Adsorption Isotherms — Revision Notes

⚡ 30-Second Revision

Adsorption Isotherm: — $x/m$ vs. $P$ (or $C$) at constant $T$.
Freundlich Isotherm: — Empirical. $x/m = kP^{1/n}$. Linear form: $log(x/m) = log(k) + (1/n)log(P)$. Slope $= 1/n$, Y-intercept $= log(k)$. Valid for intermediate pressures, fails at high pressures. $0 < 1/n < 1$.
Langmuir Isotherm: — Theoretical. $x/m = \frac{(x/m)_{max} bP}{1 + bP}$. Linear form: $rac{1}{x/m} = \frac{1}{(x/m)_{max} b} \frac{1}{P} + \frac{1}{(x/m)_{max}}$. Slope $= \frac{1}{(x/m)_{max} b}$, Y-intercept $= \frac{1}{(x/m)_{max}}$. Assumes monolayer, homogeneous surface, no interaction. Predicts saturation.

2-Minute Revision

Adsorption isotherms graphically or mathematically describe how the amount of adsorbate adsorbed ($x/m$) changes with equilibrium pressure (for gases) or concentration (for solutions) at a constant temperature. This helps characterize the adsorption process.

Freundlich Isotherm is an empirical model: $x/m = kP^{1/n}$. It's useful for describing multilayer adsorption on heterogeneous surfaces. Key points: $1/n$ is between 0 and 1, indicating varying dependence on pressure. Its linear form is $log(x/m) = log(k) + (1/n)log(P)$, where $1/n$ is the slope. A major limitation is its failure at very high pressures, where it incorrectly predicts indefinite adsorption.

Langmuir Isotherm is a theoretical model based on specific assumptions: monolayer adsorption on a homogeneous surface, fixed sites, and no interaction between adsorbed molecules. Its equation is $x/m = \frac{(x/m)_{max} bP}{1 + bP}$.

This model accurately predicts a saturation limit at high pressures, where the surface is fully covered. Its linear form is $rac{1}{x/m} = \frac{1}{(x/m)_{max} b} \frac{1}{P} + \frac{1}{(x/m)_{max}}$, allowing determination of maximum capacity ($(x/m)_{max}$) and the affinity constant ($b$).

Understanding the assumptions and limitations of both models is crucial for NEET.

5-Minute Revision

Adsorption isotherms are fundamental tools in surface chemistry, illustrating the relationship between the extent of adsorption ($x/m$) and the equilibrium pressure ($P$) or concentration ($C$) at a constant temperature ($T$). This 'constant temperature' condition is vital because temperature significantly influences adsorption, typically decreasing it for exothermic processes.

Freundlich Adsorption Isotherm:

Nature: — Empirical, derived from experimental observations.
Equation: — $x/m = kP^{1/n}$ (for gases) or $x/m = kC^{1/n}$ (for solutions).
Constants: — $k$ and $n$ are constants for a given system at a specific temperature. $1/n$ typically ranges from 0 to 1.

* If $1/n = 1$ (low pressure), $x/m propto P$. * If $1/n = 0$ (high pressure), $x/m = k$ (independent of $P$, saturation).

Linear Form: — $log(x/m) = log(k) + (1/n)log(P)$. Plotting $log(x/m)$ vs. $log(P)$ gives a straight line with slope $1/n$ and y-intercept $log(k)$.
Limitations: — Fails at very high pressures, as it predicts indefinite adsorption, which is physically impossible. Lacks a theoretical basis.

Langmuir Adsorption Isotherm:

Nature: — Theoretical, based on specific assumptions.
Assumptions:

1. Adsorption occurs at specific, fixed sites. 2. Each site can hold only one molecule (monolayer adsorption). 3. All sites are equivalent. 4. No interaction between adsorbed molecules. 5. Dynamic equilibrium between adsorption and desorption.

