The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressure; i.e., at these conditions the adsorbate's partial pressure p A {\displaystyle p_{A}} is related to its volume V adsorbed onto a solid adsorbent. The adsorbent, as indicated in the figure, is assumed to be an ideal solid surface composed of a series of distinct sites capable of binding the adsorbate. The adsorbate binding is treated as a chemical reaction between the adsorbate gaseous molecule A g {\displaystyle A_{\text{g}}} and an empty sorption site S. This reaction yields an adsorbed species A ad {\displaystyle A_{\text{ad}}} with an associated equilibrium constant K eq {\displaystyle K_{\text{eq}}} :
A g
+ S ↽ − − ⇀ A ad {\displaystyle {\ce {A_{g}{}+ S <=> A_{ad}}}} . From these basic hypotheses the mathematical formulation of the Langmuir adsorption isotherm can be derived in various independent and complementary ways: by the kinetics, the thermodynamics, and the statistical mechanics approaches respectively (see below for the different demonstrations). The Langmuir adsorption equation is
θ A = V V m = K eq A p A 1 + K eq A p A , {\displaystyle \theta _{A}={\frac {V}{V_{\text{m}}}}={\frac {K_{\text{eq}}^{A}\,p_{A}}{1+K_{\text{eq}}^{A}\,p_{A}}},}
where θ A {\displaystyle \theta _{A}} is the fractional occupancy of the adsorption sites, i.e., the ratio of the volume V of gas adsorbed onto the solid to the volume V m {\displaystyle V_{\text{m}}} of a gas molecules monolayer covering the whole surface of the solid and completely occupied by the adsorbate. A continuous monolayer of adsorbate molecules covering a homogeneous flat solid surface is the conceptual basis for this adsorption model.
Background and experiments In 1916, Irving Langmuir presented his model for the adsorption of species onto simple surfaces. Langmuir was awarded the Nobel Prize in 1932 for his work concerning surface chemistry. He hypothesized that a given surface has a certain number of equivalent sites to which a species can "stick", either by physisorption or chemisorption. His theory began when he postulated that gaseous molecules do not rebound elastically from a surface, but are held by it in a similar way to groups of molecules in solid bodies. Langmuir published two papers that confirmed the assumption that adsorbed films do not exceed one molecule in thickness. The first experiment involved observing electron emission from heated filaments in gases. The second, a more direct evidence, examined and measured the films of liquid onto an adsorbent surface layer. He also noted that generally the attractive strength between the surface and the first layer of adsorbed substance is much greater than the strength between the first and second layer. However, there are instances where the subsequent layers may condense given the right combination of temperature and pressure.
Basic assumptions of the model Inherent within this model, the following assumptions are valid specifically for the simplest case: the adsorption of a single adsorbate onto a series of equivalent sites onto the surface of the solid.
The surface containing the adsorbing sites is a perfectly flat plane with no corrugations (assume the surface is homogeneous). However, chemically heterogeneous surfaces can be considered to be homogeneous if the adsorbate is bound to only one type of functional groups on the surface. The adsorbing gas adsorbs into an immobile state. All sites are energetically equivalent, and the energy of adsorption is equal for all sites. Each site can hold at most one molecule (mono-layer coverage only). No (or ideal) interactions between adsorbate molecules on adjacent sites. When the interactions are ideal, the energy of side-to-side interactions is equal for all sites regardless of the surface occupancy.
Derivations of the Langmuir adsorption isotherm The mathematical expression of the Langmuir adsorption isotherm involving only one sorbing species can be demonstrated in different ways: the kinetics approach, the thermodynamics approach, and the statistical mechanics approach respectively. In case of two competing adsorbed species, the competitive adsorption model is required, while when a sorbed species dissociates into two distinct entities, the dissociative adsorption model need to be used.
