Langmuir adsorption model

The Langmuir adsorption model explains <a href="/facts/Adsorption/ARH9peYM">adsorption</a> by assuming an <a href="/facts/Adsorbate/ARH9peYM">adsorbate</a> behaves as an ideal gas at <a href="/facts/Isothermal_process/6mkdESuS">isothermal</a> 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 <a href="/facts/Adsorbate/ARH9peYM">adsorbate</a>'s <a href="/facts/Partial_pressure/sRw2hj4G">partial pressure</a> 
 
 
 
 
 p
 
 A
 
 
 
 
 {\displaystyle p_{A}}
 
 is related to its volume V adsorbed onto a solid <a href="/facts/Adsorbent/ARH9peYM">adsorbent</a>. 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 <a href="/facts/Chemical_kinetics/D8vyX4aE">kinetics</a>, the <a href="/facts/Thermodynamics/q4hIx3yP">thermodynamics</a>, and the <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a> 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.

Langmuir adsorption model open-in-new

Langmuir adsorption model