Freundlich and Temkin Isotherms

Introduction

Identical independent sites are the defining simplification of Langmuir adsorption, yet practical carbons, oxides and catalysts contain pores, edges and chemical groups of different energies. An isotherm can therefore rise more gradually than a single-site saturation curve. The Freundlich and Temkin descriptions capture two useful patterns without pretending that they are universally exact. Their parameters must be interpreted over the pressure or concentration range in which they were fitted.

Core explanation

The empirical Freundlich equation for gas loading is often written q = K F P^(1/n) with n > 1 in a common favourable-adsorption form. Taking logarithms gives ln q = ln K F + (1/n) ln P , after pressure is rendered dimensionless or its chosen units are incorporated into K F. A plot of ln q against ln P has slope 1/n. The model can represent a broad distribution of effective binding strengths, but it has no finite saturation limit: if extended to arbitrarily large P, q grows without bound. That is unphysical for a finite sample and is why the Freundlich form is normally a restricted-range empirical fit, not a complete adsorption theory.

The Temkin approach recognises that the incremental energy of adsorption may change as surface coverage increases. Under simplified assumptions, including a roughly linear change in adsorption energy with coverage, loading can have a logarithmic dependence such as q = B ln(K T P) across an intermediate domain. Constants B and K T depend on the precise Temkin convention. The formula should not be extrapolated to P approaching zero, where the logarithm becomes negative, nor past physical maximum coverage. Its usefulness is in describing a measured interval where adsorption heat varies approximately with loading.

Heterogeneous sites are not the only reason for non-Langmuir behaviour. Adsorbed molecules can attract or repel neighbours, solvent can compete in solution adsorption, or pore filling can produce a different shape. A Freundlich or Temkin fit alone therefore does not uniquely identify microscopic cause. Temperature-dependent measurements of differential heats, spectroscopy, porosity information and residual patterns can discriminate among explanations. A model should be judged by predictions outside the small fitting region when possible.

Step-by-step reasoning

First state whether pressure, activity or solution concentration is the independent variable and keep the convention consistent. For Freundlich, fit ln q versus ln P and obtain the exponent from the slope; calculate K F using the intercept only with the same pressure units. For Temkin, fit loading versus ln P over a specified positive-pressure range. Inspect residuals from both fits in the original loading units. Check whether either formula predicts negative uptake or unlimited uptake outside the intended range, and report that domain.

Visual explanation

Plot three possible uptake curves on q-versus-P axes. Langmuir bends toward a finite plateau; Freundlich continues rising as a power law; a Temkin relation is roughly linear against ln P over its useful interval. On a second graph draw ln q versus ln P: a straight Freundlich fit has slope 1/n. Shading a finite measurement window makes clear that different curves can be almost indistinguishable within a narrow range yet diverge elsewhere.

Real-world analogy

Think of a hotel with rooms of very different attractiveness. The best rooms fill first, then guests accept less attractive ones as demand rises. That resembles a heterogeneous surface. The analogy is only partial: real adsorption also includes interactions among guests, transport through corridors and sometimes entirely new layers of occupancy.

Real-world example

Activated carbon used to adsorb organic vapours contains a range of pore widths and surface environments. A simple power-law fit may describe loading over the moderate pressures tested in a laboratory. If engineers use the same equation at a much larger pressure, it may predict more material than the pore volume can hold. A bounded model or direct high-pressure data are then required for design.

Why?

Why might adsorption heat change with coverage? The first molecules often occupy the most favourable sites, so later molecules bind more weakly. Alternatively, attractive lateral interactions can make later uptake more favourable. Temkin-type behaviour condenses one assumed energy trend into a logarithmic equation; a measured change in heat does not by itself prove the specific linear-energy assumption.

Common misconception

"Freundlich has no plateau, so a real surface has infinite capacity" confuses an empirical fit with physics. All finite samples have finite material and space. Another misconception is to infer surface heterogeneity uniquely from a power-law slope. Multiple processes can produce similar curves, and parameter values may shift with measurement range.

Worked example

Question: A Freundlich fit has q = 0.40 P^0.5 when q is in mmol g⁻¹ and P is numerically entered in bar. Predict loading at 1.0 and 4.0 bar and state a limitation.

Reasoning: At 1.0 bar, q = 0.40(1)^0.5 = 0.40 mmol g⁻¹. At 4.0 bar, q = 0.40(4)^0.5 = 0.80 mmol g⁻¹. Doubling of loading follows because square root of four is two. The numerical coefficient embeds the bar convention. Continuing this expression to enormous pressure would give unlimited loading.

Answer: Predicted q values are 0.40 and 0.80 mmol g⁻¹; the power law should be used only within a validated pressure range.

Quick check

1. Why can a Freundlich equation not describe adsorption up to arbitrarily high pressure on a finite solid? Answer: Its power-law loading keeps increasing and has no saturation capacity.

Exam focus

Recognise the linear forms: ln q against ln P for Freundlich and q against ln P for a common Temkin convention. Keep pressure units in fitted constants. State that Freundlich is empirical and unbounded, while Temkin captures a coverage-dependent-energy pattern over a limited interval. Prefer residuals and independent physical evidence over a single high correlation coefficient.

Advanced insight

An apparent Freundlich power law can emerge from integrating ideal adsorption over a distribution of site affinities. However, extracting a unique energy distribution from one isotherm is an inverse problem: noise, pore geometry and adsorbate interactions can give similar curves. Heat-of-adsorption measurements at multiple loadings can add an independent constraint, but their interpretation depends on consistent excess or absolute loading definitions.

Summary

Freundlich describes power-law uptake and often fits heterogeneous surfaces over a finite range, but lacks a saturation limit. Temkin describes a logarithmic loading trend associated with changing adsorption energy under simplified assumptions. Neither fitted shape alone proves one microscopic mechanism. Units, pressure range, physical limits and residuals govern responsible use.

Practice questions

1. What is the slope of a ln q versus ln P plot for q = K FP^(1/n)? Answer: The slope is 1/n. 2. What happens to a Freundlich prediction as P grows without bound? Answer: Loading grows without a finite limit, which is unphysical for a finite sample. 3. Why should a Temkin logarithmic form not be used down to P = 0? Answer: ln P is undefined there and the model can predict negative loading at sufficiently low pressure. 4. Does a Temkin fit prove a particular surface energy distribution? Answer: No. It captures a trend; independent thermal and structural evidence is needed to identify the microscopic origin.

Primary research context: coverage-dependent adsorption heats and measured isosteric heats across filling.