Physical Chemistry , 1st ed.

number of sites available (that is, not covered) is given by (1 ). We can

therefore write the rate of adsorption as

Rateadskads[gas] (1 ) (22.24)

After reaction on the surface, the gas molecules have to leave the surface, or

desorb.The rate of desorption is assumed to be a zeroth-order reaction that is

related only to the coverage :

Ratedeskdes

where the subscript “des” relates the variables to the desorption process. If

the adsorption and desorption rates are equal, the reaction is occurring at

some steady pace and the adsorption and desorption rates are equal to each

other:

RateadsRatedes

kads[gas] (1 ) kdes (22.25)

We can use the above equation to algebraically solve for the coverage :



kads

k



ad

[

s

g



as

[

]

ga



s]

kdes

 (22.26)

The equilibrium constant for the adsorption/desorption process is given by

K

k

k

a

d

d

es

s

and equation 22.26 can be rewritten in terms ofKto get



K

K

[



ga

[

s

g

]

as



]

1

 (22.27)

Equation 22.27 shows that will always be less than 1 because the numerator

will always be less than the denominator.

Equations 22.26 and 22.27 define what are called Langmuir isotherms.(The

word “isotherm” is used to emphasize a constant-temperature condition in

these types of studies.) Although is difficult to measure directly, it can be

done indirectly by measuring adsorbed mass (so that the maximum mass ad-

sorbed is equivalent to a of 1) or by titration methods (so that the amount

of an acid adsorbed can be measured and related to ). A plot of(or a re-

lated variable) versus [gas] can be made, like the one in Figure 22.21. The

equilibrium constant Kcan be estimated from the graph; equation 22.27 shows

that for to equal ^12 , [gas] 1/K.

Equation 22.27 can also be rewritten in terms of the reciprocal of the cov-

erage to get





1



K

K

[



ga

[

s

g

]

as



]

1

 or 



1



K

1

[gas]

 1 (22.28)

The second equation in Equations 22.28 has the general expression for a

straight line, where yequals 1/,xis 1/[gas], and the slope is 1/K. Thus, 1/(or

variables proportional to it, like reciprocals of mass adsorbed or acid not

titrated) plotted versus 1/[gas] should give a straight line, as shown in Figure

22.22. In terms of actual measurements, if we define a change in mass of a sur-

face sample, m, as proportional to the coverage :

m

784 CHAPTER 22 Surfaces

0

1

1

[gas]

Slope ^1 K

1 

To

[gas]

0

1

 ^12

[gas] ^1 K



Figure 22.21 If adsorption of a gas (or a dis-
solved solute) on a surface follows the Langmuir
isotherm, a plot of coverage versus [gas] should
have this shape of curve. At of 0.50, [gas] should
equal 1/K, in accordance with equation 22.27.

Figure 22.22 Another way to plot the adsorp-
tion of a gas (or dissolved solute) on a surface if
it follows a Langmuir isotherm. In this case, the
straight line should have a yintercept of 1 and a
slope of 1/K. See equation 22.28.