37,700
Now, dns is equal
to minus dnl.

652
00:39:37,700 --> 00:39:39,300
This is what we did
right here.

653
00:39:39,300 --> 00:39:41,880
You take a certain number of
moles from the solid form.

654
00:39:41,880 --> 00:39:42,940
You put it to the liquid form.

655
00:39:42,940 --> 00:39:44,490
That the dn up here.

656
00:39:44,490 --> 00:39:46,970
And you've got to have the
negative of it, up here.

657
00:39:46,970 --> 00:39:49,250
So dns is minus dnl.

658
00:39:49,250 --> 00:39:53,380
Which is minus dn.

659
00:39:53,380 --> 00:40:11,510
Delta G is dn times mu
l minus mu solid.

660
00:40:11,510 --> 00:40:16,800
So now we can rephrase this,
it's all rephrasing.

661
00:40:16,800 --> 00:40:18,300
It's all basically
the same thing.

662
00:40:18,300 --> 00:40:22,490
But, we can rephrase this
process by asking the

663
00:40:22,490 --> 00:40:28,890
question, is the chemical
potential of the liquid

664
00:40:28,890 --> 00:40:33,150
greater than, equal to, or
less than the chemical

665
00:40:33,150 --> 00:40:36,320
potential of the solid?

666
00:40:36,320 --> 00:40:37,640
Of the water in the solid.

667
00:40:37,640 --> 00:40:40,530
So the chemical potential of the
water in the liquid phase

668
00:40:40,530 --> 00:40:44,160
is greater than the chemical
potential of the water in the

669
00:40:44,160 --> 00:40:49,040
solid phase, mu l is greater
than mu s, then delta G

670
00:40:49,040 --> 00:40:52,560
becomes positive.

671
00:40:52,560 --> 00:40:59,320
In that case, delta G is
greater than zero.

672
00:40:59,320 --> 00:41:03,150
And that's not going
to happen.

673
00:41:03,150 --> 00:41:06,840
On the other hand, if the
chemical potential of the

674
00:41:06,840 --> 00:41:11,520
water molecules in the liquid
phase is smaller than the

675
00:41:11,520 --> 00:41:16,450
chemical potential of the water
in the solid phase, mu s

676
00:41:16,450 --> 00:41:19,870
is bigger than mu l, this
becomes a negative number.

677
00:41:19,870 --> 00:41:24,730
Delta G is less than zero.

678
00:41:24,730 --> 00:41:27,440
And this will happen
spontaneously.

679
00:41:27,440 --> 00:41:30,200
So that illustrates this idea,
that the chemical potential of

680
00:41:30,200 --> 00:41:33,890
a species will want to go, so
the species will want to go,

681
00:41:33,890 --> 00:41:37,410
where it can minimize its
chemical potential.

682
00:41:37,410 --> 00:41:41,200
So in this case here, when we
have the spontaneous process

683
00:41:41,200 --> 00:41:44,490
of the water, of the ice cube,
melting, you can think of it

684
00:41:44,490 --> 00:41:47,240
as these water molecules
that are in the ice

685
00:41:47,240 --> 00:41:49,350
phase looking around.

686
00:41:49,350 --> 00:41:51,000
They know what their chemical
potential here

687
00:41:51,000 --> 00:41:51,670
is in the ice phase.

688
00:41:51,670 --> 00:41:53,290
They're looking around, they're
looking to see the

689
00:41:53,290 --> 00:41:54,270
water phase.

690
00:41:54,270 --> 00:41:57,960
And they see that in the water
phase, those water molecules

691
00:41:57,960 --> 00:41:59,730
have a smaller chemical
potential.

692
00:41:59,730 --> 00:42:02,320
They're happier.

693
00:42:02,320 --> 00:42:08,680
And so these solid water
molecules are jealous.

694
00:42:08,680 --> 00:42:11,500
And they want to go in
the water phase.

695
00:42:11,500 --> 00:42:13,870
And the ice cube's
going to melt.

696
00:42:13,870 --> 00:42:15,790
And it all has to do with this
difference in chemical

697
00:42:15,790 --> 00:42:19,160
potentials for the water.

698
00:42:19,160 --> 00:42:22,060
And the same thing happens for
the water molecules that are

699
00:42:22,060 --> 00:42:24,170
inside or outside of that
cell that you put in

700
00:42:24,170 --> 00:42:25,590
the distilled water.

701
00:42:25,590 --> 00:42:27,490
The water molecules in the
distilled water have a

702
00:42:27,490 --> 00:42:31,400
chemical potential which is
higher than the water

703
00:42:31,400 --> 00:42:34,580
molecules inside the cell.

704
00:42:34,580 --> 00:42:36,220
And they don't want
to be like that.

705
00:42:36,220 --> 00:42:39,740
They want to change, the system
wants to change, until

706
00:42:39,740 --> 00:42:41,960
the water molecules couldn't
care less whether they're in

707
00:42:41,960 --> 00:42:46,580
the water phase, or outside
or inside the cell.

708
00:42:46,580 --> 00:42:51,120
The system is going to change
until the water molecules have

709
00:42:51,120 --> 00:42:53,850
the same chemical potential
everywhere.

710
00:42:53,850 --> 00:43:00,540
Where they don't have to choose
one place or the other.

711
00:43:00,540 --> 00:43:02,880
And so that gives us,
immediately, what we're going

712
00:43:02,880 --> 00:43:05,530
to need when we talk
about equilibrium.

713
00:43:05,530 --> 00:43:09,450
Equilibrium, chemical
equilibrium, is going to be

714
00:43:09,450 --> 00:43:13,680
where the chemical potential
of a species is the same

715
00:43:13,680 --> 00:43:17,240
everywhere in the system.

