Question

What is meant by (i) half life Band (ii) average life of a radioactive substance? Derive expression for them. What is relation between average life and half life.

Answer 1

(i) Half life period: It is the time during which the number of atoms of a radioactive material reduces to half of the original number.

Let N₀ = number of atoms at t = 0 (at the start of the observation)

N = number of atoms after time t

So, N = N₀e^-λt ...........(1)

where λ is constant called disintegration constant

Let T = half life period

So, when t = T, N = -N₀/2.

Substituting in Eq, (1), we have

\(\frac{N_0}{2}\) = N₀e^-λt 

e^λt = 2

λt = log_e2 = 2.303 log10²

= 0.693

or T = \(\frac{0.693}{\lambda}\)

(ii) Average life: The atoms of a radioactive substance are constantly disintegrating and thus the life of every atom is different.

So average life of a radioactive substance is the ratio of sum of lives of all the atoms to the total number of atoms.

Average life = \(\frac{\text {Sum of lives of all the atoms}}{\text {Total numberof atoms}}\)

Suppose dN atoms disintegrate in a time dt, t seconds after the separation of the substance, when the actual number of atoms present is N.

Then \(\frac{dN}{dt}\) = -λN

dN = -λNdt = -λN₀e^-λt dt

where N₀ is the number of atoms in the beginning of time. Each of these dN atoms disintegrate between the time t and t + dt i.e these atoms had a life of t seconds.

∴ Total life of dN atoms = t dN

Hence total life time of all the atoms

Hence average life τ_a of the atom is the reciprocal of the radioactive disintegration constant λ.

Relation between average life and half life

Since average life of a radioactive element is given by

τ_a = \(\frac{1}{\lambda}\) ............(i)

and half life of a radioactive element is given by

T = \(\frac{0.693}{\lambda}\)

Using Eq. (i), we get

T = 0.693 τ_a.