Physical Chemistry Third Edition

28.6 Approximate Theories of Transport Processes in Liquids 1193

More Advanced Theories of Transport in Liquids

There are numerous more advanced theories of transport coefficients in liquids, mostly

based on nonequilibrium classical statistical mechanics. Some are based on approxi-

mate representations of the time-dependent reduced distribution function and others are

based on the analysis oftime correlation functions, which are ensemble averages of the

product of a quantity evaluated at time 0 and the same quantity or a different quantity

evaluated at timet.^33 For example, the self-diffusion coefficient of a monatomic liquid

is given by^34

D

1

3

∫∞

0

〈v( 0 )•v(t)〉dt (28.6-15)

wherevis the velocity of a molecule, and where the ensemble average〈v(0)•v(t)〉is

the time-correlation function of the velocity. Att0, the time-correlation function in

Eq. (28.6-15) is equal to the average of the square of the velocity, which in a classi-

cal system is equal to 3kBT/m. Evaluation of the time correlation function involves

the study of the time dependence of the probability distribution, which we have not

discussed, so we make only a few elementary comments. As time passes, the time-

correlation function eventually approaches zero, representing the fact that the velocity

of a molecule after a long time loses its “memory” of its initial velocity.

EXAMPLE28.15

Assume that the velocity time correlation function of a molecule in a liquid is given by the

formula

〈v(0)•v(t)〉

3 kBT

m

e−t/τ (28.6-16)

whereτis acorrelation time. The self-diffusion coefficient of liquid CCl 4 is equal to

1. 30 × 10 −^9 m^2 s−^1 at 298.15 K. Find the value ofτ.

Solution

D

kBT

m

∫∞

0

e−t/τdt

kBTτ

m

τ

mD

kBT



(

2. 56 × 10 −^25 kg

)(

1. 3 × 10 −^9 m^2 s−^1

)

(

1. 38 × 10 −^23 JK−^1

)

(298 K)

 8. 1 × 10 −^14 s

Other transport coefficients are expressed in terms of different time-correlation

functions. A variety of techniques have been developed to obtain approximate time-

correlation functions.^35

(^33) D. A. McQuarrie,Statistical Mechanics, Harper & Row, New York, 1976, p. 467ff.
(^34) Ibid.
(^35) Ibid.