Question

A sample contains 25% parent isotope and 75% daughter isotopes. If the half-life of the parent isotope is 72 years, how old is the sample?

Answer 1

The radioactive decay obeys first order kinetics

the rate law expression for radioactive decay is

$ln\frac{[A_{0}]}{[A_{t}]}=kt$

Where

A0 = initial concentration

At = concentration after time "t"

t = time

k = rate constant

For first order reaction the relation between rate constant and half life is:

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

Let us calculate k

k = 0.693 / 72 = 0.009625 years⁻¹

Given

At = 0.25 A0

$ln(\frac{A0}{0.25A0})=0.009625  X time$

time = 144 years

So after 144 years the sample contains 25% parent isotope and 75% daughter isotopes**

Simply two half lives

Answer 2

144 years old is the sample which is containing  25% parent isotope and 75% daughter isotopes.

What is half life period?

Half life period is the time in which the initial concentration of reactant becomes half.

The radioactive decay always obeys first order reaction and half life time of parent isotope is calculated as:

K = 0.693 / t, where

t = half life time = 72 years

K = rate constant

So, K = 0.693 / 72 = 0.009625 years⁻¹

The rate law expression for radioactive decay is expressed as:

KT = ln (A₀ / A), where

A₀ = initial concentration,

A = concentration at time T = 0.25A₀ (given)

On putting values in the above equation, we get

(0.009625) T = ln (A₀ / 0.25A₀)

T = ln (1 / 0.25) / 0.009625

T = 144 years.

Hence, 144 years old is the isotope sample.

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