Answer the question on the basis of passage given below:

Adsorption is a surface phenomenon which can be expressed by means of an emprical relationship known as Freundlich Adsorption isotherm. The relationship is given by

\(\frac{x}{m} = KP^{\frac{1}{n}}\left(n > 1\right)\)

\(x = \)Mass of the gas adsorbed

\(m = \)Mass of adsorbent

\(P = \)Pressure at which adsorption takes place

\(K\) and \(n\) are constant

Adsorption of a gas follow Freundlich isothermshown below. \(\frac{x}{m}\) is proportional to

\(p^{1/3}\)

The correct answer is option 3. \(p^{1/3}\).

The Freundlich isotherm is expressed by the equation:

\(\frac{x}{m} = K \cdot p^{1/n}\, \ -----(i)\)

Where:

\( \frac{x}{m} \) is the amount of gas adsorbed per unit mass of adsorbent,

\( p \) is the pressure of the gas,

\( K \) is a constant related to the adsorption capacity of the adsorbent, and

\( n \) is an empirical constant related to the intensity of adsorption.

Applying log in equation (i), we get

\(log\left(\frac{x}{m}\right) = log K + \frac{1}{n}log p\)

Here,  slope \(= \frac{1}{n}\)

From the graph,

slope \(= \frac{1}{n} = \frac{1}{3}\)

So, equation (i) becomes

\(\frac{x}{m} = K \cdot p^{1/3}\)

or, \(\frac{x}{m} \propto p^{1/3}\)