1
00:00:00 --> 00:00:01
2
00:00:01 --> 00:00:02
The following content is
provided under a Creative
3
00:00:02 --> 00:00:03
Commons license.
4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to
5
00:00:06 --> 00:00:10
offer high quality educational
resources for free.
6
00:00:10 --> 00:00:13
To make a donation or view
additional materials from
7
00:00:13 --> 00:00:17
hundreds of MIT courses,
visit MIT OpenCourseWare
8
00:00:17 --> 00:00:21
at ocw.mit.edu.
9
00:00:21 --> 00:00:21
PROFESSOR: Alright.
10
00:00:21 --> 00:00:24
Well, we've been looking in the
last couple lectures at a
11
00:00:24 --> 00:00:27
really important topic
in thermodynamics.
12
00:00:27 --> 00:00:31
Which is, how do you tell
what's going to happen.
13
00:00:31 --> 00:00:33
Which way does a
process want to go?
14
00:00:33 --> 00:00:36
Which way will it
go spontaneously?
15
00:00:36 --> 00:00:39
And if it goes in one direction
or another, where does is lead?
16
00:00:39 --> 00:00:42
In other words, what is
the equilibrium state?
17
00:00:42 --> 00:00:44
And this is just an incredibly
important area that
18
00:00:44 --> 00:00:48
thermodynamics allows
us to speak to.
19
00:00:48 --> 00:00:52
So we started to see this.
20
00:00:52 --> 00:01:09
Sort of direction of
spontaneous change.
21
00:01:09 --> 00:01:10
And where the equilibrium lies.
22
00:01:10 --> 00:01:18
So what we did is, remember we
started with the second law.
23
00:01:18 --> 00:01:20
Right?
24
00:01:20 --> 00:01:26
That dS is greater
than dq over T.
25
00:01:26 --> 00:01:33
And for the spontaneous change
which happens irreversibly That
26
00:01:33 --> 00:01:37
means that'll be
dq irreversible.
27
00:01:37 --> 00:01:39
It would be equal for
the reversible case.
28
00:01:39 --> 00:01:49
And we combine this with first
law, which for the case of
29
00:01:49 --> 00:01:57
pressure volume changes
we write as this.
30
00:01:57 --> 00:02:01
And so what this gave us was
a very, very useful general
31
00:02:01 --> 00:02:03
criterion for determining
whether something
32
00:02:03 --> 00:02:06
happened spontaneously.
33
00:02:06 --> 00:02:18
Namely, du plus p external dV
minus T for the surroundings
34
00:02:18 --> 00:02:26
dS, is greater than zero.
35
00:02:26 --> 00:02:31
Sorry.
36
00:02:31 --> 00:02:34
It's less than zero.
37
00:02:34 --> 00:02:43
And this is for any
spontaneous change.
38
00:02:43 --> 00:02:50
If it equals zero, then
we're at equilibrium.
39
00:02:50 --> 00:02:59
And if it's greater than
zero, then the process
40
00:02:59 --> 00:03:01
goes the other way.
41
00:03:01 --> 00:03:04
We would write the process in
the reverse to have it be less
42
00:03:04 --> 00:03:09
than 0 and it would
go spontaneously.
43
00:03:09 --> 00:03:15
And based on this one result,
we then looked under various
44
00:03:15 --> 00:03:20
constraints and said OK, what
about looking at our variables,
45
00:03:20 --> 00:03:24
volume, pressure, temperature,
other things, entropy, if we
46
00:03:24 --> 00:03:28
constrain those, what's the
condition for equilibrium?
47
00:03:28 --> 00:03:32
And that's what led us to a
number of results to determine
48
00:03:32 --> 00:03:35
what quantities we even
need to be looking at.
49
00:03:35 --> 00:03:37
To figure out equilibrium.
50
00:03:37 --> 00:03:38
And what the conditions were.
51
00:03:38 --> 00:03:41
And so what we discovered
were the following.
52
00:03:41 --> 00:03:49
This one, which we already
had seen, which is dS,
53
00:03:49 --> 00:03:50
is greater than zero.
54
00:03:50 --> 00:03:54
Change in entropy is
greater than zero, for
55
00:03:54 --> 00:04:02
an isolated system.
56
00:04:02 --> 00:04:08
We also saw that dS for
constant H and p was greater
57
00:04:08 --> 00:04:18
than zero. du, regular energy,
at constant entropy and
58
00:04:18 --> 00:04:41
volume is less then zero.
59
00:04:41 --> 00:04:43
And u is minimized
at equilibrium.
60
00:04:43 --> 00:04:47
And this is the familiar result
from ordinary mechanics, where
61
00:04:47 --> 00:04:49
you're not worrying about
something like entropy for a
62
00:04:49 --> 00:04:51
whole collection of particles.
63
00:04:51 --> 00:04:55
That is, you minimize potential
energy and you see things
64
00:04:55 --> 00:04:58
falling under the force of
gravity and so forth, going to
65
00:04:58 --> 00:05:05
potential energy minima in
conformance with this result.
66
00:05:05 --> 00:05:10
dH, S and p is less than zero.
67
00:05:10 --> 00:05:16
So our H is u plus
pV, as you know.
68
00:05:16 --> 00:05:27
And H is minimized
at equilibrium.
69
00:05:27 --> 00:05:32
And this is, of course,
with constant S V.
70
00:05:32 --> 00:05:37
This is constant S and p.
71
00:05:37 --> 00:05:41
But of course, the need to
have entropy constrained is
72
00:05:41 --> 00:05:44
never going to be the most
convenient one experimental.
73
00:05:44 --> 00:05:46
There may be circumstances
under which it's the
74
00:05:46 --> 00:05:49
case, but it's often
difficult to control.
75
00:05:49 --> 00:05:54
On the other hand, temperature,
volume and pressure are
76
00:05:54 --> 00:05:57
variables that are much easier
in the lab to keep constant.
77
00:05:57 --> 00:05:58
To keep control over.
78
00:05:58 --> 00:06:02
And so that led us to the
definitions of other energy
79
00:06:02 --> 00:06:05
quantities, the Helmholtz
and Gibbs free energy.
80
00:06:05 --> 00:06:11
We discovered that the quantity
dA, under conditions of
81
00:06:11 --> 00:06:16
constant volume and
temperature, is less than zero.
82
00:06:16 --> 00:06:20
And A is u minus TS.
83
00:06:20 --> 00:06:28
84
00:06:28 --> 00:06:34
And A is minimized at
equilibrium, under conditions
85
00:06:34 --> 00:06:39
of constant T and V.
