1
00:00:01,000 --> 00:00:04,000
The following content is
provided by MIT OpenCourseWare

2
00:00:04,000 --> 00:00:06,000
under a Creative Commons
license.

3
00:00:06,000 --> 00:00:10,000
Additional information about
our license and MIT

4
00:00:10,000 --> 00:00:15,000
OpenCourseWare in general is
available at ocw.mit.edu.

5
00:00:15,000 --> 00:00:23,000
Good afternoon.
I hope you had a nice weekend.

6
00:00:23,000 --> 00:00:30,000
Did you have a nice weekend?
Good.

7
00:00:30,000 --> 00:00:34,000
Today, we are going to start
talking about the motion of

8
00:00:34,000 --> 00:00:37,000
molecules.
We are first going to talk

9
00:00:37,000 --> 00:00:42,000
about the translational motion
of molecules today and Friday.

10
00:00:42,000 --> 00:00:46,000
And then we are going to talk
about the internal motion,

11
00:00:46,000 --> 00:00:51,000
in particular the vibrational
motion of molecules and their

12
00:00:51,000 --> 00:00:54,000
rotational motion.
That will be early next week.

13
00:00:54,000 --> 00:01:00,000
First, transitional motion.
Certainly, the quintessential

14
00:01:00,000 --> 00:01:05,000
equation that represents the
behavior of gases is,

15
00:01:05,000 --> 00:01:11,000
of course, the ideal gas law,
P equals n over V times RT.

16
00:01:11,000 --> 00:01:14,000
This equation accurately

17
00:01:14,000 --> 00:01:19,000
represents the behavior of gases
at low pressures.

18
00:01:19,000 --> 00:01:22,000
It is an empirical law,
of course.

19
00:01:22,000 --> 00:01:27,000
It is a law that Boyle and
Charles discovered by doing

20
00:01:27,000 --> 00:01:31,000
experiments.
They noted that as the

21
00:01:31,000 --> 00:01:35,000
temperature is raised,
the pressure went up.

22
00:01:35,000 --> 00:01:41,000
They noted as the amount of gas
or the number of moles of gas

23
00:01:41,000 --> 00:01:44,000
added goes up,
the pressure goes up.

24
00:01:44,000 --> 00:01:49,000
They noted that as the volume
of their gaseous container goes

25
00:01:49,000 --> 00:01:55,000
up, the pressure goes down.
Literally, this is an equation

26
00:01:55,000 --> 00:02:00,000
established by experiment and
just varying one variable at a

27
00:02:00,000 --> 00:02:04,000
time.
And since it looked like the

28
00:02:04,000 --> 00:02:09,000
pressure was directly
proportional to the temperature,

29
00:02:09,000 --> 00:02:15,000
they wrote an equation where it
was directly proportional to the

30
00:02:15,000 --> 00:02:19,000
temperature, etc.
And the proportionality

31
00:02:19,000 --> 00:02:23,000
constant here was always this
one constant R,

32
00:02:23,000 --> 00:02:27,000
no matter what gas you had.
But this ideal gas law

33
00:02:27,000 --> 00:02:33,000
describes the macroscopic
properties of a gas.

34
00:02:33,000 --> 00:02:37,000
And what I mean by macroscopic
properties are properties that

35
00:02:37,000 --> 00:02:40,000
describe a collection of
molecules.

36
00:02:40,000 --> 00:02:42,000
For example,
to really talk about a

37
00:02:42,000 --> 00:02:46,000
pressure, you have to have a
collection of molecules.

38
00:02:46,000 --> 00:02:50,000
We are going to talk about the
pressure due to one molecule,

39
00:02:50,000 --> 00:02:53,000
but that is really just a
model.

40
00:02:53,000 --> 00:02:57,000
If you want to talk about
pressure, you need to have a

41
00:02:57,000 --> 00:03:02,000
collection of molecules.
If you talk about temperature,

42
00:03:02,000 --> 00:03:06,000
it really only has meaning when
you have a collection of

43
00:03:06,000 --> 00:03:10,000
molecules.
But what we want to understand

44
00:03:10,000 --> 00:03:14,000
is what are the underlying
microscopic phenomenon that

45
00:03:14,000 --> 00:03:19,000
gives rise to this macroscopic
equation or these macroscopic

46
00:03:19,000 --> 00:03:22,000
properties.
We want to know what is going

47
00:03:22,000 --> 00:03:27,000
on in terms of the behavior of
the individual particles,

48
00:03:27,000 --> 00:03:32,000
the individual molecules that
make up this gas.

49
00:03:32,000 --> 00:03:37,000
Inquiring minds want to know
what the temperature means when

50
00:03:37,000 --> 00:03:40,000
we talk about individual
molecules.

51
00:03:40,000 --> 00:03:45,000
And that is also was Maxwell
and Boltzmann wanted to know.

52
00:03:45,000 --> 00:03:50,000
They wanted a microscopic
explanation for PV equal nRT.

53
00:03:50,000 --> 00:03:53,000
And, to do so,

54
00:03:53,000 --> 00:03:57,000
they put forth a theory called
the kinetic theory,

55
00:03:57,000 --> 00:04:03,000
or the kinetic theory for the
behavior of gases.

56
00:04:03,000 --> 00:04:07,000
And that is exactly what we are
going to take a look at here,

57
00:04:07,000 --> 00:04:11,000
this kinetic theory.
We are going to do what they

58
00:04:11,000 --> 00:04:14,000
did.
Basically, this kinetic theory

59
00:04:14,000 --> 00:04:19,000
allowed properties of gases at
low pressures to be predicted,

60
00:04:19,000 --> 00:04:24,000
and it allowed an understanding
of why some of the properties of

61
00:04:24,000 --> 00:04:28,000
real gases at higher pressures
deviated from this ideal gas

62
00:04:28,000 --> 00:04:31,000
law.
But, more importantly,

63
00:04:31,000 --> 00:04:36,000
what it did was it allowed the
quantity pressure times the

64
00:04:36,000 --> 00:04:40,000
volume **PV** to be understood
in terms of the motions of the

65
00:04:40,000 --> 00:04:43,000
molecules.
And it provided a means to

66
00:04:43,000 --> 00:04:48,000
understand this concept of a
temperature in terms of the

67
00:04:48,000 --> 00:04:52,000
motions of the molecules.
And that is what we are going

68
00:04:52,000 --> 00:04:56,000
to look at today.
We are going to see how we can

69
00:04:56,000 --> 00:05:00,000
describe pressure and volume in
terms of the motion of the

70
00:05:00,000 --> 00:05:04,000
molecules and temperature in
terms of the motion of the

71
00:05:04,000 --> 00:05:08,000
molecules.
That is our goal.

72
00:05:08,000 --> 00:05:13,000
In order to do that,
the first thing we have to do

73
00:05:13,000 --> 00:05:17,000
is to understand what we mean by
pressure.

74
00:05:17,000 --> 00:05:21,000
For example,
if you have some gas here in a

75
00:05:21,000 --> 00:05:26,000
container and you are measuring
the pressure of this gas in that

76
00:05:26,000 --> 00:05:31,000
container, --
what you are really measuring

77
00:05:31,000 --> 00:05:37,000
is the force that the gas exerts
on one of the walls of this

78
00:05:37,000 --> 00:05:41,000
container.
That is, pressure is the force

79
00:05:41,000 --> 00:05:45,000
exerted by the gas on one of
these walls.

80
00:05:45,000 --> 00:05:51,000
It is the force per unit area
that is exerted on the walls of

81
00:05:51,000 --> 00:05:56,000
that container.
That

82
00:05:56,000 --> 00:06:00,000
is what pressure is.
But what Maxwell did,

83
00:06:00,000 --> 00:06:04,000
--
-- and this is 1850,

84
00:30:55,000 --> 00:06:07,000
What Maxwell did was recognize
No surprise.

