The grand potential Φ(T, V, {μA}), also known as the Landau free energy, is a thermodynamical potential obtained by performing Legendre transformations of the internal energy U(S, V, {NA}). Use Legendre transformations to express the thermodynamical potential Φ(T, V, {μA}) in terms of the

Question

The grand potential [latex]Φ(T, V, \{μ_A\})[/latex], also known as the Landau free energy, is a thermodynamical potential obtained by performing Legendre transformations of the internal energy [latex]U(S, V, \{N_A\})[/latex]. Use Legendre transformations to express the thermodynamical potential [latex]Φ(T, V, \{μ_A\})[/latex] in terms of the thermodynamical potential F. Also determine the differential [latex]dΦ(T, V, \{μ_A\})[/latex].

Accepted Answer

To obtain the grand potential [latex]Φ(T, V, \{μ_A\})[/latex], we perform Legendre transformations on the internal energy [latex]U(S, V, \{N_A\})[/latex] with respect to the entropy S and the number of moles [latex]N_A[/latex] of every substance A,

[latex] Φ = U − \frac{\partial U}{\partial S} S - \sum\limits_{A}{\frac{\partial U}{\partial N_A} N_A } = U -T S -\sum\limits_{A}{\mu _A N_A} = F - \sum\limits_{A}{\mu _A N_A} = -p V [/latex].

Differentiating the grand potential [latex]Φ(T, V, \{μ_A\})[/latex] yields,

[latex] dΦ = dU − T dS − S dT − \sum\limits_{A}{\mu _A dN_A} -\sum\limits_{A}{ N_A} dμ_A = −S dT − p dV − \sum\limits_{A}{ N_A} dμ_A [/latex].

Question 4.4: The grand potential Φ(T, V, {μA}), also known as the Landau ...

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