Answer the question on the basis of passage given below:

Adsorption is the phenomenon of attracting and retaining the molecules of a substance on the surface of a solid resulting into a higher concentration on the surface than in the bulk. The variation in the amount of gas adsorbed by the adsorbent with pressure at constant temperature is expressed as isotherm. A catalyst is a substance, when adsorbed enhances the rate of a chemical reaction without itself getting used up in the reaction. Depending on the type of the particles of the dispersed phase, colloids are classified as multimolecular, macromolecular and associated colloids and soap is such an example of associated colloids. Colloids show various properties such as Tyndall effect, colour, Brownian movement and charge development. There are positively charged sols and negatively charged sols. Due to this, electrophoresis and electroosmosis are shown by colloids.

Following is not an equation for Freundlich adsorption isotherm-

$\log\frac{x}{m}=\log k+\frac{1}{n}\log P$

The correct answer is option 2. $\log\frac{x}{m} = \log k + \frac{1}{n}\log P$.

The variation of extent of adsorption \((\frac{x}{m})\) with pressure \((P)\) at a particular temperature was given mathematically by Freundlich in 1909. From the adsorption isotherm, the following observations can be easily made:

(i) At low pressure, the graph is almost straight line which indicates that \(\frac{x}{m}\) is directly proportional to pressure. This may be expressed as:

\(\frac{x}{m}\propto P\)

or, \(\frac{x}{m} = kP\) -------(i)

where \(k\) is a constant

(ii) At high pressure, the graph becomes almost constant which means that \(\frac{x}{m}\) becomes independent of pressure. This may be expressed as:

\(\frac{x}{m} =\, \ constant\)

or, \(\frac{x}{m} \propto P^0\)    \(∵ P^0 = 1\)

or, \(\frac{x}{m} = kP^0\) -------(ii)

(iii) Thus, in the intermediate range of pressure, \(\frac{x}{m}\) will depend upon the power of pressure which lies between 0 to 1 i.e., fractional power of pressure (probable range 0.1 to 0.5). This may be expressed as

\(\frac{x}{m} \propto P^{1/n}\)

or, \(\frac{x}{m} = kP^{1/n}\) -------(iii)

where \(n\) can take any whole number value which depends upon the nature of adsorbate and adsorbent. The above relationship is also called Freundlich's adsorption isotherm and is shown in figure above.

Calculation of \(k\) and \(n\) of adsorption isotherm:

The constants k and n can be determined as explained below:

Taking logarithms on both sides of eq. (iii), we get

\(log\frac{x}{m} = log k + \frac{1}{n}log P\)

Thus, if we plot a graph between log \((\frac{x}{m})\) on y-axis(ordinate) and log P, on x-axis (abscissa), straight line will be obtained. This also shows the validity of Freundlich isotherm. The slope of the line (Fig. below) is equal to \(1/n\) and the intercept is equal to log k.