Q: One gram of radium is reduced by 2 milligram in 5 years by α-decay. Calculate the half –life of radium.

Sol: Initial mass = 1 g , t = 5 years

Reduced mass = 2 mg = 2 x 10^{-3} g = 2/1000 g

Remaining mass = 1 – (2/1000) = 998/1000

$\large \frac{N}{N_0} = \frac{998}{1000}$ (∵ mass ∝ number of atoms)

$\large \frac{N}{N_0} = e^{- \lambda t}$ ;

$\large \frac{998}{1000} = e^{- \lambda t}$

$\large \frac{1000}{998} = e^{\lambda t}$

$\large \frac{1000}{998} = e^{ 5 \lambda }$

$\large log_e\frac{1000}{998} = 5 \lambda $

$\large 2.303 log_{10}\frac{1000}{998} = 5 \lambda $

2.303 (3.0000 – 2.9991) = 5 λ

λ = (2.303 × 1 × 0.0009)/5

T_{1/2} = 0.693/ λ

= (0.693 × 5)/(2.303 × 0.0009)

= 1671.7 years.