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Chem 300 - Ch 22/#2 Today’s To Do List l Maxwell Relations l Natural Independent Variables

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Maxwell Relations l dZ = N dx + M dy If an exact differential If Z(x,y) is a state function ( N/ y) x = ( M/ x) y Maxwell Relation l Examples: dU = TdS – PdV dH = TdS + VdP dA = -PdV – SdT dG = VdP - SdT

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Maxwell Continued ( T/ V) S = - ( P/ S) V ( T/ P) S = ( V/ S) P ( P/ T) V = ( S/ V) T S(V) ( V/ T) P = - ( S/ P) T S(P) Use the last 2 to get values of S.

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S(V) ( S/ V) T = ( P/ T) V dS T = [( P/ T) V ]dV S = ∫ [( P/ T) V ]dV l For Ideal Gas: P = nRT/V ( P/ T) V = nR/V S = ∫ nRdV/V = nRln(V 2 /V 1 ) const T For V 2 > V 1 S > 0

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S(P) - ( S/ P) T = ( V/ T) P dS T = - [( V/ T) P ]dP S = - ∫ [( V/ T) P ]dP l For Ideal Gas: V = nRT/P ( V/ T) P = nR/P S = - ∫ nRdP/P = - nRln(P 2 /P 1 ) const T For P 2 > P 1 S < 0

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U (T, V) l dU = TdS – PdV ( U/ V) T = T( S/ V) T - P From Maxwell: l dA = - PdV – SdT ( S/ V) T = ( P/ T) V subst. above. ( U/ V) T = T ( P/ T) V – P (Internal Pressure) For ideal gas: ( P/ T) V = [ (RT/V)/ T] V = R/V ( U/ V) T = T (R/V) – P = RT/V – P = P – P = 0 Thus for Ideal Gas: U = f (T only)

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H (T, P) l dH = TdS + VdP ( H/ P) T = T( S/ P) T +V From Maxwell: l dG = VdP – SdT ( S/ P) T = - ( V/ T) P subst. above. ( H/ P) T = - T( V/ T) P + V For ideal gas: ( V/ T) P = [ (RT/P)/ T] V = R/P ( H/ P) T = - T(R/P) + V = - RT/P – V = -V + V = 0 Thus for Ideal Gas: H = f (T only)

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“Natural” Independent Variables l U = f(S, V) l H = f(S, P) l A = f(V, T) l G = f(P, T)

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Next Time l Gibbs-Helmholtz Equation l Fugacity

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