ΓΌ For First
Order Reaction: First
order reactions are reactions in which the rate of reaction is proportional to first
power of the concentration of reactants.
Consider a first
order reaction R → P
The rate
expression for the above reaction is
r = – d[R] ………………. (1)
dt
Rate law for
the above reaction is
r= k[R]1
= k ………………… (2)
From equations (1) & (2), we can
write
k [R] = – d[R]
dt
d[R] = – k dt
[R]
On integrating the above equation, we get
In [R] = – kt + C ........ (3)
Where C is the constant of integration.
To calculate the value of C, consider the initial conditions. i.e., when t=0,
[R] = [R]0
Substitute these values in equation (3)
In [R0] = – k ×
0 + C
C = ln [R0]
Substituting C in equation (3), we get
ln[R] = - kt + ln[R0] ………… (4)
Rearranging above equation we get
kt = ln[R0] – ln[R]
k = 1
ln [R0]
t [R]
|
k =2.303
log[R0]
t [R]
|
………… (A)
At time t1
from equation (4)
In [R1]
= – kt1 + ln [R2]
At time t2
ln [R1]
= – kt2 + ln [R2]
where, [R1]
and [R2] are the concentrations of the reactants at time t1
and t2 respectively.
Subtracting
ln [R1]
– ln [R2] = – kt1 – (–kt2)
ln [R1] = k (t2
– t1)
[R2]
Comparing
equation (2) with y = mx + c, if we plot In [R] against t, we get a straight
line with slope = – k and intercept equal to ln [R0].
The first
order rate equation (A) can also be written in the form
Log [R0] = kt
[R]
2.303
For Example:
· Hydrogenation of ethene:
C2H4(g) + H2(g)
→ C2H6(g); r =
k[C2H4]
· All natural and artificial radioactive decay
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