Question

how are half life and radioactive decay related

Answer 1

Answer : Half life and radioactive decay are inversely proportional to each other.

Explanation :

The mathematic relationship between the half-life and radioactive decay :

$N=N_oe^{-\lambda t}$              ................(1)

where,

N = number of radioactive atoms at time, t

$N_o$ = number of radioactive atoms at the beginning when time is zero

e = Euler's constant = 2.17828

t = time

$\lambda$ = decay rate

when $t=t_{1/2}$ then the number of radioactive decay become half of the initial decay atom i.e $N=\frac{N_o}{2}$.

Now substituting these conditions in above equation (1), we get

$\frac{N_o}{2}=N_oe^{-\lambda t_{1/2}}$

By rearranging the terms, we get

$\frac{1}{2}=e^{-\lambda t_{1/2}}$

Now taking natural log on both side,

$ln(\frac{1}{2})=-\lambda \times t_{1/2}$

By rearranging the terms, we get

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

This is the relationship between the half-life and radioactive decay.

Hence, from this we conclude that the Half life and radioactive decay are inversely proportional to each other. That means faster the decay, shorter the half-life.