Freundlich and Temkin Isotherms

Charcoal pulled from a poison-gas mask does not adsorb like a textbook Langmuir surface. Real activated carbon has crevices, edges, and flat terraces, each binding a gas molecule with different strength. The Langmuir isotherm assumes every site is identical, but most real surfaces are anything but uniform. The Freundlich and Temkin isotherms were built to describe exactly this messiness. They are empirical and semi-empirical equations that let chemists fit adsorption data on heterogeneous surfaces. They connect coverage to pressure or concentration without demanding the fiction of a single, uniform binding energy. This article derives both isotherms from a heterogeneous-surface picture. You will see how site-energy distributions generate the Freundlich power law. You will see how a linearly decaying heat of adsorption generates the Temkin logarithm. Two worked examples show how to extract real parameters from real-looking data, and the article closes by pointing toward how isotherm data feed into surface-area determination.

Key takeaways

The Langmuir isotherm assumes uniform sites; the Freundlich and Temkin isotherms drop that assumption.
Freundlich: $\theta = K_F p^{1/n}$ , a power law with no saturation plateau built in.
Temkin: coverage rises logarithmically with pressure because adsorption enthalpy falls linearly with $\theta$ .
Both models emerge naturally from assuming an exponential or linear spread of site energies.
Freundlich fits low-to-moderate coverage on strongly heterogeneous surfaces; Temkin suits chemisorption with strong lateral repulsion.
Key idea: on a real, energetically heterogeneous surface, molecules fill the strongest sites first, and the shape of that site-energy distribution dictates whether the isotherm looks like a power law or a logarithm.

The core insight

Both isotherms are what you get when you average the simple Langmuir isotherm over a whole population of adsorption sites with different binding energies: an exponential spread of energies integrates to the Freundlich power law, while a uniform (linear) spread of energies integrates to the Temkin logarithm — heterogeneity, not new physics, is the common ancestor of both equations.

The physical picture

The Langmuir isotherm treats a surface as a checkerboard of identical, independent boxes, each either empty or occupied by one molecule. That picture works well for a single crystal face or a well-defined chemisorption system. It fails for oxide powders, activated carbons, and clays, where pores, defects, and crystallite edges each bind an adsorbate with a different enthalpy. Imagine dosing gas onto such a surface. The strongest sites — deep pore walls, coordinatively unsaturated metal edges — fill first, because they have the largest (most negative) adsorption enthalpy $\Delta H_{ads}$ . As pressure rises, progressively weaker sites start to fill. The apparent isotherm you measure is a weighted sum (technically an integral) of many local Langmuir isotherms, one for each site energy, weighted by how many sites exist at that energy. Two limiting shapes of that site-energy distribution give the two named isotherms. If the number of sites falls off exponentially as the adsorption energy weakens, the sum integrates into a power law in pressure — the Freundlich isotherm. If instead the sites are spread with uniform density across a finite energy range — meaning equal numbers of sites at every enthalpy value in that range — the enthalpy of adsorption itself decreases linearly with coverage, and the sum integrates into a logarithm of pressure — the Temkin isotherm. This linear decay is physically sensible: as good sites disappear, newly adsorbed molecules also repel each other electrostatically, further weakening the effective binding energy.

Figure

Site-energy distributions behind the two isotherms

Once heterogeneity is granted, the second key physical fact is coverage-dependent enthalpy. On a Temkin surface, the heat of adsorption decreases linearly with $\theta$ :

\Delta H_{ads}(\theta) = \Delta H_0 (1 - \alpha\theta)

Here $\Delta H_0$ is the (negative) enthalpy at zero coverage and $\alpha$ is a dimensionless constant describing how fast binding weakens — whether from progressively poorer sites, from repulsive interactions between adsorbed molecules, or both. This coverage-dependent enthalpy is the microscopic seed from which the Temkin logarithm grows.

Figure

Shapes of the three isotherms at fixed temperature

The core relations

\theta = K_F\, p^{1/n} \qquad (n > 1)

where $\theta$ is fractional coverage, $p$ is equilibrium pressure, $K_F$ is the Freundlich constant (units of pressure $^{-1/n}$ ), and $1/n$ is the heterogeneity exponent, typically $0 < 1/n < 1$ .

