Langmuir equation

Langmuir derived a relationship for q and C based on some quite reasonable assumptions. These are: a uniform surface, a single layer of adsorbed material, and constant temperature. The rate of attachment to the surface should be proportional to a driving force times an area. The driving force is the concentration in the fluid, and the area is the amount of bare surface. If the fraction of covered surface is F , the rate per unit of surface is:

       rate going on = k1 C ( 1 - F )

 The evaporation from the surface is proportional to the amount of surface covered:

        rate leaving = k2 F

where k1 and k2 are rate coefficients
C = concentration in the fluid
F = fraction of the surface covered

 Animation of finding adsorption sites

 At equilibrium, the two rates are equal, and we find that:

By dividing the numerator and denominator by k1, and making use of the fact that q will be proportional to  F, the useful form of the equation is:

where qm = q for a complete monolayer
 Ka = a coefficient

 Taking reciprocals and rearranging: ( details of math for sleepy people)

 A plot of  versus  should indicate a straight line of slope  and an intercept of . The graph shows data points and lines fitted to both Freundlich and Langmuir equations.

Equations Fitted to Data


Try this yourself. The equations are handled separately. The idea is to imagine a data set and to observe that either equation can usually fit pretty well if you fiddle with the coefficients.
How to relate graphs to equations
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