85
00:06:07,000 --> 00:06:14,000
that he could understand this
macroscopic pressure in terms of

86
00:06:14,000 --> 00:06:20,000
the individual forces of the
molecules when they hit the

87
00:06:20,000 --> 00:06:22,000
container.
In other words,

88
00:06:22,000 --> 00:06:29,000
he proposed that this gas was
composed of these molecules and

89
00:06:29,000 --> 00:06:34,000
that these molecules were
moving.

90
00:06:34,000 --> 00:06:38,000
That was his proposal.
And that when they moved and

91
00:06:38,000 --> 00:06:44,000
hit the walls of the container,
well, that was the force that

92
00:06:44,000 --> 00:06:48,000
was exerted by the gas.
In other words,

93
00:06:48,000 --> 00:06:52,000
the force was really the
individual forces,

94
00:06:52,000 --> 00:06:56,000
F sub i,
here, of the individual

95
00:06:56,000 --> 00:07:00,000
molecules.
That total force was the

96
00:07:00,000 --> 00:07:05,000
individual force of the
molecules hitting the walls of

97
00:07:05,000 --> 00:07:08,000
the container.
That is what led to this

98
00:07:08,000 --> 00:07:13,000
macroscopic concept,
the macroscopic quantity of

99
00:07:13,000 --> 00:07:15,000
pressure.
That was his idea.

100
00:07:15,000 --> 00:07:20,000
Well, if that was his idea,
he carried it through,

101
00:07:20,000 --> 00:07:24,000
now, to a prediction.
The idea is that this force

102
00:07:24,000 --> 00:07:29,000
arises from the individual
forces of these individual

103
00:07:29,000 --> 00:07:35,000
molecules hitting the walls of
the container.

104
00:07:35,000 --> 00:07:41,000
Let's look at what theory he
wrote down.

105
00:07:53,000 --> 00:07:56,000
His goal, here,
was to calculate these

106
00:07:56,000 --> 00:08:00,000
individual forces,
F sub i.

107
00:08:00,000 --> 00:08:03,000
And, of course,
go back to classical mechanics.

108
00:08:03,000 --> 00:08:07,000
Force is the mass times the
acceleration,

109
00:08:07,000 --> 00:08:10,000
the mass of the particle,
the molecule,

110
00:08:10,000 --> 00:08:14,000
times its acceleration.
And acceleration we can write

111
00:08:14,000 --> 00:08:17,000
just in terms of delta v over
delta t.

112
00:08:17,000 --> 00:08:21,000
That is just the mass times

113
00:08:21,000 --> 00:08:26,000
delta v over the change in time.
And if the mass is constant

114
00:08:26,000 --> 00:08:29,000
here, this numerator,
that is the change in momentum

115
00:08:29,000 --> 00:08:35,000
per unit change in time.
That is what this force is.

116
00:08:35,000 --> 00:08:39,000
What we have to do,
here, is calculate the change

117
00:08:39,000 --> 00:08:44,000
in the momentum of the wall when
the molecule hits the wall.

118
00:08:44,000 --> 00:08:50,000
That will be the force of the
individual molecule in the wall,

119
00:08:50,000 --> 00:08:55,000
that over the time between the
collisions of the molecule with

120
00:08:55,000 --> 00:09:00,000
the wall.
That is the individual forces.

121
00:09:00,000 --> 00:09:04,000
That is what we are trying to
calculate, right here.

122
00:09:04,000 --> 00:09:09,000
Let's do that and calculate
delta p, the change in the

123
00:09:09,000 --> 00:09:14,000
wall's momentum.
Let's do it in one dimension

124
00:09:14,000 --> 00:09:16,000
first.
Here is our box.

125
00:09:16,000 --> 00:09:21,000
This is just a cross-sectional
view of that box I drew over

126
00:09:21,000 --> 00:09:24,000
here.
And this is going to be the

127
00:09:24,000 --> 00:09:30,000
wall that we are going to be
interested in.

128
00:09:30,000 --> 00:09:34,000
The box has a length l.
And we have this molecule with

129
00:09:34,000 --> 00:09:38,000
some mass m.
It is coming into the wall and

130
00:09:38,000 --> 00:09:43,000
is going to collide with it,
and it has some velocity vector

131
00:09:43,000 --> 00:09:45,000
v.
But let's do it in one

132
00:09:45,000 --> 00:09:48,000
dimension because it is simpler
to do.

133
00:09:48,000 --> 00:09:54,000
And then we are going to extend
it to three dimensions in a few

134
00:09:54,000 --> 00:09:57,000
minutes.
We are just going to do it in

135
00:09:57,000 --> 00:10:01,000
one dimension now.
We are only going to be

136
00:10:01,000 --> 00:10:05,000
interested in then the
x-component of the velocity of

137
00:10:05,000 --> 00:10:10,000
this molecule coming into the
wall, and we want to know what

138
00:10:10,000 --> 00:10:14,000
is the force exerted by this
molecule when it collides with

139
00:10:14,000 --> 00:10:18,000
this wall.
First we have to get the change

140
00:10:18,000 --> 00:10:21,000
in the momentum.
This molecule comes in and hits

141
00:10:21,000 --> 00:10:26,000
the wall, and we are going to
consider it to be an elastic

142
00:10:26,000 --> 00:10:30,000
collision.
If the component of the

143
00:10:30,000 --> 00:10:36,000
velocity in the x direction
before the collision is v sub x,

144
00:10:36,000 --> 00:10:40,000
then after the collision,
it is minus v sub x.

145
00:10:40,000 --> 00:10:45,000
We have just changed the
direction of our velocity

146
00:10:45,000 --> 00:10:50,000
vector, not the magnitude.
The change in the atom's

147
00:10:50,000 --> 00:10:54,000
momentum, then,
is just the momentum after

148
00:10:54,000 --> 00:10:59,000
minus the moment before.
The momentum after was m times

149
00:10:59,000 --> 00:11:03,000
minus v sub x.
The momentum before,

150
00:11:03,000 --> 00:11:06,000
m sub v sub x.
Delta in the change in the

151
00:11:06,000 --> 00:11:09,000
atom's momentum,
is minus 2 m v sub x.

152
00:11:09,000 --> 00:11:14,000
However, we have to conserve

153
00:11:14,000 --> 00:11:17,000
momentum.
The momentum change in the atom

154
00:11:17,000 --> 00:11:22,000
plus the momentum change in the
wall has to equal zero.

155
00:11:22,000 --> 00:11:27,000
And what we want to know is the
momentum change in the wall,

156
00:11:27,000 --> 00:11:31,000
because we are after this
macroscopic quantity,

157
00:11:31,000 --> 00:11:35,000
pressure.
And so, if the change in the

158
00:11:35,000 --> 00:11:40,000
momentum of the atom is minus 2
m v sub x,

159
00:11:40,000 --> 00:11:46,000
then the change in the
momentum of the wall is 2 m v

160
00:11:46,000 --> 00:11:49,000
sub x .

161
00:11:49,000 --> 00:11:52,000
We have one quantity here.
We have delta p.

162
00:11:52,000 --> 00:11:57,000
But now, we have to calculate
how often that momentum in the

163
00:11:57,000 --> 00:12:03,000
wall changes due to this
molecule's collisions.

164
00:12:03,000 --> 00:12:07,000
We need the delta t.
And so, what happens here?

165
00:12:07,000 --> 00:12:11,000
The molecule comes in,
collides, and then reflects.

166
00:12:11,000 --> 00:12:15,000
And the molecule is now going
in this direction,

167
00:12:15,000 --> 00:12:20,000
where it hits the back wall,
and then it reflects.

168
00:12:20,000 --> 00:12:24,000
And ultimately it comes back
and hits the front wall.

169
00:12:24,000 --> 00:12:29,000
And what we want to know is
what is the time here between

170
00:12:29,000 --> 00:12:34,000
the collisions.
The time between when that

171
00:12:34,000 --> 00:12:38,000
molecule makes the momentum in
the wall change.

