Question

Show that the number of nuclei of a radioactive material decreases exponentially with time.

Answer 1

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

From the law of radioactive decay, `(dN)/dt ∝ N`

∴ `(dN)/dt = - λN`    ... (1)

which is the differential form of the law.

∴ `(dN)/N - λdt`   ... (2)

where λ is a proportionality constant also known as the disintegration constant or the radioactive decay constant. For a certain radioactive element, it is a constant. N drops as t grows, as indicated by the minus sign. Combining Equation (1),

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

∴ log_e N − log_e N₀ = − λt

∴ `log_e (N/(N_0)) = − λt`   ... (3)

∴ `N/(N_0) = e^(-λt)`

∴ N = N₀ e^−λt     ... (4)

The law of radioactive decay has an exponential form. It demonstrates how the quantity of nuclei diminishes exponentially over time.