6.1 Adsorption Isotherms

Adsorption Isotherms

Adsorption isotherms describe how molecules accumulate on surfaces as a function of pressure or concentration at constant temperature. They provide the quantitative framework you need to characterize surface properties, determine surface areas, and predict how adsorbent materials will perform in real applications like catalysis, gas storage, and pollutant removal.

This section covers the distinction between physisorption and chemisorption, the major isotherm models (Langmuir, Freundlich, BET), and how to extract useful parameters from experimental data.

Adsorption in Surface Chemistry

Concept and Significance

Adsorption is the accumulation of atoms, ions, or molecules (the adsorbate) from a gas, liquid, or dissolved solid onto a surface (the adsorbent), forming a thin film at the interface between two phases. This is distinct from absorption, where molecules penetrate into the bulk of the material.

Adsorption modifies surface properties such as reactivity, catalytic activity, and wettability. It underpins a wide range of applications:

• Gas separation and purification (e.g., pressure swing adsorption for oxygen enrichment)
• Heterogeneous catalysis (reactant molecules must adsorb before reacting)
• Environmental remediation (activated carbon filters for water and air treatment)

Factors Affecting Adsorption

The extent of adsorption depends on several variables:

• Temperature: Lower temperatures generally favor adsorption because the process is exothermic. Raising the temperature shifts the equilibrium toward desorption.
• Pressure (or concentration): Higher pressures drive more adsorbate onto the surface, up to a saturation limit.
• Surface area: More available surface means more adsorption sites. Porous materials like activated carbon (surface areas of 500–2000 $\text{m}^2/\text{g}$) are effective precisely because of their enormous internal surface area.
• Nature of adsorbent and adsorbate: Surface chemistry, pore size, and structure of the adsorbent, along with the size, polarity, and chemical properties of the adsorbate, govern the strength and selectivity of the interaction.

Physisorption vs. Chemisorption

These two categories differ fundamentally in the strength and nature of the adsorbate-surface interaction.

Physisorption Characteristics

• Weak, reversible interaction driven by van der Waals forces, dipole-dipole interactions, or hydrogen bonding
• Enthalpy of adsorption is relatively low: roughly 5–40 kJ/mol
• Can form multilayers because each adsorbed layer interacts weakly with the next
• Non-specific: a given surface can physisorb many different adsorbates
• Strongly temperature-dependent; desorption occurs readily with modest heating

Chemisorption Characteristics

• Strong interaction involving the formation of actual chemical bonds (covalent or ionic) between adsorbate and surface
• Enthalpy of adsorption is much higher: roughly 40–400 kJ/mol
• Limited to a monolayer, since bond formation requires direct contact with surface sites
• Highly selective: depends on chemical compatibility between adsorbate and surface
• Often irreversible, or reversible only at elevated temperatures
• May require an activation energy, so chemisorption can increase with temperature up to a point (unlike physisorption)

Examples and Applications

Physisorption: Nitrogen adsorption on activated carbon (the basis of BET surface area measurements), organic pollutant removal by clay minerals. Applications include gas storage, gas separation, and environmental cleanup.

Chemisorption: Hydrogen dissociatively adsorbing on palladium or platinum surfaces, oxygen binding to metal oxides like $\text{TiO}_2$ or $\text{ZnO}$. Applications include heterogeneous catalysis, chemical sensors, and surface functionalization.

Adsorption Isotherm Analysis

Types of Adsorption Isotherms

An adsorption isotherm plots the amount of adsorbate on the surface versus the equilibrium pressure (or concentration) at constant temperature. The shape of the curve tells you about the adsorption mechanism, surface homogeneity, and adsorbate-adsorbate interactions.

The three most important models for this course:

• Langmuir isotherm: Assumes monolayer adsorption on a homogeneous surface with no lateral interactions between adsorbed molecules. Produces a curve that rises steeply at low pressure and levels off to a plateau.
• Freundlich isotherm: An empirical model that accounts for surface heterogeneity and does not assume a saturation limit. Follows a power-law relationship.
• BET isotherm: Extends the Langmuir model to multilayer adsorption by treating each adsorbed layer as a new substrate for the next. Used extensively to measure specific surface areas of solids.

