Q.Observe the given figure and answer the following questions : The figure is a straight-line plot of $\log \frac{x}{m}$ (y-axis) against $\log p$ (x-axis) with a positive slope and a positive intercept on the $\log \frac{x}{m}$ axis (the Freundlich adsorption isotherm, where $\frac{x}{m}$ is the mass of gas adsorbed per gram of adsorbent and $p$ is the pressure).

Freundlich Adsorption Isotherm

Imagine you have a jar of charcoal and you pump in some gas. Some of that gas sticks to the surface of the charcoal — that's adsorption. The question is: if you increase the pressure of the gas, how much more gas will stick? Does it keep increasing forever, or does it slow down?

That's exactly what the Freundlich isotherm describes.

The Intuition

At very low pressures, there's plenty of empty surface area. So if you double the pressure, roughly twice as many gas molecules hit the surface and stick — adsorption increases almost proportionally. But as pressure rises, the surface starts getting crowded. Now doubling the pressure doesn't double the adsorption, because many spots are already taken. The increase slows down.

So the relationship is not a straight line. It's a curve that rises steeply at first, then flattens out — but never quite reaches a perfect flat ceiling (unlike the Langmuir isotherm, which does).

The Empirical Relation

Freundlich proposed a simple equation that captures this behaviour:

Where:

• $x$ = mass of the gas adsorbed
• $m$ = mass of the adsorbent (the solid)
• $\frac{x}{m}$ = amount adsorbed per unit mass of adsorbent
• $P$ = equilibrium pressure of the gas
• $k$ and $n$ are constants that depend on the adsorbent, the gas, and the temperature

The constant $n$ is always greater than 1. This is crucial — it's what makes the curve bend.

What the Constants Mean

$k$ tells you about the capacity of the adsorbent — a larger $k$ means more adsorption at a given pressure. $n$ tells you about the intensity or favourability of adsorption. When $n$ is large (say 3 or 4), the curve flattens quickly — adsorption is strong at low pressures but saturates fast. When $n$ is close to 1, the curve is nearly a straight line — adsorption keeps increasing almost linearly with pressure.

Taking Logarithms to See the Line

The real power of the Freundlich isotherm shows up when you take logs:

This is the equation of a straight line: $y = c + mx$, where $y = \log(x/m)$ and $x = \log P$. The slope is $1/n$ and the intercept is $\log k$.

So if you plot experimental data as $\log(x/m)$ vs $\log P$, and you get a straight line, the Freundlich isotherm fits your data. The slope gives you $n$, the intercept gives you $k$.

Where It Works and Where It Doesn't

The Freundlich isotherm works beautifully for many real systems — especially adsorption on rough, heterogeneous surfaces like charcoal or silica gel. It's simple, it fits a wide range of pressures, and it doesn't assume the surface is uniform (which real surfaces never are). …