Question

Write down the laws of radioactive disintegration and deduce an expression for the decay law. Hence define disintegration constant.

Or

State the laws of radioactive decay and hence derive the relation N = N₀e^-λt where the letters have their usual meanings.

Answer 1

Laws of Radioactive disintegration

1. Radioactivity is spontaneous process which does not depend upon external factors.
2. During disintegration either α or ß-particle is emitted. Both are never emitted simultaneously.
3. Emission of α-particle decreases atomic number by two and mass number by 4.
4. Emission of ß-particle increases atomic number by one but mass number remains the same.
5. Emission of γ-ray does not change atomic or mass number.
6. The number of atoms disintegrated per second is directly proportional to the number of radioactive atoms actually present at that instant. This law is called radioactive decay law.

Let at t = 0

N₀ = number of atoms present

N = number of atom left after time t.

dN = number of atoms disintegrated in time dt.

i.e. \(-\frac{dN}{dt} \) ∝ N

or \(-\frac{dN}{dt} \) = λN, ..........(i)

where λ is a constant called disintegration constant and depends upon the nature of the radioactive substance.

Now from, (i), we have

\(\frac{1}{N}\) dN = -λ dt

or ∫ \(\frac{1}{N}\) dN = -λ∫ dt

or log_eN = λt + C, .........(ii)

where C is constant of integration. To determine its value,

since N = N₀ initially,

i.e. when t = 0, N = N₀

log_eN₀ = 0 + C

Substituting the value of C in (ii), we have

log_eN = -λt + log_eN₀

or log_eN - log_eN₀ = -λt

loge \(\frac{N}{N_0}\) = -λt

or \(\frac{N}{N_0}\) = e^-λt

or N = N0 e^-λt ............(iii)

which is the required equation.

Eq. (iii) shows that the radioactive decay occurs according to exponential law and the number of radioactive atoms decreases exponentially. Since N shall become zero only when t becomes infinity, it shows that time taken by a radioactive element to disintegrate completely is infinitely long.

The curve in Fig. shows the exponential decay radioactive substance.

Disintegration constant

In Eq. (iii), if t = 1/λ

Then N = N₀ e^{-λx\(\frac{1}{\lambda}\)} = = N₀e^-1 = \(\frac{N_0}{e}\)

Thus disintegration constant is defined as the time after which the number of radioactive atoms reduce to 1/e times the original number of atoms.