Question

A first-order reaction has a half-life of 20.0 minutes. Starting with 1.00 × 1020 molecules of reactant at time t = 0, how many molecules remain unreacted after 100.0 minutes?

Answer 1

Half-life time of a reaction is time at which reactant concentration becomes half of its initial value.

Half-life of the first order reaction is 20 min. Rate constant can be calculated as follows:

$K=\frac{0.6932}{t_{1/2}}=\frac{0.6932}{20 min}=0.03466 min^{-1}$

The rate expression for first order reaction is as follows:

$k=\frac{2.303}{t}log\frac{A_{0}}{A_{t}}$

initial number of molecules of reactant are $10^{20}$, time is 100 min thus, putting the values to calculate number of reactant at time 100 min,

$0.03466 min^{-1}=\frac{2.303}{100 min}log\frac{[10^{20}]}{A_{t}}$

On rearranging,

$\frac{10^{20}}{A_{t}}=31.988$

Or,

$A_{t}=3.13\times 10^{18}$

Therefore, number of molecules unreacted will be $3.13\times 10^{18}$