Isotherms
When a gas or solute comes into contact with a solid surface, molecules accumulate at the interface in a process called adsorption. The amount of substance adsorbed does not increase indefinitely; it depends critically on the pressure (for gases) or concentration (for solutions) of the adsorbate at a fixed temperature. An adsorption isotherm is a graph that shows how the extent of adsorption varies with pressure or concentration at constant temperature — the word 'isotherm' literally means 'same temperature.' Studying these curves allows chemists and engineers to design catalysts, gas storage materials, water purification systems, and drug delivery platforms. Two mathematical models dominate introductory surface chemistry: the empirical Freundlich isotherm, proposed in 1909, and the theoretical Langmuir isotherm, derived from first principles in 1916. The Freundlich model is versatile and applies to heterogeneous surfaces with a distribution of binding energies. The Langmuir model makes idealising assumptions — a monolayer of adsorbate, uniform surface sites, and no interaction between adsorbed molecules — but it gives powerful mechanistic insight and is the foundation for more advanced models like BET theory used to measure surface areas of porous materials. Understanding both isotherms requires appreciating the difference between physisorption (weak van der Waals forces, easily reversible, ΔH ≈ −5 to −40 kJ mol⁻¹) and chemisorption (covalent/ionic bond formation, stronger, ΔH ≈ −40 to −400 kJ mol⁻¹). Both processes reach a dynamic equilibrium between adsorption and desorption. The isotherm equations quantify that equilibrium, and their linearised forms let you extract key constants from simple straight-line graphs — an essential skill both in examination questions and in real laboratory work.
- What an Isotherm Tells Us
- The Freundlich Adsorption Isotherm
- The Langmuir Adsorption Isotherm (Advanced)
- Comparison and Limitations
What an Isotherm Tells Us
Plot the amount of gas adsorbed per gram of adsorbent (x/m, in mol g⁻¹ or g g⁻¹) on the y-axis against equilibrium pressure p (in Pa or atm) on the x-axis, keeping temperature constant. The resulting curve is the adsorption isotherm. At low pressures the curve rises steeply (many vacant sites), then levels off as the surface becomes saturated. The shape of the curve — whether it stays concave, becomes linear, or shows a sharp step — tells you about the nature of adsorption and which model applies.
The Freundlich Adsorption Isotherm
Herbert Freundlich (1909) proposed an empirical equation based on experimental observations for heterogeneous surfaces: x/m = k·p^(1/n), where x is mass of adsorbate (g), m is mass of adsorbent (g), p is equilibrium pressure, k is the Freundlich constant related to adsorption capacity, and 1/n is the heterogeneity factor (0 < 1/n < 1). When 1/n = 1 the isotherm is linear; when 1/n < 1 the curve is concave (favourable adsorption, normal case). Crucially, the equation predicts that adsorption continues to increase with pressure indefinitely, which is physically unrealistic at very high pressures — a known limitation. To find k and n experimentally, take logarithms: log(x/m) = log k + (1/n)·log p. This is the equation of a straight line: plot log(x/m) vs log p; the slope gives 1/n and the y-intercept gives log k. For solution-phase adsorption, replace pressure p with equilibrium concentration c.
The Langmuir Adsorption Isotherm (Advanced)
Irving Langmuir (1916) derived his isotherm from a kinetic argument. At equilibrium, the rate of adsorption equals the rate of desorption. Let θ = fraction of surface sites occupied (0 ≤ θ ≤ 1). Rate of adsorption ∝ p·(1−θ) (pressure times fraction of vacant sites). Rate of desorption ∝ θ. Setting them equal: k_a·p·(1−θ) = k_d·θ, where k_a and k_d are the adsorption and desorption rate constants. Define the Langmuir constant K_L = k_a/k_d (units: Pa⁻¹). Solving for θ: θ = K_L·p / (1 + K_L·p). Since x/m = (x/m)_max · θ, the Langmuir equation becomes: x/m = (x/m)_max · K_L·p / (1 + K_L·p). At low pressures (K_L·p ≪ 1): x/m ≈ (x/m)_max · K_L·p — linear behaviour. At high pressures (K_L·p ≫ 1): x/m ≈ (x/m)_max — saturation (monolayer coverage). To linearise, take reciprocals: p/(x/m) = p/(x/m)_max + 1/(K_L·(x/m)_max). Plotting p/(x/m) vs p gives a straight line with slope 1/(x/m)_max and intercept 1/(K_L·(x/m)_max).
