If we consider a gas with a known temperature T and chemical potential $\mu$. A surface is in contact with a gas.Gas molecules can be adsorbed at N points on the surface. In this example it was shown that the fugacity of the adsorbed particles is equal to that of the free particles, when the system reaches equilibrium.Adsorption reduces the energy of a gas molecule by the value $\Delta$.

As we know fugacity is given by: $z=e^{\beta \mu}$.

This is possible when :

1. The temperature of the adsorbed particles is equal to the one of the free particles $T\_{adsorbed}=T\_{free}$
2. Their chemical potentials are also equal, meaning $\mu\_{adsorbed}=\mu\_{free}$

Now, I can understand the first condition happening. Whenever a particle is adsorbed it gives $\Delta$ energy in the form of heat to the surface. After this process happens for a while, the surface ends up having a temperature similar to that of the gas, in other words the adsorbed particles have the same temperature (and average energy since $\bar \epsilon=\frac 3 2KT$) as the free ones.

But I cannot think of a reason as to why the chemical potentials should be equal too.
I know that chemical potentials are taken into consideration when we have a chemical reaction or a phase transition. And only for the case when we have phase transition between two arbitray phases $A$ and $B$, I know that $\mu\_{A}(P,T)=\mu\_{B}(P,T)$. So I don't understand in which category does the above example falls? Is it a reaction, or is it a phase transition?