• ussername
In summary: You are only creating one surface, not two separate surfaces for the same interface. In summary, when deriving the Gibbs adsorption isotherm, the surface work done to change the surface of interface is neglected because there is only one surface present in the system. This is justified because in the gaseous phase, the weak intermolecular attraction forces make the surface work negligible. Additionally, in equilibrium, the differential forms of chemical potentials are equal, which can be derived from the equality of mixed derivatives of Gibbs energy with its total differential.
ussername
Let's consider an isotherm isobaric adsorption of gas (A) on the adsorbent (B). There are two phases in the system:
- volume phase (1) that consists of gas and adsorbent.
- surface phase (2) that contains a layer of adsorbed gas on the surface of adsorbent.

When deriving Gibbs adsorption isotherm, the work needed to change the surface of interface of phase (1) is neglected, i.e. the Gibbs free energy changes for both phases are:

I wonder why one can neglect
? During adsorption the gas passes from phase (1) into phase (2) and the surface of interface of (1) is changing. Why is that?

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First, are you sure the surface area term isn't supposed to be ##A_2d\gamma_2##?

To answer your question, writing out the total Gibbs free energy ##G = G_1+G_2## gives
$$G = U_1 + U_2 +P_1V_1 - T_1S_1-T_2S_2 +\mu_{1A}N_A +\mu_{1B}N_B +\mu_{2A}N_A +\mu_{2B}N_B + \gamma A$$
(NB--the ##P_2V_2## term disappears because I'm assuming the surface has zero volume) (also, sorry in advance if I mess this up: I'm used to working with an interface and two bulk phases, not one bulk phase). Since there's only one surface (the surface of phase 1 and the surface of phase 2 are the same surface), we only have a single ##\gamma A## term, not one for each phase. If we want to divide the Gibbs free energy terms into one part for the bulk and one part for the surface, it doesn't really make much sense to put the surface tension work term into the bulk Gibbs energy. So it ends up in the surface free energy expression.

Thank you for your answer, but first of all I need to clarify where can be Gibbs adsorption isotherm used.
If it would be applied to a solid adsorbent - what is the meaning of surface tension? It is the surface tension when changing the surface of the solid adsorbent? If so, I have no idea how one can measure it. I know only measuring techniques to determine surface tension of liquids.

##\gamma A## is the work done to form the interface. In a liquid-liquid system or a liquid-gas system, we equate ##\gamma## with surface tension, but in general it represents surface energy density per unit area. There are a few different ways to measure this, the most straightforward of which is probably contact angle measurements.

ussername said:
I wonder why one can neglect
?
I think a good explanation is that in gaseous phase the surface work is negligible due to weak intermolecular attraction forces.
In such case I'm nearly able to derive Gibbs adsorption isotherm.
During isotherm isobaric process (neglecting surface works in both phases) the equilibrium criteria is the Gibbs energy minimum:
G(nA,nB)=min
Also that is:
μ1A2A
μ1B2B

But, how can I derive that even differential forms of chemical potencials are equal (dμ1A=dμ2A, dμ1B=dμ2B) in equilibrium?

ussername said:
μ1A=μ2A
μ1B=μ2B

But, how can I derive that even differential forms of chemical potencials are equal (dμ1A=dμ2A, dμ1B=dμ2B) in equilibrium?
Just substitute ##\mu_{1A}## for ##\mu_{2A}## before taking the exterior derivative.

ussername said:
I think a good explanation is that in gaseous phase the surface work is negligible due to weak intermolecular attraction forces.
Again, you're only making one interface. If you include two ##\gamma A## terms, you overcount. It's like saying that you're creating a gas-solid interface as well as a solid-gas interface. It's the same interface.

TeethWhitener said:
If you include two γA terms, you overcount
I don't think so. I consider two subsystems (bulk and surface). For each of them I can express the Gibbs energy total differential and for each of them I perform a surface work when moving a particle to the interface (and those surface works generally differ).

TeethWhitener said:
Just substitute ##\mu_{1A}## for ##\mu_{2A}## before taking the exterior derivative.
It seems better for me to derive it from the equality of mixed derivatives of Gibbs energy with its total differential:
dG=μA1⋅dnA1B1⋅dnB1A2⋅dnA2B2⋅dnB2
That is:

From that the differential forms of chemical potencials are opposite but I suppose it does not matter.

ussername said:
For each of them I can express the Gibbs energy total differential and for each of them I perform a surface work when moving a particle to the interface (and those surface works generally differ).
But the surface is the interface. I'm not sure how I can explain this more clearly.

Bystander

What is the Gibbs adsorption isotherm derivation?

The Gibbs adsorption isotherm derivation is a mathematical expression that describes the relationship between the surface excess of a substance and its concentration at the interface between two phases, such as a liquid and a gas. It is often used in the study of surface chemistry and physical chemistry.

What is the equation for the Gibbs adsorption isotherm derivation?

The equation for the Gibbs adsorption isotherm derivation is: Γ = RTln(C/C°), where Γ is the surface excess of the substance, R is the gas constant, T is the temperature, C is the concentration of the substance at the interface, and C° is the bulk concentration.

The Gibbs adsorption isotherm derivation assumes that the surface is homogeneous and that the interactions between the molecules at the interface are the same as those in the bulk phase. It also assumes that the surface is in equilibrium with the bulk phase and that the temperature and pressure are constant.

How is the Gibbs adsorption isotherm derivation used in experiments?

The Gibbs adsorption isotherm derivation can be used in experiments to determine the surface excess of a substance at the interface between two phases. By measuring the concentration of the substance at different temperatures and pressures, the surface tension and other properties of the interface can also be calculated.

What are the limitations of the Gibbs adsorption isotherm derivation?

The Gibbs adsorption isotherm derivation is based on several assumptions that may not hold true in all situations. It also only applies to ideal solutions and may not accurately describe the behavior of real systems. Additionally, it does not take into account the effects of surface roughness or the presence of impurities at the interface.

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