Comparison and Limitations
Freundlich: empirical, works well for heterogeneous surfaces and moderate pressure ranges, no saturation limit (physically unrealistic at very high p). Langmuir: theoretical, predicts saturation (monolayer), assumes uniform sites and no lateral interactions between adsorbate molecules. Real surfaces are rarely perfectly uniform — zeolites, activated charcoal, and metal catalysts all have heterogeneous binding sites. The BET (Brunauer–Emmett–Teller) isotherm extends Langmuir to multilayer adsorption and is the standard method for measuring specific surface area (m² g⁻¹) of porous materials used in industry.
The Key Equation
- x — Mass of adsorbate (g): Total mass of gas or solute adsorbed on the surface
- m — Mass of adsorbent (g): Mass of solid adsorbent used (e.g. activated charcoal, silica gel)
- k — Freundlich capacity constant (g g⁻¹ Pa⁻¹ᐟⁿ): Empirical constant reflecting adsorption capacity; increases with stronger adsorbent–adsorbate affinity
- p — Equilibrium pressure (Pa or atm): Pressure of the gas phase at equilibrium with the adsorbed layer
- 1/n — Heterogeneity factor (dimensionless): Lies between 0 and 1; closer to 0 means greater surface heterogeneity; when 1/n = 1 the isotherm is linear
Worked Example
Problem. In an experiment, 0.80 g of activated charcoal adsorbs 40 mg of acetic acid from solution when the equilibrium concentration is 0.10 mol L⁻¹, and adsorbs 80 mg when the equilibrium concentration is 0.40 mol L⁻¹. (a) Use the Freundlich isotherm to find 1/n and k. (b) Predict the adsorption at c = 0.25 mol L⁻¹.
Answer. 1/n = 0.50, k = 158 mg g⁻¹ (mol L⁻¹)⁻⁰·⁵; adsorption at c = 0.25 mol L⁻¹ is 79 mg g⁻¹.
Origins · 1 minIrving Langmuir and how this was discovered
Irving Langmuir derived the theoretical adsorption isotherm bearing his name in 1916 while working at General Electric Research Laboratory. By applying the principle of dynamic equilibrium to adsorption and desorption rates on a homogeneous surface, he created one of the first quantitative molecular models of surface phenomena. His work explained catalysis at metal surfaces and earned him the 1932 Nobel Prize in Chemistry — the first Nobel awarded to an industrial chemist. Langmuir was also a pioneer of cloud seeding (artificial rainfall). He demonstrated that spraying dry ice (solid CO₂) into supercooled clouds could trigger ice crystal formation and precipitation — an early form of weather modification that he tested dramatically over New York State in the late 1940s.
Related Concepts
- Adsorption vs Absorption — Understanding the distinction between surface accumulation (adsorption) and bulk uptake (absorption) is the conceptual foundation before studying isotherms.
- Factors Affecting Adsorption — Temperature, pressure, surface area, and adsorbate nature all shift the isotherm curve; knowing these factors contextualises why k and n change between systems.
- Heterogeneous Catalysis — The Langmuir isotherm directly underpins the Langmuir–Hinshelwood mechanism for reactions on catalyst surfaces, such as the Haber process and catalytic converters.
- Le Chatelier's Principle — Adsorption equilibria obey Le Chatelier's principle: increasing pressure shifts equilibrium toward greater surface coverage, explaining the rising portion of every isotherm.
- Enthalpy of Adsorption — The value of K_L in the Langmuir equation is temperature-dependent via a van't Hoff relationship; plotting ln K_L vs 1/T yields the enthalpy of adsorption.