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The adsorption process is studied through the development of an adsorption isotherm which relates the amount of adsorbate (x) adsorbed on the surface of adsorbent (m) and pressure at constant temperature. Different adsorption isotherms have been developed by Freundlich, Langmuir, and by means of the Brunauer, Emmett, and Teller (BET) theory. Simply, the adsorption process can be represented as:

Freundlich Adsorption Isotherm

Freundlich gave an empirical expression representing the isothermal variation of adsorption of a quantity of gas adsorbed by unit mass of solid adsorbent with pressure:

In this equation, x is the mass of the gas adsorbed on mass, m, of the adsorbent at pressure p; k and n are constants whose values depend upon adsorbent and gas at particular temperature. This isotherm establishes the relationship of adsorption with lower pressures but is not always suitable for high-pressure situations.

Langmuir Adsorption Isotherm

The Langmuir adsorption isotherm is based on several assumptions, one of which is that dynamic equilibrium exists between adsorbed gaseous molecules and the free gaseous molecules:

In this equation, A(g) is the unabsorbed gaseous molecule, B(s) is unoccupied metal surface, and AB is adsorbed gaseous molecule from which a relationship between the number of active sites of the surface undergoing adsorption and pressure can be derived:

Here, θ is the number of sites of the surface which are covered with gaseous molecule, P is the pressure, and K is the equilibrium constant for distribution of adsorbate between the surface and the gas phase. However, the Langmuir adsorption equation is that it is valid at low pressure only. At lower pressure, KP is small and the factor 1+KP in denominator is close to unity which reduces the Langmuir equation to:

At high pressure, KP is large and the factor 1+KP is almost equal to KP, thereby reducing the Langmuir equation to:

BET Adsorption Isotherm

The BET theory equation invokes the concept that under the condition of high pressure and low temperature, the thermal energy of gaseous molecules decreases and more and more gaseous molecules would be available per unit surface area. As a result, multilayer adsorption will occur and can be represented by the BET equation:

Another form of the BET equation is:

In these equations, Vmono is the adsorbed volume of gas at high pressure conditions so as to cover the surface with a unilayer of gaseous molecules. Thus:

K₁ is the equilibrium constant when a single molecule is adsorbed per vacant site and K_L is the equilibrium constant to the saturated vapor liquid equilibrium.

See also: Adsorption, Adsorption Process.