Page 151 - Introduction to chemical reaction engineering and kinetics

6.4 Simple Collision Theory of Reaction Rates 133

For the bimolecular reaction 2A + products, by combining equations 6.4-12 and -13,
using equation 6.4-7a to eliminate ZAA, and converting completely to a molar basis, with
(rSCT) in mol L-i s-i, ck, = 1000 NA,,cA, where cA is in mol L-l, and k,lm* = RIM,,
where MA is the molar mass of A, we obtain

= 2000pN,,d&i-R/M,)1’2T1’2,-E”‘RTc~ (6.4-14)
TSCT = k,,,Ci
where
kSCT = 2000pN,,d&rRIM,)“2T1’2,-E*‘RT (6.4-15)

We may compare these results with a second-order rate law which exhibits Arrhenius
temperature dependence:

c2 = A~-EAIRT~~
r ohs = k ohs A A (6.1-1)
We note that the concentration dependence (ci) is the same, but that the temperature
dependence differs by the factor T1” in rscr. Although we do not have an independent
value for E* in equations 6.4-14 and -15, we may compare E* with EA by equating r,cT
and r,b,; thus,

k ohs = kSCT
d In k,,,ldT = d In k,,,/dT

and, from the Arrhenius equation, 3.1-6,

E,IRT2 = 1/2T + E*IRT2

or

EA = ~RT+E* (6.4-16)

Similarly, the pre-exponential factor AsCT can be obtained by substitution of E* from
6.4-16 into 6.4-15:

ASCT = 2000pN,,d;(~RIMA)1”e1’2T”2 (6.4-16a)

According to equations 6.4-16 and -16a, EA and A are somewhat dependent on T. The
calculated values for A,,, usually agree with measured values within an order of mag-
nitude, which, considering the approximations made regarding the cross-sections, is sat-
isfactory support for the general concepts of the theory. SCT provides a basis for the
estimation of rate constants, especially where experimental values exist for related reac-
tions. Then, values of p and E* can be estimated by comparison with the known system.

For the reaction 2HI + H, + I,, the observed rate constant (2k in r,, = 2kc&) is 2.42 X
10e3 L mol-’ s-l at 700 K, and the observed activation energy, EA, is 186 kJ mol-’
(Moelwyn-Hughes, 1957 p. 1109). If the collision diameter, dHI, is 3 5 X lo-lo m for HI
(M = 128), calculate the value of the (“steric”) p factor necessary for’agreement between
the observed rate constant and that calculated from the SCT.