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

3. 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

4. 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.

5. 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

6. 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

7. 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)

8. 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)

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

10. Next Time l Gibbs-Helmholtz Equation l Fugacity