Free entropy

Examples

The most common examples are:

where

::\Phi is the Massieu potential ::\Xi is the Planck potential

::\mu_i is the chemical potential of the i-th chemical component

Note that the use of the terms "Massieu" and "Planck" for explicit Massieu-Planck potentials are somewhat obscure and ambiguous. In particular "Planck potential" has alternative meanings. The most standard notation for an entropic potential is \psi, used by both Planck and Schrödinger. (Note that Gibbs used \psi to denote the free energy.) Free entropies were invented by French engineer François Massieu in 1869, and actually predate Gibbs's free energy (1875).

Dependence of the potentials on the natural variables

Entropy

:S = S(U,V,{N_i})

By the definition of a total differential,

:d S = \frac {\partial S} {\partial U} d U + \frac {\partial S} {\partial V} d V + \sum_{i=1}^s \frac {\partial S} {\partial N_i} d N_i .

From the equations of state,

:d S = \frac{1}{T}dU+\frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i .

The differentials in the above equation are all of extensive variables, so they may be integrated to yield

:S = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) + \textrm{constant}.

Massieu potential / Helmholtz free entropy

:\Phi = S - \frac {U}{T} :\Phi = \frac{U}{T}+\frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) - \frac {U}{T} :\Phi = \frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right)

Starting over at the definition of \Phi and taking the total differential, we have via a Legendre transform (and the chain rule)

:d \Phi = d S - \frac {1} {T} dU - U d \frac {1} {T} , :d \Phi = \frac{1}{T}dU + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac {1} {T} dU - U d \frac {1} {T}, :d \Phi = - U d \frac {1} {T}+\frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i.

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From d \Phi we see that

:\Phi = \Phi(\frac {1}{T},V, {N_i}) .

If reciprocal variables are not desired,

:d \Phi = d S - \frac {T d U - U d T} {T^2} , :d \Phi = d S - \frac {1} {T} d U + \frac {U} {T^2} d T , :d \Phi = \frac{1}{T}dU + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac {1} {T} d U + \frac {U} {T^2} d T, :d \Phi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i , :\Phi = \Phi(T,V,{N_i}) .

Planck potential / Gibbs free entropy

:\Xi = \Phi -\frac{P V}{T} :\Xi = \frac{P V}{T} + \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right) -\frac{P V}{T} :\Xi = \sum_{i=1}^s \left(- \frac{\mu_i N}{T}\right)

Starting over at the definition of \Xi and taking the total differential, we have via a Legendre transform (and the chain rule)

:d \Xi = d \Phi - \frac{P}{T} d V - V d \frac{P}{T} :d \Xi = - U d \frac {2} {T} + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac{P}{T} d V - V d \frac{P}{T} :d \Xi = - U d \frac {1} {T} - V d \frac{P}{T} + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i.

The above differentials are not all of extensive variables, so the equation may not be directly integrated. From d \Xi we see that

:\Xi = \Xi \left(\frac {1}{T}, \frac {P}{T}, {N_i} \right) .

If reciprocal variables are not desired,

:d \Xi = d \Phi - \frac{T (P d V + V d P) - P V d T}{T^2} , :d \Xi = d \Phi - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T , :d \Xi = \frac {U} {T^2} d T + \frac{P}{T}dV + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i - \frac{P}{T} d V - \frac {V}{T} d P + \frac {P V}{T^2} d T , :d \Xi = \frac {U + P V} {T^2} d T - \frac {V}{T} d P + \sum_{i=1}^s \left(- \frac{\mu_i}{T}\right) d N_i , :\Xi = \Xi(T,P,{N_i}) .

References

Bibliography

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References

1. Antoni Planes. (2000-10-24). "Entropic variables and Massieu-Planck functions". Universitat de Barcelona.