Adsorption and Isotherms

01 / UNDERSTAND & APPLY

1. A surface has a finite population of places to bind

Adsorption is enrichment of a substance at an interface. The adsorbate is the species held there, and the adsorbent is the material providing the interface. Gas molecules on a solid and dissolved molecules associated with a solid surface are common examples. Absorption instead refers to uptake into a bulk phase. Both can occur in the same material, so an observed loss from a solution does not by itself identify the mechanism.

An interface need not be the visible outside of a particle. A porous solid may have extensive internal pore walls, which are still surfaces. Crushing or preparing a porous solid can increase accessible surface area per gram, but a molecule too large to enter a pore cannot use all of that nominal area. Surface chemistry, pore size, diffusion and solvent competition can matter as much as geometric area.

Let θ be the fraction of a specified set of adsorption sites occupied. For a simple one-molecule-per-site monolayer model, θ ranges from zero to one. If q is adsorbed amount per mass of solid and q_max is its monolayer capacity under that model, q=q_maxθ. The units of q might be mmol g⁻¹, whereas θ is dimensionless.

An adsorption isotherm relates equilibrium uptake to pressure or dissolved concentration at fixed temperature. It does not describe how fast equilibrium is reached. A time-dependent uptake curve and an equilibrium isotherm answer different questions, even if they use the same sample. Always say which measurement is being interpreted.

Sorption data require a material balance. If a closed solution loses solute from concentration C₀ to C_e, a common ideal balance is q=(C₀−C_e)V_solution/m_solid. That assumes the volume stays effectively constant and no reaction, precipitation, evaporation or other loss removes the solute. The measured concentration difference alone cannot establish adsorption without those controls.

02 / UNDERSTAND & APPLY

2. Classify bonding without turning trends into absolute rules

Physisorption is binding dominated by intermolecular interactions without formation of a new strong chemical bond to the surface. Chemisorption involves substantial electronic rearrangement and chemical bonding between adsorbate and substrate. The distinction is based on bonding, not on a universal numerical heat cutoff or the mere fact that adsorption happens on a metal.

Physisorption is often less specific and more readily reversible, and it can form multiple layers. Chemisorption is typically more site- and substance-specific; its directly chemically bound layer has a finite capacity. Physical adsorption can still occur above a chemisorbed layer. Strong binding does not make desorption logically impossible, and weak binding does not remove every kinetic obstacle in a porous material.

Gas adsorption is commonly exothermic: binding lowers enthalpy. Immobilizing a gas molecule also usually lowers its translational freedom, giving a negative adsorption entropy. For favourable adsorption at specified conditions, the relevant free-energy balance must favour the adsorbed state. In solution, release or rearrangement of solvent can complicate the entropy and enthalpy picture, so “all adsorption has negative entropy” is too broad.

For exothermic adsorption, raising temperature usually lowers equilibrium binding strength at fixed pressure. Chemisorption may nevertheless be observed to increase initially with temperature during a finite-time experiment because crossing an adsorption activation barrier becomes faster. An uptake increase before equilibrium is reached does not prove that the equilibrium adsorption process is endothermic.

Pressure can increase gas uptake by supplying a larger chemical potential, until the relevant capacity is approached or another adsorption regime occurs. The effect is not unlimited in a single-layer site model. Dissolved solutes instead require their solution activity or a suitable concentration approximation, and the solvent may compete for the same surface.

03 / UNDERSTAND & APPLY

3. Derive molecular Langmuir adsorption from available sites

Write A(g)+*⇌A*, where * denotes a vacant site and A* an occupied site. For the simplest Langmuir model, all sites are equivalent, each molecule occupies one site, only one layer is counted, and lateral interactions do not make the binding constant depend on coverage. These assumptions are the content of the model, not optional details.

At fixed temperature, use r_ads=k_ads P(1−θ) and r_des=k_des θ. Adsorption requires both an arriving gas molecule and a vacancy; desorption requires an occupied site. At equilibrium the forward and reverse rates are equal, so k_ads P(1−θ)=k_des θ. Define K=k_ads/k_des in this pressure convention and solve θ=KP/(1+KP).

In this elementary expression K has inverse-pressure units so that KP is dimensionless. A rigorous thermodynamic equilibrium constant uses dimensionless gas activity P/P° under ideal-gas conditions and is dimensionless. The two conventions describe the same algebra after including the chosen standard pressure; do not compare numerical constants expressed in bar⁻¹ and Pa⁻¹ without conversion.

The half-coverage pressure follows immediately: θ=1/2 implies KP=1. Therefore P_half=1/K. A larger K at the same temperature corresponds to stronger binding and a smaller pressure needed for half occupancy. It does not necessarily imply a larger q_max, because binding strength and total site count are different parameters.

At KP≪1, θ≈KP, so uptake is approximately linear in pressure. At KP≫1, θ approaches one and uptake approaches q_max. The approach is asymptotic: finite pressure does not produce θ exactly one in the reversible model. Checking these two limits is a good way to catch an inverted fraction.

