L. Glasser Rhodes University Grahamstown. 6140, South Africa
1
I
Rates of Bimolecular Heterogeneous Reactions following the Langmuir- Hinshelwood Mechanism
The Langmuir-Hinshelwood equation is fundamental to the study of heterogeneous ~atalysis.~~'Since it is derived from an idealized model of bimolecular surface-mediated reactions, it involves certain considerable limitations in its application to real systems; nevertheless, it deserves some attention because this model forms a basis to the study of such reaction rates. The model assumes a uniform solid surface with no interaction between adsorbed s ~ e c i e s(other than reaction!), requires equilibrium hetween gas and ndwrhrd species st all products. The times. and oermits oino adsort)tion of rr:~cti ng at equal "terra s of 5 lrom me va e . 2, from cft-lo r ghtmml c m r g Milxom n lhe rate 0cc.r when
where k11 is the bimolecular rate constant. This is the Langmnir-Hinshelwood equation. The variation in reaction rate with pressure is not easily visualized because of the presence of the two independent variables ( p ~ , p ~and ) , recourse is normally had to consideration of limiting hehavior to illustrate this variation. Thus, if p~ is maintained constant then, at sufficiently low values of .D .. A rate 1. ~TIKAKRPR (1 + ~ ~ p ~ ) " ~ i.e., the rate is first-order in A. The same result follows if A is weakly adsorbed, i.e., if KAis sufficiently small. At sufficiently high values of PA, with p~ kept constant rate (kuK~pnK~-')lp~ i.e., the rate is of order -1 in A. Hence, the rate passes through a maximum as p~ is varied, attaining that maximum when KAPA= KBPB 1;conversely, if PA is kept constant and p~ varied. a maximum rate is attained when K n ~ =n K A D A 1. Theseeonditions for maximal rates are fougdby pa&l differentiation of the rate ex~ressionwith resDect to the variable pressure and setting the results equal to zero. Many texts mistakenly assert that maximal rates are found when K A ~ A = K B ~ BHowever, . only if p~ a n d p are ~ not independent, so that K A p+~KBPB= const
+
+
'Thomas, J. M., and Thomas, W. J., "Introductionto the Principles of Heteroreneous Catalvsis." Academic Press. London. 1967.
Figure 2. Contour diagram of reaction rate as a function of K*pa and of Kspe; the scalesalong each axisare equal, and the contours are platted at equal intervals of 0.02 k,,, increasing from 0.04 k,, for the contour nearest the origin. Lines representing the conditions: K,pe = constant: Kap, = Kepe; and K A ~ A KBps = constant, are indicated on the figure.
+
22 / Journal of Chemical Education
a t successive values). The appearance of a maximum in the rate is obvious; it has the value rate (rnax)
k =I
K'pB
4 (1 + KBPB)
and rises towards a saturation value, corresponding to complete surface coverage, as pe (and PA) increases: viz. rate
-
kt114
11' 1 ~ 1 t preisurrs h are \.ari(.d i i n i u l t u n c ~ d yand indt.prndrntlv thm. at IOU, ort'sures tor smnll values id K .A. and Ku. -. corresponding to weak adsorption rate = ( ~ I I K A K ~ P A P B i.e., the rate is second-order overall, and first-order in each reactant. These limiting results and orders, though quantitative and descriptive, may not he readily assimilable. Their significance is more easily appreciated with the aid of the following two diagrams of the Lannmuir-Hinshelwood eauation. Fieure 2 is acontour plot of the equation surface, thk variablesbeing K i p ~ a n dKnp, . with rate as a parameter. The contours are plottt.il at equal incrrmenrs of rare. 'l'hr i~pproarhto a pl:ttvnu nf high prvssure is appnrent, ;IS is t he first.nrcl~,rkinetiri at low pressures. The &ror symmetry about the line K@A = K ~ p is n evident. Figure 3 displays the same surface in three dimensions, where the various lines represent pressure variation of one comoonent a t constant oressure of the other. and thus correspbnd to the curves in Figure 1.Saturation ofthe adsorption sites a t high pressure exolains, as before. the slow increase in rate with pressure a t hiih reactant pre&ures. ~
rate K.P.
I(. P.
Figure 3. Surface diagram of reaction rate as a functionof K A ~and A of Kspe. me two sets of lines (vertical sections through the surface)represent variation of K ~ p constant d values of Kern (cf.Fig. 1)and vice versa. The ridge from the from of the figure lies an the line Kapa = KBW:the maxima inthevertical sections at K,p, = Ksps = 1 are clearly visible. Figure 2 correspondsto a plan view of the region of this figure near its origin. Acknowledgment
The computer plotting programs used in the preparation of the accompanying figures were provided by Dr. P. D. Terry, adapted from available contour (J. Suowden, Ohio State University) and surface"1otting programs. q a t k i n s , S. L., Commun. Assue. Compt. Mach., 17(a), 570 (1974).
Volume 56, Number 1, January 1979 1 23
