An organic compound undergoes first order decomposition. If the time taken for the $60 %$ decomposition is $540 s$ , then the time required for $90 %$ decomposition will be is _____ s. (Nearest integer).
Given : $\ln 10 = 2.3; \log 2 = 0.3$

Question

An organic compound undergoes first order decomposition. If the time taken for the $60 %$ decomposition is $540 s$ , then the time required for $90 %$ decomposition will be is _____ s. (Nearest integer).

Given : $\ln 10 = 2.3; \log 2 = 0.3$

TestHub · Accepted Answer

For the First order reaction, integrated rate equation can be written as follows,

$t = \frac{1}{k} \ln \frac{a_{0}}{a_{t}} t = time k = rate constant a_{0} = intial concentration a_{t} = concentration at time t$

$\frac{t_{1}}{t_{2}} = \frac{\frac{1}{k} \ln \frac{a_{0}}{0.4 a_{0}}}{\frac{1}{k} \ln \frac{a_{0}}{0.1 a_{0}}}$

$\frac{540}{t_{2}} = \frac{\ln \frac{10}{4}}{\ln 10}$

$\frac{540}{t_{2}} = \frac{\log 10 - \log 4}{\log 10}$

$\frac{540}{t_{2}} = \frac{1 - 0.6}{1}$

$\Rightarrow \frac{540}{t_{2}} = 0.4$

$\Rightarrow t_{2} = \frac{540}{0.4} = 1350 \sec$

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