86
00:06:39 --> 00:06:44
And finally, and in many cases
the most important of the
87
00:06:44 --> 00:06:46
results, because of the
conditions it applies to, we
88
00:06:46 --> 00:06:53
saw that this Gibbs free
energy is less than zero,
89
00:06:53 --> 00:06:57
that's our condition
for spontaneous change.
90
00:06:57 --> 00:07:05
Where the Gibbs free energy, u
plus pV minus TS is H minus TS.
91
00:07:05 --> 00:07:08
92
00:07:08 --> 00:07:22
Also A plus pV and G is
minimized at equilibrium
93
00:07:22 --> 00:07:26
with constant temperature
and pressure.
94
00:07:26 --> 00:07:27
And that's why the Gibbs
free energy is just so
95
00:07:27 --> 00:07:28
enormously important.
96
00:07:28 --> 00:07:31
Because so much of what we do
in chemistry does take place
97
00:07:31 --> 00:07:34
with constant temperature
and pressure.
98
00:07:34 --> 00:07:37
So we have this condition
that's established in a
99
00:07:37 --> 00:07:40
quantity that we know
how to calculate.
100
00:07:40 --> 00:07:44
That tells us the direction of
spontaneous change for ordinary
101
00:07:44 --> 00:07:47
processes, chemical processes,
mixing and you name it, under
102
00:07:47 --> 00:07:54
conditions that are easy
to achieve in the lab.
103
00:07:54 --> 00:08:01
OK, now what we'd like to do is
be able to calculate any of
104
00:08:01 --> 00:08:04
these quantities in terms of
temperature, pressure,
105
00:08:04 --> 00:08:05
volume properties.
106
00:08:05 --> 00:08:08
That is, in terms of
equations of state.
107
00:08:08 --> 00:08:09
For any material.
108
00:08:09 --> 00:08:12
Then we would really be able to
essentially calculate anything.
109
00:08:12 --> 00:08:14
Anything thermodynamic.
110
00:08:14 --> 00:08:15
About a material.
111
00:08:15 --> 00:08:18
Of course, that's assuming we
know the equation of state.
112
00:08:18 --> 00:08:20
We may or may not.
113
00:08:20 --> 00:08:25
But because in many cases we
can reasonably either model or
114
00:08:25 --> 00:08:28
measure equations of state,
collect data for a material for
115
00:08:28 --> 00:08:32
its temperature, pressure,
volume relations, then in fact
116
00:08:32 --> 00:08:35
if we can relate all these
quantities to those, then in
117
00:08:35 --> 00:08:37
fact we really can calculate
essentially all of
118
00:08:37 --> 00:08:38
the thermodynamics.
119
00:08:38 --> 00:08:42
For the material.
120
00:08:42 --> 00:08:58
So let's relate the
thermodynamic quantities to
121
00:08:58 --> 00:09:07
equation of state p, V, T data.
122
00:09:07 --> 00:09:12
And we can do that by going
through and deriving what we'll
123
00:09:12 --> 00:09:16
call the fundamental equations
of thermodynamics. that'll
124
00:09:16 --> 00:09:17
provide these relations.
125
00:09:17 --> 00:09:19
And at this point we know
enough to do this in a
126
00:09:19 --> 00:09:21
straightforward way.
127
00:09:21 --> 00:09:38
So if we start with a relation
for energy, T dS minus p dV.
128
00:09:38 --> 00:09:43
Where u is written as a
function of entropy and volume.
129
00:09:43 --> 00:09:47
And we've seen that that's
generally the case.
130
00:09:47 --> 00:09:59
It comes from the fact that dq
reversible is T dS, and dw
131
00:09:59 --> 00:10:04
reversible is minus p dV.
132
00:10:04 --> 00:10:08
And of course du is
the some of those.
133
00:10:08 --> 00:10:14
So, this is generally true.
134
00:10:14 --> 00:10:15
Since these are all
state functions.
135
00:10:15 --> 00:10:19
That is, this is derived in the
case for reversible paths.
136
00:10:19 --> 00:10:23
But since these are all simply
state functions and quantities,
137
00:10:23 --> 00:10:25
this is generally true.
138
00:10:25 --> 00:10:28
Now we can use it to derive
differential relations for
139
00:10:28 --> 00:10:29
all of the thermodynamics
quantities.
140
00:10:29 --> 00:10:32
So let's just go
through and do that.
141
00:10:32 --> 00:10:38
So H is u plus pV.
142
00:10:38 --> 00:10:41
143
00:10:41 --> 00:10:49
So dH is just du plus
p dV plus V dp.
144
00:10:49 --> 00:10:53
And now we're just going
to substitute du in here.
145
00:10:53 --> 00:10:58
And the p dV terms
are going to cancel.
146
00:10:58 --> 00:11:08
So we have the result that
dH is T dS plus V dp.
147
00:11:08 --> 00:11:11
Right?
148
00:11:11 --> 00:11:17
And that shows us that H is
written naturally as a function
149
00:11:17 --> 00:11:23
of entropy and pressure.
150
00:11:23 --> 00:11:26
And now let's keep going.
151
00:11:26 --> 00:11:29
A is u minus TS.
152
00:11:29 --> 00:11:31
153
00:11:31 --> 00:11:41
dA is du minus T dS minus S dT.
154
00:11:41 --> 00:11:42
We're going to do
the same thing.
155
00:11:42 --> 00:11:45
Substitute this for du.
156
00:11:45 --> 00:11:51
This time, the T dS terms
are going to cancel.
157
00:11:51 --> 00:11:58
So we have dA is minus
S dT minus T dS.
158
00:11:58 --> 00:12:02
159
00:12:02 --> 00:12:03
That can't be right.
160
00:12:03 --> 00:12:06
And it isn't.
161
00:12:06 --> 00:12:12
Minus S dT, that's the p dV
term that's left, minus p dV.
162
00:12:12 --> 00:12:15
And it shows us that A is
written naturally as a
163
00:12:15 --> 00:12:24
function of T and V.
164
00:12:24 --> 00:12:27
G, we can write in any
of a number of ways.
165
00:12:27 --> 00:12:31
Let's write it as H minus TS.
166
00:12:31 --> 00:12:40
So dG is dH minus
T dS minus S dT.
167
00:12:40 --> 00:12:43
Here's dH.
168
00:12:43 --> 00:12:46
We'll substitute that
in, and the T dS terms
169
00:12:46 --> 00:12:48
are going to cancel.
170
00:12:48 --> 00:12:56
So dG is minus S dT plus V dp.
171
00:12:56 --> 00:12:59
172
00:12:59 --> 00:13:02
And this shows that G is
written naturally as a
173
00:13:02 --> 00:13:10
function of T and p.
