1 · Data

no data

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Paste two columns: time and q_t Tab, comma, semicolon or space separated. A header row is detected automatically. Column 1 = time, column 2 = qt. Optional column 3 = error bars on qt.

Time unit q unit

2 · Experiment

Temperature (K) C₀ (mg L⁻¹) Adsorbent dose (g L⁻¹) Particle radius (cm) For Boyd and Crank diffusion coefficients.

Model advisor

Run the analysis to see which models your data can actually support.

3 · Models

all none

4 · Fitting options

Residual weighting Ranking criterion Also compute classical linearised fits for comparison

1 · Data

no data

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Paste two columns: C_e and q_e Column 1 = equilibrium concentration Ce, column 2 = equilibrium loading qe. Optional column 3 = error bars. Optional column 4 = initial concentration C0 (enables the Langmuir separation factor RL).

C unit q unit

2 · Experiment

Temperature (K) Molar mass (g mol⁻¹) Of the adsorbate, not the adsorbent. Only Dubinin–Radushkevich and the thermodynamics tab use it. Solubility C_s (mg L⁻¹) How much dissolves before the water saturates. Only the BET model uses it. Adsorbent dose (g L⁻¹)

Model advisor

Run the analysis to see which models your data can actually support.

3 · Models

all none

4 · Fitting options

Residual weighting Ranking criterion Also compute classical linearised fits

1 · Isotherms at several temperatures

Enter one isotherm dataset per temperature. AdsorpFit fits your chosen isotherm model at each temperature, converts its constant into a properly dimensionless K°, and runs the van't Hoff analysis.

2 · Equilibrium constant

this choice changes ΔG°

Isotherm model to fit at each T Route to a dimensionless K°

ℹ

Adsorbate molar mass (g mol⁻¹)

3 · Optional analyses

Non-linear van't Hoff with a ΔC_p term (needs ≥ 4 temperatures) Isosteric heat of adsorption vs. surface loading Arrhenius activation energy from kinetic rate constants

How to use AdsorpFit: and how to read what it tells you

Every model AdsorpFit can fit

AdsorpFit fits 21 adsorption isotherm models and 13 adsorption kinetic models to your own measurements, then runs a full van 't Hoff thermodynamic analysis, entirely inside your browser. Fitting is by bounded non-linear least squares, with linearised forms offered alongside for comparison. Every fit returns parameter standard errors, 95% confidence intervals, AICc ranking against the other models you selected, checks that the data actually fall inside the model's valid domain, and a written interpretation of what the numbers mean. Nothing you paste is uploaded anywhere.

Adsorption isotherm models (21)

