Question

Samples of two radioactive nuclides A and B are taken. λ_A and λ_B are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?

1. Initial rate of decay of A is twice the initial rate of decay of B and λ_A = λ_B.
2. Initial rate of decay of A is twice the initial rate of decay of B and λ_A > λ_B.
3. Initial rate of decay of B is twice the initial rate of decay of A and λ_A > λ_B.
4. Initial rate of decay of B is the same as the rate of decay of A at t = 2h and λ_B < λ_A.

Options

• a and c

• a and d

• b and d

• a and b

Answer 1

b and d

Explanation:

Law of radioactive disintegration: According to Rutherford and Soddy law for radioactive decay is as follows:

At any instant, the rate of decay of radioactive atoms is proportional to the number of atoms present at that instant.” i.e.

dN/dt ∞ N ⇒ dN/dt = – λN

it can be proved that N = N₀e^–λ1

In terms of mass M – M₀e^–λ1

where N = Number of atoms that remain undecayed after time t,

N₀ = Number of atoms present initially (i.e., at t = 0),

M = Mass of radioactive nuclei at time t,

M₀ = Mass ofradioactive nuclei at time t = 0,

N₀ – N= Number of the disintegrated nuclei in time t,

dN/dt= rate of decay, λ = Decay constant or disintegration constant or radioactivity constant or Rutherford Soddy’s constant or the probability of decay per unit time of a nucleus.

The samples of the two radioactive nuclides A and B can simultaneously have the same decay rate at any time if the initial rate of decay of A is twice the initial rate of decay of B and λ_A > λ_B.

Also, when the initial rate of decay of B is the same as the rate of decay of A at t = 2h and λ_B < λ_A.