Maximizing Entropy vs Minimizing Gibbs Function - Why?

[Thermodave]

It occurs to me that some people create codes to maximize the entropy of a system in order to predict equilibrium concentrations. However, others minimize the Gibbs function. I understand the relationship, so why the two methods? Aren't these results always the same? Is there a practical reason for this?

 

[Gerenuk]

There is a difference in what these energy apply to.

Basically maximizing the total entropy of system+reservoir is (under special circumstances only!) equivalent to minimizing the Gibbs energy of the system alone.

It is incorrect to minimize the entropy of the system alone. Also the contraints (pressure or temperature kept constant?) matter for the decision whether to use Gibbs energy or one of the other three potentials (Helmholtz, Energy, Enthalpy).

 

[Mapes]

There is a difference in what these energy apply to.

Basically minimizing the total entropy of system+reservoir is (under special circumstances only!) equivalent to minimizing the Gibbs energy of the system alone.

It is incorrect to minimize the entropy of the system alone. Also the contraints (pressure or temperature kept constant?) matter for the decision whether to use Gibbs energy or one of the other three potentials (Helmholtz, Energy, Enthalpy).

Surely you mean maximize entropy?

We start with the differential expression for energy, dU=T\,dS-P\,dV+\sum_i \mu_i\,dN_i+\dots. For systems at constant entropy, volume, and mass (and all other extensive variables constant), we minimize the potential U (the internal energy).

We can rewrite the equation as -dS=-1/T\,dU-P/T\,dV+\sum_i \mu_i/T\,dN_i+\dots. So for systems at constant energy, volume, and mass, we maximize the potential S (the entropy).

We can always derive various different potentials for variations on the system. If the entropy, pressure, and mass are constant, for example, we use the Legendre transform H=U+PV to get dH=-T\,dS-V\,dP+\sum_i \mu_i\,dN_i. Thus, we minimize the enthalpy H for these systems.

If the temperature, pressure, and mass are constant (a frequent scenario), we minimize the Gibbs potential G=U+PV-TS. If the temperature, volume, and chemical potential of species i are constant, we minimize the potential \Lambda=U-TS-\mu_i N_i. And so on.

Callen's Thermodynamics has a nice discussion of this. It all comes from the tendency for total entropy to be maximized (i.e., Second Law).