Problems and Solutions on Thermodynamics and Statistical Mechanics

78 Problems d Sddiond on Thermodynamics d %atidt&d Mechanics

where a is a constant, Lo is the length at zero tension, and L(T) is a
function of temperature T only.

(a) The cylinder is stretched reversibly and isothermally from L = Lo
to L = 2Lo. Find the heat transferred to the cylinder, Q, in terms of
a, TI LO and ao, the thermal expansion coefficient at zero tension, being

ao= -~^1 dLo(T)

Lo(T) dT '

(b) When the length is changed adiabatically, the temperature of the

cylinder changes. Derive an expression for the elastocaloric coefficient,

(dT/aL)s where S is the entropy, in terms of a, T, L, LO, ao, and Ct, the

heat capacity at constant length.

(c) Determine whether CL is a function of T alone, CL(T), or whether

it must also depend on the length, CL(T, L), for this system.

Solution:

(MITI

Let @ be the free energy. From d@ = -SdT + FdL, we get

Thus

and

-aTLo

Fig. 1.26.