264
4 Thermodynamics and Statistical Physics
P − =
dAdt
2λ
∞
0
νdn
∞
0
e
−r/λ dr
π/2
0
sin θ cos θ
mu + r cos θ
dmu
dz
dθ
The factor e
−r/λ is included to ensure that the molecule in traversing the
distance r toward dA does not get scattered and prevented from reaching dA.
Similarly, transport of momentum upward, from molecules in the lower
hemisphere through dA in time dt is
P + =
dAdt
2λ
∞
0
νdn
∞
0
e
−r/λ dr
π/2
0
sin θ cos θ
mu − r cos θ
dmu
dz
dθ
Hence net momentum transfer to the reference plane through an area dA in
time dt is
P = P − − P + =
dAdt
λ
mdu
dz
∞
0
νdn
∞
0
re
−r/λ dr
π/2
0
cos
2
θ sin θ dθ
=
dAdt
λ
m
du
dz
n < ν >
λ
2
3
=
m
3
dAdt
du
dz
λn < ν >
(the first integral gives n < ν >, the second one λ
2 and the third one a factor
1/3)
Momentum transported per second is force
F =
λ
3
dAn < ν > m
du
dz
The viscous force is
ηdA
du
dz
=
λ
3
dAn < ν > m
du
dz
or η =
1
3
mn < ν > λ =
1
3
ρ < ν > λ
where mn = ρ = density of molecules.
4.16 λ =
1
√
2π nσ 2
=
1
√
2π × 3 × 10 25 × (2.5 × 10 −10 ) 2
= 1.2 × 10
−7 m
f =
ν
λ
=
1, 000
1.2 × 10 −7 = 8.33 × 10
9 s
−1
4.17 T 1 V
γ −1
1
= T 2 V
γ −1
2
,
V 2
V 1
γ −1
=
T 1
T 2
or 2
γ −1
= 1.32, γ = 1.4
Number of degrees of freedom,
f =
2
γ − 1
=
2
1.4 − 1
= 5