The Freundlich Isotherm

What the Freundlich isotherm describes

An adsorption isotherm answers one question: at a fixed temperature, how much of a substance sticks to a surface as a function of how much of it is floating around in the gas or solution? Plot the amount adsorbed per gram of solid, q, against the equilibrium concentration or pressure left over, C, and you get a curve whose shape reveals what the surface is doing.

The Freundlich isotherm is the empirical curve that fits an enormous fraction of real-world data:

    q = K_F · C^(1/n)          with   0 < 1/n < 1

  q    = amount adsorbed per gram of adsorbent at equilibrium   (mg/g)
  C    = equilibrium concentration of adsorbate remaining       (mg/L, or partial pressure p)
  K_F  = Freundlich capacity constant  (how much sticks at C = 1)
  1/n  = heterogeneity / intensity exponent  (dimensionless, usually 0.2–0.6)

The signature feature is the fractional exponent. Because 1/n is less than one, coverage rises sublinearly: double the concentration and the loading goes up by a factor of 2^1/n < 2. The curve climbs steeply at first — the surface's best sites grab molecules greedily — then bends over and keeps rising more and more gently, but it never flattens into a plateau. That single missing plateau is what separates Freundlich from the Langmuir model, and it is the fingerprint of a surface whose sites are not all the same.

The mechanism: a surface with a spread of site energies

Langmuir's model imagines an idealized surface: every site is identical, holds exactly one molecule, and none of the bound molecules talk to each other. Real surfaces — activated carbon, soil minerals, a sintered metal catalyst — are nothing like that. They are rough, defect-riddled, and chemically varied. Some sites are deep energy wells (edges, corners, micropores, dangling bonds); most are shallow. Adsorption on such a surface follows an "electron-arrow" logic of energetics rather than of curly arrows:

1. The strongest sites fill first. At the lowest concentrations, molecules find and occupy the deepest wells — the high-binding-energy sites. These give the isotherm its steep initial slope. A little adsorbate produces a lot of loading.
2. Progressively weaker sites take over. As those best sites saturate, the next arriving molecules can only settle onto shallower, less favorable sites. Each increment of concentration now buys less loading — the curve bends over. This is the physical origin of the sublinear power law.
3. The distribution of site energies sets the exponent. If the number of sites falls off exponentially with increasing binding energy — many weak sites, exponentially fewer strong ones — then summing (integrating) a local Langmuir response over the whole energy distribution yields exactly q ∝ C^1/n. The heterogeneity parameter 1/n is literally the width of that exponential energy spread.
4. No global saturation. Because there is always some even-weaker site or a second adsorbed layer available, the surface never truly runs out, so the model shows no ceiling. That is a feature over the fitting range and a bug outside it.

So the Freundlich equation is not a lucky fit. It is what you get when you take Langmuir's clean single-site picture and smear the binding energy over an exponential distribution — the honest description of a messy surface.

Reading the data: the log–log plot

You almost never fit the power law directly. Take the base-10 logarithm of both sides:

    q = K_F · C^(1/n)

  take log₁₀ of both sides:

    log q = log K_F + (1/n) · log C
             └──┬──┘    └─┬─┘  └─┬─┘
            intercept   slope   x-axis

  Plot  log q  (y)  against  log C  (x):
      • a straight line confirms Freundlich behavior
      • slope        = 1/n            (the heterogeneity exponent)
      • y-intercept  = log K_F   →   K_F = 10^(intercept)

The straightness of the log–log line is the standard diagnostic. If your bench data plot as a line on log–log axes, Freundlich fits; if they curve toward a horizontal asymptote, you have a real saturation limit and should reach for Langmuir instead. This is why an environmental engineer's isotherm report always shows log q versus log C, not q versus C.

Worked example: dye removal on activated carbon

A common laboratory exercise: fit methylene-blue adsorption onto powdered activated carbon. Suppose a batch experiment at 25 °C gives these equilibrium points:

C (mg/L)	q (mg/g)	log C	log q
1.0	12.0	0.00	1.08
10.0	34.0	1.00	1.53
100.0	95.0	2.00	1.98

The log q values increase by ≈ 0.45 for every 1.0 increase in log C, so the slope 1/n ≈ 0.45, giving n ≈ 2.2 (favorable, moderately heterogeneous). The intercept at log C = 0 is log q ≈ 1.08, so K_F = 10^1.08 ≈ 12 mg/g per (mg/L)^1/n. The fitted isotherm is therefore:

    q = 12 · C^0.45      (mg/g)

  Sanity check at C = 100 mg/L:
    q = 12 · 100^0.45 = 12 · 7.94 ≈ 95 mg/g   ✓ matches the data

An engineer sizing a carbon column would use exactly these two constants. To knock a dye concentration down from C = 100 to C = 1 mg/L, the carbon's working capacity drops from 95 to 12 mg/g — a reminder that the last traces are the expensive ones, because the steep part of the isotherm lives at low C.