172
00:12:38,000 --> 00:12:40,000
What is this time,
delta t?

173
00:12:40,000 --> 00:12:46,000
Well, if we know the length of
the box and we know the value of

174
00:12:46,000 --> 00:12:49,000
v sub x, we can calculate that
time.

175
00:12:49,000 --> 00:12:54,000
That time is just two times the
length of the box because that

176
00:12:54,000 --> 00:13:00,000
molecule is traveling back and
then forth, --

177
00:13:00,000 --> 00:13:04,000
-- that is 2l divided by the
velocity component in the x

178
00:13:04,000 --> 00:13:09,000
direction, v sub x.
That

179
00:13:09,000 --> 00:13:13,000
is delta t.
Now we have delta p and delta

180
00:13:13,000 --> 00:13:18,000
t, and we can plug that in and
make it simple a little bit.

181
00:13:18,000 --> 00:13:23,000
And so, here is the force
exerted by the collision of one

182
00:13:23,000 --> 00:13:28,000
molecule on that one wall of the
container, m v sub x squared

183
00:13:28,000 --> 00:13:33,000
over l.

184
00:13:33,000 --> 00:13:38,000
But we said that Maxwell's idea
was that the macroscopic total

185
00:13:38,000 --> 00:13:43,000
force was the sum of these
individual forces.

186
00:13:43,000 --> 00:13:48,000
What we have to do is take the
force of each individual

187
00:13:48,000 --> 00:13:52,000
molecule and add them up over
all the molecules.

188
00:13:52,000 --> 00:13:56,000
There is that force for
molecule one,

189
00:13:56,000 --> 00:13:59,000
molecule two,
molecule three,

190
00:13:59,000 --> 00:14:05,000
all the way up to molecule n.
I am going to pull out an m

191
00:14:05,000 --> 00:14:10,000
over l out of this.
I just pull out an m over l,

192
00:14:10,000 --> 00:14:16,000
and I have left the sum of the
squares of the velocities in the

193
00:14:16,000 --> 00:14:20,000
x direction for each one of the
molecules.

194
00:14:20,000 --> 00:14:23,000
Notice here,
in this treatment,

195
00:14:23,000 --> 00:14:27,000
we have identical m's.
The particles are the same,

196
00:14:27,000 --> 00:14:33,000
but, and this is important,
the velocities of the molecules

197
00:14:33,000 --> 00:14:38,000
are not the same.
That is going to be important.

198
00:14:38,000 --> 00:14:44,000
Now, this expression here,
I want to simplify a little

199
00:14:44,000 --> 00:14:45,000
bit.
In particular,

200
00:14:45,000 --> 00:14:51,000
I want to simplify the sum of
the squares of the velocity

201
00:14:51,000 --> 00:14:56,000
components in the x direction
for each one of the molecules.

202
00:14:56,000 --> 00:15:02,000
To do that simplification,
I am going to introduce this

203
00:15:02,000 --> 00:15:06,000
quantity.
This quantity is the average of

204
00:15:06,000 --> 00:15:10,000
the square of the velocity in
the x direction.

205
00:15:10,000 --> 00:15:15,000
What this means is I take the
velocity in the x direction,

206
00:15:15,000 --> 00:15:18,000
I square it,
and then I take the average.

207
00:15:18,000 --> 00:15:23,000
This is not the square of the
average velocity in the x

208
00:15:23,000 --> 00:15:26,000
direction.
That is different.

209
00:15:26,000 --> 00:15:31,000
This is the average of the
square of the velocity in the x

210
00:15:31,000 --> 00:15:35,000
direction.
How am I going to evaluate that

211
00:15:35,000 --> 00:15:38,000
quantity?
Well, I am going to take the

212
00:15:38,000 --> 00:15:43,000
velocity in the x direction for
molecule one and square it and

213
00:15:43,000 --> 00:15:47,000
add to that the square of the
velocity in the x direction for

214
00:15:47,000 --> 00:15:52,000
molecule two and add to that the
square of the velocity in the x

215
00:15:52,000 --> 00:15:58,000
direction for molecule three,
all the way up to molecule n.

216
00:15:58,000 --> 00:16:03,000
And then, if I want the
average, I am going to divide by

217
00:16:03,000 --> 00:16:06,000
n, the number of molecules there
are.

218
00:16:06,000 --> 00:16:11,000
I am just going to bring n,
here, over to the other side.

219
00:16:11,000 --> 00:16:17,000
I have n times the average of
the velocity in the x direction

220
00:16:17,000 --> 00:16:21,000
squared as this sum,
and this is exactly the sum

221
00:16:21,000 --> 00:16:25,000
that I had in my expression for
the total force.

222
00:16:25,000 --> 00:16:31,000
I can simplify that now.
There is that same expression.

223
00:16:31,000 --> 00:16:36,000
Here is my total force.
What I am going to do is I am

224
00:16:36,000 --> 00:16:42,000
going to substitute n times the
average of the velocity in the x

225
00:16:42,000 --> 00:16:46,000
direction squared in for this
whole sum, and I have now

226
00:16:46,000 --> 00:16:52,000
something that is much tidier.
That is the total force exerted

227
00:16:52,000 --> 00:16:57,000
by all the collisions of the
molecules in the container on

228
00:16:57,000 --> 00:17:03,000
that front wall.
But now I want the pressure.

229
00:17:03,000 --> 00:17:08,000
And the pressure is just force
per unit area.

230
00:17:08,000 --> 00:17:15,000
And so, I am going to take my
expression for the force and

231
00:17:15,000 --> 00:17:21,000
divide it by the unit area.
The area is the area of this

232
00:17:21,000 --> 00:17:25,000
wall, here, in my initial
example.

233
00:17:25,000 --> 00:17:31,000
I am going to call that area A.
And if this is a Q,

234
00:17:31,000 --> 00:17:36,000
and I am going to make it a Q
because it is easier to do it

235
00:17:36,000 --> 00:17:42,000
that way, the area times the
length then of this box is just

236
00:17:42,000 --> 00:17:44,000
the volume.
Here is the volume.

237
00:17:44,000 --> 00:17:49,000
There is my expression for the
pressure due to all of the

238
00:17:49,000 --> 00:17:54,000
molecule colliding with the
front wall of that box.

239
00:17:54,000 --> 00:17:58,000
However, this is an expression
for the pressure in one

240
00:17:58,000 --> 00:18:03,000
dimension only,
the x dimension.

241
00:18:03,000 --> 00:18:05,000
And we know,
in real life,

242
00:18:05,000 --> 00:18:10,000
we have three dimensions.
We have to take care of that.

243
00:18:10,000 --> 00:18:14,000
Let's do that now.
Let's extend this problem to

244
00:18:14,000 --> 00:18:18,000
three dimensions.
To do so, I am just going to

245
00:18:18,000 --> 00:18:23,000
realize right here that the
square of the velocity is the

246
00:18:23,000 --> 00:18:26,000
sum of the squares of the
components.

247
00:18:26,000 --> 00:18:31,000
That, you understand.
That is okay.

248
00:18:31,000 --> 00:18:37,000
What is not so obvious is this.
The average of the square of

249
00:18:37,000 --> 00:18:43,000
the velocity is the sum of the
average of the squares of each

250
00:18:43,000 --> 00:18:47,000
one of the components.
That is true.

251
00:18:47,000 --> 00:18:52,000
You can prove that.
We are not going to prove that.

