74.
(a) When n = 1, V = V
m
= RT /p, where V
m
is the molar volume of the gas. So
V
m
=
RT
p
=
(8.31 J/mol
·K)(273.15K)
1.01
× 10
5
Pa
= 22.5 L .
(b) We use v
rms
=
3RT /M . The ratio is given by
v
rms,He
v
rms,Ne
=
M
Ne
M
He
=
20 g
4.0 g
= 2.25 .
(c) We use λ
He
= (
√
2πd
2
N/V )
−1
, where the number of particles per unit volume is given by N/V =
N
A
n/V = N
A
p/RT = p/kT . So
λ
He
=
1
√
2πd
2
(p/kT )
=
kT
√
2πd
2
p
=
(1.38
× 10
−23
J/K)(273.15K)
1.414π(1
× 10
−10
m)
2
(1.01
× 10
5
Pa)
= 0.84 µm .
(d) λ
Ne
= λ
He
= 0.84 µm.