174
00:13:10 --> 00:13:22
So these, which we will exalt
and celebrate by our
175
00:13:22 --> 00:13:30
sparingly-used colored chalk,
are our fundamental equations
176
00:13:30 --> 00:13:54
of thermodynamics.
177
00:13:54 --> 00:13:58
So what they do is, they're
describing how these
178
00:13:58 --> 00:14:03
thermodynamic properties
change, in terms of only state
179
00:14:03 --> 00:14:07
functions and state variables.
180
00:14:07 --> 00:14:08
Very, very useful.
181
00:14:08 --> 00:14:11
And that's what it means, when
we say well, it's natural then,
182
00:14:11 --> 00:14:14
to express say, G as a function
of T and p, that's
183
00:14:14 --> 00:14:14
what we're saying.
184
00:14:14 --> 00:14:17
Is that we can express
its changes in terms
185
00:14:17 --> 00:14:19
of these variables.
186
00:14:19 --> 00:14:25
Related only through quantities
that are functions of state.
187
00:14:25 --> 00:14:29
I don't need to know about
a specific path here.
188
00:14:29 --> 00:14:33
If I know about the states
involved, I just need to
189
00:14:33 --> 00:14:40
know what the volume
was in each of them.
190
00:14:40 --> 00:14:45
Now, before, of course, in the
first part of the class we
191
00:14:45 --> 00:14:49
started out looking at u and
then looking at H not as
192
00:14:49 --> 00:14:53
functions of S and V or S and
p, but as functions of
193
00:14:53 --> 00:14:55
temperature, mostly.
194
00:14:55 --> 00:14:58
In general, temperature
and volume or pressure.
195
00:14:58 --> 00:15:01
And it doesn't mean that
something was somehow
196
00:15:01 --> 00:15:04
wrong with that.
197
00:15:04 --> 00:15:07
It certainly is, it still
is going to be useful
198
00:15:07 --> 00:15:08
to do thermochemistry.
199
00:15:08 --> 00:15:11
To ask questions like how much
heat is released in a chemical
200
00:15:11 --> 00:15:15
reaction that takes place
at constant temperature.
201
00:15:15 --> 00:15:17
Not one of these variables.
202
00:15:17 --> 00:15:19
And we can calculate that.
203
00:15:19 --> 00:15:22
So it's not that we're
somehow throwing away
204
00:15:22 --> 00:15:23
our ability to do that.
205
00:15:23 --> 00:15:28
However, the thing to remember
is, when you look at heats of
206
00:15:28 --> 00:15:32
reaction under those conditions
it's all well and good.
207
00:15:32 --> 00:15:35
But it doesn't tell you, this
is the direction that the
208
00:15:35 --> 00:15:37
reaction is going to go.
209
00:15:37 --> 00:15:41
It doesn't tell you, this is
the equilibrium concentration
210
00:15:41 --> 00:15:43
that you'll end up with.
211
00:15:43 --> 00:15:46
That doesn't come out of
what we calculated before
212
00:15:46 --> 00:15:47
in thermochemistry.
213
00:15:47 --> 00:15:51
What does come out, which is
very useful is, if you do run
214
00:15:51 --> 00:15:54
the reaction, here's how
much heat comes out.
215
00:15:54 --> 00:15:57
And if you want to run a
furnace and provide energy,
216
00:15:57 --> 00:16:01
that's an extremely important
thing to be able to calculate.
217
00:16:01 --> 00:16:03
Because you're going to run it
and you'll probably find
218
00:16:03 --> 00:16:05
conditions under which you can
run it more or less
219
00:16:05 --> 00:16:07
to completion.
220
00:16:07 --> 00:16:10
But it doesn't tell you, by
itself, which direction
221
00:16:10 --> 00:16:14
things run in.
222
00:16:14 --> 00:16:17
Whereas under these conditions,
these quantities, if you look
223
00:16:17 --> 00:16:20
at free energy change, for
example, at constant
224
00:16:20 --> 00:16:24
temperature and pressure, you
can still calculate H.
225
00:16:24 --> 00:16:26
You can still calculate
the heat that's released.
226
00:16:26 --> 00:16:28
This is what will tell you
under some particular
227
00:16:28 --> 00:16:31
conditions what will
actually happen.
228
00:16:31 --> 00:16:33
Where will you end up.
229
00:16:33 --> 00:16:34
Very, very important,
of course, to be able
230
00:16:34 --> 00:16:39
to understand that.
231
00:16:39 --> 00:16:55
Now, it's also very useful to
look at some of the relations
232
00:16:55 --> 00:16:57
that come out of these
fundamental equations.
233
00:16:57 --> 00:16:59
And they're straightforward
to derive.
234
00:16:59 --> 00:17:04
So, all I want to do now is
look at the derivatives of the
235
00:17:04 --> 00:17:06
free energies with respect
to temperature and
236
00:17:06 --> 00:17:10
volume and pressure.
237
00:17:10 --> 00:17:15
So for example, if I look at A,
which we now have written as
238
00:17:15 --> 00:17:19
the function of T and V, of
course, in general I can always
239
00:17:19 --> 00:17:25
write dA as partial of A, with
respect to T at constant volume
240
00:17:25 --> 00:17:29
dT, plus partial of A with
respect to V, at constant
241
00:17:29 --> 00:17:32
temperature dV.
242
00:17:32 --> 00:17:35
And I know what those
turn out to be.
243
00:17:35 --> 00:17:41
It's minus S dT minus p dV.
244
00:17:41 --> 00:17:42
So what does that tell me?
245
00:17:42 --> 00:17:47
It tells me that the partial
of A with respect to T at
246
00:17:47 --> 00:17:51
constant V is minus S.
247
00:17:51 --> 00:17:52
Right?
248
00:17:52 --> 00:17:55
In other words, now I know how
to tell how the Helmholtz
249
00:17:55 --> 00:17:59
free energy changes as a
function of temperature.
250
00:17:59 --> 00:18:04
Or as a function of volume.
dA/dV, at constant T,
251
00:18:04 --> 00:18:08
must be negative p.
252
00:18:08 --> 00:18:12
Things that I can measure.
253
00:18:12 --> 00:18:14
So I can in a very
straightforward way say,
254
00:18:14 --> 00:18:19
OK, well, here is my
Helmholtz free energy.
255
00:18:19 --> 00:18:22
If I'm working under conditions
of constant temperature and
256
00:18:22 --> 00:18:23
volume, that's very useful.