Kinetic derivation This section provides a kinetic derivation for a single-adsorbate case. The kinetic derivation applies to gas-phase adsorption. The multiple-adsorbate case is covered in the competitive adsorption sub-section. The model assumes adsorption and desorption as being elementary processes, where the rate of adsorption rad and the rate of desorption rd are given by
r ad = k ad p A [ S ] , {\displaystyle r_{\text{ad}}=k_{\text{ad}}p_{A}[S],}
r d = k d [ A ad ] , {\displaystyle r_{\text{d}}=k_{d}[A_{\text{ad}}],}
where pA is the partial pressure of A over the surface, [S] is the concentration of free sites in number/m2, [Aad] is the surface concentration of A in molecules/m2 (concentration of occupied sites), and kad and kd are constants of forward adsorption reaction and backward desorption reaction in the above reactions. At equilibrium, the rate of adsorption equals the rate of desorption. Setting rad = rd and rearranging, we obtain
[ A ad ] p A [ S ] = k ad k d = K eq A . {\displaystyle {\frac {[A_{\text{ad}}]}{p_{A}[S]}}={\frac {k_{\text{ad}}}{k_{\text{d}}}}=K_{\text{eq}}^{A}.}
The concentration of sites is given by dividing the total number of sites (S0) covering the whole surface by the area of the adsorbent (a):
[ S 0 ] = S 0 / a . {\displaystyle [S_{0}]=S_{0}/a.}
We can then calculate the concentration of all sites by summing the concentration of free sites [S] and occupied sites:
[ S 0 ] = [ S ] + [ A ad ] . {\displaystyle [S_{0}]=[S]+[A_{\text{ad}}].}
Combining this with the equilibrium equation, we get
[ S 0 ] = [ A ad ] K eq A p A + [ A ad ] = 1 + K eq A p A K eq A p A [ A ad ] . {\displaystyle [S_{0}]={\frac {[A_{\text{ad}}]}{K_{\text{eq}}^{A}p_{A}}}+[A_{\text{ad}}]={\frac {1+K_{\text{eq}}^{A}p_{A}}{K_{\text{eq}}^{A}p_{A}}}[A_{\text{ad}}].}
We define now the fraction of the surface sites covered with A as
θ A = [ A ad ] [ S 0 ] . {\displaystyle \theta _{A}={\frac {[A_{\text{ad}}]}{[S_{0}]}}.}
This, applied to the previous equation that combined site balance and equilibrium, yields the Langmuir adsorption isotherm:
θ A = K eq A p A 1 + K eq A p A . {\displaystyle \theta _{A}={\frac {K_{\text{eq}}^{A}p_{A}}{1+K_{\text{eq}}^{A}p_{A}}}.}
Thermodynamic derivation In condensed phases (solutions), adsorption to a solid surface is a competitive process between the solvent (A) and the solute (B) to occupy the binding site. The thermodynamic equilibrium is described as
Solvent (bound) + Solute (free) ↔ Solvent (free) + Solute (bound). If we designate the solvent by the subscript "1" and the solute by "2", and the bound state by the superscript "s" (surface/bound) and the free state by the "b" (bulk solution / free), then the equilibrium constant can be written as a ratio between the activities of products over reactants:
K = a 1 b × a 2 s a 2 b × a 1 s . {\displaystyle K={\frac {a_{1}^{\text{b}}\times a_{2}^{\text{s}}}{a_{2}^{\text{b}}\times a_{1}^{\text{s}}}}.}
For dilute solutions the activity of the solvent in bulk solution a 1 b ≃ 1 , {\displaystyle a_{1}^{\text{b}}\simeq 1,} and the activity coefficients ( γ {\displaystyle \gamma } ) are also assumed to ideal on the surface. Thus, a 2 s = X 2 s = θ , {\displaystyle a_{2}^{\text{s}}=X_{2}^{\text{s}}=\theta ,} a 1 s = X 1 s , {\displaystyle a_{1}^{\text{s}}=X_{1}^{\text{s}},} , and X 1 s + X 2 s = 1 , {\displaystyle X_{1}^{\text{s}}+X_{2}^{\text{s}}=1,} where X i {\displaystyle X_{i}} are mole fractions. Re-writing the equilibrium constant and solving for θ {\displaystyle \theta } yields
θ = K a 2 b 1 + K a 2 b . {\displaystyle \theta ={\frac {Ka_{2}^{\text{b}}}{1+Ka_{2}^{\text{b}}}}.}
Note that the concentration of the solute adsorbate can be used instead of the activity coefficient. However, the equilibrium constant will no longer be dimensionless and will have units of reciprocal concentration instead. The difference between the kinetic and thermodynamic derivations of the Langmuir model is that the thermodynamic uses activities as a starting point while the kinetic derivation uses rates of reaction. The thermodynamic derivation allows for the activity coefficients of adsorbates in their bound and free states to be included. The thermodynamic derivation is usually referred to as the "Langmuir-like equation".
