Chemisorption. The Langmuir Equation

Chemisorption may be studied as a separate phenomenon from catalysis simply by using a single adsorbate species which is stable toward reaction. Examples would be the chemisorption of nitrogen on tungsten, of hydrogen on nickel, and of ammonia on carbon black. The adsorption process can be slow and temperature-dependent, that is, increase in rate with increasing temperature, indicating an activation energy for adsorption; it may not always be possible to attain a reversible equilibrium. The better-behaved systems, however, show rapid adsorp­

tion and for each pressure of adsorbate gas there will be an equilibrium amount of adsorption. The data are conventionally plotted as an adsorption isotherm, that is, as amount adsorbed ν (usually measured as cubic centimeters STP of gas adsorbed per gram of adsorbent) versus pressure, as shown in Fig. 14-16. The

SPECIAL TOPICS, SECTION 1 589

- 2 3 . 5 ° C

—I 1 l^ r— Γ I I ι ' 150 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0

Γ, Κ (b)

FIG. 14-16. (a) Adsorption isotherms for ammonia on charcoal, (b) Adsorption isosteres for ammonia on charcoal. [From Vol. /, "Physical Adsorption" of Stephen Brunauer, "The Adsorption of Gases and Vapors" (copyright 1943 © 1971 by Princeton University Press). Reprinted by permission of Princeton University Press, Princeton, New Jersey.]

normal behavior is for ν to approach a limiting value v^m with increasing pressure, and, as also shown in the figure, for the amount adsorbed to decrease with increasing temperature at constant pressure.

A very straightforward treatment of chemisorption was given by I. Langmuir in 1918. The picture is one of a surface having a certain number of adsorption sites S of which 5² may be occupied by adsorbate and S⁰ = S — .S^ is then bare.

Adsorption equilibrium is treated as a dynamic state in which the rate of adsorption is equal to the rate of desorption. The former rate is taken to be nonactivated and just proportional to the surface collision frequency of the gaseous adsorbate on the bare sites, which, by Eq. (2-45), is proportional to the pressure P:

rate of adsorption = k²PS⁰ = k²P(S — SJ.

F I G . 14-17. Plot of adsorption data in the form of Eq. (14-93).

J I I 1 2 4 6 8

Λ Torr

The rate of desorption is proportional to the number of occupied sites S¹ : rate of desorption = k¹S¹.

The two rates are set equal, and on solving for S¹ we obtain

^1 = θ = ^{b P} (14-91)

S 1+bP^{9 { }}

where θ is the fraction of surface covered and b = k²\k^x. The limiting value v^m of Fig. 14-16 corresponds to Θ = 1, hence v/v^m = 0, and we can write

V m b P (14-92)

1 + bP

Equations (14-91) and (14-92) are known as the Langmuir adsorption isotherm.

The latter may be rearranged to the form

ρ l ρ

— = r - + — , (14-93) ν bv^m v^m

which states that a plot of P/v versus Ρ should be linear. Some data obeying Eq.

(14-92) are plotted in this manner in Fig. 14-17.

Although the adsorption process is not considered to be activated, that for desorption must be, since there will be an adsorption energy Q which the adsorbate must gain if it is to desorb; k^x must therefore be of the form k± = k^e-^Ql^RT, and the constant b consequently can be written

b = b⁰e°/^RT. (14-94)

Equation (14-91) reduces to the form

θ = bP, or ν = v^mbP (14-95)

at low pressure, so that the Langmuir model requires that the adsorption isotherm be linear at low pressures; the isosteres for this region obey a Clausius-Clapeyron type of equation:

\-w-\„r-*F'

⁽¹⁴

-

⁹⁶⁾

SPECIAL TOPICS, SECTION 1 591 From its nature, Q must be a positive number, and so pressure must increase with temperature. Alternatively, at constant pressure,

l^L\

= ⁼__Q_

\ dT

I

^{ρ dT RT2 ' y }}

Finally, if more than one species is competing for adsorption, the adsorption isotherm for some one of them is given by the Langmuir derivation as

b Ρ

V i

l + Z W

where the summation is over all species.

