Question

A substance decomposes by following first order kinetics. If $50$% of the compound is decomposed in $120$ minutes, how long will it take for $90$% of the compound to decompose?

Answer 1

Given: Half life $\left(T_{1/2}\right)=120$ minutes
For 1st order reaction, Half life $\left(T_{1/2}\right)=\frac{0.693}{rateconstant\left(K\right)}$
Here, $K=\frac{0.693}{120}=0.00578{min}^{-1}$
Let initial concentration a mol and time $t$ in minutes required for $90\%$ decomposition of the compound
Now, $t=\frac{2.303}{K}\log \frac{a}{a-0.9a}$
$=\frac{2.303}{0.00578}\log 10=398$ minutes
Thus, $398$ minutes are required for $90\%$ decomposition of the compound.