-1

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?

imbAF
  • 1,344

2 Answers2

1

I know that chemical potentials are taken into consideration when we have a chemical reaction or a phase transition.

The chemical potential µ is (well, should be) taken into consideration whenever matter moves, as unconstrained matter moves to the point of lowest chemical potential. This is analogous to heat flowing from hotter to colder objects and volumes shifting to expand high-pressure areas into low-pressure areas. The chemical potential is the partial molar Gibbs free energy G (i.e., $\mu_i\equiv\left(\frac{\partial G}{\partial N_i}\right)$ for material $N_i$), and G is minimized under the familiar setting of constant temperature and pressure. At equilibrium, G for the system is stationary, and µ for any particular material is spatially uniform. This holds regardless of whether the process is considered a phase change or a reaction (or both).

Chemomechanics
  • 25,546
  • But if we consider the gas and surface system, or rather the free and adsorbed particles, in equilibrium, that means that the nr. of free and adsorbed particles is fixed, but that doesn't imply equal in numbers. Shouldn't that play a role in whether the chemical potentials of each group of particles are equal in eq. or not? – imbAF Jan 30 '22 at 19:41
  • Why would that play a role? – Chemomechanics Jan 30 '22 at 21:23
  • if it wouldn't then what is the use of the index "i"in this case? Because you are clearly differentiating between the different components – imbAF Jan 31 '22 at 15:38
  • Outside equilibrium, the chemical potentials aren’t equal, and the numbers $N_i$ generally aren’t equal, as you note; that’s why you need the index. Am I interpreting your question correctly? – Chemomechanics Jan 31 '22 at 16:33
  • Yes, but in equilibrium, are they? Does equilibrium imply equal amount of particles? Or only in this case, since we are dealing with the same type ? – imbAF Jan 31 '22 at 16:34
  • Equilibrium doesn’t generally imply the same number of particles. That would occur only in a specially contrived example (such as two sides of a permeable membrane separating equal volumes at equal temperatures, for instance). The constraint here is instead something like $dN_\mathrm{adsorbed}=-dN_\mathrm{gas}$; a particle that desorbs from the surface (and doesn’t diffuse into the material or do anything else) must enter the gas phase. Mass conservation, in other words. – Chemomechanics Jan 31 '22 at 19:42
0

Adsorbtion is a phase change for sure. A substance from a gas phase is transferred into a solid phase.

The underlying mechanism can be a chemical reaction. Or not.

fraxinus
  • 7,906