\ln\theta = \ln K_F + \frac{1}{n}\ln p

where this linearized form lets $K_F$ and $n$ be read off from a plot of $\ln\theta$ against $\ln p$ .

\theta = \frac{RT}{\Delta H_{max}}\ln\left(A_0\, p\right)

where $\Delta H_{max} = \alpha|\Delta H_0|$ is the total spread of adsorption enthalpy across the surface, and $A_0$ is a constant combining the pre-exponential factor and $\Delta H_0$ .

\theta = c_1 \ln p + c_2, \qquad c_1 = \frac{RT}{\Delta H_{max}}

where $c_1$ and $c_2$ are the slope and intercept obtained by plotting $\theta$ against $\ln p$ directly — the standard way Temkin data are analyzed.

Derivation AFrom a heterogeneous surface to the Freundlich and Temkin laws

Goal. Show that averaging the Langmuir isotherm over a distribution of site energies yields the Freundlich power law for an exponential distribution and the Temkin logarithm for a uniform distribution.

Step 1. Start from the local Langmuir isotherm. For a patch of sites all sharing one adsorption energy $\varepsilon$ , the local coverage obeys

\theta(\varepsilon,p) = \frac{K(\varepsilon)\,p}{1+K(\varepsilon)\,p}, \qquad K(\varepsilon)=K_0 e^{-\varepsilon/RT}

where $K(\varepsilon)$ is the local equilibrium (Langmuir) constant, growing exponentially as the binding energy $\varepsilon$ becomes more negative (stronger binding), by the van 't Hoff-type relation between $K$ and adsorption enthalpy.

Step 2. Introduce a distribution of site energies. Let $f(\varepsilon)\,d\varepsilon$ be the fraction of surface sites with energy between $\varepsilon$ and $\varepsilon+d\varepsilon$ , normalized so $\int f(\varepsilon)\,d\varepsilon = 1$ . The observed (average) coverage is the integral

\theta(p) = \int_{-\infty}^{0} \theta(\varepsilon,p)\, f(\varepsilon)\, d\varepsilon

Step 3. Choose an exponential distribution and recover Freundlich. Take $f(\varepsilon) = (1/\varepsilon_0)\,e^{\varepsilon/\varepsilon_0}$ for $\varepsilon<0$ , meaning strongly binding sites are exponentially rarer. This is the Sips/Zeldowitsch condensation approximation: replace the sharp Langmuir step $\theta(\varepsilon,p)\approx 1$ for $K(\varepsilon)p>1$ and $\theta\approx0$ otherwise. The integral over this step function, weighted by the exponential $f(\varepsilon)$ , evaluates to

\theta(p) \propto p^{RT/\varepsilon_0}

Identifying $1/n = RT/\varepsilon_0$ and absorbing constants into $K_F$ reproduces

\theta = K_F\, p^{1/n}

which is exactly the Freundlich isotherm — a direct consequence of exponentially decaying site strength.

Step 4. Choose a uniform distribution and recover Temkin. Now take $f(\varepsilon) = 1/\Delta H_{max}$ , constant across a finite window of width $\Delta H_{max}$ , and zero outside it. Using the same condensation approximation (each site is either full or empty depending on whether $K(\varepsilon)p$ exceeds 1), the coverage equals the fraction of the energy window for which sites have already condensed:

\theta(p) = \frac{RT}{\Delta H_{max}}\ln\left(K_0\, p\right) + \text{const}

Step 5. Collect constants. Writing $A_0 = K_0\,e^{\text{const}\cdot \Delta H_{max}/RT}$ absorbs the additive constant into the argument of the logarithm, giving the compact form

\boxed{\theta = \frac{RT}{\Delta H_{max}}\ln(A_0\, p)}

This is the Temkin isotherm: coverage grows logarithmically with pressure because the marginal site being filled has an enthalpy that decreases linearly with $\theta$ , and the logarithm is the mathematical fingerprint of integrating a linearly-varying exponent.

