# Finding the Freundlich adsorption isotherm from the graph of log(x/m) v/s log(p)

The following problem was asked in JEE Mains 2020 (Sept 2, Shift 1),

The mass of gas adsorbed, $$x$$, per unit mass of adsorbate, $$m$$, was measured at various pressures, $$p$$. A graph between $$\log\frac xm$$ and $$\log p$$ gives a straight line with slope equal to $$2$$ and the intercept equal to $$0.4771$$. The value of $$\frac xm$$ at a pressure of $$\pu{4 atm}$$ is: (Given $$\log 3 = 0.4771$$)

I know the Freundlich adsorption isotherm formula, but I decided to write the equation of given line $$( y = c + mx)$$ as,

$$\log \frac xm = \log 3 + 2\log p$$

So, the relation comes out to be,

$$\frac xm = 3p^2$$

On plugging $$p = 4$$, we get,

$$\frac xm = 3(16) = 48$$

But, the answer was wrong as per the official key, which mostly has genuine answers. Where am I wrong?

6

• $1/n$ lies between 0 and 1.. Sep 8 '20 at 14:38
• @Safdar: Yes, I'd doubt over this during the exam. But, if I take $n=2$ and then, draw for $\log p$ v/s $\log x/m$, then it goes very complex. Sep 8 '20 at 14:41
• I think the question has a mistake. You should take $n=2$ then equation is: $$\log \frac{x}{m} = \log 3 + \frac{1}{n} \log p$$ and you got the given answer. Sep 8 '20 at 15:20
• @MathewMahindaratne, Great, I never thought that. Actually, I have to challenge the given key, that's why I asked. Thanks for pointing this thing :) Sep 8 '20 at 15:30

One version of Freundlich adsorption isotherm equation is: $$\frac xm = Kp^{\frac12},$$ which can also be written as: $$\log \frac xm = \log K + \frac12\log p$$ This is a straight line equation of type $$( y = c + mx)$$ as given in the question. However, I think the question has made a mistake saying the slope is $$2$$, but instead it should be $$\frac12$$. Accordingly, the equation with given numeric values should be:

$$\log \frac xm = \log 3 + \frac12\log p$$

Or without logarithm:

$$\frac xm = 3p^\frac12$$

On plugging $$p = 4$$, You'll get:

$$\frac xm = 3 \times 4^\frac12 = 3 \times 2 = 6$$

Hence, you get the given answer.