10.3 The Gas Kinetic Theory of Transport Processes in Hard-Sphere Gases 463
Chapter 9 we haveD 1
1
2
(
8 kBT
πm) 1 / 2
1
√
2 πd^2 Ntot1
πd^2 Ntot(
kBT
πm) 1 / 2
(approximate
expression)(10.3-8)
where we have used Eq. (9.4-6) for the mean speed and Eq. (9.8-18) for the mean
free path.
Our result can be improved by taking into account the fact that all molecules do not
arrive at a given plane having had their last collision at the same location. When this is
done, the resulting expression for the diffusion coefficient has the same dependence on
density, temperature, mass, and hard-sphere diameter as the expression of Eq. (10.3-8),
but differs by 18% from that expression:^5D 1
3 π
16λ〈v〉3
8 d^2 Ntot(
kBT
πm) 1 / 2
(more accurate
expression)(10.3-9)
We will use this equation for numerical calculations instead of Eq. (10.3-8).
Table A.19 in the appendix gives some experimental values for self-diffusion coef-
ficients, obtained from tracer diffusion experiments. When these data are used to cal-
culate effective hard-sphere diameters the calculated hard-sphere diameters depend
on temperature, with smaller diameters corresponding to higher temperatures. This is
explained by the fact that the actual intermolecular repulsive potential is not infinitely
steep like the hard-sphere potential. When two molecules strike together more strongly,
as they more often do at higher temperature, the distance of closest approach is smaller
and the effective hard-sphere diameter is smaller.EXAMPLE10.12
Calculate the self-diffusion coefficient of N 2 gas at 298 K and 1.00 atm.
Solution
We use the ideal gas law to calculate the number density:Ntot
N
V
P
kBT
101325 N m−^2
(
1. 3807 × 10 −^23 JK−^1)
(298 K) 2. 46 × 1025 m−^3From Table A.15,d 3. 7 × 10 −^10 m.D 1
3
8 d^2 Ntot(
kBT
πm) 1 / 2
3
8 d^2 Ntot(
RT
πM) 1 / 2
3
8(
3. 7 × 10 −^10 m) 2 (
2. 46 × 1025 m−^3)⎛
⎝(
8 .3145 J K−^1 mo1−^1)
(298 K)π(
0 .028 kg mol−^1)⎞
⎠1 / 2 1. 9 × 10 −^5 m^2 s−^1(^5) R. D. Present,Kinetic Theory of Gases, McGraw-Hill, 1958, Section 8-3.