04 / UNDERSTAND & APPLY

4. Fit a model only over a range where its assumptions make sense

Rearranging the Langmuir equation gives P/q=1/(Kq_max)+P/q_max. A plot of P/q against P is linear for exact data obeying the model, with slope 1/q_max and intercept 1/(Kq_max). Thus q_max is the reciprocal slope and K is slope divided by intercept. Track units through each reciprocal rather than reporting a bare number.

Linearization can distort experimental error because the measured uptake appears in a denominator. For actual data analysis, fitting q directly with an appropriate uncertainty model is often preferable. A visually straight transformed plot is supporting evidence, not proof that every microscopic site is identical.

The empirical Freundlich relation is q=k P^(1/n) for a gas pressure convention, or q=k C_e^(1/n) for a solution concentration convention, over a stated range at fixed temperature. Taking logarithms gives log q=log k+(1/n)log P. The slope is the exponent 1/n. The units of k depend on the units chosen for q and P and on the exponent.

A sublinear exponent between zero and one can describe decreasing incremental uptake on a heterogeneous surface over a useful interval. The formula has no finite saturation capacity as pressure grows without bound. Therefore it should not be extended to arbitrarily high pressures when a monolayer or pore capacity is known to exist.

A logarithm formally acts on a dimensionless ratio to a reference unit. In classroom straight-line plots, the numerical values in explicitly fixed units are often used. Changing those units changes the intercept and the numerical k, while the exponent remains unchanged. Comparing intercepts from inconsistent units is not a chemical conclusion.

05 / UNDERSTAND & APPLY

5. Change the site balance when the chemistry changes

If A₂ adsorbs dissociatively as two A atoms on two sites, use A₂(g)+2*⇌2A*. In a simple mean-field equilibrium model, K P=θ²/(1−θ)², giving θ=√(KP)/(1+√(KP)). The square root arises because one gas molecule corresponds to two occupied sites; it is not an arbitrary modification of the molecular formula.

This expression assumes equivalent sites and the simple treatment of vacancy pairs. Real surfaces can impose neighbouring-site geometry or interactions that alter the kinetics and equilibrium model. State which idealized site counting is being used. At low coverage the elementary model gives θ proportional to √P rather than P.

For two molecular species A and B competing for the same one-site pool, let θ_*+θ_A+θ_B=1. The equilibrium relations are θ_A=K_A P_A θ_* and θ_B=K_B P_B θ_*. Solving yields θ_*=1/(1+K_A P_A+K_B P_B), with each adsorbed fraction given by its corresponding numerator over that denominator.

Increasing a strongly bound spectator’s pressure can lower reactant coverage even when reactant pressure is unchanged. This provides a simple model for competitive blocking. It does not require the spectator to be converted chemically. Conversely, separate noncompeting site families need a different balance; one shared denominator is then not automatically appropriate.

In a closed vessel, gas pressure can fall as adsorption proceeds. The equilibrium pressure in an isotherm is the final pressure, not the initial pressure before uptake. Combine the isotherm with total material conservation when the adsorbed amount is not negligible relative to the free gas inventory.

06 / UNDERSTAND & APPLY

6. Translate molecular coverage into an area with explicit assumptions

If a measured monolayer contains n_m moles of adsorbate and each adsorbed molecule occupies an assumed area σ, the estimated surface area is S=n_m N_A σ. Dividing by the adsorbent mass gives specific area. This is a geometric counting model: it assumes the chosen probe forms the stated monolayer and its molecular footprint is applicable.

Convert square nanometres carefully: 1 nm²=10⁻¹⁸ m². A millimole contains about 6.022×10²⁰ molecules, so a molecular footprint of a fraction of a square nanometre can produce tens or hundreds of square metres per gram. Such large areas can arise from pore walls rather than an implausibly large external particle face.

BET analysis extends adsorption modelling to multilayer uptake and is widely used in surface-area characterization. Its fitting range and assumptions matter; a BET area is not obtained simply by calling every high-pressure adsorption value a monolayer. Pore filling and capillary condensation can introduce additional structure into real isotherms.

An adsorbent can have a high measured area but a low useful catalytic rate if the active chemical sites are sparse, inaccessible, poisoned or incorrectly structured. Surface area is one material descriptor, not a complete measure of catalytic performance. Likewise, very strong adsorption can trap products and obstruct repeated turnover.

07 / UNDERSTAND & APPLY

7. A catalyst changes the route and rate, while equilibrium thermodynamics stays fixed

A catalyst participates in reaction steps and is regenerated in the net cycle. It provides an alternative pathway with more favourable kinetic barriers under the operating conditions. Saying that it is regenerated is more accurate than saying it never reacts. Real catalysts can nevertheless lose activity through poisoning, decomposition, fouling or structural change.

A catalyst does not change the free-energy difference between the same reactant and product states, the reaction enthalpy, or the equilibrium constant at a fixed temperature. It accelerates approach to equilibrium by affecting compatible forward and reverse pathways. If the uncatalysed reaction is thermodynamically uphill under the conditions, a catalyst alone does not make the equilibrium favour products.

