Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

2.3 Kinetics of reversible first-order reactions (relaxation processes) 21
I
Solution of the rate equations for [A] and [B]:
Returning to the time dependence of the reaction
A
1 À
−1
B (2.56)
we now solve the rate equation
[A]
= − 1 [A] + −1 [B] (2.57)
• Mass balance:
[B] = [A] 0
− [A] (2.58)
• Integration:
[A]
= − 1 [A] + −1 [B] (2.59)
= − 1 [A] + −1 ([A] 0
− [A]) (2.60)
= − ( 1 + −1 )[A]+ −1 [A] 0
(2.61)
y
[A]
( 1 + −1 )[A]− −1 [A] 0
= − (2.62)
• Solution by substitution:
=( 1 + −1 )[A]− −1 [A] 0
(2.63)
y
y
[A] =( 1 + −1 ) (2.64)
[A] =
( 1 + −1 )
(2.65)
= −
( 1 + −1 )
(2.66)
= − ( 1 + −1 ) (2.67)
ln = − ( 1 + −1 ) + (2.68)
• Initial value condition at =0:
y
y
= 0 (2.69)
=ln 0 (2.70)
0
=exp[− ( 1 + −1 ) ] (2.71)