Adsorption Isotherms
Freundlich's Limitation, Langmuir's Theory, and the Mathematical Traps.
An Adsorption Isotherm is a mathematical equation or a graph that expresses the variation in the amount of gas adsorbed by a given mass of solid adsorbent with pressure, strictly at a constant temperature ("iso" = same, "therm" = temperature). Mastering these isotherms is arguably the most mathematically intensive and heavily tested section of Surface Chemistry in JEE and NEET.
Let $x$ be the mass of the adsorbate (the gas).
Let $m$ be the mass of the adsorbent (the solid).
Then, $\frac{x}{m}$ represents the Extent of Adsorption.
1. Freundlich Adsorption Isotherm (The Empirical Approach)
In 1909, Freundlich proposed a purely empirical (observation-based, no theoretical derivation) mathematical relationship between the extent of adsorption ($\frac{x}{m}$) and the equilibrium pressure ($P$).
(Where $k$ and $n$ are constants that depend on the nature of the gas, the solid, and the temperature).
Figure 1: Adsorption Isotherm Curve showing the three pressure dependence regions.
Understanding the Exponent ($\frac{1}{n}$):
The value of $\frac{1}{n}$ lies between 0 and 1. Looking at the graph, the behavior depends on the pressure range:
- At Low Pressures: The graph is almost a straight line. Adsorption is directly proportional to pressure. Therefore, $\frac{1}{n} = 1$, and $\frac{x}{m} = kP^1$.
- At High Pressures: The surface becomes completely covered. Further increases in pressure do not increase adsorption. The graph becomes horizontal (independent of pressure). Therefore, $\frac{1}{n} = 0$, and $\frac{x}{m} = kP^0 = k$.
- At Intermediate Pressures: This is where the Freundlich equation applies. $\frac{x}{m}$ depends on a fractional power of $P$.
2. The Logarithmic Plot (Testing the Isotherm)
To verify if an adsorption process follows the Freundlich isotherm, we take the logarithm on both sides of the equation.
This is the equation of a straight line: $y = c + mx$. If we plot $\log(\frac{x}{m})$ on the Y-axis and $\log(P)$ on the X-axis, we must get a straight line if the Freundlich isotherm holds true.
Figure 2: Verification of the Freundlich Isotherm via a linear logarithmic plot.
- Slope: Gives the value of $\frac{1}{n}$.
- Y-Intercept: Gives the value of $\log(k)$.
3. Langmuir Adsorption Isotherm (The Theoretical Masterpiece)
Because Freundlich's empirical formula failed at high pressures, Irving Langmuir derived a completely new isotherm based on kinetic gas theory.
The Fundamental Assumptions of Langmuir:
- The surface of the solid is perfectly uniform, and all adsorption sites are energetically equivalent.
- There are no interactions (attraction/repulsion) between adsorbed molecules on adjacent sites.
- Adsorption strictly forms a Uni-molecular layer. Once a site is covered, a second molecule cannot stick on top of it. (Note: This means Langmuir primarily describes Chemisorption, but at low pressures, it applies well to physisorption too).
- Adsorption is a dynamic equilibrium between two opposing processes:
- Condensation: Gas molecules striking the bare surface and sticking. (Rate $\propto$ Pressure $\times$ Bare Surface Area).
- Evaporation/Desorption: Adsorbed molecules leaving the surface. (Rate $\propto$ Covered Surface Area).
The Langmuir Equation:
(Where $a$ and $b$ are Langmuir parameters depending on temperature and the gas/solid pair).
Case 1: At Low Pressures ($bP \ll 1$)
The denominator $(1 + bP) \approx 1$.
$\frac{x}{m} = aP$
(Adsorption is directly proportional to Pressure; First-order kinetics).
Case 2: At High Pressures ($bP \gg 1$)
The denominator $(1 + bP) \approx bP$.
$\frac{x}{m} = \frac{aP}{bP} = \frac{a}{b} = \text{Constant}$.
(Adsorption is completely independent of Pressure; Zero-order kinetics).
Conclusion: The Langmuir equation perfectly mathematically derives the saturation plateau that Freundlich could not!
4. Adsorption from Solutions
Solids can also adsorb solutes directly from a liquid solution (e.g., activated charcoal removing colored impurities from sugar solutions, or removing acetic acid from water).
- The extent of adsorption from a solution decreases with an increase in temperature.
- It increases with an increase in the surface area of the adsorbent.
- The Isotherm: The exact same Freundlich isotherm applies, but we replace the Pressure ($P$) of the gas with the Equilibrium Concentration ($C$) of the solute in the liquid phase.
$\log\left(\frac{x}{m}\right) = \log(k) + \frac{1}{n}\log(C)$
Mastery Check: Adsorption Isotherms
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