# Old exam questions Most probable speed Mean speed Root mean ```Old exam questions
Most probable speed
3
2004
HE 3.1, 3.2
HE 4.1, 4.2
HE 5.1
F3
2005
HE 3.1, 3.2
HE 4.1, 4.2
HE 5.1, 5.2
F 3, 5a-c
2007
E1 4, 5
E2 1, 2, 3, 4
... -
⎛ m ⎞ 2 2 − mu 2 2 k BT
⎟⎟ u e
F (u ) = 4π ⎜⎜
⎝ 2πk BT ⎠
2006
M 3, 4
2008
E2 1, 2, 3, 4, 5
2
⌧ Mean speed
⎛ m
F (u ) = 4π ⎜⎜
⎝ 2πk BT
... -
⌧ ... -
3
⌧ Root mean square speed
3
⎞ 2 2 − mu 2 2 k BT
⎟⎟ u e
⎠
4
⎛ m
F (u ) = 4π ⎜⎜
⎝ 2πk BT
... -
⌧ 3
⎞ 2 2 − mu 2 2 k BT
⎟⎟ u e
⎠
5
Example: T of a single molecule?
Maxwell-Boltzmann distribution
u mp =
2 k BT
m
u av =
8k BT
πm
ump =
2 k BT
m
8k B T
uav =
πm
urms
ump =
Suppose you have a single molecule of N2 in a box. Its
speed is found to be 575 m/sec. What is its
temperature?
3k BT
=
m
u mp =
2 k BT
m
2 k BT
m
u av =
8 k BT
πm
urms =
3k BT
m
Thermodynamic temperature is the property that
determines if two systems are in thermal equilibrium
with each other.
... -
⌧ 6
7
⌧ ... -
continued…
Example: Average KE of a gas molecule?
Revised: At what temperature would a single molecule of
N2 moving at 575 m/sec have kinetic energy equal to
kBT?
Suppose you have a mole of N2 in a box. The average
speed for gas molecules is found to be 575 m/sec.
What is the average kinetic energy of a molecule of the
gas?
u mp =
... -
⌧ 8
2 k BT
m
u av =
... -
8 k BT
πm
urms =
⌧ 3k BT
m
9
Molecular collisions in a gas
Collisions per sec for one molecule
z=4 π
• How many collisions does
a gas molecule make per
sec?
• How far does a gas
molecule travel between
collisions?
• How manyy total collisions
per sec are there in a
sample of gas?
⌧ ... -
10
Total # of collisions in a sample
Z tot =
... -
N
2
... -
N 2 RT
σ
V
M
⌧ 11
Mean free path
⎛
N
RT ⎞
⎜4 π σ 2
⎟
⎜
⎟
V
M
⎝
⎠
⌧ 12
... -
⌧ 13
Example: Nitrogen transport kinetics
Diffusion coefficient and molecular shape
Suppose you have a mole of N2 in a box at STP. What
are 1) the number of collisions per second for a N2
molecule, 2) the total number of molecular collisions in
the box in a millisecond,
millisecond 3) the characteristic time
between collisions for a typical N2 molecule and 4) the
typical distance a molecule flies between collisions?
σ = 3.74 &times; 10-10 m
... -
ux
M = 28 g/mol
⌧ 14
Microscopic diffusion constant
... -
⌧ 15
Einstein relationship
d 2x
dx
m 2 = Fc , x (t ) − f
dt
dt
&lt; x 2 &gt;= 2 Dt
ux
... -
⌧ 16
... -
⌧ 17
Diffusion of a sphere
Diffusion of nonspherical particles
f 0 = 6πηa
ux
What is f for a
sphere?
h ?
2
2⎛a⎞
⎜ ⎟
3⎝b⎠
f
=
f0
⎛ a⎞
ln⎜ 2 ⎟ − 0.30
⎝ b⎠
3
2
aeff = 3 3ab
... -
⌧ 18
Example: diffusion coefficient of a nanotube
... -
η = 1.0 &times; 10-3 Pa&middot;sec
a/b = 1 &times; 104
A carbon nanotube is 5 nm in diameter and has a lengthto-diameter ratio of 10,000. What is the diffusion
coefficient of this nanotube in water at T = 25 &deg;C?
C?
2
19
⌧ continues
2
aeff = 3 3ab
2
f 0 = 6πη a
a = 2.5 &times; 10-5 m
b = 2.5 &times; 10-9 m
η = 1.0 &times; 10-3 Pa&middot;sec
2
2⎛a⎞
⎜ ⎟
3⎝b⎠
f = f0
⎛ a⎞
ln⎜ 2 ⎟ − 0.30
⎝ b⎠
3
... -
⌧ 20
... -
⌧ 21
```