Animated Solution for Chemistry - Surface Chemistry: In Freundlich adsorption isotherm, slope of $AB$ line is

Imagine you are conducting an experiment to see how much gas gets adsorbed onto a solid surface at a constant temperature. You plot your data, and you get a beautiful straight line.

In our problem, we are given a graph where the y-axis represents $\log \left(\frac{x}{m}\right)$ and the x-axis represents $\log p$.

Our goal is to decode this graph, find the slope of the line segment $AB$, and determine the valid mathematical range for this slope.

To understand the geometry of this graph, we need to look at the physical law governing it. The relationship between the extent of adsorption and pressure is given by the Freundlich Adsorption Isotherm.

The empirical equation is:

Here, $x$ is the mass of the gas adsorbed, $m$ is the mass of the adsorbent, $p$ is the pressure, and $k$ and $n$ are constants that depend on the nature of the gas and the solid surface.

I know this power equation doesn't look like a straight line yet, but let's take a breath and use a simple mathematical tool: logarithms.

By taking the logarithm on both sides of the equation, we can linearize it.

Using the properties of logarithms, the exponent $\frac{1}{n}$ comes down as a multiplier:

Now, look at the equation we just derived. It perfectly matches the standard equation of a straight line, $y=mx+c$.

In our case, $y=\log \left(\frac{x}{m}\right)$ and $x=\log p$.

By direct comparison, the y-intercept $c$ is $\log k$, and the slope $m$ is exactly $\frac{1}{n}$.

But there is a catch here! The constant $n$ is experimentally found to be always greater than or equal to $1$ ($n\ge 1$).

Because of this constraint, the value of the slope $\frac{1}{n}$ must strictly lie in the interval between $0$ and $1$.

Did you get the feel of it? Let's see what this range actually means in the real world.

At very low pressures, $\frac{1}{n}$ approaches $1$. This means $\frac{x}{m}\propto p^1$, so the adsorption increases linearly with pressure.

At very high pressures, $\frac{1}{n}$ approaches $0$. This means $\frac{x}{m}\propto p^0$, making the adsorption independent of pressure. The surface is completely saturated!

This beautiful transition is exactly what the Freundlich isotherm captures, and it is a favorite concept for JEE.