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.