Freundlich adsorption isotherm — explain with graph and equation

Question

State the Freundlich adsorption isotherm. Write its equation and explain the significance of the constants. How can the equation be verified graphically?

(NCERT Class 12, Chapter 5)

Solution — Step by Step

The Freundlich isotherm gives an empirical relationship between the amount of gas adsorbed per unit mass of adsorbent ( $x/m$ ) and the pressure ( $p$ ) at constant temperature:

\frac{x}{m} = k \cdot p^{1/n}

where $k$ and $n$ are constants that depend on the nature of the adsorbent and the gas, and on temperature. Here $n > 1$ always.

The value of $1/n$ lies between 0 and 1:

At low pressure: adsorption is directly proportional to pressure ( $x/m \propto p$ , i.e., $1/n \to 1$ )
At high pressure: adsorption becomes independent of pressure ( $x/m$ = constant, i.e., $1/n \to 0$ )
At intermediate pressure: $x/m = kp^{1/n}$ with $0 < 1/n < 1$

The Freundlich isotherm works well at intermediate pressures but fails at very high pressures (where it should show saturation).

Taking logarithm of both sides:

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

This has the form $y = c + mx$ . A plot of $\log(x/m)$ vs $\log p$ gives a straight line with:

Slope = $1/n$
y-intercept = $\log k$

If the experimental data gives a straight line on this log-log plot, the Freundlich isotherm is verified.

Why This Works

The Freundlich isotherm is an empirical equation — it fits experimental data without a rigorous theoretical derivation. It assumes a heterogeneous adsorbent surface where different sites have different adsorption energies. The most energetic sites fill first (at low pressure), and progressively weaker sites fill as pressure increases.

The constant $k$ reflects the adsorption capacity (larger $k$ means more adsorption), while $1/n$ reflects the intensity of adsorption. When $1/n = 0$ , adsorption is independent of pressure (surface is saturated).

Alternative Method — Comparison with Langmuir Isotherm

The Langmuir isotherm, $\frac{x}{m} = \frac{ap}{1 + bp}$ , is theoretically more rigorous and properly accounts for saturation at high pressures. The Freundlich isotherm can be seen as an approximation of the Langmuir isotherm at intermediate pressures.

For JEE, remember three isotherms: (1) Freundlich — empirical, works at moderate pressures, (2) Langmuir — assumes monolayer, uniform surface, (3) BET — extends Langmuir to multilayer adsorption. Board exams focus on Freundlich; JEE may ask conceptual comparisons.

Common Mistake

Students often write $n$ in the exponent instead of $1/n$ . The correct form is $x/m = kp^{1/n}$ where $n > 1$ , making the exponent $1/n$ a fraction between 0 and 1. Writing $kp^n$ with $n > 1$ would imply adsorption increases faster than linearly with pressure, which contradicts the physical picture of surface saturation.

Question

Solution — Step by Step

Why This Works

Alternative Method — Comparison with Langmuir Isotherm

Common Mistake

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