Question

Derive an expression for the law of radioactive decay.

Derive the decay law of radioactivity.

Answer 1

At any instant, the rate of radioactive disintegration is directly proportional to the number of nuclei of the radioactive element present at that time.

Let N₀ be the number of nuclei at time t = 0, and N be the number of nuclei present at time t.

From the law of radioactive decay,

`(dN)/(dt) ∝ N`

∴ `(dN)/(dt) = -λN`

which is the differential form of the law.

∴ `(dN)/N = -λdt`   ...(1)

where λ is a proportionality constant known as the radioactive decay constant or disintegration constant. It is the constant for a specific radioactive element. The negative sign indicates that N decreases as t increases. Integrating Equation (1),

`int_(N_0)^N (dN)/(N) = int_0^t λdt = - λ int_0^tdt`

∴ `log_eN - log_eN_0 = -λχint`

∴ `log_e(N/N_0) = -λt`

∴ `N/N_0 = e^(-λt)`

∴ N = N₀ e^−λt

This is the exponential version of the law of radioactive decay. It indicates that the number of nuclei declines exponentially with time.