Freundlich Adsorption Isotherm

Adsorption isotherms can also be examined quantitatively: a plot of the amount of adsorbate adsorbed against the pressure (or, for adsorption from solution, the concentration of the adsorbate) at constant temperature is called an adsorption isotherm, and several mathematical equations have historically been proposed to fit its shape — the Freundlich isotherm being the earliest and most widely used.

According to Freundlich, the amount x of adsorbate adsorbed per gram m of adsorbent, at a pressure p and constant temperature, follows the empirical relationship

where k and n are empirical constants (introduced by Freundlich) that depend on the particular adsorbent–adsorbate pair and on temperature, and n is always found to be greater than unity, so the exponent $1/n$ is always a positive fraction less than one. This equation applies directly to the adsorption of gases on solid surfaces; when instead applied to adsorption from solution, pressure p is simply replaced by the concentration c of the adsorbate in solution, giving the equivalent form $x/m = Kc^{1/n}$. Either version quantitatively predicts how the amount adsorbed responds to changing pressure (or concentration) at fixed temperature — and in particular predicts that x/m keeps increasing as pressure rises, though with steadily DIMINISHING returns, since raising p to a fractional power less than one produces ever-smaller proportional gains as p grows.

Because the equation as written is not linear, it is normally tested and used in its logarithmic form, obtained by taking logarithms of both sides:

Plotting $\log(x/m)$ on the vertical axis against $\log p$ on the horizontal axis therefore gives a straight line, whose SLOPE equals $1/n$ and whose Y-INTERCEPT equals $\log k$ — precisely the two constants of the original Freundlich equation, now readable directly off a straight-line graph constructed from a handful of experimental (p, x/m) readings. …