# Effect of pressure on adsorption of a gas

I am currently studying about the basic terms of surface chemistry, and am facing some conceptual doubt regarding the following:

What effect does pressure have on the adsorption of a gas (adsorbate) on a adsorbent? I don't understand the fact that the adsorption of a gas increases very rapidly as the pressure is increased from small values at low temperature.

I have the idea that increasing the pressure brings the molecules of the gas close together so that they have more intermolecular forces of attraction, but how do I correlate this with adsorption?

The following equilibrium roughly represents what goes on during physisorption of a gases onto a a surface. $$\ce{Gas + Surface <=>[adsorption][desorption] Gas--Surface}$$

The effect of pressure can be understood on the basis of Le Chatelier's principle, which states that if a stress (such as a change in pressure) is applied to a system in equilibrium, the equilibrium will shift so as to tend to counteract the effect of the constraint.

In this case, we are increasing the pressure, and the system would respond by wanting to reduce the number of gas phase molecules, and in this case it can do so by favouring the forward reaction wherein the gas phase molecules are immobilised on the surface.

Adsorption is usually described through isotherms, that is, the amount of adsorbate on the adsorbent as a function of its pressure (if gas) or concentration (if liquid) at constant temperature. Several isotherm models exist to describe this phenomenon.

Bonus Round: Additional Commentary

One of the earliest mathematical fits for experimental data was the Freundlich adsorption isotherm. Experimentally it was determined that extent of adsorption varies directly with pressure and then it directly varies with pressure raised to the power 1/n until saturation pressure $P_s$ is reached. Beyond that point rate of adsorption saturates even after applying higher pressure. Thus Freundlich adsorption isotherm failed at higher pressure.

$$\frac{x}{m} = Kp^{\frac{1}{n}}$$

where $x$ is the quantity of adsorbate adsorbed in moles, $m$ is the mass of the adsorbent (this is basically a normalisation to allow for comparison between different materials), $P$ is the pressure of adsorbate and $K$ and $n$ are empirical constants at the given temperature.

The function is not adequate at very high pressure because in reality, as I have already mentioned, because $\frac{x}{m}$ has an asymptotic maximum as pressure increases without bound. As the temperature increases, the constants $k$ and $n$ change to reflect the empirical observation that the quantity adsorbed rises more slowly and higher pressures are required to saturate the surface.

Irving Langmuir was the first to derive a semi-empirical model with a kinetic basis and was derived based on statistical thermodynamics.

It is quite simple, (and elegant in my opinion), and finds extensive use. it is based on the following four assumptions

1. The surface containing the adsorbing sites is a perfectly flat plane with no corrugations (assume the surface is homogeneous) .
2. The adsorbing gas adsorbs into an immobile state.
3. All sites are equivalent.
4. Each site can hold at most one molecule of A (mono-layer coverage only).
5. There are no interactions between adsorbate molecules on adjacent sites.
6. There are no phase transitions.

Unfortunately,these four assumptions are seldom all true: there are always imperfections on the surface, adsorbed molecules are not necessarily inert, and the mechanism is clearly not the same for the very first molecules to adsorb to a surface as for the last. The fourth condition is the most troublesome, as frequently more molecules will adsorb to the monolayer; this problem is addressed by the BET isotherm for relatively flat (non-microporous) surfaces.

Anyway, the surface coverage is given as follows $$\theta = \frac{K_{eq}p}{1+ K_{eq}p}$$