Relatively few chemisorption systems obey the simple Langmuir model really well; the data may not fit the Langmuir equation, especially at the extremes, or if they do, the variation of the b parameter with temperature may fail to obey the simple exponential law. The basic model seems sound, however, and such diffi­

culties can often be attributed to nonuniformity of the surface so that Q varies with degree of surface occupancy. The adsorbate adsorbs first on the more active sites, with highest Q⁹ and then on progressively less active ones. Hydrogen appears to be present on the surface as individual atoms, so that the desorption process requires a recombination of two surface atoms, and is therefore bimolecu­

lar. Analysis leads to an isotherm of the form

* = t t W -

⁽¹⁴

"

⁹⁹⁾

The derivation assumes that the adsorbing gas requires two adjacent sites, which has the effect of making the rate of adsorption proportional to S⁰².

As may well be imagined, there are a number of sophisticated treatments both of the Langmuir derivation and of variants such as that just given. Equation (14-91) may be obtained, for example, by means of a statistical thermodynamic derivation in which the adsorbed gas has the partition function

— V s i t e V i n t ^e >

where e^Qi^RT allows for the change in zero-point energy on adsorption; Qf^{n t} may be taken to be the same as for the free gas, and Q |^{i t e} is given by the number of distinguishable ways of arranging Ν molecules on S sites. Our present use of the Langmuir model, however, is simply as a convenient basis for treating the kinetics

of heterogeneous catalysis.

β. Kinetics of Heterogeneous Catalysis

A number of simple catalytic systems can be reasonably well treated on the basis that the reactants adsorb on the catalyst surface, to react according to a mass action type of rate law.

Suppose that the reaction is A —• C + D and that the surface reaction proceeds according to the rate law dnjdt = —k6^AS, where 6^AS is the surface covered by

adsorbed A, or, by Eq. (14-98),

3t ^- ~

^kS

^H - t f , S v . + M , '

^{( 1 4}

^-

^{, 0 0 )}

If the products C and D are weakly adsorbed, then Eq. (14-100) reduces to dn^{A =} _ ^{k s} b^AP^A

dt - 1 + b^AP^A '

which means that at low P^A the rate should be proportional to P^A , while at high P^A the rate levels off at the maximum value kS, and hence is independent of P^A . This last situation is then one of a zero-order reaction. Behavior of this type has been observed for the decomposition of HI on gold and platinum and of N²0 on indium sesquioxide.

If the reaction is of the type A + Β -> C + D and the surface reaction obeys the rate law dnjdt = —kd^AV^BS, then the general expression for the apparent

rate law is

DNA _ _ / . c 6 A W B Π4-10Π

dt " + b^AP^A + b^BP^B + b^cP^c + b^OP^O)²' ^U*^{1 U 1 ;} If both products and reactants are weakly adsorbed, this reduces to

= -kSb^Ab^BP^AP^B = -k^&OOP^AP^B . dn^A

—fa - — —^u^Au^Br^Ar^B = — / v a p p i -^A^ B

Examples are the reaction between nitric oxide and oxygen on glass and the low-pressure, low-temperature hydrogenation of ethylene on active carbon. If reactant A and the products are weakly adsorbed but Β is strongly adsorbed, then

dn^A __ b^AP^A _ k^ai>vP^A dt ' ^v~ b^BP^B P^B *

It is thus possible for a reactant to inhibit a reaction.

We return to Eq. (14-100) to make a final observation. At low P^A , the equation becomes

^ ⁼ —kSb^AP^A = — A :^{a p p}P^A ·

The reaction is first order, and its temperature dependence could be written as

d(ln &^{a p}p ) £ a p p

dT ^- RT² '

However, &^{a p}p contains the product of b^A and the true rate constant k and by Eq. (14-97), it follows that

£ a p p = ^ t r u e QA ·

Thus the apparent activation energy is less than the true one by the heat of

SPECIAL TOPICS, SECTION 2 593 adsorption. There is not only this effect, of course, but the true activation energy may be less than that for the bulk-phase reaction. Thus the true activation energy for the tungsten-catalyzed decomposition of ammonia is only 39 kcal m o l e^{- 1} as compared to the value of about 90 kcal m o l e^{- 1} for the gas-phase reaction. One reason a catalyst is effective, then, is because of a reduction in the activation energy for reaction.

14-ST-2 Statistical Thermodynamic

In document 14-2 Rate Laws and Simple Mechanisms (Pldal 46-51)