Question

Derive the formula for the change of entropy in polytropic process

Answer 1

hope this is correct:

dQ=dU-dW (1) ->1st law (the minus sign in dW will be useful later);

dQ=TdS (2) ->from definition of entropy for reversible processes;

dW=PdV (3) -> Compresion-expansion work;

dU=CvdT (4) -> for ideal gases U=U(T);

substituting (2) and (3) in (1):

dS=(dU/T)-(PdV/T),

inserting (4) in this expresion, and using ideal gases (PV=NRT) yields:

dS=(CvdT/T)-(NRdV/V)

integrating:

S2-S1=Cvln(T2/T1)-NRln(V2/V1)

now, using T2/T1=(V2/V1)^(1-n) (from polytropic process definition coupled with ideal gas eq.), and knowing that Cp-Cv=NR (for ideal gases):

S2-S1=ln(V2/V1)*(Cv(1-n)-NR)=ln(V2/V1)*(Cp-nCv)

the definition of adiabatic factor is k=Cp/Cv, so:

S2-S1=Cv*ln(V2/V1)*(k-n)

if Cv' is the constant-volume specific heat per unit mass:

S2-S1=mCv'(k-n)*ln(V2/V1)