Change in entropy in a polytropic process

how can we show that the change in entropy of an ideal gas undergoing a polytropic process 1-2 is given by
S2 - S1 = mCv (k - n) ln(V2/V1)
where the symbols have their usual meanings. ...

any help please???

Replies

  • The myth buster
    The myth buster
    well i believe that instead of (k-n) term you should be having k^2-1 term in the expression..rest of it is fine..!!
  • Kyle Schieber
    Kyle Schieber
    Hello, for those who are interested in the answer:
    It all begins with these four expressions:

    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)

    regards!

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