257
00:18:23 --> 00:18:26
Now, if I want to change those
quantities; change the
258
00:18:26 --> 00:18:28
temperature, change the
volume, how will it change?
259
00:18:28 --> 00:18:32
Well, I can, for any given
case, measure the pressure,
260
00:18:32 --> 00:18:35
determine the entropy and
I'll know what the slope
261
00:18:35 --> 00:18:38
of change will be.
262
00:18:38 --> 00:18:45
Similarly for G as a function
of temperature and pressure, I
263
00:18:45 --> 00:18:47
can go through the
same procedure.
264
00:18:47 --> 00:18:54
That is, it's easy to write
down straight away that dG,
265
00:18:54 --> 00:18:59
with respect to temperature at
constant pressure is minus S.
266
00:18:59 --> 00:19:03
That is, this is, dG/dT
at constant pressure.
267
00:19:03 --> 00:19:18
And this is dG/dp at
constant temperature.
268
00:19:18 --> 00:19:22
So again with the Gibbs free
energy, now I see how to
269
00:19:22 --> 00:19:25
determine, if I change the
pressure, if I change the
270
00:19:25 --> 00:19:29
temperature by some modest
amount, how much is the Gibbs
271
00:19:29 --> 00:19:30
free energy going to change?
272
00:19:30 --> 00:19:39
Well, it's easy to see.
273
00:19:39 --> 00:19:43
These two relations involving
entropy are also useful because
274
00:19:43 --> 00:19:48
they'll let us see how entropy
depends on volume and pressure.
275
00:19:48 --> 00:20:09
And let me show you
how that goes.
276
00:20:09 --> 00:20:22
Now, you've already seen how
entropy depends on temperature.
277
00:20:22 --> 00:20:26
We've already seen that,
going to write dS as
278
00:20:26 --> 00:20:32
dq reversible over T.
279
00:20:32 --> 00:20:42
And it's Cv dT over T
at constant volume.
280
00:20:42 --> 00:20:51
It's Cp dT over T at
constant pressure.
281
00:20:51 --> 00:20:56
So we know that dS/dT at
constant volume is Cv over
282
00:20:56 --> 00:21:03
T, and dS/dT at constant
pressure is Cp, over T.
283
00:21:03 --> 00:21:08
And we've seen that on
a number of occasions.
284
00:21:08 --> 00:21:11
So that tells us what to
do to know the entropy as
285
00:21:11 --> 00:21:13
the temperature changes.
286
00:21:13 --> 00:21:19
But now, what happens if,
instead we look at what happens
287
00:21:19 --> 00:21:21
when we go to some state one
to some other state two
288
00:21:21 --> 00:21:22
and it's the pressure.
289
00:21:22 --> 00:21:24
Or the volume, that changes.
290
00:21:24 --> 00:21:28
And by the way, just to be
explicit about this, let's take
291
00:21:28 --> 00:21:32
this example, it means that
delta S, if we undergo a
292
00:21:32 --> 00:21:35
change from, say, T1 to T2.
293
00:21:35 --> 00:21:42
There's Cp over T dT.
294
00:21:42 --> 00:21:51
So it's Cp log of T2 over
T1, and we saw this before.
295
00:21:51 --> 00:21:56
So now, instead, let's
look at some process.
296
00:21:56 --> 00:21:59
State one goes to state two.
297
00:21:59 --> 00:22:04
Let's have constant T.
298
00:22:04 --> 00:22:09
And look at what happens if
pressure goes from p1 to p2.
299
00:22:09 --> 00:22:14
Or volume goes from V1 to V2.
300
00:22:14 --> 00:22:17
And see what happens there.
301
00:22:17 --> 00:22:20
We looked at pressure change
before, actually, in discussing
302
00:22:20 --> 00:22:24
the third law, the fact that
the entropy goes to zero as the
303
00:22:24 --> 00:22:27
absolute temperature goes to
zero for a pure,
304
00:22:27 --> 00:22:28
perfect crystal.
305
00:22:28 --> 00:22:31
But, actually, we didn't
do that in a general way.
306
00:22:31 --> 00:22:34
We just treated the one case
of an ideal gas as the
307
00:22:34 --> 00:22:36
temperature is reduced.
308
00:22:36 --> 00:22:39
But we can do this, generally,
by using what are called
309
00:22:39 --> 00:22:41
Maxwell relations.
310
00:22:41 --> 00:22:46
And all this is, is saying that
when you take a mixed second
311
00:22:46 --> 00:22:50
derivative, it doesn't matter
in which order you take
312
00:22:50 --> 00:22:54
the two derivatives.
313
00:22:54 --> 00:23:03
So, let's, we're going to
use this relationship.
314
00:23:03 --> 00:23:13
And we're going to
use these two.
315
00:23:13 --> 00:23:18
So, using those, now, what
happens if we take the second
316
00:23:18 --> 00:23:22
derivative of A, the mixed
derivative, partial with
317
00:23:22 --> 00:23:27
respect to T and the
partial with respect to V.
318
00:23:27 --> 00:23:35
So let's leave these off for a
moment, and now let's try that.
319
00:23:35 --> 00:23:41
And the point is that the
second derivative of A, with
320
00:23:41 --> 00:23:48
respect to V and T in this
order is the same as the second
321
00:23:48 --> 00:23:52
derivative of a with respect
to T and V in this order.
322
00:23:52 --> 00:23:56
It doesn't matter which order.
323
00:23:56 --> 00:23:58
But that turns out
to be useful.
324
00:23:58 --> 00:24:01
So let's do this explicitly.
325
00:24:01 --> 00:24:04
Which means we're going to take
the derivative with respect
326
00:24:04 --> 00:24:06
to volume of dA/dT.
327
00:24:06 --> 00:24:09
328
00:24:09 --> 00:24:15
Now, the dA/dT isn't
constant volume.
329
00:24:15 --> 00:24:18
The derivative we're taking
with respect to volume,
330
00:24:18 --> 00:24:23
when we take that it's
at constant temperature.
331
00:24:23 --> 00:24:24
But what is it?
332
00:24:24 --> 00:24:30
Well, we already know what
dA/dT at constant V is.
333
00:24:30 --> 00:24:32
It's negative S.
334
00:24:32 --> 00:24:35
So this is negative dS/dV.
335
00:24:35 --> 00:24:38
336
00:24:38 --> 00:24:43
At constant temperature.
337
00:24:43 --> 00:24:46
Now let's take it in
the other order.
338
00:24:46 --> 00:24:57
So d/dT of dA/dV,
just like this.
339
00:24:57 --> 00:25:01
The dA/dV is calculated
at constant temperature.
340
00:25:01 --> 00:25:02
We know it.
