# The Langmuir Isotherm

In this applet we explore the Langmuir isotherm.

An adsorption isotherm is a plot of the fractional coverage of the surface, , against pressure. The simplest physically plausible isotherm is based on three assumptions:

• Adsorption cannot proceed beyond monolayer coverage.
• All sites are equivalent and the surface is uniform.
• The ability of a molecule to adsorb at a given site is independent of the occupation of neighbouring sites.

The resulting isotherm is called the Langmuir isotherm:

Where K = ka / kd (Pa-1), ka is the rate constant of adsorption, kd is the rate constant for desorption, and p is the partial pressure of the adsorbate (Pa).

Select the Simple Adsorption tab to plot this equation.

The Data, Plot and Least Squares Analysis tabs consider the adsorption of CO on charcoal at 273 K. The fractional coverage is calculated as

Where Vinfinity is the volume corresponding to complete coverage.

We rewrite the Langmuir isotherm as

It follows that a plot of p/V against V should be a straight line, and

where slope( p, pV_ratio ) and intercept( p, pV_ratio ) are the slope and y-intercept, respectively, of a plot of pV_ratio against p.

When a molecule dissociates upon adsorption, the Langmuir isotherm is

Select the with Dissociation tab to plot this equation.

Different isotherms are obtained at different temperatures on account of the temperature dependence of K. It follows that we can use the van 't Hoff equation to determine the isosteric enphalpy of adsorption, , the standard enthalpy of adsorption at a fixed surface coverage:

We again consider the adsorption of CO on charcoal. The pressures of CO needed for the volume of adsorption (corrected to 1.00 atm and 273 K) to be 10.0 cm3 are entered on the Data tab below.

The Langmuir isotherm can be rearranged to

Therefore, a plot (Plot tab) of ln(p) against 1/T should be a straight line of slope / R. It follows that the isosteric enthalpy of adsorption is

This value is calculated on the Least Squares Analysis tab below.