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336
Rate = –
d[A]
= k[A]
dt
...(i)
(a) Where [A] is the concentration of the reactant A that remains unreacted at time
t and –d[A] / dt is the rate of the reaction measured at time t at which A is converted
to the products. By rearranging and integrating the equation (i) between the limits
[A] = [A]0 at t = 0 and [A] = [A]t at t = t, we write,
[A] t
∫
[A]0
t
d[A]
–k dt
A =
∫
0
where [A]0 is the initial concentration of A at t = 0 and [A]t is the concentration of A
that remains unreacted at t.
[A]
On performing the integration, we get {In[A]}[A]0t = –k(t) 0t
Or ln [A]t – ln[A]0 = –kt
[A] t
and ln [A] = –kt
0
[A] 0
Hence, kt = ln [A]
t
1
...(ii)
[A] 0
that is kt = t ln [A]
t
Converting ln to log10 the integrated rate law becomes
k=
[A] 0
2.303
log10
t
[A] t
...(iii)
Note :
The rate law equation can also be written in the following alternative forms :
[A] t
ln
(a) The equation (ii),
[A] 0 = –kt
can be written by taking antilog of both sides as
[A] t
–kt
[A] 0 = e
This is called as exponential form
of first order reaction.
Or
[A] t = [A]0e–kt
(b) If we let ‘a’ mol dm–3 (that is M) as the initial concentration of A at t = 0 and
x as the concentration of A that decreases during time t from the beginning of the
Unique Solutions ®
S.Y.J.C. Science - Chemistry - Part I
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