341
00:25:02 --> 00:25:05
Then we can take the derivative
of that quantity, when we vary
342
00:25:05 --> 00:25:07
the temperature, holding
the volume constant.
343
00:25:07 --> 00:25:11
But again, dA/dV
dT, there it is.
344
00:25:11 --> 00:25:14
It's negative p.
345
00:25:14 --> 00:25:24
So this is just negative
dp/dT at constant volume.
346
00:25:24 --> 00:25:31
These things have to be
equal to each other.
347
00:25:31 --> 00:25:36
Because these mixed second
derivatives are the same thing.
348
00:25:36 --> 00:25:37
But that's very useful.
349
00:25:37 --> 00:25:41
Because this is what
comes directly out of an
350
00:25:41 --> 00:25:43
equation of state, right?
351
00:25:43 --> 00:25:46
You know how pressure changes
with temperature at constant
352
00:25:46 --> 00:25:48
volume if you know the
equation of state.
353
00:25:48 --> 00:25:50
It relates the
pressure, volume, and
354
00:25:50 --> 00:26:03
temperature together.
355
00:26:03 --> 00:26:06
So from measured equation of
state data, or from a model
356
00:26:06 --> 00:26:09
like the ideal gas or the van
der Waal's gas or another
357
00:26:09 --> 00:26:11
equation of state,
you know this.
358
00:26:11 --> 00:26:14
Can determine how entropy
is going to behave as
359
00:26:14 --> 00:26:31
the volume changes.
360
00:26:31 --> 00:26:42
If we try that for an
ideal gas, pV is nRT.
361
00:26:42 --> 00:26:54
So dp/dT at constant volume,
it's just nR over V.
362
00:26:54 --> 00:27:03
And that, now, we know
must equal dS/dV,
363
00:27:03 --> 00:27:04
with a positive sign.
364
00:27:04 --> 00:27:08
At constant temperature.
365
00:27:08 --> 00:27:11
So now let's try looking
at something where
366
00:27:11 --> 00:27:13
are V1 goes to V2.
367
00:27:13 --> 00:27:17
The volume is going to
change, and we can see
368
00:27:17 --> 00:27:20
how the entropy changes.
369
00:27:20 --> 00:27:29
So, if one goes to two and V1
goes to V2, and it's constant
370
00:27:29 --> 00:27:33
temperature, just what
we've specified there.
371
00:27:33 --> 00:27:48
Delta S is S(T, V2) minus S(T,
V1), T's staying the same.
372
00:27:48 --> 00:27:57
So it's just the integral
from V1 to V2 of dS/dV At
373
00:27:57 --> 00:27:59
constant temperature dV.
374
00:27:59 --> 00:28:00
And now we know what that is.
375
00:28:00 --> 00:28:08
So it's nR integral from
V1 to V2 dV over V.
376
00:28:08 --> 00:28:16
So it's nR log V2 over V1.
377
00:28:16 --> 00:28:18
There's our delta S.
378
00:28:18 --> 00:28:22
So we know how to calculate it.
379
00:28:22 --> 00:28:30
Make sense?
380
00:28:30 --> 00:28:35
Now, we can do the same
procedure for the
381
00:28:35 --> 00:28:38
pressure change.
382
00:28:38 --> 00:28:41
And all we do is, I'll just
outline this, I think.
383
00:28:41 --> 00:28:45
I won't write it
all on the board.
384
00:28:45 --> 00:28:48
But, of course, it's going to
come from the fact that these
385
00:28:48 --> 00:28:52
second derivatives
are also equal.
386
00:28:52 --> 00:29:05
So d squared G dT dp is
equal to d squared G dp dT.
387
00:29:05 --> 00:29:08
In other words, the order of
taking the derivatives with
388
00:29:08 --> 00:29:13
respect to pressure and
temperature doesn't matter.
389
00:29:13 --> 00:29:20
And what this will show is that
dS/dp at constant temperature,
390
00:29:20 --> 00:29:24
here we saw how entropy varies
with volume, this is going to
391
00:29:24 --> 00:29:27
show us how it varies
with pressure.
392
00:29:27 --> 00:29:34
Is equal to minus dV/dT
at constant pressure.
393
00:29:34 --> 00:29:38
And again, this is something
that comes from an
394
00:29:38 --> 00:29:40
equation of state.
395
00:29:40 --> 00:29:44
We know how the volume and
temperature vary with
396
00:29:44 --> 00:29:46
respect to each other
at constant pressure.
397
00:29:46 --> 00:29:53
That's what the equation
of state tells us.
398
00:29:53 --> 00:29:57
So, again, I can just
use that result.
399
00:29:57 --> 00:30:03
So, if we do a process where
one goes to two at constant
400
00:30:03 --> 00:30:11
temperature, and now the
pressure, p1, goes to p2, well
401
00:30:11 --> 00:30:21
then delta S is just the
integral from p1 to p2 of dS/dp
402
00:30:21 --> 00:30:26
times dS, so it's just this.
403
00:30:26 --> 00:30:30
And so of course it's
still pV equals nRT.
404
00:30:30 --> 00:30:38
So now we just have
nR over p dp.
405
00:30:38 --> 00:30:39
Right?
406
00:30:39 --> 00:30:41
So we're going to
see the same story.
407
00:30:41 --> 00:30:49
It's nR log of p2 over p1 for
the process where there's
408
00:30:49 --> 00:31:01
a pressure change.
409
00:31:01 --> 00:31:04
Any questions about this part?
410
00:31:04 --> 00:31:07
So what we've done is
take one step further.
411
00:31:07 --> 00:31:15
We've used the fundamental
equations that are hiding
412
00:31:15 --> 00:31:19
down here, out of sight
but never out of mind.
413
00:31:19 --> 00:31:23
And what we've done is look at
the derivatives of the new free
414
00:31:23 --> 00:31:28
energies that we've just
recently introduced, A and G.
415
00:31:28 --> 00:31:33
And then, the only thing we've
done beyond that is say, OK,
416
00:31:33 --> 00:31:35
well now let's just take the
mixed second derivatives, they
417
00:31:35 --> 00:31:37
have to be equal to each other.
418
00:31:37 --> 00:31:42
And what's fallen out when we
do that, because in each case,
419
00:31:42 --> 00:31:46
one of the first derivatives
gives us the entropy.
420
00:31:46 --> 00:31:48
Then the second derivative
gives the change in entropy
421
00:31:48 --> 00:31:51
with respect to the variable
that we're differentiating,
422
00:31:51 --> 00:31:54
with respect to which is
either pressure or volume.