Statistical mechanical derivation This derivation based on statistical mechanics was originally provided by Volmer and Mahnert in 1925. The partition function of the finite number of adsorbents adsorbed on a surface, in a canonical ensemble, is given by
Z ( N A ) = [ ζ L N A N S ! ( N S − N A ) ! ] 1 N A ! , {\displaystyle Z(N_{A})=\left[\zeta _{L}^{N_{A}}{\frac {N_{S}!}{(N_{S}-N_{A})!}}\right]{\frac {1}{N_{A}!}},}
where ζ L {\displaystyle \zeta _{L}} is the partition function of a single adsorbed molecule, N S {\displaystyle N_{S}} is the number of adsorption sites (both occupied and unoccupied), and N A {\displaystyle N_{A}} is the number of adsorbed molecules which should be less than or equal to N S {\displaystyle N_{S}} . The terms in the bracket give the total partition function of the N A {\displaystyle N_{A}} adsorbed molecules by taking a product of the individual partition functions (refer to Partition function of subsystems). The 1 / N A ! {\displaystyle 1/N_{A}!} factor accounts for the overcounting arising due to the indistinguishable nature of the adsorbates. The grand canonical partition function is given by
Z ( μ A ) = ∑ N A = 0 N S exp ( N A μ A k B T ) ζ L N A N A ! N S ! ( N S − N A ) ! . {\displaystyle {\mathcal {Z}}(\mu _{A})=\sum _{N_{A}=0}^{N_{S}}\exp \left({\frac {N_{A}\mu _{A}}{k_{\text{B}}T}}\right){\frac {\zeta _{L}^{N_{A}}}{N_{A}!}}\,{\frac {N_{S}!}{(N_{S}-N_{A})!}}.}
μ A {\displaystyle \mu _{A}} is the chemical potential of an adsorbed molecule. As it has the form of binomial series, the summation is reduced to
Z ( μ A ) = ( 1 + x ) N S , {\displaystyle {\mathcal {Z}}(\mu _{A})=(1+x)^{N_{S}},}
where x = ζ L exp ( μ A k B T ) . {\displaystyle x=\zeta _{L}\exp \left({\frac {\mu _{A}}{k_{\rm {B}}T}}\right).}
The grand canonical potential is
Ω = − k B T ln ( Z ) = − k B T N S ln ( 1 + x ) , {\displaystyle \Omega =-k_{\rm {B}}T\ln({\mathcal {Z}})=-k_{\rm {B}}TN_{S}\ln(1+x),}
based on which the average number of occupied sites is calculated
⟨ N A ⟩ = − ( ∂ Ω ∂ μ A ) T , area , {\displaystyle \langle N_{A}\rangle =-\left({\frac {\partial \Omega }{\partial \mu _{A}}}\right)_{T,{\text{area}}},}
which gives the coverage
θ A = ⟨ N A ⟩ N S = x 1 + x . {\displaystyle \theta _{A}={\frac {\langle N_{A}\rangle }{N_{S}}}={\frac {x}{1+x}}.}
Now, invoking the condition that the system is in equilibrium, that is, the chemical potential of the adsorbed molecules is equal to that of the molecules in gas phase, we have
μ A = μ g , {\displaystyle \mu _{A}=\mu _{\text{g}},}
The chemical potential of an ideal gas is
μ g = ( ∂ A g ∂ N ) T , V {\displaystyle \mu _{\text{g}}=\left({\frac {\partial A_{\text{g}}}{\partial N}}\right)_{T,V}}
where A g = − k B T ln Z g {\displaystyle A_{g}=-k_{\rm {B}}T\ln Z_{g}} is the Helmholtz free energy of an ideal gas with its partition function
Z g = q N N ! . {\displaystyle Z_{g}={\frac {q^{N}}{N!}}.}
q {\displaystyle q} is the partition function of a single particle in the volume of V {\displaystyle V} (only consider the translational freedom here).
