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The reaction 𝐴 → 𝑃 is of order “𝑥”, i.e. 𝑅𝑎𝑡𝑒 = 𝑘[𝐴] 𝑥 . When [𝐴]𝑜 = 0.2 𝑀, the half-life of the reaction is 200 𝑠. When [𝐴]𝑜 = 0.4 𝑀, the half-life of the reaction is 25 𝑠. Calculate the order of the reaction, i.e. solve for “𝑥”. Show detailed workings.
Accepted Answer
The half-life of a reaction is the time it takes for the concentration of a reactant to decrease to half its initial value. For a reaction of order 'x', the half-life is given by:
t1/2 = (0.693) / (k[A]^(x-1))
where:
t1/2 is the half-life
k is the rate constant
[A] is the initial concentration of the reactant
x is the order of the reaction
We are given two sets of data:
1. [A]o = 0.2 M, t1/2 = 200 s
2. [A]o = 0.4 M, t1/2 = 25 s
Let's use these data to solve for 'x'.
From the first set of data:
200 = (0.693) / (k(0.2)^(x-1))
From the second set of data:
25 = (0.693) / (k(0.4)^(x-1))
Dividing the first equation by the second equation, we get:
8 = (0.4)^(x-1) / (0.2)^(x-1)
8 = (2)^(x-1)
Taking the logarithm of both sides (base 2):
log2(8) = x - 1
3 = x - 1
x = 4
Therefore, the order of the reaction (x) is 4.
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