Problems and Solutions on Thermodynamics and Statistical Mechanics

336 Problems d Solutions on Thermodynamics d Statistical Mechanics

On the other hand, from

U=F-T(g) V , P=-(%) T ,

3 3

n

we have U = 3 (1 - - - k) NkT + ;pV = apV + bNkT giving a = -, b =

2147

(a) Given JTz exp(-ax2)dx = fi, show that

P 1

$ +

(b) Given that ~ << 1 and that a is of the order of -, show

(c) Two atoms interact through a potential

U(x) = uo [ (y2 - 2 (37 ,

where x is their separation. Sketch this potential. Calculate the value of x
for which U(x) is minimum.
(d) Given a row of such atoms constrained to move only on the x
axis, each assumed to interact only with its nearest neighbors, use classical
statistical mechanics to calculate the mean interatomic separation Z(T).
To do this, expand U about its minimum, keeping as many terms
as necessary to obtain the lowest order temperature dependence of z(T).
Assume that kT << U,, and in the relevant integrals extend the limits of
integration to koo where appropriate. Explain clearly the justification for
extending the limits. Also calculate