q = V ( 2 π m k B T h 2 ) 3 / 2 . {\displaystyle q=V\left({\frac {2\pi mk_{\rm {B}}T}{h^{2}}}\right)^{3/2}.}
We thus have μ g = − k B T ln ( q / N ) {\displaystyle \mu _{g}=-k_{\rm {B}}T\ln(q/N)} , where we use Stirling's approximation. Plugging μ g {\displaystyle \mu _{g}} to the expression of x {\displaystyle x} , we have
θ A 1 − θ A = x = ζ L N q {\displaystyle {\frac {\theta _{A}}{1-\theta _{A}}}=x=\zeta _{L}{\frac {N}{q}}}
which gives the coverage
θ A = ζ L / ( q / N ) 1 + ζ L / ( q / N ) {\displaystyle \theta _{A}={\frac {\zeta _{L}/(q/N)}{1+\zeta _{L}/(q/N)}}}
By defining
P 0 = k B T ζ L ( 2 π m k B T h 2 ) 3 / 2 {\displaystyle P_{0}={\frac {k_{\text{B}}T}{\zeta _{L}}}\left({\frac {2\pi mk_{\text{B}}T}{h^{2}}}\right)^{3/2}}
and using the identity P V = N k B T {\displaystyle PV=Nk_{\rm {B}}T} , finally, we have
θ A = P P + P 0 . {\displaystyle \theta _{A}={\frac {P}{P+P_{0}}}.}
It is plotted in the figure alongside demonstrating that the surface coverage increases quite rapidly with the partial pressure of the adsorbants, but levels off after P reaches P0.
Competitive adsorption The previous derivations assumed that there is only one species, A, adsorbing onto the surface. This section considers the case when there are two distinct adsorbates present in the system. Consider two species A and B that compete for the same adsorption sites. The following hypotheses are made here:
All the sites are equivalent. Each site can hold at most one molecule of A, or one molecule of B, but not both simultaneously. There are no interactions between adsorbate molecules on adjacent sites. As derived using kinetic considerations, the equilibrium constants for both A and B are given by
[ A ad ] p A [ S ] = K eq A {\displaystyle {\frac {[A_{\text{ad}}]}{p_{A}\,[S]}}=K_{\text{eq}}^{A}}
and
[ B ad ] p B [ S ] = K eq B . {\displaystyle {\frac {[B_{\text{ad}}]}{p_{B}\,[S]}}=K_{\text{eq}}^{B}.}
The site balance states that the concentration of total sites [S0] is equal to the sum of free sites, sites occupied by A and sites occupied by B:
[ S 0 ] = [ S ] + [ A ad ] + [ B ad ] . {\displaystyle [S_{0}]=[S]+[A_{\text{ad}}]+[B_{\text{ad}}].}
Inserting the equilibrium equations and rearranging in the same way we did for the single-species adsorption, we get similar expressions for both θA and θB:
θ A = K eq A p A 1 + K eq A p A + K eq B p B , {\displaystyle \theta _{A}={\frac {K_{\text{eq}}^{A}\,p_{A}}{1+K_{\text{eq}}^{A}\,p_{A}+K_{\text{eq}}^{B}\,p_{B}}},}
θ B = K eq B p B 1 + K eq A p A + K eq B p B . {\displaystyle \theta _{B}={\frac {K_{\text{eq}}^{B}\,p_{B}}{1+K_{\text{eq}}^{A}\,p_{A}+K_{\text{eq}}^{B}\,p_{B}}}.}
Dissociative adsorption The other case of special importance is when a molecule D2 dissociates into two atoms upon adsorption. Here, the following assumptions would be held to be valid:
D2 completely dissociates to two molecules of D upon adsorption. The D atoms adsorb onto distinct sites on the surface of the solid and then move around and equilibrate. All sites are equivalent. Each site can hold at most one atom of D. There are no interactions between adsorbate molecules on adjacent sites. Using similar kinetic considerations, we get
[ D ad ] p D 2 1 / 2 [ S ] = K eq D . {\displaystyle {\frac {[D_{\text{ad}}]}{p_{D_{2}}^{1/2}[S]}}=K_{\text{eq}}^{D}.}
The 1/2 exponent on pD2 arises because one gas phase molecule produces two adsorbed species. Applying the site balance as done above,
θ D = ( K eq