The Helmholtz Free Energy (A) is the Legendre transform of which energy with temperature replacing entropy as the independent variable?

The Helmholtz Free Energy (A) is the Legendre transform of which energy with temperature replacing entropy as the independent variable?

• A. Internal energy (U)
• B. Enthalpy (H)
• C. Gibbs free energy (G)
• D. Entropy (S)

Answer: (A)

The main idea is swapping the entropy variable for its conjugate temperature through a Legendre transform. Internal energy U is a function of S, V, N, with dU = T dS - P dV + μ dN. The Helmholtz free energy is defined as A = U - T S, and using dU you get dA = - S dT - P dV + μ dN. This shows A depends on temperature, volume, and particle number (T, V, N) rather than entropy, so temperature has taken the place of entropy as the independent variable. Therefore, Helmholtz free energy is the Legendre transform of internal energy with temperature replacing entropy.

The main idea is swapping the entropy variable for its conjugate temperature through a Legendre transform. Internal energy U is a function of S, V, N, with dU = T dS - P dV + μ dN. The Helmholtz free energy is defined as A = U - T S, and using dU you get dA = - S dT - P dV + μ dN. This shows A depends on temperature, volume, and particle number (T, V, N) rather than entropy, so temperature has taken the place of entropy as the independent variable. Therefore, Helmholtz free energy is the Legendre transform of internal energy with temperature replacing entropy.