For the first order reaction the Rate constant
is given 5 min, hence
Let initial concentration ‘a’ of A be 100 mol L–1. To reach 25% of initial concentration means, (a – x) = 25 mol L–1.
Or
or
Ostwald Isolation Method: In this method, the concentration of all the reactants are taken in large excess except that of one. The concentration change only for this reactant is significant as other are so much in excess that practically there is no change in their concentrations. The constant terms may be combined with the rate constant and we may write
The value of ‘a’, i.e., the order of reaction with respect to A can be determined by the methods given above.
For first order reaction
and
......(ii)
Dividing (i) by (ii), we get
or
t/s |
0 |
100 |
200 |
300 |
p/pa |
4.00 x 103 |
3.50 x 103 |
3.00 x 103 |
2.5 x103 |
0–100 s, rate = – [3.50 – 400] x 103 Pa/100 s = 5 Pa / s
100-200 s, rate = – [3.00 – 3.50] x 103 Pa/100 s = 5 Pa / s
200-300 s, rate = – [2.50 – 3.00] x 103 Pa / 100 s = 5 Pa / s
We notice that the rate remains constant, therefore, reaction is of zero order.
k = rate = 5 Pa / s
t1 / 2 = initial concentration or pressure/2K
= 4.00 x 103 Pa / 2 x 5 Pa s–1 = 400 s.
Half-life of a reaction: The half-life of a reaction represented as t1/2 is the time required for the reactant concentration to drop to one half of its initial value. Consider the zeroth order reaction.
R → Products
Integrate Zeroth order rate equation.
[R] = – kt + [R]0
where at time t = t1/2, the fraction of [R] that remains [R] / [R]0, therefore above equation can be written
[R]0/2 = – k t1/2 + [ R]0
k1/2 = [R]0 /2
for the first order integrated rate equation
at time
The half life depends on reactant concentration in different order of reactions as follows.
For zero order raction t1/2 ∝ [R]0. For first order reaction t1/2 is independent of R0, for second order reaction t1/2 ∝ 1/[R]0.
For nth order reaction t1/2 ∝ 1/[R]0n–1.
Catalytic decomposition of nitrous oxide by gold at 900°C at an initial pressure of 200 mm was 50% in 53 minutes and 73% in 100 minutes.
(i) What is the order of reaction?
(ii) How much will it decompose in 100 minutes at the same temperature but at an initial pressure of 600 mm?
(i) Let [A]0 = 100.
Then, [A]t at 53 minutes = (100 – 50) = 50 and [A], at 100 minutes = (100 – 73) = 27.
Substituting t and concentration values in the integrated rate equation for first-order reaction.
At t = 53 min,
At t = 100 min,
Since the value of k is constant, the order of reaction is 1.
(ii) For a first order reaction, the time required to complete any fraction is independent of the initial concentration of reactant.
∴ 73% of N2O will decompose when the initial concentration is 600 mm which corresponds to a pressure of