716
00:43:17,240 --> 00:43:21,900
So at 0 degrees Celsius, one
bar, which is the melting

717
00:43:21,900 --> 00:43:26,530
point of water, the chemical
potential of a molecule of

718
00:43:26,530 --> 00:43:30,120
water in the ice phase and in
the liquid phase is the same.

719
00:43:30,120 --> 00:43:31,620
That's the definition of
the melting point.

720
00:43:31,620 --> 00:43:32,065
It doesn't care.

721
00:43:32,065 --> 00:43:33,460
It could go either way.

722
00:43:33,460 --> 00:43:34,950
It's an equilibrium.

723
00:43:34,950 --> 00:43:39,420
You take an ice cube, water,
liquid water, 0 degrees

724
00:43:39,420 --> 00:43:40,950
Celsius, one bar.

725
00:43:40,950 --> 00:43:42,910
You come back three
days later.

726
00:43:42,910 --> 00:43:43,900
It's the same.

727
00:43:43,900 --> 00:43:45,610
Come back a week later,
it's the same.

728
00:43:45,610 --> 00:43:49,200
It's an equilibrium.

729
00:43:49,200 --> 00:43:52,050
Chemical potential of the water
species is the same

730
00:43:52,050 --> 00:43:52,920
everywhere.

731
00:43:52,920 --> 00:43:54,000
It's an equilibrium.

732
00:43:54,000 --> 00:43:59,200
And I'm just repeating that
because this is so important.

733
00:43:59,200 --> 00:44:04,810
OK, any questions?

734
00:44:04,810 --> 00:44:08,720
The last thing we're going to
do is to illustrate also the

735
00:44:08,720 --> 00:44:19,150
importance of mixing to the
chemical potential.

736
00:44:19,150 --> 00:44:27,210
So I'm going to set up sort of
an arbitrary system here.

737
00:44:27,210 --> 00:44:32,390
This is kind of like the cell,
or the fish, also, idea.

738
00:44:32,390 --> 00:44:35,790
I'm going to put a system where
on one side I have a

739
00:44:35,790 --> 00:44:41,295
pure gas, A. On the other side,
I have a mixture of A

740
00:44:41,295 --> 00:44:46,110
and B. And here is going to be
a membrane that only allows A

741
00:44:46,110 --> 00:44:51,450
to go through.

742
00:44:51,450 --> 00:44:54,710
Only A can go through
that membrane.

743
00:44:54,710 --> 00:44:57,215
They're going to be partial
pressures in here, p prime B,

744
00:44:57,215 --> 00:45:01,930
and p prime A. For the gas
pressures on that side, and

745
00:45:01,930 --> 00:45:05,340
pressure on that side
is p sub A.

746
00:45:05,340 --> 00:45:11,240
And my goal in this example
here is to show that if I

747
00:45:11,240 --> 00:45:16,970
compare the chemical potential
of a species in a mixture,

748
00:45:16,970 --> 00:45:23,320
where the temperature and the
pressure total are T and p,

749
00:45:23,320 --> 00:45:26,150
and I compare that to the
chemical potential of the same

750
00:45:26,150 --> 00:45:31,040
species when it's pure, when
it's not mixed with anything

751
00:45:31,040 --> 00:45:33,560
else, under the same temperature
and pressure

752
00:45:33,560 --> 00:45:41,110
conditions, that, in fact that
that equals sign is not there.

753
00:45:41,110 --> 00:45:42,240
That's not what I'm
trying to show.

754
00:45:42,240 --> 00:45:43,230
I'm trying to show
that there's a

755
00:45:43,230 --> 00:45:47,370
less-than sign right here.

756
00:45:47,370 --> 00:45:51,640
To show that if you take, again,
this is the cell idea.

757
00:45:51,640 --> 00:45:55,050
If you take the water and the
distilled water, under

758
00:45:55,050 --> 00:45:56,140
constant temperature
and pressure

759
00:45:56,140 --> 00:45:59,400
conditions, it's pure.

760
00:45:59,400 --> 00:46:01,240
And it's looking at the cell.

761
00:46:01,240 --> 00:46:03,356
And inside the cell is
there, boy, is there

762
00:46:03,356 --> 00:46:04,820
a mixture of things.

763
00:46:04,820 --> 00:46:06,730
There's salts, there are
proteins, there are all sorts

764
00:46:06,730 --> 00:46:10,670
of things, right?

765
00:46:10,670 --> 00:46:12,350
It's a mixed system.

766
00:46:12,350 --> 00:46:14,230
The water in that mixed
system, under the same

767
00:46:14,230 --> 00:46:16,910
temperature and pressure
conditions, the chemical

768
00:46:16,910 --> 00:46:19,970
potential of that water
molecule is less.

769
00:46:19,970 --> 00:46:23,560
And that's just an
entropy thing.

770
00:46:23,560 --> 00:46:25,080
Entropy wants to increase.

771
00:46:25,080 --> 00:46:32,940
It just wants to be in a
place with high energy.

772
00:46:32,940 --> 00:46:34,240
It's the Gibbs free energy.

773
00:46:34,240 --> 00:46:36,240
Gibbs free energy has
enthalpy and entropy

774
00:46:36,240 --> 00:46:37,230
incorporated into it.

775
00:46:37,230 --> 00:46:39,860
The enthalpy's not
doing anything.

776
00:46:39,860 --> 00:46:43,820
It's all driven by entropy.

777
00:46:43,820 --> 00:46:49,250
So this is what we're going
to try to show.

778
00:46:49,250 --> 00:46:51,960
And I'm not going to
get to it today.

779
00:46:51,960 --> 00:46:56,180
We'll start with it
on Wednesday.

780
00:46:56,180 --> 00:47:01,250
And I'll let your ruminate on
this for the next few days.