Questions: Adsorption Isotherms: Langmuir and BET Models

Question 1

A catalyst surface is exposed to increasing reactant pressure. At low pressures, the reaction rate doubles when pressure doubles. At very high pressures, further increases in pressure have no effect on the rate. What explains this transition?

• A. At high pressures, the reactant molecules begin to repel each other on the surface, blocking adsorption
• B. The surface sites become fully occupied (θ → 1), so all available sites are already used and adding more gas cannot increase the rate
• C. High pressure causes desorption to dominate over adsorption, reversing the equilibrium
• D. The reaction shifts from chemisorption to physisorption at high pressure, which is slower

Answer: (B)

This is the hallmark of Langmuir saturation behavior. At low pressure (Kp ≪ 1), θ ≈ Kp and rate ∝ θ ∝ p — first-order in pressure. At high pressure (Kp ≫ 1), θ → 1 — essentially all sites are occupied and adding more gas finds no empty sites, so rate becomes independent of pressure (zero-order). The surface has a fixed number of sites; once saturated, it cannot adsorb more. This is the direct physical meaning of the denominator (1 + Kp) in the Langmuir equation.

Question 2

Which combination of assumptions is essential to the Langmuir adsorption model?

• A. Multilayer adsorption, heterogeneous surface sites, and strong adsorbate-adsorbate interactions
• B. Monolayer coverage only, equivalent and independent binding sites, and dynamic equilibrium between adsorption and desorption
• C. Physisorption only, uniform temperature across the surface, and irreversible binding
• D. Covalent bonding to the surface, sites with varying binding energies, and no desorption at equilibrium

Answer: (B)

The three core Langmuir assumptions are: (1) monolayer — each site holds at most one molecule, no stacking; (2) equivalent sites — every site has the same binding energy; and (3) dynamic equilibrium — adsorption and desorption occur at equal rates. Violating any of these sends you toward a different isotherm model. The BET model relaxes (1) to allow multilayers. The Freundlich isotherm is used for heterogeneous surfaces.

Question 3

At low gas pressure, the Langmuir isotherm predicts that fractional surface coverage θ increases approximately linearly with pressure.

• T. True
• F. False

Answer: True

At low pressure where Kp ≪ 1, the denominator (1 + Kp) ≈ 1, so θ = Kp/(1+Kp) ≈ Kp. Coverage is directly proportional to pressure — linear behavior. This is also called the Henry's law region of the isotherm. Physically, nearly all sites are empty, so every molecule that hits the surface finds an empty site, and coverage grows in direct proportion to the number of collisions (which is ∝ p).

Question 4

The BET model assumes adsorption is complete after one monolayer forms, at which point the isotherm levels off just like the Langmuir model.

• T. True
• F. False

Answer: False

The BET model explicitly allows multilayer adsorption — this is its whole purpose and what distinguishes it from Langmuir. Once the first monolayer forms, additional molecules can adsorb on top of it through weaker van der Waals forces, with the second and subsequent layers behaving like condensation of the bulk liquid. The BET isotherm therefore does not level off but continues to rise with pressure, eventually diverging near the saturation vapor pressure. The Langmuir model levels off; BET does not.

Question 5

The Langmuir adsorption equation θ = Kp/(1+Kp) is mathematically identical to the Michaelis-Menten enzyme kinetics equation v = Vmax[S]/(Km+[S]). What does this structural similarity reveal about the two systems?

Model answer: Both equations describe reversible, saturable binding to a fixed number of equivalent, independent sites at equilibrium. In both cases, the denominator (1+Kp or Km+[S]) arises from the competition between occupied and unoccupied sites; the numerator reflects occupancy. The similarity reveals that the mathematics of surface adsorption and enzyme-substrate binding are governed by the same equilibrium logic: a fixed number of sites, reversible occupancy, and saturation when all sites are filled. The equilibrium constant K (adsorption) and 1/Km (enzyme affinity) play analogous roles.

This mathematical equivalence is not a coincidence — it reflects the same underlying physics of bimolecular reversible binding. Henri and Michaelis-Menten derived their equation independently of Langmuir, but both derived it from the same equilibrium condition: rate of binding = rate of unbinding. Recognizing this connection helps you transfer intuition across domains: enzyme inhibition curves, receptor binding studies, and adsorption isotherms all follow the same hyperbolic saturation form.