Equation: — $x/m = \frac{(x/m)_{max} bP}{1 + bP}$. $(x/m)_{max}$ is the maximum adsorption capacity (monolayer capacity), $b$ is the Langmuir constant related to affinity.
Linear Form: — $rac{1}{x/m} = \frac{1}{(x/m)_{max} b} \frac{1}{P} + \frac{1}{(x/m)_{max}}$. Plotting $1/(x/m)$ vs. $1/P$ gives a straight line with slope $rac{1}{(x/m)_{max} b}$ and y-intercept $rac{1}{(x/m)_{max}}$.
Strengths: — Accurately predicts saturation at high pressures. Provides mechanistic insight.
Limitations: — Assumptions (homogeneous surface, no interaction) are often idealizations.

Comparison: Freundlich is empirical and describes multilayer, heterogeneous adsorption. Langmuir is theoretical and describes monolayer, homogeneous adsorption. Both are crucial for understanding adsorption phenomena in various applications like catalysis and purification.

Worked Mini-Example (Langmuir): If $(x/m)_{max} = 20,\text{mg/g}$ and $b = 0.1,\text{atm}^{-1}$, what is $x/m$ at $P=5,\text{atm}$? $x/m = \frac{20 \times 0.1 \times 5}{1 + (0.1 \times 5)} = \frac{10}{1 + 0.5} = \frac{10}{1.5} approx 6.67,\text{mg/g}$.

Prelims Revision Notes

Adsorption isotherms are graphical or mathematical relationships between the extent of adsorption ($x/m$) and equilibrium pressure ($P$) or concentration ($C$) at constant temperature ($T$).

Freundlich Adsorption Isotherm:

Equation: — $x/m = kP^{1/n}$ (for gases) or $x/m = kC^{1/n}$ (for solutions).
Nature: — Empirical (experimental).
Range of $1/n$: — Typically $0 < 1/n < 1$.

* Low pressure: $1/n approx 1 implies x/m propto P$. * High pressure: $1/n approx 0 implies x/m = k$ (saturation).

Linear Plot: — $log(x/m)$ vs. $log P$.

* Slope $= 1/n$. * Y-intercept $= log k$.

Limitations: — Fails at very high pressures (predicts infinite adsorption). Lacks theoretical basis.

Langmuir Adsorption Isotherm:

Equation: — $x/m = \frac{(x/m)_{max} bP}{1 + bP}$ (for gases) or $x/m = \frac{(x/m)_{max} bC}{1 + bC}$ (for solutions).
Nature: — Theoretical (mechanistic).
Assumptions (CRITICAL):

1. Fixed, specific adsorption sites. 2. Monolayer adsorption (one molecule per site). 3. Homogeneous surface (all sites equivalent). 4. No interaction between adsorbed molecules. 5. Dynamic equilibrium.

Constants:

* $(x/m)_{max}$: Maximum adsorption capacity (monolayer capacity). * $b$: Langmuir constant, related to affinity.

Linear Plot: — $1/(x/m)$ vs. $1/P$.

* Slope $= \frac{1}{(x/m)_{max} b}$. * Y-intercept $= \frac{1}{(x/m)_{max}}$.

Strengths: — Predicts saturation accurately. Provides insight into surface properties.
Applicability: — Often better for chemisorption due to specific site and monolayer assumptions.

Key Differences to Remember:

Freundlich: Empirical, multilayer, heterogeneous surface, fails at high P.
Langmuir: Theoretical, monolayer, homogeneous surface, predicts saturation.

Numerical Tips: Be careful with units and algebraic manipulation, especially for linear forms to find constants. Practice interpreting graphs to identify slopes and intercepts.

Vyyuha Quick Recall

To remember Langmuir's assumptions: My Homework For Science Is Done.

Monolayer adsorption
Homogeneous surface
Fixed sites
Specific sites
Interaction (No interaction between adsorbed molecules)
Dynamic equilibrium