Question

A first order reaction is half completed in 45 minutes. How long does it need 99.9% of the reaction to be completed
(a) 5 hours
(b) 7.5 hours
(c) 10 hours
(d) 20 hours

Answer 1

It will take 7.5 hours for 99.9% of the reaction to be completed

Explanation:

Expression for rate law for first order kinetics is given by:

$t=\frac{2.303}{k}\log\frac{a}{a-x}$

where,

k = rate constant  

t = age of sample

a = let initial amount of the reactant  

a - x = amount left after decay process  

a) for completion of half life:  

Half life is the amount of time taken by a radioactive material to decay to half of its original value.

$t_{\frac{1}{2}}=\frac{0.69}{k}$

$k=\frac{0.69}{45}=0.015min^{-1}$

b) for completion of 99.9 % of reaction  

$t=\frac{2.303}{k}\log\frac{100}{100-99.9}$

$t=\frac{2.303}{k}\times 3$

$t_{99.9}=\frac{6.90}{k}$

$t_{99.9}=\frac{6.90}{0.015}=460min=7.5hours$      (1hour=60min)

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