252
00:18:52,000 --> 00:18:57,000
I won't hold you responsible
for proving that,

253
00:18:57,000 --> 00:19:01,000
but that is true.
That is correct,

254
00:19:01,000 --> 00:19:05,000
but now here comes a critical
assumption in Boltzmann's

255
00:19:05,000 --> 00:19:08,000
treatment.
The critical assumption is that

256
00:19:08,000 --> 00:19:12,000
the motion of the molecules in
this gas, here,

257
00:19:12,000 --> 00:19:16,000
is random in the sense that the
molecules don't have a preferred

258
00:19:16,000 --> 00:19:19,000
direction.
They are going in the x

259
00:19:19,000 --> 00:19:23,000
direction as often as they are
going in the y direction as

260
00:19:23,000 --> 00:19:28,000
often as they are going in the z
direction, so the motion is

261
00:19:28,000 --> 00:19:32,000
random.
If that motion is random,

262
00:19:32,000 --> 00:19:36,000
then the average of the
velocity squared in the x

263
00:19:36,000 --> 00:19:42,000
direction is going to be equal
to that in the y direction.

264
00:19:42,000 --> 00:19:47,000
It is going to be equal to that
in the z direction if that

265
00:19:47,000 --> 00:19:52,000
motion is random.
And that is great because it is

266
00:19:52,000 --> 00:19:56,000
going to make things a little
simpler for us up here.

267
00:19:56,000 --> 00:20:00,000
If that is right,
then the average of the

268
00:20:00,000 --> 00:20:06,000
velocity squared is three times
the average of the velocity

269
00:20:06,000 --> 00:20:11,000
squared in any one of the
dimensions.

270
00:20:11,000 --> 00:20:16,000
That is going to make it easy
to extrapolate this to three

271
00:20:16,000 --> 00:20:22,000
dimensions because now I am
going to be able to substitute,

272
00:20:22,000 --> 00:20:28,000
which had the average of the
velocity squared only in the x

273
00:20:28,000 --> 00:20:34,000
direction, I am going to be able
to substitute in an expression

274
00:20:34,000 --> 00:20:40,000
for the average velocity in
three dimensions.

275
00:20:40,000 --> 00:20:43,000
That is just going to be
one-third that.

276
00:20:43,000 --> 00:20:48,000
This is going to be one-third
the average of the velocity

277
00:20:48,000 --> 00:20:50,000
squared.
That is great.

278
00:20:50,000 --> 00:20:56,000
Now, I am going to do that
substitution way up into there.

279
00:20:56,000 --> 00:21:00,000
And when I do that,
look at this.

280
00:21:00,000 --> 00:21:04,000
I have a result.
This is the kinetic theory

281
00:21:04,000 --> 00:21:07,000
result.
We just did exactly what

282
00:21:07,000 --> 00:21:11,000
Maxwell did.
We have an expression for the

283
00:21:11,000 --> 00:21:16,000
pressure times the volume,
which is written here in terms

284
00:21:16,000 --> 00:21:21,000
of the average of the velocity
squared of the molecules.

285
00:21:21,000 --> 00:21:28,000
It is written in terms of the
motion of the molecules.

286
00:21:28,000 --> 00:21:31,000
For the first time,
there is an understanding,

287
00:21:31,000 --> 00:21:36,000
here, of what gives rise to
pressure, and that is the

288
00:21:36,000 --> 00:21:41,000
velocity or the motion of these
molecules hitting the wall.

289
00:21:41,000 --> 00:21:45,000
That is great.
That is the kinetic theory

290
00:21:45,000 --> 00:21:48,000
result.
But now, Boltzmann also knew

291
00:21:48,000 --> 00:21:53,000
from experiment that P times V
is equal to nRT.

292
00:21:53,000 --> 00:21:57,000
That is the experimental
result, which had been known

293
00:21:57,000 --> 00:22:02,000
already for over a hundred
years.

294
00:22:02,000 --> 00:22:07,000
That is the experiment.
If his theory is correct,

295
00:22:07,000 --> 00:22:14,000
if PV is equal to N m average
velocity squared over three,

296
00:22:14,000 --> 00:22:20,000
it better be equal to nRT.

297
00:22:20,000 --> 00:22:26,000
That will give us,
here, a prediction for what the

298
00:22:26,000 --> 00:22:33,000
velocity of the molecules ought
to be in terms of something

299
00:22:33,000 --> 00:22:40,000
experimentally controllable.
We can see if this kinetic

300
00:22:40,000 --> 00:22:45,000
theory model is correct.
We can solve this for the

301
00:22:45,000 --> 00:22:52,000
average of the velocity squared.
It is equal to 3n RT N over m.

302
00:22:52,000 --> 00:22:57,000
We can go in the laboratory,

303
00:22:57,000 --> 00:23:02,000
vary T and see if,
in fact, the average of the

304
00:23:02,000 --> 00:23:07,000
velocity squared of the
molecules is equal to this

305
00:23:07,000 --> 00:23:11,000
expression here.
That is great.

306
00:23:11,000 --> 00:23:16,000
We have a way to experimentally
check this theory.

307
00:23:16,000 --> 00:23:21,000
And you also see,
here, now, a relationship

308
00:23:21,000 --> 00:23:26,000
between the velocity of the
molecules and this macroscopic

309
00:23:26,000 --> 00:23:32,000
quantity, temperature.
Temperature is related to the

310
00:23:32,000 --> 00:23:37,000
motion of these molecules.
But, before we go on,

311
00:23:37,000 --> 00:23:41,000
this is kind of a messy
expression here.

312
00:23:41,000 --> 00:23:44,000
It has too many n's and m's in
it.

313
00:23:44,000 --> 00:23:48,000
Let me simplify that a little
bit for you.

314
00:23:48,000 --> 00:23:54,000
I am going to simplify this so
that this is 3RT over capital M,

315
00:23:54,000 --> 00:24:00,000
where the following is true.

316
00:24:00,000 --> 00:24:04,000
Over here, n is the number of
moles in the gas.

317
00:24:04,000 --> 00:24:08,000
That is little n.
Big N was the number of

318
00:24:08,000 --> 00:24:12,000
molecules in the gas.
Little m was the mass per

319
00:24:12,000 --> 00:24:15,000
molecule.
All of this is equivalent to

320
00:24:15,000 --> 00:24:20,000
one over big M,
where big M was kilograms per

321
00:24:20,000 --> 00:24:23,000
mole.
You can convince yourselves of

322
00:24:23,000 --> 00:24:27,000
this equality.
I am taking all these N's and

323
00:24:27,000 --> 00:24:31,000
m's and making one big M.

324
00:24:36,000 --> 00:24:38,000
That is my expression,
here.

325
00:24:38,000 --> 00:24:42,000
I have the average of the
velocity squared equal to 3RT

326
00:24:42,000 --> 00:24:47,000
over M.
But this quantity is the

327
00:24:47,000 --> 00:24:52,000
average of the velocity squared.
It is more convenient for us to

328
00:24:52,000 --> 00:24:57,000
talk about a quantity
proportional to the velocity and

329
00:24:57,000 --> 00:25:02,000
not the velocity squared.
What I am going to do is take

330
00:25:02,000 --> 00:25:04,000
the square root of it.
That is simple.

331
00:25:04,000 --> 00:25:09,000
I now have the square root of
the average of the velocity

332
00:25:09,000 --> 00:25:12,000
squared.
That is the square root of 3RT

333
00:25:12,000 --> 00:25:14,000
over M.
I am going to call that the

334
00:25:14,000 --> 00:25:18,000
root mean square velocity.
I am going to put an rms here

335
00:25:18,000 --> 00:25:23,000
as a subscript for the velocity.

336
00:25:23,000 --> 00:25:25,000
It is the root mean square
velocity.

337
00:25:25,000 --> 00:25:29,000
I wanted to talk about a
quantity proportional to the

338
00:25:29,000 --> 00:25:34,000
velocity, instead of the
velocity squared.

339
00:25:34,000 --> 00:25:38,000
That is all I did there.
That is the root mean square

340
00:25:38,000 --> 00:25:41,000
velocity.
But the other big thing about

341
00:25:41,000 --> 00:25:45,000
it is you can see,
for the first time,

342
00:25:45,000 --> 00:25:47,000
now we have got,
and Maxwell had,

343
00:25:47,000 --> 00:25:51,000
an understanding of what
temperature was.

