Energies, Thermodynamic Identities, Maxwell Relations, and the Magic Square Professor A. J. Mallinckrodt, Physics Department, Cal Poly Pomona, November 2001

Thermodynamic identity for energy Magic Square VAT, UG, SHP (“Vat Ugh Ship”) = − ()+µ + dU TdS PdV dN ... Notice that 1) each side has V A T an energy, 2) each energy is Additional energies flanked by its two most H = U + PV (Enthalpy) important natural variables, U G A = U − TS (Helmholtz free energy) 3) conjugate variables G = U + PV − TS (Gibbs free energy) occupy opposite ends of each S H P diagonal, 4) crossed arrows Additional thermodynamic identities point upward toward V and T and away from S and P. dH = TdS + VdP()+µdN +... Partial derivatives from the Magic Square =− − ()+µ dA SdT PdV dN +... Starting at any energy, read toward one of the natural dG = −SdT + VdP()+µdN +... variables “Partial of – wrt –” and away from the other “at constant –” and look across for the result and its sign (as Natural variables indicated by the arrow head) “is —.” For instance: U ⇒ S & V (& N & ...) =+V A T V A ∂T H ⇒ S & P (& N & ...) A ⇒ T & V (& N & ...) ∂ G ⇒ T & P (& N & ...) U G U G

Conjugate variables (S) ∂H ∂P =-S H (P) S & T, P & V, (µ & N, ...) 1) “Partial of H wrt P at constant S is +V” Partial derivative relations 2) “Partial of G wrt T at constant P is –S”  ∂U  ∂U   = ,   = − , ... Thermodynamic identities from the Magic Square  ∂  T  ∂  P S V ,N,... V S, N,... Intimately related to the partial derivative relations. For any  ∂H  ∂H   = ,   = , ... energy read toward both of its natural variables and across for  ∂  T  ∂  V S P, N,... P S, N,... the conjugate variables and signs. For instance,  ∂A  ∂A 1) “dH is +V times dP plus +T times dS”   = − ,   =− , ...  ∂  S  ∂  P 2) “dG is -S times dT plus +V times dP” T V , N,... V T, N ,...  ∂G  ∂G Maxwell Relations from the Magic Square   = −S,   = V , ...  ∂   ∂  Using only corners, start at any corner and read clockwise or T P, N,... P T ,N ,... counterclockwise “Partial of — wrt — at constant —”. Maxwell relations Proceed in the same direction to the next corner and then (obtained by equating reordered second derivatives of each reverse direction to obtain the other side of the equation. energy with respect to its natural variables) Insert a minus sign if the arrows do not point toward or away  ∂   ∂  from both starting points. For example:  T = − P  ∂V  ∂S  S V V A T  ∂   ∂   V =  T  ∂   ∂  S P P S  ∂   ∂  U G  S  = − V   ∂   ∂  P T T P  ∂   ∂  S H P  P =  S   ∂   ∂  T V V T “Partial of T wrt V at constant S and partial of P wrt S at constant V are opposites.” (Since T has an arrowhead and P doesn’t.)