An energy profile for a catalysed mechanism can contain several intermediates and transition states rather than one smaller hump. The relevant overall rate can depend on surface coverages, concentration, catalyst resting state and the highest effective free-energy cost from that resting state. Therefore a single universal “activation energy reduction” is often an oversimplification, even though it is a useful introductory picture.

Homogeneous catalysis uses catalyst and reactants in the same phase. Acid-catalysed ester hydrolysis, for example, uses proton transfers and carbonyl activation within a solution-phase mechanism, with the acid regenerated. Heterogeneous catalysis uses an interface between phases, such as gaseous reactants reacting on a solid metal or oxide.

Catalytic selectivity means favouring a particular product pathway among competing possibilities. It does not mean changing the equilibrium constant of an otherwise unchanged overall reaction. Shape-selective porous catalysts can restrict access or transition-state geometry; a molecule may be excluded from a pore even if its adsorption on an exposed surface would be favourable.

A promoter improves a catalyst’s activity, stability or selectivity through structural or electronic effects without necessarily serving as the main catalytic component. A poison suppresses activity, often by strongly binding to active sites; not every inhibitor is irreversible. An autocatalytic reaction has a product that itself accelerates the reaction, so rate can rise as product accumulates before depletion eventually matters.

08 / UNDERSTAND & APPLY

8. Adsorption, surface reaction and desorption must all work together

A simplified heterogeneous cycle contains transport to the surface, adsorption, reaction of bound species, product desorption and transport away. If external transport or pore diffusion is limiting, the observed rate need not report the intrinsic chemical step. Making catalyst particles smaller can change measured rate by improving transport as well as changing exposed area.

In a Langmuir–Hinshelwood description, both reacting species are adsorbed before the surface reaction. In an Eley–Rideal description, a gas- or solution-phase reactant reacts directly with a surface-bound species. These labels describe mechanistic possibilities; a rate law consistent with one is not always unique proof of that mechanism.

For an illustrative one-reactant model with rapid molecular adsorption equilibrium and a slow unimolecular conversion A*→products+*, the rate per site is r=kθ_A=kKP/(1+KP). It is first order in pressure at low KP and approaches zero order at high KP. Saturation can therefore change the apparent reaction order without changing the written slow elementary step.

For two adsorbed reactants sharing the same site pool, a simple mean-field reaction rate may be proportional to θ_Aθ_B. A high pressure of one reactant can then suppress the other’s coverage and reduce the rate. The most strongly adsorbing surface is not automatically the fastest catalyst: binding must be sufficient for activation but weak enough to permit turnover and product release.

Industrial examples illustrate this balance rather than a universal recipe. Iron-based catalysts are associated with ammonia synthesis, vanadium-oxide systems with sulfur dioxide oxidation, and metals such as nickel, palladium or platinum with hydrogenation in suitable contexts. Their effective formulations, promoters and operating windows differ. A catalyst name alone does not specify all reaction conditions or guarantee selectivity.

09 / UNDERSTAND & APPLY

9. Enzyme catalysis adds binding structure and biological constraints

Enzymes are biological catalysts, commonly proteins, with active-site environments that bind substrates and stabilize catalytic configurations. Some require cofactors or metal ions. Specificity can arise from shape, charge, hydrogen bonding and the positioning of reactive groups. The active site is not necessarily a rigid lock; conformational adjustment and dynamics can contribute to catalysis.

A basic model is E+S⇌ES→E+P. Under initial-rate conditions with negligible product reaction and the usual steady-state assumptions, v=V_max[S]/(K_M+[S]). At low substrate concentration the rate is approximately proportional to [S]; at high concentration it approaches V_max. This saturation resembles a Langmuir curve algebraically, but K_M is not generally a simple equilibrium dissociation constant.

For this model K_M=(k_−1+k_cat)/k_1, while the ES dissociation constant is k_−1/k_1. They become close only under additional conditions such as k_cat being small compared with k_−1. V_max=k_cat[E]_total, so doubling active enzyme concentration doubles V_max if the other assumptions remain valid.

Temperature can accelerate chemical steps but can also destabilize enzyme structure; pH affects active-site ionization and substrate binding. A measured optimum is specific to the enzyme, medium and timescale. “All enzymes work best at one fixed temperature or pH” is not a valid general rule.

A simple competitive inhibitor raises the apparent K_M without changing V_max in the elementary reversible competitive model, because sufficiently high substrate concentration can outcompete it. Other inhibition patterns have different parameter effects. Denaturation, irreversible inactivation and loss of a required cofactor should not be forced into that simple competition formula.

10 / UNDERSTAND & APPLY

10. Solved practice and a model-choice checklist

Before calculating, identify what is being counted: adsorbed molecules, occupied sites, adsorbent mass, catalyst active sites or enzyme molecules. Next check whether the supplied constant belongs to a pressure-based, concentration-based or dimensionless activity convention. Finally state whether equilibrium or a rate model is being used.

The original exercises below deliberately separate site count, binding strength and reaction rate. A good answer includes a domain check: coverages must lie between zero and one, competitive fractions must sum to one, and a one-layer capacity must not be exceeded by the model.