MIT 3.00 Fall 2002 154 c W.C Carter ° Lecture 22 Mathematical Relations and Changing Variables Last Time Reaction Equilibria Exact Differentials Legendre Transformations Lechatelier’s Principle MIT 3.00 Fall 2002 155 c W.C Carter ° Maxwell’s Relations µ df = ∂f ∂x ¶ µ dx + y ∂f ∂y ¶ dy (22-1) x A property of a perfect differential is: ∂ 2f ∂ 2f = ∂x∂y ∂y∂x (22-2) If Equation 22-2 is applied to dU = T dS − P dV : µ ¶ µ ¶ ∂ 2U ∂P ∂ 2U ∂T =− = ∂V ∂S ∂S V ∂S∂V ∂V S (22-3) This can be summarized in the following tables (you should be able to derive these tables on your own): Second Law Formulation dU = T dS −P dV P + C i=1 µi dNi Internal Energy U Independent Conjugate Variables Variables ¡ ¢ S T = ∂U ∂S ¡ ∂U ¢V,Ni V −P = ∂V S,Ni ³ ´ ∂U Ni µi = ∂N i S,V,Nj 6=Ni Maxwell ¡ ∂T ¢ Relations ¡ ∂P ¢ = − ∂V ∂S ¡ ∂µi ¢ S,Ni ³ ∂P ´ V,Ni = − ∂Ni ∂V S,Ni ³ ´S,V,Nj 6=Ni ¡ ∂µi ¢ ∂P = − ∂N ∂V S,Ni i S,V,Nj 6=Ni MIT 3.00 Fall 2002 156 c W.C Carter ° Enthalpy H H = U + PV H = G + TS H = F + PV − TS Second Law Independent Conjugate Maxwell Formulation Variables Variables Relations ¡ ¢ ¡ ∂T ¢ ¡ ¢ dH = T dS S T = ∂H = ∂V ∂S V,Ni ∂P S,Ni ³ ∂S ´ P,Ni ¡ ∂H ¢ ¡ ∂µi ¢ ∂V +V dP P V = ∂P S,Ni = ∂Ni ∂P S,Ni ³ ´ ³ ´S,P,Nj 6=Ni ¡ ∂µi ¢ PC ∂H ∂T + i=1 µi dNi Ni µi = ∂Ni = ∂N ∂S P,Ni i S,P,Nj 6=Ni S,P,Nj 6=Ni Helmholtz Free Energy F F = U − TS F = H − PV − TS F = G + PV Second Law Independent Conjugate Maxwell Formulation Variables Variables ¡ ∂F ¢ ¡ ∂S ¢ Relations ¡ ∂P ¢ dF = −SdT T −S = ∂T V,Ni = ∂V T,Ni ³ ∂T´ V,Ni ¡ ∂F ¢ ¡ ∂µi ¢ ∂P −P dV V −P = ∂V T,Ni = − ∂N ∂V T,Ni ³ ´ ³ i ´T,V,Nj 6=Ni ¡ ¢ PC ∂µ ∂F ∂S i + i=1 µi dNi Ni µi = ∂N = − ∂N ∂T V,Ni i i T,V,Nj 6=Ni T,V,Nj 6=Ni Gibbs Free Energy G G = U − TS + PV G = F + PV G = H − TS Second Law Independent Conjugate Maxwell Formulation Variables Variables Relations ¡ ¢ ¡ ∂S ¢ ¡ ¢ dG = −SdT T −S = ∂G = − ∂V ∂T P,Ni ∂P T,Ni ³ ∂T ´ P,Ni ¡ ∂G ¢ ¡ ∂µi ¢ ∂S +V dP P V = ∂V T,Ni = − ∂Ni ∂T P,Ni ³ ´ ³ ´ T,P,Nj 6=Ni ¡ ∂µi ¢ PC ∂G ∂V + i=1 µi dNi Ni µi = ∂Ni = ∂N ∂P T,Ni i T,P,Nj 6=Ni T,P,Nj 6=Ni Change of Variable Sometimes it is more useful to be able to measure some quantity, such as µ ¶ ∂S CP = T = f1 (T, P ) ∂T P or µ ¶ ∂S CV = T = f2 (T, V ) ∂T V (22-4) (22-5) under different conditions than those indicated by their natural variables. It would be easier to measure CV at constant P , T , so a change of variable would be useful. MIT 3.00 Fall 2002 157 c W.C Carter ° To change variables, a useful scheme using Jacobians can be employed:23 ¯ ∂u ∂u ¯ ¯ ¯ ∂(u, v) ∂y ¯ ≡ det ¯¯ ∂x ∂v ∂v ¯ ∂(x, y) ∂x ∂y ∂u ∂v ∂u ∂v = − ∂x ∂y ∂y ∂x µ ¶ µ ¶ µ ¶ µ ¶ ∂u ∂v ∂u ∂v = − ∂x y ∂y x ∂y x ∂x y = (22-6) ∂u(x, y) ∂v(x, y) ∂u(x, y) ∂v(x, y) − ∂x ∂y ∂y ∂x ∂(u, v) ∂(v, u) ∂(v, u) =− = ∂(x, y) ∂(x, y) ∂(y, x) µ ¶ ∂(u, v) ∂u = ∂(x, v) ∂x v ∂(u, v) ∂(u, v) ∂(r, s) = ∂(x, y) ∂(r, s) ∂(x, y) (22-7) To see where the last rule comes from: For example, µ CV = T 23 ∂S ∂T ¶ =T V ∂(S, V ) ∂(T, V ) An alternative scheme is presented in Denbigh, Sec. 2.10(c) (22-8) MIT 3.00 Fall 2002 158 c W.C Carter ° Using the Maxwell relation: ¡ ∂S ¢ ∂P T =− ¡ ∂V ¢ ∂T P : ¡ ¢ 2 [ ∂V ] P ¢ CP − CV = −T ¡∂T ∂V ∂P T (22-9)