Question

Which sample, A or B shown in figure has shorter mean-life?

Answer 1

Mean (or average) life (ґ): The time for which a radioactive material remains active is defined as the mean (average) life of that material.

• It is defined as the sum of lives of all atoms divided by the total number of atoms.

i.e., `r = "Sum of the lives of all the atoms"/"Total number of atoms" = 1/λ`

• From `N = N_0e^(-λt)` ⇒ In `(N/N_0)/t = - λ`, the slope of the line shown in the graph, i.e., the magnitude of the inverse of slope of `N/N_0` vs t curve is known as mean life (τ).

• From `N = N_0e^(-λt)`, if `t = 1/λ = τ`

⇒ `N = N_0e^(-t) = N_0(1/e)` = 0.37 N₀

= 37% of N₀

i.e., mean life is the time interval in which the number of undecayed atoms (N) becomes `1/e` times or 0.37 times or 37% of the original number of atoms

• It is the time in which the number of decayed atoms (N₀ – N) becomes `(1 - 1/e)` times or 0.63 times or 63% of the original number of atoms.

From the given figure, we can say that

At t = 0, `((dN)/(dt))_A = ((dN)/(dt))_B`

As `(dN)/(dt) = - λN`

⇒ (N₀)_A = (N₀)_B

Now considering any instant t by drawing a line perpendicular to the time axis, we find that `((dN)/(dt))_A > ((dN)/(dt))_B`

Hence, λ_AN_A = λ_BN_B

∴ N_A > N_B ⇒ λ_B > λ_B  ......[Rate of decay of B is slower]

But average life `τ = 1/λ` ⇒ τ_A > τ_B