How do you measure Gibbs free energy?

You can't measure it directly, and `DeltaG` is non-arbitrarily defined only for isothermal processes. Here are two ways to investigate the possibilities.

SCENARIO 1: DISSOLVING SOLUTE INTO SOLVENT

One way to measure it indirectly is through the chemical potential at constant temperature.

Let's say we had a solvent `j` and we add solute to it. Then:

`mu_j = mu_j^"*" + RTlnchi_j`

where:

• `chi_j = (n_j)/(n_(t ot))` is the mol fraction of solvent `j` in solution.
• `mu_j = G_j/n_j` is the chemical potential, or the molar Gibbs' free energy of substance `j`.
• `mu_j^"*"` is the chemical potential for some reference point. In this case, it is for an unmixed solvent `j`.

We define the change in molar Gibbs' free energy as:

`Deltamu_j = mu_j - mu_j^"*" = RTlnchi_j`

`= DeltabarG_j = barG_j - barG_j^"*" = (DeltaG_j)/n_j`,

where the bar signifies molar quantities.

So, the change in the Gibbs' free energy of the solvent for adding solute into solution (starting with no solute) would be given by:

`barul(|stackrel(" ")(" "DeltaG_j = n_jRTlnchi_j" ")|)`

SCENARIO 2: MEASUREMENT OF NATURAL VARIABLES

Let's attempt to derive a functional form using only natural variables (`V, T, P`, etc). Consider the Maxwell Relation of `G` in a thermodynamically-closed system, i.e. a conservative system:

`dG = -SdT + VdP` `" "bb((1))`

Now, if we write the total differential of `G`, there is another form of the Maxwell Relation:

`dG = ((delG)/(delT))_PdT + ((delG)/(delP))_TdP` `" "bb((2))`

We know the second term to be `V` by inspection of `(1)`. For ideal gases we can use the ideal gas law, `V = (nRT)/P`, at constant temperature.

But unfortunately the first term has no physical meaning in thermodynamics, as the entropy at absolute zero is arbitrarily defined (Physical Chemistry by Levine). This is referring to...

`DeltaG = DeltaH - Delta(TS)`

`= DeltaH - T_1DeltaS - S_1DeltaT - DeltaSDeltaT`

However, `S_1` is arbitrary in the field of thermodynamics. As such, `DeltaG` is only defined in a non-arbitrary way for isothermal processes, i.e. in

`barul(|stackrel(" ")(" "DeltaG = DeltaH - TDeltaS" ")|)`

Otherwise, it cannot be measured.