344
00:25:51,000 --> 00:25:56,000
Temperature is related to the
motions of the molecules.

345
00:25:56,000 --> 00:25:59,000
Temperature is related,
in this way,

346
00:25:59,000 --> 00:26:06,000
to the speed of the molecules.
Those are the two important

347
00:26:06,000 --> 00:26:11,000
results.
And this kinetic theory makes a

348
00:26:11,000 --> 00:26:19,000
prediction for what those
velocities ought to be.

349
00:26:26,000 --> 00:26:30,000
In addition,
the temperature is a measure of

350
00:26:30,000 --> 00:26:34,000
the kinetic energy of the
molecules.

351
00:26:34,000 --> 00:26:37,000
How is that?
Well, it is for this reason.

352
00:26:37,000 --> 00:26:43,000
Here is the expression we
derived from the kinetic theory.

353
00:26:43,000 --> 00:26:47,000
And then here is an expression
that I just wrote down,

354
00:26:47,000 --> 00:26:53,000
that says the average kinetic
energy of a molecule is one-half

355
00:26:53,000 --> 00:26:58,000
M, where M is kilograms per
mole, times the average of the

356
00:26:58,000 --> 00:27:04,000
velocity squared.

357
00:27:04,000 --> 00:27:09,000
If I substitute the average of
the velocity squared into here,

358
00:27:09,000 --> 00:27:13,000
I get three-halves RT.

359
00:27:13,000 --> 00:27:16,000
This is telling us,
right here, that the

360
00:27:16,000 --> 00:27:22,000
temperature is also a measure of
the kinetic energy of these

361
00:27:22,000 --> 00:27:25,000
molecules.
We are getting a microscopic

362
00:27:25,000 --> 00:27:30,000
view, here, of what temperature
is.

363
00:27:30,000 --> 00:27:34,000
It is related to the motions of
these molecules.

364
00:27:34,000 --> 00:27:40,000
Now, before I go on talking
about this, let me make one big

365
00:27:40,000 --> 00:27:43,000
point.
That is, this expression here,

366
00:27:43,000 --> 00:27:48,000
for the average energy,
notice that it is one-half M

367
00:27:48,000 --> 00:27:53,000
times the average of the
velocity squared.

368
00:27:53,000 --> 00:27:59,000
It is not one-half M times the
square of the average velocity

369
00:27:59,000 --> 00:28:03,000
This is important.

370
00:28:03,000 --> 00:28:09,000
The average energy is not the
square of the average velocity.

371
00:28:09,000 --> 00:28:14,000
Rather, the average energy is
the average of the velocity

372
00:28:14,000 --> 00:28:17,000
squared.
There is a big distinction.

373
00:28:17,000 --> 00:28:23,000
This is because the average
energy is the second moment of

374
00:28:23,000 --> 00:28:26,000
the velocity distribution
function.

375
00:28:26,000 --> 00:28:31,000
Variables don't always
correspond in a one-to-one

376
00:28:31,000 --> 00:28:36,000
manner.
You don't have to understand

377
00:28:36,000 --> 00:28:43,000
that, if this is foreign to you,
but I do want you to know this

378
00:28:43,000 --> 00:28:48,000
is correct.
Now, let me pick up back here.

379
00:28:48,000 --> 00:28:54,000
What I want you to notice is
that the root mean square

380
00:28:54,000 --> 00:29:00,000
velocity has a mass dependence
in it.

381
00:29:00,000 --> 00:29:04,000
What does that mean?
Well, it means the following.

382
00:29:04,000 --> 00:29:09,000
For some constant temperature,
say we pick 300 degrees Kelvin,

383
00:29:09,000 --> 00:29:14,000
the velocity of the molecule is
going to depend on its mass.

384
00:29:14,000 --> 00:29:18,000
And it is inversely
proportional to the mass,

385
00:29:18,000 --> 00:29:23,000
so heavier molecules move more
slowly, lighter molecules move

386
00:29:23,000 --> 00:29:25,000
more quickly.
For example,

387
00:29:25,000 --> 00:29:30,000
helium at 300 degrees Kelvin,
it is cruising along at 3

388
00:29:30,000 --> 00:29:36,000
miles per hour at
room temperature.

389
00:29:36,000 --> 00:29:40,000
Xenon, on the other hand,
which is much more massive,

390
00:29:40,000 --> 00:29:44,000
is moving at a measly 534 miles
per hour.

391
00:29:44,000 --> 00:29:47,000
There is a mass dependence
here.

392
00:29:47,000 --> 00:29:52,000
However, there is no mass
dependence to the kinetic

393
00:29:52,000 --> 00:29:54,000
energy.
The kinetic energy,

394
00:29:54,000 --> 00:29:58,000
we saw, was three-halves RT.

395
00:29:58,000 --> 00:30:04,000
You don't see a mass dependence
in here, do you?

396
00:30:04,000 --> 00:30:09,000
The kinetic energy is only
dependent on the temperature.

397
00:30:09,000 --> 00:30:15,000
Whether or not you have helium
or xenon, the kinetic energy of

398
00:30:15,000 --> 00:30:21,000
those atoms is 3.74 kilojoules
per mole at 300 degrees Kelvin.

399
00:30:21,000 --> 00:30:26,000
It does not matter that helium
is moving six times as fast as

400
00:30:26,000 --> 00:30:30,000
xenon.
They both have the same kinetic

401
00:30:30,000 --> 00:30:34,000
energy.
There is no mass dependence in

402
00:30:34,000 --> 00:30:36,000
kinetic energy.
Now you say,

403
00:30:36,000 --> 00:30:39,000
oh, but look at this,
here is a mass,

404
00:30:39,000 --> 00:30:43,000
there is a mass dependence.
No, because you have to

405
00:30:43,000 --> 00:30:46,000
remember that you substitute in
here.

406
00:30:46,000 --> 00:30:49,000
If you square this,
there is an M here,

407
00:30:49,000 --> 00:30:53,000
and that cancels.
There is no mass dependence in

408
00:30:53,000 --> 00:30:57,000
the kinetic energy,
but the velocity is dependent

409
00:30:57,000 --> 00:31:01,000
on the mass.
That is important.

410
00:31:01,000 --> 00:31:07,000
Well, I told you that this was
the kinetic theory result.

411
00:31:07,000 --> 00:31:12,000
That is, that the root mean
square velocity of these

412
00:31:12,000 --> 00:31:16,000
molecules was represented by
this equation.

413
00:31:16,000 --> 00:31:21,000
And, from this equation,
if you calculate at 300 degrees

414
00:31:21,000 --> 00:31:25,000
Kelvin, these are,
in fact, the velocities of

415
00:31:25,000 --> 00:31:30,000
those atoms.
But how do we know this is

416
00:31:30,000 --> 00:31:33,000
right?
How do we go and measure the

417
00:31:33,000 --> 00:31:38,000
velocities or the speeds of
molecules or atoms?

418
00:31:38,000 --> 00:31:44,000
Well, this is the way we do it.
It is called the time-of-flight

419
00:31:44,000 --> 00:31:47,000
technique.
Are we all on board,

420
00:31:47,000 --> 00:31:48,000
here?
Questions?

