Question

You are given an ideal gas which undergoes a process where the internal energy U is a function of the volume, V according to U = bV^ f , where b and f are known constants. You know the ratio of specific heats, γ.

(a) Suppose the internal energy is increase by a finite amount ∆. What work is performed by the gas.

(b) What amount of heat is transferred to the gas in the situation (a).

(c) What is the heat capacity per mole in this process?

Answer 1

a) Work done W = PdV (assuming that the pressure change is small for small change in internal enrgy

We know that U = bV^f

dU = fbV^f-1dV

So, W = P*dU/(fb*V^f-1 )

b) According to first law of thermodynamics,

dU = Q-W

So, heat produced

Q = dU + W = dU*(1+ P/(fb*V^f-1 ))

c) The molar heat capacity is given by

c = Q/dT

According to the ideal gas equation, for one mole,

PV = RT

dT = PdV/R

So, c = dU*(1+ P/(fb*V^f-1 ))/dT = R*dU*(1+ P/(fb*V^f-1 ))/PdV = R*fbV^f-1(1+ P/(fb*V^f-1 )/P = R((fbV^f-1/P) -1)