Question

From a radioactive substance n_1 nuclei decay per second at an instant when total number of nuclei are n_2. Find half-life of the radioactive substance.

Answer 1

From a radioactive substance, $n_{1}$ nuclei decay per second at an instant when total number of particles emitted are $n_{2}$.

Hence, $\frac{dn_{1} }{dt} = - n_{2}$   ---- Equation 1

As radioactive reactions are first order, hence decay of $n_{1}$ is also first order.

    $\frac{dn_{1} }{dt} = - An_{1}$   ----- Equation 2

Equation both equations, we get:

    $n_{2} = A n_{1}$

⇒$A=\frac{n_{2}}{n_{1}}$

Half life of a radioactive substance = $\frac{0.693}{A}$ = $0.693 \frac{n_{1} }{n_{2}}$

Answer 2

Thus the half life of radioactive substance is t1/2 = (n2 / n1)1n2

Explanation:

Using the equation:

−(dN / dt) = λN

We have, n1 = λn2

:. λ = n1 / n2

Now, half-life is given by

t1/2 = 1n2 / λ = 1n2 (n1 / n2)

t1/2 = (n2 / n1)1n2

Thus the half life of radioactive substance is t1/2 = (n2 / n1)1n2