Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodynamic n 147

5. NONEQUILIBRIUM THERMODYNAMICS (1148-1159)

1148
A tube of length L contains a solution with sugar concentration at
time t = 0 given by

7rx 1 3TX 1

n(z, 0) = no + n1 cos - + - cos - + - cos -

{L9 L25 L

Assume that n(x, t) obeys a one-dimensional diffusion equation with

(a) Write down the diffusion equation for n(z, t).

(b) Calculate n(x, t) for t > 0.

diffusion constant D.

(MITI

0 L

Fig. 1.45.

Solution:

(a) The diffusion equation is

and the condition for existence of solutions are

(b) Let n(z, t) = X(z)T(t). We then have

X"(2) + XX(2) = 0,

T'(t) + DXT(t) = 0, with X # (^0) and X'(0) = X'(L) = 0.
The conditions require X = (ICT/L)~, k = 1,2,3,.... The general solution
is