Isotherm Shape Interpretation (IUPAC Classification)

The IUPAC classification identifies several characteristic isotherm shapes:

• Type I: Monolayer adsorption on a microporous solid. Sharp rise at low pressure, then a plateau. Described well by the Langmuir model.
• Type II: Multilayer adsorption on a non-porous or macroporous surface. An inflection point marks the transition from monolayer completion to multilayer buildup. Described by the BET model.
• Type III: Weak adsorbate-adsorbent interactions (weaker than adsorbate-adsorbate interactions). Convex curve with no identifiable monolayer.
• Type IV: Similar to Type II at low pressures, but exhibits a hysteresis loop at higher pressures due to capillary condensation in mesopores (pore diameters 2–50 nm).
• Type V: Similar to Type III but with hysteresis, again indicating mesoporous structure with weak adsorbate-surface interactions.

Adsorption Isotherm Modeling

Langmuir Adsorption Isotherm

The Langmuir model relates fractional surface coverage $\theta$ to equilibrium pressure $P$:

$\theta = \frac{KP}{1 + KP}$

where $K$ is the adsorption equilibrium constant (ratio of rate constants for adsorption and desorption, $K = k_a / k_d$).

Key assumptions:

1. The surface is energetically uniform (all sites are equivalent)
2. Each site holds at most one adsorbate molecule (monolayer only)
3. No interactions between adsorbed molecules on neighboring sites

Linearization for data analysis:

Rearranging to a linear form makes it possible to extract $K$ and the maximum adsorption capacity $q_m$ from experimental data:

$\frac{P}{q} = \frac{1}{q_m K} + \frac{P}{q_m}$

where $q$ is the amount adsorbed and $q_m$ is the monolayer capacity.

Steps to use this:

1. Measure $q$ at several equilibrium pressures $P$
2. Plot $P/q$ on the y-axis versus $P$ on the x-axis
3. Fit a straight line: the slope gives $1/q_m$ and the y-intercept gives $1/(q_m K)$
4. Calculate $q_m$ from the slope, then $K$ from the intercept

Freundlich Adsorption Isotherm

The Freundlich model is empirical and works well for heterogeneous surfaces:

$q = K_F P^{1/n}$

where $K_F$ is the Freundlich capacity constant and $n$ is a parameter related to adsorption intensity. When $1/n < 1$, adsorption is favorable; when $1/n > 1$, it is unfavorable.

Linearization:

Taking the logarithm of both sides:

$\log q = \log K_F + \frac{1}{n} \log P$

1. Plot $\log q$ versus $\log P$
2. The slope gives $1/n$ and the y-intercept gives $\log K_F$

Note that the Freundlich isotherm does not predict a saturation plateau. It works well at intermediate pressures but can break down at very low or very high pressures.

BET Adsorption Isotherm

The BET model handles multilayer adsorption. The linearized BET equation is:

$\frac{P}{q(P_0 - P)} = \frac{1}{q_m C} + \frac{C - 1}{q_m C} \cdot \frac{P}{P_0}$

where $P_0$ is the saturation vapor pressure of the adsorbate and $C$ is the BET constant, which reflects the strength of interaction between the adsorbate and the surface relative to adsorbate-adsorbate interactions in subsequent layers.

Steps to determine surface area using BET:

1. Measure the volume of gas adsorbed at several relative pressures $P/P_0$ (typically in the range 0.05–0.35)

2. Plot $P/[q(P_0 - P)]$ versus $P/P_0$

3. From the slope $s = (C-1)/(q_m C)$ and intercept $i = 1/(q_m C)$, solve for $q_m$ and $C$:

   • $q_m = 1/(s + i)$
   • $C = (s/i) + 1$

4. Calculate the specific surface area: $A = q_m \cdot N_A \cdot \sigma$, where $N_A$ is Avogadro's number and $\sigma$ is the cross-sectional area of one adsorbate molecule (for $\text{N}_2$ at 77 K, $\sigma = 0.162 \text{ nm}^2$)

Applications of Adsorption Isotherm Models

• Process optimization: Selecting the right adsorbent and operating conditions for gas storage, separation, or purification
• Material characterization: Determining specific surface area (BET), pore size distribution, and surface heterogeneity from isotherm data
• Performance prediction: Modeling how adsorption-based systems will behave under different pressures and temperatures
• Adsorbent design: Guiding the development of new materials for carbon capture, water treatment, and catalytic applications