What is an adsorption isotherm? Describe Freundlich adsorption isotherm.

The plot between the extent of adsorption $\left(\frac{x}{m}\right)$ against the pressure of gas $(P)$ at constant temperature $(T)$ is called the adsorption isotherm.
Freundlich adsorption isotherm:
Freundlich adsorption isotherm gives an empirical relationship between the quantity of gas adsorbed by the unit mass of solid adsorbent and pressure at a specific temperature.
From the given plot it is clear that at pressure $P_{s},$ $\frac{x}{m}$ reaches the maximum value. $P_{s}$ is called the saturation pressure. Three cases arise from the graph now.
Case $I$ - At low pressure:
The plot is straight and sloping,indicating that the pressure is directly proportional to $\frac{x}{m}$:
$\frac{x}{m} \propto P$
$\frac{x}{m} = k P$ ($k$ is a constant)
Case $II$ - At high pressure:
When pressure exceeds the saturation pressure,$\frac{x}{m}$ becomes independent of $P$ values.
$\frac{x}{m} \propto P^{0}$
$\frac{x}{m} = k P^{0}$
Case $III$ - At intermediate pressure:
At intermediate pressure,$\frac{x}{m}$ depends on $P$ raised to the powers between $0$ and $1.$ This relationship is known as the Freundlich adsorption isotherm.
$\frac{x}{m} \propto P^{\frac{1}{n}}$
$\frac{x}{m} = k P^{\frac{1}{n}}$ $(n > 1)$
Now,taking log:
$\log \left(\frac{x}{m}\right) = \log k + \frac{1}{n} \log P$
On plotting the graph between $\log \left(\frac{x}{m}\right)$ and $\log P,$ a straight line is obtained with the slope equal to $\frac{1}{n}$ and the intercept equal to $\log k.$