Animated Solution for Chemistry - Surface Chemistry: In Freundlich adsorption isotherm at moderate pressure, the extent of adsorption $\left(\frac{x}{m}\right)$ is directly proportional to $p^x$. The value of $x$ is

Imagine you are wearing a gas mask in a toxic environment. The activated charcoal inside the mask acts like a magnet for poisonous gases, trapping them on its surface while letting clean air pass through. This surface phenomenon, where molecules of a gas accumulate on the surface of a solid, is called adsorption.

To understand exactly how much gas gets trapped at a given pressure, scientists use mathematical models called isotherms. One of the most famous and widely used models is the Freundlich Adsorption Isotherm.

Freundlich observed that if you plot the extent of adsorption—represented by $\frac{x}{m}$ (where $x$ is the mass of the gas adsorbed and $m$ is the mass of the solid adsorbent)—against the pressure $p$ at a constant temperature, you get a very specific curve.

To describe this curve mathematically, he proposed an empirical equation:

Here, $k$ and $n$ are constants that depend on the nature of the gas and the solid surface at a particular temperature. Crucially, the value of $n$ is always greater than 1 ($n>1$).

If you look closely at the isotherm graph, it doesn't behave the same way everywhere. It can be divided into three distinct pressure zones:

1. Low Pressure: At very low pressures, the curve is almost a straight line. This means the extent of adsorption is directly proportional to the pressure. Mathematically, this is when $\frac{1}{n}=1$, giving us $\frac{x}{m}\propto p^1$.

2. High Pressure: At very high pressures, the solid surface becomes completely saturated with gas molecules. No matter how much more pressure you apply, no more gas can be adsorbed. The curve flattens out, becoming independent of pressure. Here, $\frac{1}{n}=0$, giving us $\frac{x}{m}\propto p^0$.

3. Moderate Pressure: This is the intermediate zone where the curve is bending. The extent of adsorption depends on pressure, but not linearly. It follows the fractional power relationship given by the Freundlich equation:

Now, let's look at the problem at hand. The question states that at moderate pressure, the extent of adsorption is directly proportional to $p^x$:

We already know from the standard Freundlich isotherm that in the moderate pressure region, the relationship is:

By simply comparing the two proportionalities, it becomes crystal clear that the exponents of the pressure $p$ must be identical.

Equating the powers, we get our final answer:

This elegant little problem is a perfect test of your conceptual clarity regarding the different pressure regimes of the Freundlich adsorption isotherm. Always remember to visualize the graph—it holds the key to the equations!