421
00:31:48,000 --> 00:31:51,000
Okay.
How are we going to do this

422
00:31:51,000 --> 00:31:56,000
time-of-flight technique?
What we are going to do is we

423
00:31:56,000 --> 00:32:02,000
are going to have a little
pinhole here that we can open

424
00:32:02,000 --> 00:32:08,000
and shut really quickly.
We are going to let out a

425
00:32:08,000 --> 00:32:12,000
little pulse of gas.
To measure the velocity of the

426
00:32:12,000 --> 00:32:16,000
molecules, we are literally
going to measure the time it

427
00:32:16,000 --> 00:32:21,000
takes the molecules to fly from
where we let them out to some

428
00:32:21,000 --> 00:32:24,000
detector.
And since we know the distance,

429
00:32:24,000 --> 00:32:28,000
we are going to be able to
calculate the velocity from

430
00:32:28,000 --> 00:32:32,000
that.
The idea is at time t equals 0,

431
00:32:32,000 --> 00:32:38,000
we let out a little pulse of
gas, and then we start a clock

432
00:32:38,000 --> 00:32:41,000
running.
Then we just measure how long

433
00:32:41,000 --> 00:32:46,000
it takes the molecules to fly
from this origin here to this

434
00:32:46,000 --> 00:32:49,000
detector.
And since we built the

435
00:32:49,000 --> 00:32:54,000
apparatus, and we know what L
is, we can calculate the

436
00:32:54,000 --> 00:32:56,000
velocity.
Time-of-flight,

437
00:32:56,000 --> 00:33:02,000
that is what is done.
However, when we let this

438
00:33:02,000 --> 00:33:07,000
little pulse of gas out,
and now we let the molecules

439
00:33:07,000 --> 00:33:12,000
fly to that detector over here,
what happens as a function of

440
00:33:12,000 --> 00:33:15,000
time?
What will happen is that pulse

441
00:33:15,000 --> 00:33:21,000
of gas will spread out because
not all of the molecules or

442
00:33:21,000 --> 00:33:25,000
atoms in that pulse of gas have
the same velocity.

443
00:33:25,000 --> 00:33:32,000
Some of those atoms are moving
faster than the other atoms.

444
00:33:32,000 --> 00:33:36,000
And so, what is going to happen
is that the molecules or atoms

445
00:33:36,000 --> 00:33:41,000
that are moving faster are going
to hit the detector first.

446
00:33:41,000 --> 00:33:46,000
The molecules or atoms that are
moving more slowly are going to

447
00:33:46,000 --> 00:33:48,000
hit the detector at a later
time.

448
00:33:48,000 --> 00:33:53,000
And that is what we also want
to know, this distribution of

449
00:33:53,000 --> 00:33:55,000
velocities.
But in the measurement,

450
00:33:55,000 --> 00:34:01,000
what we are going to measure is
a distribution of times.

451
00:34:01,000 --> 00:34:04,000
Out of our detector,
we are going to have a plot

452
00:34:04,000 --> 00:34:08,000
that looks like this.
This is going to be f of t,

453
00:34:08,000 --> 00:34:12,000
essentially the number of
molecules hitting the detector

454
00:34:12,000 --> 00:34:15,000
at a certain time t versus the
time.

455
00:34:15,000 --> 00:34:20,000
When we first let our pulse of
gas out, that is time t equals

456
00:00:00,000 --> 00:34:22,000
Then, for a while,

457
00:34:22,000 --> 00:34:26,000
there are no molecules hitting
the detector because it takes a

458
00:34:26,000 --> 00:34:32,000
while for them to get over here
to this detector.

459
00:34:32,000 --> 00:34:35,000
But then, all of a sudden,
they start reaching the

460
00:34:35,000 --> 00:34:38,000
detector.
And this is essentially just a

461
00:34:38,000 --> 00:34:43,000
number of molecules that hit the
detector as a function of time.

462
00:34:43,000 --> 00:34:47,000
That number of molecules
increases and becomes a maximum

463
00:34:47,000 --> 00:34:50,000
here at some time,
and then it exponentially

464
00:34:50,000 --> 00:34:53,000
decays, here.
So, this is what we measure.

465
00:34:53,000 --> 00:34:58,000
This is a distribution here of
flight times of the molecules in

466
00:34:58,000 --> 00:35:02,000
this pulse of gas.
Well, that is nice,

467
00:35:02,000 --> 00:35:05,000
but this is a distribution of
flight times.

468
00:35:05,000 --> 00:35:10,000
It is not a distribution of
velocities, and we wanted a

469
00:35:10,000 --> 00:35:14,000
distribution of velocities.
We want to know the velocity,

470
00:35:14,000 --> 00:35:19,000
here, of these molecules.
You know how to convert time,

471
00:35:19,000 --> 00:35:22,000
given the path length,
to velocity,

472
00:35:22,000 --> 00:35:25,000
but it is not so
straightforward because we have

473
00:35:25,000 --> 00:35:31,000
a distribution function.
We have a distribution in time,

474
00:35:31,000 --> 00:35:36,000
and we want to convert that to
a distribution in velocity.

475
00:35:36,000 --> 00:35:41,000
We have to change the variable
here in a distribution function.

476
00:35:41,000 --> 00:35:45,000
How do we do that?
Well, we want this f of t to be

477
00:35:45,000 --> 00:35:50,000
an f of v.
We recognize here that this

478
00:35:50,000 --> 00:35:54,000
distribution in time,
the probability of finding a

479
00:35:54,000 --> 00:35:59,000
molecule between t and t plus
dt, has got to be equivalent to

480
00:35:59,000 --> 00:36:04,000
the probability of finding a
molecule with a velocity between

481
00:36:04,000 --> 00:36:09,000
v and v plus dv.
But to get from one

482
00:36:09,000 --> 00:36:13,000
distribution function to
another, for example,

483
00:36:13,000 --> 00:36:20,000
if we want f of v,
what we have to know is how one

484
00:36:20,000 --> 00:36:25,000
variable changes with respect to
another.

485
00:36:30,000 --> 00:36:33,000
We have this distribution
function f of t,

486
00:36:33,000 --> 00:36:37,000
but we need to know how t
changes with v.

487
00:36:37,000 --> 00:36:40,000
We need dt by dv,
we need that,

488
00:36:40,000 --> 00:36:44,000
so let's get it.
I will tell you why in a

489
00:36:44,000 --> 00:36:47,000
moment.
We know how v changes with t.

490
00:36:47,000 --> 00:36:52,000
We are going to take the
derivative of v with respect to

491
00:36:52,000 --> 00:36:58,000
t and turn things around.
So, dt / dv is proportional to

492
00:36:58,000 --> 00:37:04,000
minus t squared over L.

493
00:37:04,000 --> 00:37:06,000
This is telling us,
essentially,

494
00:37:06,000 --> 00:37:09,000
how the variable t changes with
v.

495
00:37:09,000 --> 00:37:13,000
To every point in our
time-of-flight distribution,

496
00:37:13,000 --> 00:37:17,000
we are going to multiply this
by this, what is called the

497
00:37:17,000 --> 00:37:20,000
Jacobean.
We need this because the time

498
00:37:20,000 --> 00:37:24,000
and the velocity do not
correlate in a one-to-one

499
00:37:24,000 --> 00:37:27,000
manner.
That often happens with two

500
00:37:27,000 --> 00:37:33,000
distribution functions.
If you do not understand what I

501
00:37:33,000 --> 00:37:38,000
just said, it is okay.
This was just some extra.

502
00:37:38,000 --> 00:37:42,000
I do not hold you responsible
for it.

503
00:37:42,000 --> 00:37:48,000
I just changed my variable in
the distribution function.

504
00:37:48,000 --> 00:37:50,000
Yes?
I think in the notes,

505
00:37:50,000 --> 00:37:54,000
I might have had as a
proportionality.

506
00:37:54,000 --> 00:38:00,000
Up here I actually have the
equal sign.

507
00:38:00,000 --> 00:38:04,000
That actually won't matter in
this transformation.

508
00:38:04,000 --> 00:38:06,000
Pardon?
I understand that.

