© A.W.Marczewski 2002
A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces
Reload Adsorption Guide
ADSORPTION: φfunction
General Integral Equation /
GL (Generalized Langmuir) /
All equations (preview)
Adsorption type (
Linear Langmuir plot /
Graham plot /
Consistency /
Henry constant )
Popular isotherms
(
Mono,
Multilayer,
Experimental,
Micro,
Mesoporous
)
Data analysis: (
LSq data fitting /
Global heterogeneity /
Linear plots /
φfunction
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
φfunction method:
Use a socalled:
φfunction method (where
φ = δ log(a)/
δ log(c)) (derivative of adsorption isotherm in logarithmic coordinates).
 A.W. Marczewski, M. Jaroniec, Mh. Chem, 114, 711, (1983),
(doi).
 M. Jaroniec, A.W. Marczewski, Mh. Chem. 115, 997 (1984),
(doi).
 M. Jaroniec, A.W. Marczewski, Mh. Chem. 115, 1013 (1984),
(doi).
 M. Jaroniec, A.W. Marczewski, J. Colloid Interface Sci., 101, 280 (1984),
(doi).
φfunction is a dimensionless variable (independent of adsorption and concentration/pressure units as well as independent of type of logarithm you use). Various plots of φfunction versus adsorption, pressure/concentration etc. allow to determine isotherm type and isotherm parameters (or sometimes at least to reject some existing choices)
(Here are model pictures)
NOTE. This method allows to determine equation type and parameters very precisely, however your data should be very good  evenly spaced, smooth and in a wide range of relative adsorption φ)  method is sensitive to data scatter.

for Langmuir (L) and LangmuirFreundlich (LF) isotherms:
( φ = m / [1 + (Kc)^{m}] = m [1  θ] )
(special case is Langmuir, m=1) a simple dependence of φ vs. a is obtained

for Generalized Langmuir (GL) isotherm
( φ = m / [1 + (Kc)^{n}] = m [1  θ^{n/m}] )
in order to obtain simple linear dependences involving only experimental data (e.g. φ, a, c) calculation of 2^{nd} derivative would be required. In such a case method becomes much more sensitive to experimental data scatter.

for Generalized Langmuir with FG lateral interactions (GLFG) isotherm
( φ = m [1  θ^{n/m}] / {1  mαθ [1  θ^{n/m}]} )
where α is lateral interaction coefficient.

RadkePrausnitz aka. RedlichPeterson (RP)
( φ = 1  n {(Kc)^{n} / [1 + (Kc)^{n}]} )
(3step method):
 determine parameter m from linear dependence 1φ vs. a/c
1  φ = m  (m/a_{m}K)(a/c)
 determine parameter product a_{m}K from linear dependence (a/c) vs. (a/c)c^{m}
(a/c) = a_{m}K  K^{m} [(a/c)c^{m}]
 determine parameter K and a_{m} from linear dependence (c/a) vs. (c^{m}) (a_{m}K and m are already estimated).
(c/a) = 1/(a_{m}K)  K^{m}/(a_{m}K) (c^{m})

Jossens
( φ = 1 / [1 + b m a^{m}] = 1 / [1 + m ln(Kc/a)] )
(2step to 3step method):
 determine parameter m from linear dependence (1/φ  1) vs. ln(a/c)
(1/φ  1) = m ln(K)  m ln(a/c)
 determine parameters b and K from linear dependence ln(a/c) vs. a^{m}
ln(a/c) = ln(K)  b (a^{m})
 improve Kvalue by using one of dependencies (use earlier obtained m and b):
 determine parameter K from linear dependence (a/c) vs. exp(b a^{m})
(a/c) = K exp(b a^{m})
 determine parameter K from linear dependence (c/a) vs. exp(b a^{m})
(c/a) = K exp(b a^{m})

linear dependence (approximate!): φ vs. ln(a)
It corresponds to φ = n [ln(a_{m})  ln(a)] or isotherm equation: a = a_{m} exp(A/c^{n}). This isotherm does not reduce to Henry isotherm for low concentrations and is not consistent with local Langmuir behaviour, however it may be treated as an approximation of GL equation for high adsorption values (close to monolayer) especially if strong lateral interactions are also involved (e.g. GLFG). Then estimated parameters ( n = n_{GL}, a_{m} = a_{m,GL} and A = m_{GL}/(n K^{n}) ) may be used in GL equation.

for Jovanovic and JovanovicFreundlich (JF/Jovm) isotherms
( φ = m (Kc)^{m} / {exp[(Kc)^{m}]  1} = m [(1θ)/θ] [ ln (1θ)] )
linear dependences are difficult to use (at least a_{m} is required in order to calculate θ):
 estimate m from: φ vs. [(1θ)/θ] [ ln(1θ)]
φ = m [(1θ)/θ] [ ln(1θ)]
much better idea is to use simple linear relationship not involving φ  both m and K are determined in one step (a_{m} is still required)

Freundlich (F or DA with n=1): φ = const (and equal to B_{1}RT = m ).

DubininRadushkevich (DR or DA with n=2), linear dependences:

estimate ln(c_{o}) (and B_{2}(RT)^{2}) from: φ vs. ln(c)
φ = [2B_{2}(RT)^{2}] [ln(c_{o})  ln(c)]

estimate ln(a_{m}) (and B_{2}(RT)^{2}) from: φ^{2} vs. ln(a)
φ^{2} = [4B_{2}(RT)^{2}]^{2} [ln(a_{m})  ln(a)]

DubininAstakhov (DA; n must be known in advance) linear dependences:
 estimate ln(c_{o}) (and B_{n}(RT)^{n}) from: φ^{1/(n1)} vs. ln(c)
φ^{1/(n1)} = [n B_{n}(RT)^{n}]^{1/(n1)} [ln(c_{o})  ln(c)]
 estimate ln(a_{m}) (and B_{n}(RT)^{n}) from: φ^{n/(n1)} vs. ln(a)
φ^{n/(n1)} = [n^{n} B_{n}(RT)^{n}]^{1/(n1)} [ln(a_{m})  ln(a)]
Adsorption type (
Linear Langmuir plot /
Graham plot /
Consistency /
Henry constant )
Popular isotherms
(
Mono,
Multilayer,
Experimental,
Micro,
Mesoporous
)
Data analysis: (
LSq data fitting /
Global heterogeneity /
Linear plots /
φfunction
)
Prediction/Description of
Multicomponent adsorption /
Wastewater adsorption
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
General Integral Equation /
GL (Generalized Langmuir) /
All equations (preview)
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