Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.13 EXERCISES

O

R

x

y

θ

θ

2 θ

Figure 5.5 The reflecting mirror discussed in exercise 5.24.

5.26 FunctionsP(V,T),U(V,T)andS(V,T) are related by

TdS=dU+PdV,

where the symbols have the same meaning as in the previous question. The

pressurePis known from experiment to have the form

P=

T^4

3

+

T

V

,

in appropriate units. If

U=αV T^4 +βT ,

whereα,β, are constants (or, at least, do not depend onTorV), deduce thatα

must have a specific value, but thatβmay have any value. Find the corresponding

form ofS.

5.27 As in the previous two exercises on the thermodynamics of a simple gas, the
quantitydS=T−^1 (dU+PdV) is an exact differential. Use this to prove that
(
∂U
∂V

)

T

=T

(

∂P

∂T

)

V

−P.

In the van der Waals model of a gas,Pobeys the equation

P=

RT

V−b

−

a

V^2

,

whereR,aandbare constants. Further, in the limitV→∞, the form ofU

becomesU=cT ,wherecis another constant. Find the complete expression for

U(V,T).

5.28 The entropyS(H, T), the magnetisationM(H, T) and the internal energyU(H,T)
of a magnetic salt placed in a magnetic field of strengthH, at temperatureT,
are connected by the equation

TdS=dU−HdM.