509
00:38:06,000 --> 00:38:10,000
That is fine.
I don't have the equal sign

510
00:38:10,000 --> 00:38:13,000
there.
That is why I left it out

511
00:38:13,000 --> 00:38:17,000
there, I think.
Anyway, this is the velocity

512
00:38:17,000 --> 00:38:20,000
distribution.
You don't have to understand

513
00:38:20,000 --> 00:38:24,000
how I got there.
This is what the velocity

514
00:38:24,000 --> 00:38:29,000
distribution looks like.
It is what is called the

515
00:38:29,000 --> 00:38:32,000
Maxwell-Boltzmann velocity
distribution.

516
00:38:32,000 --> 00:38:36,000
And the bottom line is that
Maxwell and Boltzmann predicted

517
00:38:36,000 --> 00:38:40,000
this, about 1855.
They actually predicted this

518
00:38:40,000 --> 00:38:43,000
distribution function.
We did not predict it.

519
00:38:43,000 --> 00:38:46,000
We did not go through that part
of kinetic theory,

520
00:38:46,000 --> 00:38:50,000
but they predicted it.
However, it was only until

521
00:38:50,000 --> 00:38:54,000
that the technology existed,
fast enough timing and

522
00:38:54,000 --> 00:38:57,000
electronics existed,
to actually measure this

523
00:38:57,000 --> 00:39:02,000
experimentally.
This took a hundred years or so

524
00:39:02,000 --> 00:39:07,000
in order for this distribution
function to actually be

525
00:39:07,000 --> 00:39:10,000
measured, but here it is,
f of v.

526
00:39:10,000 --> 00:39:13,000
First of all,
there is all of this stuff,

527
00:39:13,000 --> 00:39:16,000
which is proportionality
constants.

528
00:39:16,000 --> 00:39:19,000
We will talk about that in a
moment.

529
00:39:19,000 --> 00:39:23,000
But the variable,
here, is v squared

530
00:39:23,000 --> 00:39:28,000
times an exponentially decaying
function with a v squared in

531
00:39:28,000 --> 00:39:31,000
there.
What does that mean?

532
00:39:31,000 --> 00:39:35,000
Well, if you look at the form
of f of v, this v squared is

533
00:39:35,000 --> 00:39:38,000
what gives rise to this increase
in f of v.

534
00:39:38,000 --> 00:39:42,000
Right here, at low velocities,
that is a quadratic,

535
00:39:42,000 --> 00:39:45,000
v squared.
But you are multiplying it by

536
00:39:45,000 --> 00:39:49,000
an exponentially decaying
function with this v squared in

537
00:39:49,000 --> 00:39:52,000
the argument.
And so that is what gives you

538
00:39:52,000 --> 00:39:54,000
this tail.
If you are multiplying a

539
00:39:54,000 --> 00:39:58,000
function that is going up and
one decreasing,

540
00:39:58,000 --> 00:40:03,000
you are going to get a maximum
at some value of v.

541
00:40:03,000 --> 00:40:09,000
That is where the shape of the
Maxwell-Boltzmann distribution

542
00:40:09,000 --> 00:40:14,000
function comes from.
And what this is telling you is

543
00:40:14,000 --> 00:40:21,000
the probability here of finding
a molecule in a gas with a speed

544
00:40:21,000 --> 00:40:26,000
between v and v plus dv.
That is what that is telling

545
00:40:26,000 --> 00:40:30,000
you.
We often characterize these

546
00:40:30,000 --> 00:40:35,000
distribution functions by some
quantities, and one of those

547
00:40:35,000 --> 00:40:39,000
quantities is what we call the
most probable speed.

548
00:40:39,000 --> 00:40:44,000
Here is the distribution
function, and I have the most

549
00:40:44,000 --> 00:40:48,000
probable speed labeled.
The most probable speed,

550
00:40:48,000 --> 00:40:53,000
(v)mp, is simply the value of v
at which the probability is the

551
00:40:53,000 --> 00:40:56,000
largest.
That was like our most probable

552
00:40:56,000 --> 00:41:02,000
value of r in the radial
distribution functions.

553
00:41:02,000 --> 00:41:04,000
That is what the most probable
speed is.

554
00:41:04,000 --> 00:41:07,000
If you wanted to get that
mathematically,

555
00:41:07,000 --> 00:41:11,000
what you would do is take this
distribution function,

556
00:41:11,000 --> 00:41:14,000
take the derivative,
set it equal to zero,

557
00:41:14,000 --> 00:41:18,000
and then solve for v.
That makes that derivative

558
00:41:18,000 --> 00:41:21,000
equal to zero.
The derivative is zero at

559
00:41:21,000 --> 00:41:24,000
maxima or minima.
And then, you would find that

560
00:41:24,000 --> 00:41:29,000
the value of the most probable
speed is the square root of 2RT

561
00:41:29,000 --> 00:41:33,000
over M.

562
00:41:33,000 --> 00:41:36,000
You are not responsible for
taking this derivative and

563
00:41:36,000 --> 00:41:41,000
setting it equal to zero.
You are responsible for knowing

564
00:41:41,000 --> 00:41:44,000
physically what the most
probable speed is.

565
00:41:44,000 --> 00:41:48,000
The fact that it is this value
here, where the probability is

566
00:41:48,000 --> 00:41:51,000
the largest.
It is the most probable value

567
00:41:51,000 --> 00:41:54,000
of v.
And you are responsible for

568
00:41:54,000 --> 00:41:57,000
recognizing this.
I don't ask you to memorize it

569
00:41:57,000 --> 00:42:01,000
or to write it down.
But, if you see it,

570
00:42:01,000 --> 00:42:07,000
you should know what it is.
That is one quantity that we

571
00:42:07,000 --> 00:42:11,000
use to characterize this
distribution function.

572
00:42:11,000 --> 00:42:16,000
Another quantity that we use is
the average speed,

573
00:42:16,000 --> 00:42:21,000
v bar, average speed.
And the first thing that you

574
00:42:21,000 --> 00:42:26,000
see is that the average speed is
a little bit higher than the

575
00:42:26,000 --> 00:42:31,000
most probable speed.
It is a little bit larger.

576
00:42:31,000 --> 00:42:35,000
Why is that?
Well, it is a little bit larger

577
00:42:35,000 --> 00:42:38,000
because on these
Maxwell-Boltzmann distribution

578
00:42:38,000 --> 00:42:43,000
functions, there are molecules
way out here that have very high

579
00:42:43,000 --> 00:42:45,000
speeds.
There are not a lot of

580
00:42:45,000 --> 00:42:50,000
molecules that have very high
speeds, but there are molecules

581
00:42:50,000 --> 00:42:54,000
with very high speeds.
And so, when you average over

582
00:42:54,000 --> 00:42:58,000
this distribution function,
because there is such a long

583
00:42:58,000 --> 00:43:01,000
Boltzmann tail here,
is what it is called,

584
00:43:01,000 --> 00:43:05,000
the average velocity is going
to be a little higher than the

585
00:43:05,000 --> 00:43:10,000
most probable velocity.
That is physically why the

586
00:43:10,000 --> 00:43:15,000
average velocity is a little bit
larger than the most probable

587
00:43:15,000 --> 00:43:17,000
velocity.
That is important.

588
00:43:17,000 --> 00:43:20,000
If I wanted to calculate what
the average velocity is,

589
00:43:20,000 --> 00:43:24,000
I would take the distribution
function, multiply it by v,

590
00:43:24,000 --> 00:43:28,000
and then integrate over the
range of v, which is zero to

591
00:43:28,000 --> 00:43:32,000
infinity.
You don't have to do that,

592
00:43:32,000 --> 00:43:36,000
but the quantity that you would
get, if you had done that,

593
00:43:36,000 --> 00:43:41,000
is the square root of 8RT over
pi M.