423
00:31:54 --> 00:31:59
And the useful outcome of all
that is that we get to see how
424
00:31:59 --> 00:32:03
entropy changes with one of
those variables in terms of
425
00:32:03 --> 00:32:09
only V, T, and p, which come
out of some equation of state.
426
00:32:09 --> 00:32:11
And all we did, further, is
take that second derivative.
427
00:32:11 --> 00:32:13
That mixed second derivative.
428
00:32:13 --> 00:32:15
And, of course, see that
either way we do that
429
00:32:15 --> 00:32:24
we'll have an equality.
430
00:32:24 --> 00:32:34
Now, let's go back to our
older friends u and H.
431
00:32:34 --> 00:32:46
Which we've expressed now in
terms of S and V, S and p.
432
00:32:46 --> 00:32:52
So, so far we don't have a way
to just write off, relate them
433
00:32:52 --> 00:32:53
to equation of state data.
434
00:32:53 --> 00:32:57
Which also would
be very useful.
435
00:32:57 --> 00:33:02
Here, A and G, we've already
got as functions of these
436
00:33:02 --> 00:33:05
easily controlled, conveniently
controlled state variables.
437
00:33:05 --> 00:33:13
Let's look at those
quantities. u and H.
438
00:33:13 --> 00:33:17
And look at, for example,
the V dependence of u.
439
00:33:17 --> 00:33:35
The volume dependence.
440
00:33:35 --> 00:33:44
And in particular let's look
at, for example, du/dV
441
00:33:44 --> 00:33:46
at constant temperature.
442
00:33:46 --> 00:33:53
Now, we can immediately
see what du/dV at
443
00:33:53 --> 00:33:57
constant entropy is.
444
00:33:57 --> 00:33:59
Experimentally, though,
that's not such an easy
445
00:33:59 --> 00:34:02
situation to arrange.
446
00:34:02 --> 00:34:07
Of course, this is a much
more practical one.
447
00:34:07 --> 00:34:10
But it doesn't just fall out
immediately from the one
448
00:34:10 --> 00:34:14
fundamental equation for du.
449
00:34:14 --> 00:34:16
But we can start there.
450
00:34:16 --> 00:34:24
So, du is T dS minus p dV.
451
00:34:24 --> 00:34:34
And I can take this derivative.
du/dV at constant T.
452
00:34:34 --> 00:34:35
And so, what is it?
453
00:34:35 --> 00:34:39
Well, it's not just p
because there's some
454
00:34:39 --> 00:34:43
dS/dV at constant T.
455
00:34:43 --> 00:34:44
This isn't zero.
456
00:34:44 --> 00:34:50
There's some variation, dS/dV,
at constant temperature.
457
00:34:50 --> 00:34:53
That's going to matter.
458
00:34:53 --> 00:35:00
This part, of course,
is just minus p.
459
00:35:00 --> 00:35:15
But we just figured out what
dS/dV at constant T is.
460
00:35:15 --> 00:35:19
This is dp/dT at constant V.
461
00:35:19 --> 00:35:20
So that's neat.
462
00:35:20 --> 00:35:24
So in other words, we can
write this as T, dp/dT
463
00:35:24 --> 00:35:29
at constant V, minus p.
464
00:35:29 --> 00:35:35
Let's just check T, p, T, V, p.
465
00:35:35 --> 00:35:37
Right?
466
00:35:37 --> 00:35:38
In other words, we
just have pressure,
467
00:35:38 --> 00:35:40
temperature and volume.
468
00:35:40 --> 00:35:42
Again, if we know the
equation of state, we
469
00:35:42 --> 00:35:45
know all this stuff.
470
00:35:45 --> 00:35:47
So again, we can measure
equation of state data.
471
00:35:47 --> 00:35:50
Or, if we know the equation of
state from a model, ideal gas,
472
00:35:50 --> 00:35:56
van der Waal's gas, whatever,
now we can determine u.
473
00:35:56 --> 00:35:58
From equation of state data.
474
00:35:58 --> 00:36:17
Terrific, right?
475
00:36:17 --> 00:36:22
So let's take our one model
that we keep going back to.
476
00:36:22 --> 00:36:25
Equation of state, and
just see how it works.
477
00:36:25 --> 00:36:28
That is, ideal gas.
478
00:36:28 --> 00:36:30
And see how it works with that.
479
00:36:30 --> 00:36:37
Now, we saw before, or really I
should say we accepted before,
480
00:36:37 --> 00:36:40
that for an ideal gas, u was a
function of temperature only.
481
00:36:40 --> 00:36:43
Well, now let's try it.
482
00:36:43 --> 00:36:50
So, dp/dT, for our ideal
gas, at constant volume,
483
00:36:50 --> 00:36:54
remember pV is nRT.
484
00:36:54 --> 00:36:59
So this nR over V.
485
00:36:59 --> 00:37:01
And then, using the relation
again, we can just
486
00:37:01 --> 00:37:05
write this as p over T.
487
00:37:05 --> 00:37:07
In other words, we're taking
advantage of the fact that
488
00:37:07 --> 00:37:10
we now know that quantity.
489
00:37:10 --> 00:37:12
In the case of the ideal
gas, we just have a
490
00:37:12 --> 00:37:13
simple model for it.
491
00:37:13 --> 00:37:14
More generally, we
could measure it.
492
00:37:14 --> 00:37:16
We could just collect
a bunch of data.
493
00:37:16 --> 00:37:17
For a material.
494
00:37:17 --> 00:37:21
What's the volume it
occupies at some pressure
495
00:37:21 --> 00:37:23
and temperature?
496
00:37:23 --> 00:37:25
Now let's change the pressure
and temperature and sweep
497
00:37:25 --> 00:37:27
through a whole range of
pressures and temperatures and
498
00:37:27 --> 00:37:28
measure the volume in
every one of them.
499
00:37:28 --> 00:37:31
Well, then, we could just use
that for our equation of state.
500
00:37:31 --> 00:37:33
One way or another, we can
determine this quantity.
501
00:37:33 --> 00:37:35
For the ideal gas it's this.
502
00:37:35 --> 00:37:45
So now our du/dV, at constant T
is just T times dp/dT, which is
503
00:37:45 --> 00:37:52
just p over T minus
p, it's zero.
504
00:37:52 --> 00:37:55
Remember the Joule expansion.
505
00:37:55 --> 00:37:58
And we saw that, you saw that
the Joule coefficient for
506
00:37:58 --> 00:38:01
an ideal gas was zero.
507
00:38:01 --> 00:38:06
So that you could see that for
the ideal gas, u would not be a
508
00:38:06 --> 00:38:08
function of volume, but
only of temperature.
