Question

Use the definition of enthalpy, the first law, Model 2, and that entropy is a state function to show that for the reversible expansion of one more of an ideal gas: deltaS = C_P ln(T₂/T₁) – R ln(P₂/P₁) where C_P is not equal to f(T).

Answer 1

S= Entropy; Q= heat transfer; H= Enthalpy; V= Volume; P=pressure; T=temperature; Cp = heat capacity at constant pressure; R= gas constant

By second law,

By differentiating,     ........................1

Differntial first law, dQ= dH - P.dV ......................2

Enthalpy equation, dH=Cp*dT    .........................3

Equation of state, P.V=R.T         .........................4

By substituting eq.4 and 3 in 2

Divide by T and substitute eq 1.,

Integrating

By second law,

By differentiating,     ........................1

Differntial first law, dQ= dH - P.dV ......................2

Enthalpy equation, dH=Cp*dT    .........................3

Equation of state, P.V=R.T         .........................4

By substituting eq.4 and 3 in 2

Divide by T and substitute eq 1.,

Integrating