Langmuir 1918: Adsorption is confined to a monolayer, with no stacking of adsorbate. Fitted parameters: q_max, K_L. Langmuir, I. (1918) J. Am. Chem. Soc. 40, 1361-1403.
Freundlich 1906: The surface is energetically heterogeneous, with an exponential distribution of site energies. Fitted parameters: K_F, n. Freundlich, H.M.F. (1906) Z. Phys. Chem. 57, 385-470.
Temkin 1940: The heat of adsorption of all molecules in the layer decreases linearly with surface coverage, because of adsorbate–adsorbate repulsion. Fitted parameters: A_T, b_T. Temkin, M.J. & Pyzhev, V. (1940) Acta Physiochim. URSS 12, 217-222.
Dubinin–Radushkevich 1947: Adsorption proceeds by pore filling rather than layer-by-layer coverage. Fitted parameters: q_s, K_ad. Dubinin, M.M. & Radushkevich, L.V. (1947) Proc. Acad. Sci. USSR 55, 331-337.
Jovanović 1969: Monolayer coverage as in Langmuir, but allowing for mechanical contact between adsorbing and desorbing molecules. Fitted parameters: q_max, K_J. Jovanović, D.S. (1969) Kolloid-Z. Z. Polym. 235, 1203-1214.
Halsey 1948: Describes multilayer adsorption at a relatively large distance from the surface. Fitted parameters: K_H, n_H. Halsey, G. (1948) J. Chem. Phys. 16, 931-937.
Harkins–Jura 1944: Multilayer adsorption on a heterogeneous pore distribution. Fitted parameters: A, B. Harkins, W.D. & Jura, G. (1944) J. Chem. Phys. 12, 112-113.
BET (liquid phase) 1938: Multilayer adsorption: molecules adsorb on top of already-adsorbed molecules. Fitted parameters: q_s, C_BET. Brunauer, S., Emmett, P.H. & Teller, E. (1938) J. Am. Chem. Soc. 60, 309-319.
Elovich (isotherm) 1962: Adsorption sites increase exponentially with coverage, implying multilayer adsorption. Fitted parameters: q_m, K_E. Elovich, S.Y. & Larinov, O.G. (1962) Izv. Akad. Nauk. SSSR, Otd. Khim. Nauk 2, 209-216.
Sips (Langmuir–Freundlich) 1948: A hybrid of Langmuir and Freundlich: Freundlich-like at low Ce, Langmuir-like at high Ce. Fitted parameters: q_max, K_s, m_s. Sips, R. (1948) J. Chem. Phys. 16, 490-495.
Tóth 1971: Derived from potential theory for heterogeneous adsorption. Fitted parameters: q_max, K_T, n_T. Tóth, J. (1971) Acta Chim. Acad. Sci. Hung. 69, 311-328.
Redlich–Peterson 1959: An empirical hybrid of Langmuir and Freundlich. Fitted parameters: K_RP, a_RP, g. Redlich, O. & Peterson, D.L. (1959) J. Phys. Chem. 63, 1024-1026.
Khan 1996: A general model for pure solutions, intended for multi-component systems. Fitted parameters: q_max, b_K, a_K. Khan, A.R., Al-Waheab, I.R. & Al-Haddad, A. (1996) Environ. Technol. 17, 13-23.
Radke–Prausnitz 1972: Performs particularly well at dilute concentrations, its original purpose. Fitted parameters: q_max, K_RP, m_RP. Radke, C.J. & Prausnitz, J.M. (1972) Ind. Eng. Chem. Fundam. 11, 445-451.
Hill 1910: Derived for binding of a ligand to a homogeneous substrate. Fitted parameters: q_SH, K_D, n_H. Hill, A.V. (1910) J. Physiol. 40, iv-vii.
Koble–Corrigan 1952: An empirical combination of the Langmuir and Freundlich forms. Fitted parameters: A, B, n. Koble, R.A. & Corrigan, T.E. (1952) Ind. Eng. Chem. 44, 383-387.
Brouers–Sotolongo (deformed Weibull) 2005: Derived from a statistical distribution of adsorption energies (a deformed exponential / Weibull form). Fitted parameters: q_max, K_BS, α. Brouers, F. et al. (2005) J. Hazard. Mater. 138, 591-597.
Vieth–Sladek 1965: Two populations of adsorbate coexist: one dissolved into the solid following Henry's law, one bound to discrete sites following Langmuir. Fitted parameters: k_VS, q_max, b. Vieth, W.R. & Sladek, K.J. (1965) J. Colloid Sci. 20, 1014-1033.
Fritz–Schlünder (IV) 1974: A flexible empirical equation with no single mechanistic derivation. Fitted parameters: A, B, α, β. Fritz, W. & Schlünder, E.U. (1974) Chem. Eng. Sci. 29, 1279-1282.
Baudu 1990: An extension of Langmuir in which the affinity b₀ is itself allowed to vary with coverage. Fitted parameters: q_max, b₀, x, y. Baudu, M. (1990) PhD thesis, Université de Rennes; see Limousin, G. et al. (2007) Appl. Geochem. 22, 249-275.
Marczewski–Jaroniec 1983: Derived from a generalised (quasi-Gaussian) distribution of adsorption energies, with m and n controlling each tail independently. Fitted parameters: q_max, K, m, n. Marczewski, A.W. & Jaroniec, M. (1983) Monatsh. Chem. 114, 711-715.

