2004/11/13 | 质量酌定律[Law of Mass Action]
类别(℃〖化学〗) | 评论(0) | 阅读(152) | 发表于 22:02
Given a chemical equation of the form
(1)

then
(2)

(3)

In equilibrium,
(4)

(5)

For a chemical reaction
(6)

the law of mass action gives the rate as
(7)

where brackets denote concentrations and the reverse reaction is assumed to be negligible. Denote initial values with a 0 subscript and define
(8)

(9)

(10)

Since the amounts remaining of A and B after N molecules of C have been formed are and , we can rewrite (7) as

(11)

This is a nonlinear differential equation, but it can be separated and directly integrated
(12)

This integral is of the form
(13)

for q < 0, where , , and . The "discriminant" q is


(14)

(15)

Plugging the values for our problem into (13) gives
(16)

where C is a constant. Simplifying,
(17)

In order to find the value of C, we use the initial condition that , so plugging into (17) gives
(18)

Multiplying (17) through and exponentiating gives
(19)

Plugging (18) into (19) then gives
(20)

Solving for N,
(21)



gives the answer
(22)

Checking, we see that does indeed give 0, and gives .
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