Freundlich vs Langmuir vs BET

	Freundlich	Langmuir	BET
Equation	q = KF·C1/n	q = qmax·KC/(1+KC)	multilayer extension of Langmuir
Nature	Empirical (derivable)	Mechanistic (kinetic)	Mechanistic (multilayer)
Surface	Heterogeneous, distributed energies	Uniform, identical sites	Uniform, stacked layers
Layers	Multilayer / effective	Single monolayer	Infinite multilayers
Saturation plateau	None — rises forever	Yes — q → qmax	Diverges near p/p₀ → 1
Low-C limit	Slope → ∞ (violates Henry's law)	Linear (obeys Henry's law)	Linear
Linearized as	log q vs log C	1/q vs 1/C or C/q vs C	1/[v(p₀/p−1)] vs p/p₀
Key constants	KF, 1/n	qmax, K	vm, c
Best used for	Activated carbon, soils, dyes	Chemisorption, clean single-crystal surfaces	Surface-area measurement (m²/g)

A useful mental model: Freundlich is Langmuir integrated over a distribution of surfaces. When the heterogeneity vanishes (all sites equal), the distribution collapses to a spike and Freundlich degenerates back toward simpler behavior. The Sips (Langmuir–Freundlich) equation, q = q_max(KC)^1/n/[1+(KC)^1/n], is the deliberate hybrid: it behaves like Freundlich at low C and like Langmuir (with a genuine plateau) at high C, curing both endpoint failures at the cost of a third parameter.

What the two constants mean physically

• K_F — the capacity constant. Numerically it is the loading q when C = 1 in whatever units you chose. A bigger K_F means a hungrier adsorbent overall. Its units are awkward — mg/g·(L/mg)^1/n — because they depend on 1/n, which is exactly why you should never compare K_F values measured with different n or different concentration units. In soil science the same constant is written K_F or K_d (the linear special case) and used to predict how far a pesticide migrates.
• 1/n — the intensity or heterogeneity exponent. Bounded between 0 and 1 for favorable adsorption. Smaller 1/n ⇒ steeper initial uptake and a more heterogeneous surface (a wider spread of site energies). As 1/n → 1 (n → 1) the surface becomes effectively uniform and the isotherm collapses to the linear partition law q = K_F·C. Values of 1/n > 1 signal cooperative, "unfavorable," S-shaped adsorption and are uncommon.
• The dimensionless separation factor. Some workers report R_L or use 1/n directly to classify the isotherm: 1/n between 0 and 1 is favorable, 1/n = 1 is linear, 1/n > 1 is unfavorable. This one-number verdict is why Freundlich fits are so popular in applied reports.

Real-world applications

• Drinking-water and wastewater treatment. Granular activated carbon (GAC) removal of trace organics — atrazine, geosmin, PFAS precursors, taste-and-odor compounds, textile dyes — is almost universally fitted with Freundlich constants. Design software (like the US EPA's adsorption models) takes K_F and 1/n from a bench isotherm and predicts breakthrough time for a full-scale column.
• Soil and pesticide fate. The Freundlich K_F (soil sorption coefficient) tells regulators how strongly a herbicide binds soil organic matter, and therefore how mobile it is toward groundwater. Registration dossiers for agrochemicals report Freundlich fits by regulatory requirement.
• Biochar and remediation. Heavy-metal (Pb²⁺, Cd²⁺, Cu²⁺) and dye uptake on biochar, chitosan, and low-cost biosorbents is reported with Freundlich parameters in thousands of papers per year, because these materials are maximally heterogeneous.
• Heterogeneous catalysis kinetics. When a catalyst surface is heterogeneous, the coverage of a reactant that feeds a rate law is better described by a Freundlich-type dependence than a Langmuir one, which changes the apparent reaction order with respect to pressure.
• Chromatography. Nonlinear, overloaded chromatographic peaks tail because the stationary-phase isotherm is Freundlich-shaped: the steep low-concentration part elutes faster than the shallow high-concentration part, producing the characteristic tailing band.

Limitations and where it breaks

• No high-concentration ceiling. The power law predicts loading rising without bound. Real surfaces saturate, so above the fitting range Freundlich over-predicts. Switch to Langmuir or Sips when a plateau is visible.
• Violates Henry's law at C → 0. Because 1/n < 1, the slope dq/dC = (K_F/n)·C^{(1/n − 1)} diverges to infinity as C → 0. Thermodynamics demands a finite linear (Henry) limit, so Freundlich is physically wrong at the very lowest concentrations — a real problem for ultra-trace contaminant modeling.
• K_F units are not portable. Because the units of K_F carry the fractional power 1/n, you cannot legitimately compare K_F across studies with different n or different units. Always report 1/n alongside.
• It is not truly temperature-independent. Both K_F and 1/n drift with temperature; an "isotherm" is only valid at the temperature it was measured. Fitting one set of constants across a temperature range is a common mistake.
• Purely descriptive. A good Freundlich fit tells you the surface is heterogeneous but does not identify the sites, the mechanism (physisorption vs chemisorption), or the adsorption enthalpy. It is an engineering interpolation, not a mechanistic proof.

Historical note: Herbert Freundlich, 1906

The relationship was published in 1906 by Herbert Max Finlay Freundlich (1880–1941), a German physical chemist who spent much of his career at the Kaiser Wilhelm Institute in Berlin before emigrating — first to London, then to the University of Minnesota — after the Nazi dismissals of 1933. He set out the power law q = k·C^1/n as a compact empirical summary of adsorption data for solutes on charcoal, building on an even earlier 1895 observation by Boedecker; the equation is sometimes therefore called the Freundlich–Boedecker isotherm.

Crucially, Freundlich's formula predates Irving Langmuir's mechanistic derivation of 1916–1918 by a decade. Langmuir's single-site kinetic model was the theoretical breakthrough (and won the 1932 Nobel Prize in Chemistry), yet Freundlich's older empirical curve often fits real, dirty surfaces better — a recurring lesson that a mechanistic model of an idealized system can be less useful than an empirical fit to the messy real one. The reconciliation came later, when it was shown that integrating Langmuir over an exponential site-energy distribution reproduces Freundlich exactly, giving the empirical law a firm statistical-mechanical footing.