Physical Chemistry , 1st ed.

Figure 22.23 shows a plot of this data, with 1/ con the y-axis and 1/[acid]

on the x-axis. A best-fit straight line is drawn, showing that the yintercept is

approximately 141. According to equation 22.29, this is the value of 1/.

Taking the reciprocal, we find that is  1141 , or 0.00709. Our best-fit straight

line also has a slope of 8.45, which equals 1/(Keq). Having solved for ,

we use algebra to find a value for K, and get K16.7. If we had additional

data—say, the size of the acetic acid molecule or the surface area of the

charcoal—we could calculate the surface area or the molecule size, respectively.

The Langmuir isotherm is a common way of modeling gas- or liquid-phase

molecules adsorbing on surfaces, but it isn’t the only way. Another way to

model the coverage versus the concentration of the adsorbing species is the

Freundlich isotherm,which follows the equation

K[adsorbing species]c (22.30)

where Kand care experimentally determined constants. This equation is usu-

ally plotted in terms of its logarithm, which makes it

log log Kclog [adsorbing species]

and has the form of a straight line. There are other isotherms defined for differ-

ent heterogeneous systems. (Consult a text on surface science for more details.)

What if a reaction involves the adsorption of two different gas-phase species,

A and B, onto a solid surface? In order to model those processes, we will have

to define two different coverage variables Aand B. If we assume that the ad-

sorption and desorption of each process are in equilibrium, then we can rewrite

equation 22.25, the equality of the adsorption and desorption rates, for each

gas-phase species:

kads,A[A] (1 AB) kdes,AA

kads,B[B] (1 AB) kdes,BB (22.31)

By defining two equilibrium constants KAand KBin terms of the adsorption

and desorption rate constants like we did earlier, we can solve for the two cov-

erages. We give their expressions without showing the algebra:

A (22.32)

B

These expressions define Langmuir-Hinshelwood isotherms.(Cyril Norman

Hinshelwood was a British chemist who won a 1956 Nobel Prize for studying

chemical reaction mechanisms.)

Finally, suppose a diatomic gas is adsorbed on a surface and the first step is

for the molecule to dissociate and occupy two sites, one by each atom:

surface surface

A 2 (g) 2A 2A (adsorbed)

If these processes are in equilibrium, then the stoichiometry of the reaction

would affect the expression for the coverage by the A atoms. Instead of equa-

tion 22.27, one would get for a Langmuir isotherm

JQPJ JQPJ

KB[B]



KA[A] KB[B]  1

KA[A]



KA[A] KB[B]  1

786 CHAPTER 22 Surfaces

700

100

080

1

 c

1

[acid]

70605040302010

600

500

400

300

200

Figure 22.23 Plot of data for Example 22.8.
The yintercept equals 1/, the inverse of the pro-
portionality constant, whereas the slope equals
1/K. See Example 22.8 for details.