Question

A first order reaction is half-completed in $45$ minutes. How long does it need for $99.9\%$ of the reaction to be completed?

• A. $20$ hours
• B. $10$ hours
• C. $7\frac{3}{10}$ hours
• D. $5$ hours

Answer 1

The correct option is C $7\frac{3}{10}$ hours

$t_{1/2}=45$ minutes

For 1st order reaction $t_{1/2}=\frac{0.693}{k}\therefore k=\frac{0.693}{45}{min}^{-1}=0.0154{min}^{-1}$

Again for 1st order reaction we have

$k=\frac{2.303}{t}\log \left[\frac{100}{100-99.9}\right]$

(considering original concentration to be 100)

$t=\frac{2.303}{k}\log \left[\frac{100}{0.1}\right]$

$t=\frac{2.303}{0.0154}\log 1000$

$t=\frac{2.303}{0.0154}\times 3=448.636minutes$

$t=7\frac{3}{10}hours.$

Hence, the correct option is $C$