Question

The differential for the Internal energy, U, at constant composition is
?U = −P?V + T?S
(a) What are the natural independent variables of U? [2]
(b) Derive an expression for the change in internal energy at constant Volume starting with the
above differential for the internal function, U, at constant composition. [3]
(c) Using the criterion for exact differentials, write the Maxwell relation that is derived from this
equation. [2]
(d) Based on your answer in part (a), write the total differential of U in terms of partial
differentials. [2]
(e) By comparison of the above equation (equation at the beginning of the question) and the total
differential of U written in part (d), obtain appropriate expressions (partial differentials) for −P
and T.

Answer 1

The differential for the Internal energy, U, at constant composition is
?U = −P?V + T?S
(a) What are the natural independent variables of U?

Ans: Fundamental equations of Thermodynamics

(1) The combined first and second law From the first law: dU = dq + dW

From the second law: T dq dS ≥ Where, for irreversible system T dq dS > and,

for reversible system dq dS = T dq dS = For a closed system in which only reversible pV work is involved

dW = − pdV and T dq dS =

∴ therefore dU = TdS − pdV .......(equation 1)

we will take the differential of U

dU = (dU/dS)v dS + (dU/dV)s dV..........(equation 2)

by equation 1 & 2

T= (dU/dS)v & p=- (dU/dV)s

s and V are the natural variables of U represented as U(S,V)

(b) Derive an expression for the change in internal energy at constant Volume starting with the above differential for the internal function, U, at constant composition.

According to the first law of thermodynamics, u=q+w

u=q+w, where u is changing in internal energy, qq is heat liberated and ww is the work done in the process.

Now at constant volume, w=0w=0, hence u=q

u=q. Since qq is n⋅Cv⋅Tn⋅Cv⋅T, where nn is the amount of substance in mole, CvCv is the molar heat capacity at constant volume and TT is the temperature change. uu comes out to be n⋅Cv⋅Tn⋅Cv⋅T.

but, this expression is valid for all other cases too whether volume is constant or not.

Same is the case with H=n⋅Cp⋅TH=n⋅Cp⋅T.

1)The change in internal energy should be written as ΔU=nCvΔTΔU=nCvΔT,

not nCvTnCvT. This equation is valid for any temperature change (irrespective of whether the volume or pressure changes) only for an ideal gas. The equation for the change in enthalpy should be

ΔH=ΔU+Δ(PV)

For an ideal gas, this equation reduces to

ΔH=nCvΔT+nRΔT=nCpΔT

ΔH=nCvΔT+nRΔT=nCpΔT

This equation is valid for any temperature change only for an ideal gas.

(c)Using the criterion for exact differentials, write the Maxwell relation that is derived from this
equation.

Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable ( temp T and entropy S) and mechanical natural variables ( P and V)