Question

For an ideal gas, consider the molar volume Vm = (V/n) = Vm(T,P). In other words, the molar volume is a function of temperature and pressure.

a) Write the total differential dVm.

b) Show that dVm is exact.

c) Derive an expression for the differential work dw performed in a reversible process by expansion/compression of the gas.

d) Show that dw is inexact.

e) What is the thermodynamic significance of having an exact differential?

Answer 1

(a) Total diferntial

dVm = (dV/dT)_p dT + (dV/dP)_T dP

Now we have to calculate (dV/dT)_p and (dV/dP)_T

V_m = RT/P (Ideal gas equation)

(dV/dP)_T = -RT/P²   (T is constant here)

(dV/dT)_p = R/P (here P is constant)

Therefore the expression becomes-

dVm = R/P dT + (-RT/P²) dP

(b) To prove dVm exact, take second second partial differential

Let M = R/P and N = -RT/P²

Now do the partial differentiation again as shown below

(dM/dP)_T = -R/P²

(dN/dT)_P = -R/P²

Since they have comes out to be equal It is proved that dVm is a exact differential

(c) For a finite reversible change in volume of n moles of an ideal gas

w_rev = dw_rev = -PdV = -nRT/V dV

Temperature is constant for isothermal process

w_rev = -nRT1/V dV = nRTln(V₂/V₁)

(d) Work done is a path function (i.e., depends on the path followed between initial and final point), therefore it is an inexact differential

(e) If the differential of any variable is exact it means it is a state function means depends only on final and initial state of the system and not on the path followed.