# How to calculate Gibbs free energy of adsorption at constant temperature with varying pressures

I want to calculate the Gibbs free energy of adsorption for an oxidized surface. The literature I consulted used the equations attached to this post. I am trying hard to find my way through. Any idea how to go about it would be deeply appreciated. I am supposed to have a linear plot with pressure as abscissa and delta G as ordinate.

IMO the equations you show use too many "$$\Delta$$" which makes it more confusing. I will hence try to avoid the use of "$$\Delta$$" in the equations below.

To get the adsorption free energy for the oxidized surface, you need to calculate $$$$G_{\mathrm{ads}} = G_{\mathrm{slab(oxidized)}} - G_{\mathrm{slab}} - N_{\mathrm{O}}\mu_{\mathrm{O}}(T,p) \tag{1} \label{1}$$$$

To avoid computationally expensive phonon calculations, the common assumption is that the free energy corrections for the surface and the oxidized surface will cancel out by $$G_{\mathrm{slab(oxidized)}} - G_{\mathrm{slab}}$$, except for the vibrational free energy term for the oxygens adsorbed on the surface, $$F^{\mathrm{vib}}_{\mathrm{O*}}$$. We can then rewrite the equation as $$$$G_{\mathrm{ads}} = E_{\mathrm{slab(oxidized)}} + F^{\mathrm{vib}}_{\mathrm{O*}} - E_{\mathrm{slab}} - N_{\mathrm{O}}\mu_{\mathrm{O}}(T,p) \tag{2} \label{2}$$$$ where $$E$$ represents the electronic energy which can be calculated by e.g. DFT.

In general, we can calculate $$F^{\mathrm{vib}}$$ by $$$$F^{\mathrm{vib}} = E^{\mathrm{ZPE}} + \int_0^TC_V\mathrm{d}T - TS \tag{3} \label{3}$$$$ where $$E^{\mathrm{ZPE}}$$ is the zero-point energy, $$C_V$$ is the heat capacity which is separable into translational, rotational, vibrational and electronic parts, $$S$$ is the entropy which is separable into translational, rotational, vibrational and configurational parts. You can calculate these terms in the harmonic limit using e.g. ase.thermochemistry.HarmonicThermo. Note that this could be expensive if you have high O* coverage. A simplification would be to calculate using only one O* adsorbate, and multiply the single-O* vibrational free energy term by the number of O*.

Next, we can assume the oxidized surface is in equilibrium with the gas phase O$$_2$$, and calculate $$\mu_{\mathrm{O}}(T,p)$$ as $$\mu_{\mathrm{O_2(g)}}(T,p) / 2$$. The $$\mu_{\mathrm{O_2(g)}}(T,p)$$ term can then be determined in the ideal gas limit by $$$$\mu_{\mathrm{O_2(g)}}(T, p) = E_{\mathrm{O_2(g)}} + \Delta\mu_{\mathrm{O_2(g)}}^{0,T} + k_{\mathrm{B}}T\ln(p_{\mathrm{O_2(g)}}/p^0) \tag{4} \label{4}$$$$ where $$E_{\mathrm{O_2(g)}}$$ is the electronic energy of the O$$_2$$ molecule which can be calculated by DFT, $$\Delta\mu^{0,T}_{\mathrm{O_2(g)}}$$ is the chemical potential correction term (including zero-point energy) obtained from vibrational calculations at standard pressure $$p^0=1$$ atm, $$k_{\mathrm{B}}$$ is the Boltzmann constant, and $$p_{\mathrm{O_2(g)}}$$ is the fugacity of the gas phase O$$_2$$.

In general, we can calculate $$\Delta\mu^{0,T}$$ as $$$$\Delta\mu^{0,T} = E^{\mathrm{ZPE}} + \int_0^TC_P\mathrm{d}T - TS \tag{5} \label{5}$$$$ where $$C_P = C_V + k_{\mathrm{B}}$$ (switching from constant-volume to constant-pressure). These terms can again be calculated by ASE using ase.thermochemistry.IdealGasThermo.

It is not hard to see that the adsorption free energy $$G_{\mathrm{ads}}$$ is effectively a function of $$T$$ and $$p_{\mathrm{O_2(g)}}$$. At constant temperature and various pressures, the expensive terms only need to be calculated once.

• Thank you once again for the rescue. Considering about an earlier answer you gave for the oxidized surface would ase be able to do the calculation for me? If not I have access to a supercomputer can I use DFT to calculate all? I saw in literature some referring to JANAF tables in determining the standard chemical potential in your last equation. Any information about that too? Commented Jul 31, 2023 at 15:19
• Yes you can use ASE to calculate all, but you still need DFT which needs supercomputer. And yes, instead of calculating the standard chemical potential of O$_2$(g) by yourself, you can refer to the NIST JANAF tables, but there are many other terms that still needs DFT. Commented Jul 31, 2023 at 17:07
• Thanks. I will do phonon calculations to determine the Fvib and use the JANAF table to determine the standard chemical potential. Thank you once again. Commented Jul 31, 2023 at 18:09
• The equation of chemical potential, I mean the last but one. The last term on the right-hand side. Please is it kb or R? I am seeing R everywhere. Any clarification? Commented Aug 4, 2023 at 10:47
• @NanaKofiBoakye They are essentially the same thing. $R = N_{\mathrm A}k_{\mathrm B}$ Commented Aug 4, 2023 at 12:19