Worked examples

Example 1 — Extracting Freundlich parameters from adsorption data. Suppose acetic acid adsorbs from aqueous solution onto activated carbon, and the equilibrium concentration $c$ (mol dm⁻³) and the mass adsorbed per gram of carbon $x/m$ (mmol g⁻¹) are measured:

$c$ / mol dm⁻³	0.01	0.05	0.10	0.50
$x/m$ / mmol g⁻¹	0.44	0.88	1.15	2.00

The Freundlich form for adsorption from solution replaces $p$ with $c$ :

\frac{x}{m} = K_F\, c^{1/n}

Take logarithms of both columns (using $\ln$ ):

$\ln c$	−4.605	−2.996	−2.303	−0.693
$\ln(x/m)$	−0.821	−0.128	0.140	0.693

A linear least-squares fit of $\ln(x/m)$ against $\ln c$ gives, working from the two extreme points as a check,

\text{slope} = \frac{0.693-(-0.821)}{-0.693-(-4.605)} = \frac{1.514}{3.912} = 0.387

A full four-point regression gives slope $1/n = 0.39$ and intercept $\ln K_F = 0.99$ , so

K_F = e^{0.99} = 2.7 \ \text{mmol g}^{-1}\,(\text{mol dm}^{-3})^{-0.39}

Sanity check. At $c = 0.10$ mol dm⁻³, the fitted line predicts

x/m = 2.7 \times (0.10)^{0.39} = 2.7 \times 0.407 = 1.10 \ \text{mmol g}^{-1}

close to the measured 1.15 mmol g⁻¹, confirming the fit is reasonable. Because $1/n < 1$ , the isotherm rises less steeply than linearly with concentration — consistent with progressively weaker sites being occupied, exactly as the derivation predicts. Note there is no saturation plateau in this model; extrapolating to very high $c$ would (unphysically) predict unbounded uptake, a known limitation of the Freundlich law addressed further below.

Example 2 — Temkin analysis of ammonia synthesis catalyst data. Nitrogen chemisorption on an iron catalyst (relevant to the Haber process) is a classic Temkin system, because dissociative N₂ adsorption on iron shows a heat of adsorption falling roughly linearly with coverage. Suppose at $T = 400\,\text{K}$ the fractional coverage of adsorbed nitrogen atoms is measured as a function of $\text{N}_2$ pressure:

$p$ / Pa	10	100	1000	10000
$\theta$	0.18	0.35	0.52	0.69

Plot $\theta$ against $\ln p$ :

$\ln p$	2.303	4.605	6.908	9.210
$\theta$	0.18	0.35	0.52	0.69

The data fall on a good straight line. Using the two extreme points,

c_1 = \text{slope} = \frac{0.69-0.18}{9.210-2.303} = \frac{0.51}{6.907} = 0.0738

c_2 = \text{intercept} = 0.18 - (0.0738)(2.303) = 0.18-0.170=0.010

Since $c_1 = RT/\Delta H_{max}$ , solve for the enthalpy spread:

\Delta H_{max} = \frac{RT}{c_1} = \frac{(8.314\ \text{J K}^{-1}\text{mol}^{-1})(400\ \text{K})}{0.0738}

\Delta H_{max} = \frac{3325.6\ \text{J mol}^{-1}}{0.0738} = 45{,}060\ \text{J mol}^{-1} \approx 45.1\ \text{kJ mol}^{-1}

This says the heat of adsorption falls by about 45 kJ mol⁻¹ across the full range of surface coverage — a substantial heterogeneity, typical of dissociative chemisorption on a polycrystalline metal with many distinct facet and step-edge sites. From the intercept, $A_0$ can also be extracted:

c_2 = c_1 \ln A_0 \implies \ln A_0 = \frac{0.010}{0.0738}=0.135 \implies A_0 = 1.14\ \text{Pa}^{-1}

Sanity check. At $p=1000$ Pa, the fitted line predicts

\theta = 0.0738\ln(1.14\times1000) = 0.0738\times6.937 = 0.512

matching the measured 0.52 well within experimental scatter. As expected for chemisorption on a heterogeneous metal surface, coverage rises steeply at low pressure (strongest sites fill fast) and much more slowly at high pressure — the logarithmic hallmark of the Temkin isotherm.

