Question

If the half-life of 37Rb is 4.7x101 years, how long would it take for 0.5 grams of a 2 gram sample to radioactively decay?

Answer 1

Answer: The time required will be 19.18 years

Explanation:

All the radioactive reactions follows first order kinetics.

The equation used to calculate half life for first order kinetics:

$k=\frac{0.693}{t_{1/2}}$

We are given:

$t_{1/2}=4.7\times 10^1yrs$

Putting values in above equation, we get:

$k=\frac{0.693}{4.7\times 10^1yr}=0.015yr^{-1}$

Rate law expression for first order kinetics is given by the equation:

$k=\frac{2.303}{t}\log\frac{[A_o]}{[A]}$

where,  

k = rate constant  = $0.015yr^{-1}$

t = time taken for decay process = ?

$[A_o]$ = initial amount of the reactant = 2 g

[A] = amount left after decay process =  (2 - 0.5) = 1.5 g

Putting values in above equation, we get:

$0.015yr^{-1}=\frac{2.303}{t}\log\frac{2}{1.5}\\\\t=19.18yrs$

Hence, the time required will be 19.18 years