509
00:38:08 --> 00:38:13
But actually, when you saw
that before, you weren't
510
00:38:13 --> 00:38:15
given any proof of that.
511
00:38:15 --> 00:38:19
It was just that when the
good Mr. Joule made the
512
00:38:19 --> 00:38:23
measurements, to the precision
that he could measure, he
513
00:38:23 --> 00:38:26
discovered that for some gases
it was extremely small.
514
00:38:26 --> 00:38:29
At least, smaller than
anything he could detect.
515
00:38:29 --> 00:38:32
So it sure seemed like it
was going to zero, under
516
00:38:32 --> 00:38:33
ideal gas conditions.
517
00:38:33 --> 00:38:36
And that was the result
that we came to accept.
518
00:38:36 --> 00:38:38
Here, though, you can just
derive straight away.
519
00:38:38 --> 00:38:41
That for an ideal gas it has to
be the case that there's no
520
00:38:41 --> 00:38:43
volume dependence
of the energy.
521
00:38:43 --> 00:38:53
Only a temperature dependence.
522
00:38:53 --> 00:39:21
It's the same for H.
523
00:39:21 --> 00:39:25
Just like u, we'd like to be
able to express it in a way
524
00:39:25 --> 00:39:28
that allows us to calculate
what happens only from
525
00:39:28 --> 00:39:30
equation of state data.
526
00:39:30 --> 00:39:36
But, again, our fundamental
equations show us how it
527
00:39:36 --> 00:39:45
changes as a function of
entropy and pressure.
528
00:39:45 --> 00:39:53
So, dH is T dS plus V dp.
529
00:39:53 --> 00:39:55
So let's look at dH/dp.
530
00:39:55 --> 00:39:58
531
00:39:58 --> 00:40:00
We know how to get it
immediately if we keep
532
00:40:00 --> 00:40:01
entropy constant.
533
00:40:01 --> 00:40:04
But we'd like to relate it
to what happens if we keep
534
00:40:04 --> 00:40:06
the temperature constant.
535
00:40:06 --> 00:40:09
So then, just like we saw,
analogous to what saw just
536
00:40:09 --> 00:40:16
before, it's T dS/dp
at constant T.
537
00:40:16 --> 00:40:18
Plus V.
538
00:40:18 --> 00:40:24
But now we've seen from the
Maxwell relations that dS/dp is
539
00:40:24 --> 00:40:26
minus dV/dT, for constant p.
540
00:40:26 --> 00:40:29
Again, this is this quantity,
one of these quantities that
541
00:40:29 --> 00:40:31
again we can determine from
equation of state data.
542
00:40:31 --> 00:40:35
Only V, p and T appear.
543
00:40:35 --> 00:40:44
So it's minus T dV/dT
at constant p, plus V.
544
00:40:44 --> 00:41:03
And so, again, this can come
from equation of state data.
545
00:41:03 --> 00:41:09
And if you do this again for
an ideal gas, let me see.
546
00:41:09 --> 00:41:11
So we have pV is nRT.
547
00:41:11 --> 00:41:24
So dV/dT at constant
pressure is just nR over p.
548
00:41:24 --> 00:41:26
But we can plug that in again
just like we did before.
549
00:41:26 --> 00:41:31
It's just equal to V over T.
550
00:41:31 --> 00:41:37
And so dH/dp under our
condition of constant
551
00:41:37 --> 00:41:46
temperature is just minus
T times V over T plus V,
552
00:41:46 --> 00:41:50
everything cancels,
and that's zero.
553
00:41:50 --> 00:41:54
That's our Joule -
Thompson expansion.
554
00:41:54 --> 00:41:56
That was a constant
enthalpy change.
555
00:41:56 --> 00:42:01
And again there, too, you saw
an experimental result you were
556
00:42:01 --> 00:42:04
presented with that says, well
at least to the extent that it
557
00:42:04 --> 00:42:06
could be measured, it was
obviously getting very small.
558
00:42:06 --> 00:42:11
For gases that approach
ideal gas conditions.
559
00:42:11 --> 00:42:13
Well, there you can see it.
560
00:42:13 --> 00:42:16
Sure better have gotten
small because in fact
561
00:42:16 --> 00:42:20
it has to be zero.
562
00:42:20 --> 00:42:30
Now let's take just one
somewhat more complicated case.
563
00:42:30 --> 00:42:37
Let's look at a van
der Waal's gas.
564
00:42:37 --> 00:42:40
Let's try it with a different
equation of state, that
565
00:42:40 --> 00:42:47
isn't quite as simple
as the ideal gas case.
566
00:42:47 --> 00:42:58
So, then p plus a over molar
volume squared times V minus
567
00:42:58 --> 00:43:07
b molar volume V minus b
is equal to RT, remember?
568
00:43:07 --> 00:43:14
This was back from the first or
second lecture in the course.
569
00:43:14 --> 00:43:18
So, we can separate out p.
570
00:43:18 --> 00:43:25
It's RT over molar volume
minus b minus a over
571
00:43:25 --> 00:43:31
molar volume V squared.
572
00:43:31 --> 00:43:33
And then we can take the
derivative with respect to
573
00:43:33 --> 00:43:38
temperature, it's just R
over molar volume minus b.
574
00:43:38 --> 00:43:53
So it's dp/dT at constant V is
just R over V bar minus b.
575
00:43:53 --> 00:44:04
Well, let's now look, given
this, let's now look in that
576
00:44:04 --> 00:44:08
case, at what happens to
u as a function of V.
577
00:44:08 --> 00:44:12
For the ideal gas, we know
that u is volume independent.
578
00:44:12 --> 00:44:16
It only depends on
the temperature.
579
00:44:16 --> 00:44:22
But for the van der Waal's gas,
now it's going to be different.
580
00:44:22 --> 00:44:27
And that's because this is
different from what it is
581
00:44:27 --> 00:44:28
in the ideal gas case.
582
00:44:28 --> 00:44:49
Namely, now du/dV at constant
T, for the van der Waal's gas.
583
00:44:49 --> 00:44:57
So it's this.
584
00:44:57 --> 00:45:04
So it's RT over molar
volume minus b.
585
00:45:04 --> 00:45:09
Minus p, right?
586
00:45:09 --> 00:45:13
But in fact, if you go back to
the van der Waal's equation of
587
00:45:13 --> 00:45:17
state, here's RT
over v minus b.
588
00:45:17 --> 00:45:22
If we put it as minus b, that's
just equal to a over V squared.
589
00:45:22 --> 00:45:28
Equals a over molar
volume squared.