Limiting cases & common traps

Situation	Behavior	Trap to avoid
Low coverage, near-ideal gas limit	Freundlich reduces to $\theta\propto p^{1/n}$ , never linear in $p$ unless $n=1$	Do not force Henry's-law linearity onto Freundlich fits; $n\ne1$ is the whole point
High coverage / high pressure	Freundlich predicts unbounded $\theta$ ; physically wrong beyond monolayer	Never extrapolate Freundlich far past the fitted pressure range
$\theta$ very close to 0 or 1	Temkin's logarithm diverges (undefined at $\theta=0$ , breaks down near $\theta=1$ )	Temkin is a mid-coverage approximation, not valid at the extremes
$n=1$ in Freundlich	Recovers Henry's law / linear partitioning	A fitted $n\approx1$ signals near-uniform, weakly interacting sites — check if Langmuir fits better
Comparing $K_F$ across data sets	$K_F$ has fractional units depending on $n$ ; not directly comparable between systems with different $n$	Always quote $n$ alongside $K_F$ ; never compare $K_F$ values alone

The chemist behind it

Herbert Freundlich, a German-Baltic physical chemist working in colloid science in the early twentieth century, proposed the power-law relationship in 1906 while systematically fitting adsorption data that Langmuir's later (1918) uniform-site theory could not yet explain. Freundlich's equation predates a mechanistic justification by more than a decade; it was pure curve-fitting empiricism that turned out, once statistical thermodynamics matured, to correspond exactly to an exponential distribution of adsorption energies. Mikhail Temkin, a Soviet physical chemist, derived his isotherm in the 1930s specifically to describe nitrogen chemisorption on iron catalysts used in ammonia synthesis — the reaction that, thanks to the Haber–Bosch process, feeds a large fraction of humanity through synthetic fertilizer.

Origins · 1 minHerbert Freundlich and how this was discovered

Herbert Freundlich

1880–1941 · German-Baltic colloid chemist who systematized empirical adsorption isotherms decades before their statistical-mechanical basis was understood

Ask the tutor: How does the BET isotherm extend these ideas to handle multilayer adsorption and give a real surface area in square meters per gram?

Why don't Freundlich and Temkin isotherms reduce exactly to Langmuir at some limit

They can approximate Langmuir behavior in limiting cases, but they are built from a different starting assumption. Langmuir assumes one adsorption energy for every site. Freundlich and Temkin assume a whole distribution of energies. When that distribution collapses to a single value (zero spread), both reduce toward Langmuir-like behavior, but a Freundlich exponent of exactly $n=1$ or a Temkin $\Delta H_{max}\to0$ are the mathematical, not chemical, routes there. In practice, real heterogeneous surfaces rarely reach that idealized limit.

Can the Freundlich and Temkin isotherms be used for liquids as well as gases

Yes. Both were originally developed and are still routinely applied to adsorption from solution, replacing pressure $p$ with equilibrium solute concentration $c$ . This is common in environmental chemistry, for example fitting dye or heavy-metal ion adsorption onto activated carbon or clay minerals. The mathematics is identical; only the physical interpretation of the independent variable changes from gas pressure to solution concentration.

How do you decide which isotherm to fit to a data set

Plot the data in linearized form for each candidate: $\ln\theta$ vs $\ln p$ for Freundlich, $\theta$ vs $\ln p$ for Temkin, and $p/\theta$ vs $p$ for Langmuir. Whichever plot gives the straightest line over your measured range is the best empirical description for that range. Always report the correlation coefficient and the pressure window used, since real data sets often fit more than one model equally well over a limited range.

What comes after fitting a Freundlich or Temkin isotherm to my data

Once you have chosen an isotherm and its parameters, the next natural application is estimating the surface area of your adsorbent, typically via the BET extension of the Langmuir picture. Freundlich and Temkin fits are less commonly used for area determination directly, since they lack an explicit monolayer capacity term. They remain invaluable, though, for characterizing heterogeneity and for practical engineering calculations in catalysis and separations.