594
00:43:41,000 --> 00:43:43,000
It is larger than the most

595
00:43:43,000 --> 00:43:46,000
probable value.
And then, finally,

596
00:43:46,000 --> 00:43:50,000
the other way we characterize
the distribution function is by

597
00:43:50,000 --> 00:43:55,000
this root mean square speed that
we have already talked about a

598
00:43:55,000 --> 00:43:57,000
lot.
That root mean square speed,

599
00:43:57,000 --> 00:44:02,000
here, is even a larger value
than the average velocity or the

600
00:44:02,000 --> 00:44:07,000
average speed.
And the reason for that is,

601
00:44:07,000 --> 00:44:10,000
again, because of this
Maxwell-Boltzmann tail.

602
00:44:10,000 --> 00:44:14,000
We have molecules here that
have very high speeds.

603
00:44:14,000 --> 00:44:18,000
We don't have a lot of
molecules with high speeds,

604
00:44:18,000 --> 00:44:22,000
but we have very high speeds.
When we take that speed and we

605
00:44:22,000 --> 00:44:27,000
square it, and then take the
average, they make a big

606
00:44:27,000 --> 00:44:30,000
contribution,
and they push this root mean

607
00:44:30,000 --> 00:44:35,000
square speed even higher than
the average speed.

608
00:44:35,000 --> 00:44:38,000
Again, if you wanted to
calculate what that is,

609
00:44:38,000 --> 00:44:43,000
you would take v squared,
multiply it by the distribution

610
00:44:43,000 --> 00:44:49,000
function, integrate it over all
values of E, and you would get

611
00:44:49,000 --> 00:44:53,000
what we got before,
the square root of 3RT over M.

612
00:44:53,000 --> 00:44:57,000
Bottom line here,

613
00:44:57,000 --> 00:45:01,000
for argon at 300 degrees
Kelvin, the most probable speed

614
00:45:01,000 --> 00:45:07,000
is 353 meters per second.
For argon at 300 degrees

615
00:45:07,000 --> 00:45:11,000
Kelvin, the average speed is
meters per second.

616
00:45:11,000 --> 00:45:17,000
And the root mean square speed
is 433 meters per second.

617
00:45:17,000 --> 00:45:22,000
Here, you can see how these
three quantities increase as you

618
00:45:22,000 --> 00:45:28,000
go from most probable to the
root mean square speed.

619
00:45:28,000 --> 00:45:31,000
And, in fact,
no surprise,

620
00:45:31,000 --> 00:45:38,000
if you actually go and measure
the argon speeds distribution

621
00:45:38,000 --> 00:45:44,000
function and then evaluate these
characteristics,

622
00:45:44,000 --> 00:45:51,000
you find that indeed those
measurements agree with what Mr.

623
00:45:51,000 --> 00:45:56,000
Maxwell and Mr.
Boltzmann predicted in 1850,

624
00:46:00,000 --> 00:46:02,000
Back to this distribution
function.

625
00:46:02,000 --> 00:46:06,000
We saw how to characterize it,
but I want to talk about a

626
00:46:06,000 --> 00:46:10,000
couple of other parameters that
it has in it.

627
00:46:10,000 --> 00:46:13,000
It has in it the mass,
and it has it in the

628
00:46:13,000 --> 00:46:16,000
temperature.
Let's talk about what this

629
00:46:16,000 --> 00:46:20,000
distribution function looks like
for different masses and

630
00:46:20,000 --> 00:46:24,000
different temperatures.
That is the general form,

631
00:46:24,000 --> 00:46:29,000
but different masses and
different temperatures.

632
00:46:29,000 --> 00:46:32,000
Let's start by keeping the
temperature constant,

633
00:46:32,000 --> 00:46:35,000
300 degrees Kelvin.
And now, we are going to look

634
00:46:35,000 --> 00:46:39,000
at the distribution functions
for three different masses,

635
00:46:39,000 --> 00:46:43,000
and I am going to plot that.
Here it is for xenon.

636
00:46:43,000 --> 00:46:46,000
What you can see is xenon,
very narrow distribution.

637
00:46:46,000 --> 00:46:49,000
Here it is for argon,
lighter mass,

638
00:46:49,000 --> 00:46:52,000
broader distribution.
Here it is for helium,

639
00:46:52,000 --> 00:46:55,000
really broad distribution.
Well, first of all,

640
00:46:55,000 --> 00:46:59,000
you can see that the average
speed of helium is much greater

641
00:46:59,000 --> 00:47:04,000
than it is for argon,
than it is for xenon.

642
00:47:04,000 --> 00:47:09,000
Because we already saw that it
was inversely proportional to

643
00:47:09,000 --> 00:47:15,000
the square root of the mass.
But the other thing to note is

644
00:47:15,000 --> 00:47:21,000
how broad the distribution is
for helium compared to xenon.

645
00:47:21,000 --> 00:47:26,000
And that is the general case.
The lighter masses have broader

646
00:47:26,000 --> 00:47:30,000
distributions.
Helium is so broad,

647
00:47:30,000 --> 00:47:36,000
meaning that there are helium
atoms here that are so fast that

648
00:47:36,000 --> 00:47:41,000
this is the reason why there is
relatively little helium and

649
00:47:41,000 --> 00:47:45,000
hydrogen in our atmosphere.
That is because,

650
00:47:45,000 --> 00:47:49,000
at our temperatures,
there are enough molecules with

651
00:47:49,000 --> 00:47:55,000
high enough velocities to escape
the earth's gravitational pull,

652
00:47:55,000 --> 00:48:00,000
because they are here at the
tail end.

653
00:48:00,000 --> 00:48:05,000
And so, the helium and the
hydrogen leave our atmosphere.

654
00:48:05,000 --> 00:48:09,000
Unlike Jupiter,
which is 300 times more massive

655
00:48:09,000 --> 00:48:14,000
than the earth,
where the gravitational pull is

656
00:48:14,000 --> 00:48:16,000
greater.
And, in that case,

657
00:48:16,000 --> 00:48:22,000
then those helium atoms don't
have enough velocity to escape

658
00:48:22,000 --> 00:48:26,000
the earth's gravitational pull.
So, that is mass.

659
00:48:26,000 --> 00:48:32,000
What about temperature?
Now we are going to keep the

660
00:48:32,000 --> 00:48:35,000
mass constant and we are going
to look at temperature.

661
00:48:35,000 --> 00:48:39,000
Here is the distribution
function at 100 degrees Kelvin,

662
00:48:39,000 --> 00:48:42,000
pretty narrow.
Here is the distribution

663
00:48:42,000 --> 00:48:45,000
function for argon at
degrees Kelvin,

664
00:48:45,000 --> 00:48:47,000
broader.
Here is the distribution

665
00:48:47,000 --> 00:48:50,000
function at 1000 degrees kelvin,
broader again.

666
00:48:50,000 --> 00:48:55,000
You see that as we increase the
temperature, the average speed

667
00:48:55,000 --> 00:48:58,000
increases.
As we increase the temperature,

668
00:48:58,000 --> 00:49:03,000
the width of that distribution
function increases.

669
00:49:03,000 --> 00:49:07,000
Now, one thing to note here is
this is a probability.

670
00:49:07,000 --> 00:49:12,000
The areas under these curves
have to all add up to one.

671
00:49:12,000 --> 00:49:17,000
If this distribution function
is going to move to higher

672
00:49:17,000 --> 00:49:21,000
velocities as we increase the
temperature, well,

673
00:49:21,000 --> 00:49:26,000
of course this maximum
probability is going to have to

674
00:49:26,000 --> 00:49:31,000
go down because we cannot lose
any molecules.

675
00:49:31,000 --> 00:49:35,000
All the area under this curve,
at 100 degrees Kelvin,

676
00:49:35,000 --> 00:49:40,000
has to equal the area under
this curve, at 1000 degrees

677
00:49:40,000 --> 00:49:45,000
Kelvin, because this is a
probability that we are plotting

678
00:49:45,000 --> 00:49:48,000
here.
That is our description of the

679
00:49:48,582 --> 00:49:51,000
Maxwell-Boltzmann distribution.
See you on Friday.