CHAP. 9: CHEMICAL KINETICS [CONTENTS] 299
A + B →k^2 D.9.4.6.2 Kinetic equations.
−
dcA
dτ= k 1 cA+k 2 cAcB, (9.113)−dcB
dτ= k 2 cAcB, (9.114)
dcC
dτ= k 1 cA, (9.115)
dcD
dτ= k 2 cAcB. (9.116)Using the material balance equations
cA=cA0−x−y , cB=cB0−y , cC=cC0+x , cD=cD0+y , (9.117)we may rewrite the kinetic equations (9.115) and (9.116) into the form
dx
dτ= k 1 (cA0−x−y), (9.118)
dy
dτ= k 2 (cA0−x−y)(cB0−y). (9.119)9.4.6.3 Integrated forms of the kinetic equations.
From (9.118) and (9.119) it follows that
y=cB0[
1 −exp(
−k 2
k 1x)]
, (9.120)τ=1
k∫x0dx
cA0−x−cB0[
1 −exp(
−kk^21 x)]. (9.121)
The integral does not have an analytical solution and hence it has to be solved numerically
when calculating the reactants concentrations, see basic course of mathematics.
Note:Wegscheider’s principle does not apply for this type of parallel reactions.