# Grand potential vs. Gibbs free energy of adsorption for ab initio thermodynamics

I have seen descriptions of ab initio thermodynamics where the phase energy diagram for the oxidation of metal surfaces can be constructed in terms of minimizing the grand potential or the Gibbs free energy of adsorption. I was wondering what the difference is between the two and which is the "more correct" description for ab initio thermodynamics?

In principle you could use either thermodynamic potential to describe the adsorption process, but the grand potential is a bit more natural to use. Which one you select in the end kind of depends on what you are trying to describe and what thermodynamic parameters you want to consider to be fixed.

Typically, when working with Gibbs free energies of adsorption, this will be plotted against a composition axis (such as surface coverage) since the Gibbs free energy depends directly on particle number $$G \equiv G(T, P, N) \to dG = -SdT + VdP + \mu dN$$ Note that at fixed temperature and pressure, we have $$dG = \mu dN$$. In this case, the chemical potential is a quantity to be measured or computed, and is a function of $$T$$, $$P$$, and $$N$$ $$\mu(T, P, N) = \left(\frac{\partial G}{\partial N} \right)_{T, P}$$

When working with grand potentials, the data will be presented as grand potential vs chemical potential since chemical potential is a natural variable of the grand potential $$\Phi \equiv \Phi(T, V, \mu) \to d\Phi = -SdT -PdV - Nd\mu$$ Note that at fixed temperature and volume, we have $$d\Phi = -N d\mu$$. Hence, the number of adsorbates on the surface is a quantity to be computed or measured and is a function of $$T$$, $$V$$, and $$\mu$$ $$-N(T, V, \mu) = \left(\frac{\partial \Phi}{\partial \mu} \right)_{T, V}$$

The reason why I say the grand potential is more natural to use is that experimentally, we can control the partial pressure of oxygen gas (and hence its chemical potential) much more easily than we can directly control the surface coverage of adsorbates. On the other hand, computationally, it is easier to directly sample surface coverage than it is to control the chemical potential of adsorbates.