Question

Rutherford and Soddy’s laws of radioactivity explain the rate of decay of radioactive material.

1. Arrive at the expression for the number of radio active atoms of a radioactive material remaining after an interval of time.

2. Draw the curve showing the variation of log \(\Big(\frac{N}{N_0}\Big)\) with time.

3. Two radioactive substances P and Q have half life 6 months and 3 months respectively. Find the ratio of the activity of these two materials after one year.

Answer 1

1. According to Law of Radioactive decay,

\(\frac{dN}{dt}=λN\)  \(\frac{dN}{dt} = -λdt\)

Integrating

In N = -λt + C ………(1)

C is the constant of integration. To get value of C, let us assume that initially (t = 0) the number of nuclei be N₀.

∴ C = In N₀

Substituting for C in equation (1) we get,

In N – In N₀ = -λt

In \(\frac{N}{N_0}\) = -λt

\(\frac{N}{N_0}\) e^-λt

N = N₀e^-λt

3. Activity

R = λN

R₁ = λ₁ N₁ 

R₂ = λ₂ N₂

λ₁ = \(\frac{0.693}{T_{1/2}}=\frac{0.693}{6}\)

λ₂ = \(\frac{0.693}{T_{1/2}}=\frac{0.693}{3}\)

N₁ = \(\frac{N_0}{2},N_2=\frac{N_0}{4}\)

\(\frac{R_1}{R_2}=\frac{λ_1N_1}{λ_2N_2}\)

R₁ : R₂ = 1: 1