Question

A radioactive substance of 1 g is reduced to 2.1 mg in 5 years by alpha decay. Calculate the half-life of the substance.

Answer 1

yearsnotice that the number of nuclei is proportional to the mass in a given sample

Answer 2

Answer: The half-life of a substance is 0.561 years.

Explanation:

All radioactive decay processes follow first-order reactions.

The integrated rate law equation for first-order kinetics:

$k=\frac{2.303}{t}\log \frac{a}{a-x}$      ...(1)

Given values:

a = initial concentration of reactant = 1 g

a - x = concentration of reactant left after time 't'  = 2.1 mg = 0.0021 g      (Conversion factor: 1 g = 1000 mg)

t = time period = 5 yrs

Putting values in equation 1:

$k=\frac{2.303}{5yr}\log (\frac{1}{0.0021})\\\\k=1.2334 yr^{-1}$

Calculating rate constant for first order reaction using half life:

$t_{1/2}=\frac{0.693}{k}$     ...(2)

$t_{1/2}$ = half life period

k = rate constant = $1.2334 yr^{-1}$

Putting values in equation 2:

$t_{1/2}=\frac{0.693}{1.2334yr^{-1}}\\\\t_{1/2}=0.561yr$

Hence, the half-life of a substance is 0.561 years.