Adsorption kinetic models (13)

Pseudo-first-order (Lagergren) 1898: The uptake rate is proportional to the number of unoccupied sites, d q_t/dt = k₁(q_e − q_t). Fitted parameters: q_e,cal, k₁. Lagergren, S. (1898) Kungliga Svenska Vetenskapsakademiens Handlingar 24, 1-39.
Pseudo-second-order (Ho & McKay) 1999: The uptake rate is proportional to the square of the number of unoccupied sites, d q_t/dt = k₂(q_e − q_t)². Fitted parameters: q_e,cal, k₂. Ho, Y.S. & McKay, G. (1999) Process Biochem. 34, 451-465.
Elovich 1934: The activation energy for adsorption increases linearly with coverage. Fitted parameters: α, β. Roginsky, S. & Zeldovich, Y.B. (1934) Acta Physicochim. URSS 1, 554-594; Chien, S.H. & Clayton, W.R. (1980) Soil Sci. Soc. Am. J. 44, 265-268.
Avrami (fractional order) 1940: Adapted from nucleation and crystal-growth theory. Fitted parameters: q_e, k_AV, n_AV. Avrami, M. (1940) J. Chem. Phys. 8, 212-224; Lopes, E.C.N. et al. (2003) J. Colloid Interface Sci. 263, 542-547.
Mixed 1,2-order (MOE) 2010: Interpolates continuously between PFO and PSO rather than forcing a choice between them. Fitted parameters: q_e, k_MOE, f₂. Marczewski, A.W. (2010) Langmuir 26, 15229-15238.
Pseudo-nth-order 2007: Generalises PFO and PSO by letting the order n be fitted. Fitted parameters: q_e, k_n, n. Özer, A. (2007) J. Hazard. Mater. 141, 753-761.
Ritchie nth-order 1977: Derived from the fraction of surface sites occupied, not from a concentration-driven rate law. Fitted parameters: q_e, k_R, n. Ritchie, A.G. (1977) J. Chem. Soc. Faraday Trans. 1 73, 1650-1653.
Fractal-like pseudo-first-order 2012: The rate 'constant' is not constant: on a fractal or energetically disordered surface it decays as a power of time, k(t) = k′t^(−h). Fitted parameters: q_e, k₁', h. Kopelman, R. (1988) Science 241, 1620-1626; Haerifar, M. & Azizian, S. (2012) J. Phys. Chem. C 116, 13111-13119.
Weber–Morris (intraparticle diffusion) 1963: Uptake varies with the square root of time when intraparticle diffusion controls the rate. Fitted parameters: k_id, C. Weber, W.J. & Morris, J.C. (1963) J. Sanit. Eng. Div. ASCE 89, 31-60.
Liquid film diffusion (Boyd–Adamson) 1947: The rate is controlled by diffusion of adsorbate across the stagnant liquid film around the adsorbent particle. Fitted parameters: q_e, k_fd. Boyd, G.E., Adamson, A.W. & Myers, L.S. (1947) J. Am. Chem. Soc. 69, 2836-2848.
Bangham (pore diffusion) 1924: Pore diffusion is the rate-controlling step. Fitted parameters: k₀, α. Bangham, D.H. & Burt, F.P. (1924) Proc. R. Soc. Lond. A 105, 481-488.
Homogeneous surface diffusion (Crank) 1975: Spherical, homogeneous particles of uniform radius r. Fitted parameters: q_e, D, r. Crank, J. (1975) The Mathematics of Diffusion, 2nd ed., Oxford.
Double exponential 1993: Adsorption proceeds in two parallel or sequential steps with distinct rate constants: typically a fast external surface step and a slow internal diffusion step. Fitted parameters: q_e, a₁, k_D1, k_D2. Wilczak, A. & Keinath, T.M. (1993) Water Environ. Res. 65, 238-244.

Adsorption thermodynamics

From isotherms measured at three or more temperatures, AdsorpFit derives the standard Gibbs free energy change, the standard enthalpy change and the standard entropy change by the van 't Hoff method, using a genuinely dimensionless equilibrium constant. Six documented routes to that dimensionless constant are offered, because a van 't Hoff plot built on a constant that still carries units is the single most common error in the adsorption literature. Isosteric heat of adsorption as a function of surface loading, Arrhenius activation energy from fitted rate constants, and sticking probability are also available.