Problem Sheet 2 solution - WebRing

On average, molecule M travels distance λ before colliding with any other molecule. Before
collision there should no other molecule except M in the cylinder of volume π ×d 2 × λ.
We assume that the number density of molecules nV = N/V is constant everywhere in the gas.
In the cylinder it is equal nV =
1
. Therefore in this approximation
d λ
d n
2
1
λ = .
π
π 2
This is slightly bigger than an exact value λ =
1
2π
d nV
2
(b) Derive expression for collision frequency of molecules in an ideal gas. Estimate the collision
frequency of hydrogen molecules in a container containing hydrogen gas at T= 1000 K and
P = 1 atm. (Assume that hydrogen molecules are spherical with the effective diameter equal to
twice the diameter of the 1s orbit in the hydrogen atom.)
Collision frequency is the number of collisions per unit time. Time between collisions can be
estimated as mean free path divided by the average velocity of molecules in the gas ν , i.e.
t = λ/ν . Therefore the collision frequency is f = 1/t = V n
2
2 πd ν .
From the ideal gas law
n
V
P
= =
k T
−20
k BT
1.
38×
10
v = 1 . 60 × = 1. 60
−27
m 3.
346×
10
d 2 = (4×0.0529×10 -9 ) 2 = 0.0448×10 -18 m 2
f = 2.96×10 9 s -1
B
1.
01
1.
38×
10
5
× 10
−23
Pa
= 0.
732 × 10
× 1000
× = 2.030×10 3 m/s
25
V
molecules/m 3
3. Calculate the most probable speeds of H2 and O2 molecules at 20 o C. On a single diagram sketch
the Maxwell-Boltzmann distribution of molecular speeds for H2 and O2 molecules at this
temperature.
vmp = kB T / m
2 = 1.41 kB T / m
vmp(H2) = 1560 m/s
vmp(O2) = 389 m/s
2
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