science

Thermodynamic functions

ver.1: Nov. 1, 2021

ver.2: Nov. 9, 2021

T. Yamato@NU

graph LR
subgraph Grand potential
EC["Ω = F - μ N [C]"]-->EIII["Ω = -PV [III]"]
EB["G = F + PV [B]"]-->EIII
EIIdash["G = μN [II] or F = -PV + μ N [II']"]-->EIII
end
subgraph Euler
EI'["F(T,λV,λN) = λF(T, V, N) [I']"]-->|"∂/∂λ, λ = 1"|EII'["F = -PV + μN [II']"]
end
subgraph Gibbs-Duhem
EI["G(T,P,λN) = λG(T, P, N) [I]"]-->|"∂/∂λ, λ = 1"|EII["G = μN [II]"]
end
subgraph Thermodynamic functions
E1["dU = Sdt - PdV + μdN [1]"]-.-|"F = U - TS [A]"|E2["dF = -TdS - PdV + μdN [2]"]
E2-.-|"G = F + PV [B]"|E3["dG = -TdS + VdP + μdN [3]"]
E2-.-|"Ω = F - μN [C]"|E4["dΩ = -TdS - PdV - Ndμ [4]"]
end

Proof of Gibbs-Duhem

\[\left(\frac{\partial G}{\partial {(\lambda N)}} \frac{\partial (\lambda N)}{\partial \lambda}\right)_{\lambda=1} = \mu N = G(T, P, N)\]