Question

The half-life of a first-order reaction is $100$ seconds. What is the time required in seconds for $90\%$ completion of the reaction?

• A. $100$
• B. $200$
• C. $333$
• D. $500$

Answer 1

The correct option is C $333$

Half life time $\left(\frac{t_1}{2}\right)=$ 100 sec. (given)

$K=\frac{0.693}{100}$

Time required for 90% completion?

Here let initial concentration $=a$

Concentration decomposed is 90%

$x=\frac{90\times a}{100}$

$=0.9a$

Now,

$t=\frac{2.303}{\frac{0.693}{t_1/2}}\log \frac{a}{a-x}\Rightarrow \frac{2.303}{100}\log \frac{a}{a-0.9a}$

$\Rightarrow \frac{2.303}{0.693}\log \frac{a}{0.1a}\Rightarrow \frac{2.303}{100}\log 10$

$\Rightarrow \frac{2.303}{0.693}\times 100\times \log 10$

$\Rightarrow 339.32\sec$

Approx. $333$ seconds