590
00:45:28 --> 00:45:33
But the point is, the main
point is, it's not zero.
591
00:45:33 --> 00:45:37
It's some number. a over the
molar volume squared. a is a
592
00:45:37 --> 00:45:40
positive number in the van der
Waal's equation of state.
593
00:45:40 --> 00:45:44
So this is greater than zero.
594
00:45:44 --> 00:45:52
In other words, u is a
function of T and V.
595
00:45:52 --> 00:45:59
If we don't have an ideal gas.
596
00:45:59 --> 00:46:01
By the way, just to think
about it a little bit,
597
00:46:01 --> 00:46:02
it's a positive number.
598
00:46:02 --> 00:46:08
What that means is, I've got my
ideal gas in some container.
599
00:46:08 --> 00:46:12
There's some energy,
some internal energy.
600
00:46:12 --> 00:46:13
Now I make the volume bigger.
601
00:46:13 --> 00:46:17
I allow it to expand.
602
00:46:17 --> 00:46:22
And the energy
changes, it goes up.
603
00:46:22 --> 00:46:25
In some sense it's less
favorable energetically.
604
00:46:25 --> 00:46:31
What's happening there, that a
term in the van der Waal's
605
00:46:31 --> 00:46:35
equation of state, is
describing interactions,
606
00:46:35 --> 00:46:38
favorable attractions,
between gas molecules.
607
00:46:38 --> 00:46:42
The b is describing repulsions.
608
00:46:42 --> 00:46:44
Effectively, the
volume changes.
609
00:46:44 --> 00:46:47
The molar volume is being
changed a little bit by the
610
00:46:47 --> 00:46:50
fact that if you're really
trying to make things collide
611
00:46:50 --> 00:46:54
with each other, they can't
occupy the same volume.
612
00:46:54 --> 00:46:56
And that is being
expressed here.
613
00:46:56 --> 00:47:02
The a, though is expressing
attraction between molecules
614
00:47:02 --> 00:47:04
at somewhat longer range.
615
00:47:04 --> 00:47:07
So now, if you make the
volume bigger, those
616
00:47:07 --> 00:47:08
attractions die out.
617
00:47:08 --> 00:47:11
Because the molecules are
farther apart from each other.
618
00:47:11 --> 00:47:13
So the energy goes up.
619
00:47:13 --> 00:47:17
Now, you might ask, well why
does it do that, right?
620
00:47:17 --> 00:47:22
I mean, if the energy is lower
to occupy a smaller volume,
621
00:47:22 --> 00:47:26
then if I have this room and a
bunch of molecules of oxygen,
622
00:47:26 --> 00:47:29
and nitrogen and what have you
in the air, and there are weak
623
00:47:29 --> 00:47:31
attractions between them, why
don't they all just sort
624
00:47:31 --> 00:47:34
of glum together and find
whatever volume they like.
625
00:47:34 --> 00:47:37
So that the attractive
forces can exert
626
00:47:37 --> 00:47:38
themselves a little bit.
627
00:47:38 --> 00:47:39
Not too close, right?
628
00:47:39 --> 00:47:42
Not so close that the
repulsions dominate.
629
00:47:42 --> 00:47:44
Why don't they do that?
630
00:47:44 --> 00:47:54
What else matters besides any
of those considerations?
631
00:47:54 --> 00:47:57
What else matters that I
haven't considered in
632
00:47:57 --> 00:48:02
this little discussion?
633
00:48:02 --> 00:48:04
Yeah.
634
00:48:04 --> 00:48:06
What about entropy, right?
635
00:48:06 --> 00:48:08
If I only worry about
minimizing the
636
00:48:08 --> 00:48:10
energy, it's true.
637
00:48:10 --> 00:48:13
They'll stick together
a little bit.
638
00:48:13 --> 00:48:15
But entropy also matters.
639
00:48:15 --> 00:48:18
And there's disorder achieved
by occupying the full
640
00:48:18 --> 00:48:19
available volume.
641
00:48:19 --> 00:48:21
Many more states possible.
642
00:48:21 --> 00:48:24
And that will end up winning
out at basically any realistic
643
00:48:24 --> 00:48:28
temperature where the
stuff really is a gas.
644
00:48:28 --> 00:48:29
OK.
645
00:48:29 --> 00:48:32
Next time, what you're going
to see is the following.
646
00:48:32 --> 00:48:37
It turns out we can express all
these functions in terms of G,
647
00:48:37 --> 00:48:40
we wouldn't need to choose G,
but it's a very useful
648
00:48:40 --> 00:48:40
function to choose.
649
00:48:40 --> 00:48:46
Because of its natural
expression in terms of T and p.
650
00:48:46 --> 00:48:49
So we can write any of these
functions in terms of G.
651
00:48:49 --> 00:48:51
Which means that we can
really calculate all
652
00:48:51 --> 00:48:55
the thermodynamics
in terms of only g.
653
00:48:55 --> 00:48:59
It's not necessary to do that,
but it can be quite convenient.
654
00:48:59 --> 00:49:03
And then we can say, OK, if we
have many constituents, what if
655
00:49:03 --> 00:49:04
we have a mixture of stuff?
656
00:49:04 --> 00:49:08
We can take the derivative
of G with respect to how
657
00:49:08 --> 00:49:10
much material there is.
658
00:49:10 --> 00:49:13
With respect to n,
the number of moles.
659
00:49:13 --> 00:49:15
And if there are one, and two,
and three constituents with
660
00:49:15 --> 00:49:18
respect to n1, and n2, and n3.
661
00:49:18 --> 00:49:21
Each individual
amount of stuff.
662
00:49:21 --> 00:49:23
What that's going to allow us
to do is, if we say, OK I have
663
00:49:23 --> 00:49:26
a mixture of stuff, how does
the free energy change?
664
00:49:26 --> 00:49:29
If I change the composition
of the mixture?
665
00:49:29 --> 00:49:32
If I take something away,
or put something else in?
666
00:49:32 --> 00:49:33
And we'll be able to
determine equilibrium
667
00:49:33 --> 00:49:35
under those conditions.
668
00:49:35 --> 00:49:38
That's very useful for things
like chemical reactions,
669
00:49:38 --> 00:49:41
where this constituent
changes to this one.
670
00:49:41 --> 00:49:43
And I can calculate what
happens to G under
671
00:49:43 --> 00:49:44
those conditions.
672
00:49:44 --> 00:49:46
And that's what you'll
see starting next time.
673
00:49:46 --> 00:49:48
And professor Blendi
will be taking over for
674
00:49:48 --> 00:49:49
that set of lectures.
675
00:49:49 --> 00:49:50