Law of Radioactive Decay

Law of Radioactive Decay

According to law of radioactive decay, the rate of decay of a radioactive substance is directly proportional to the number of atoms present.
Hence, − dN/dt ∝ N
or, − dN/dt = λ N     ----- Equation-1
where, λ is proportionality constant called decay constant is characteristic of the emitter measure the rate of decay. The negative sign indicates that the number of radioactive nuclei decreases over the time.
or, λ = (− dN/dt) / N

Thus, we can say that the decay constant is the ratio of the number of atoms of a radioactive substance in unit time to the number of atoms present. It follows the first order kinetics and is expressed in per second (sec⁻).
From equation-1, we have-
− dN/dt = λ N
or, dN/dt = − λN
on integrating this equation we get-
∫dN/dt = − ∫λ N
or, ln N = − λt + I     ----- Equation-2
where I is integrating constant.
When t = t_o then N = N_o,
Hence, ln N_o = I

Putting the value of I in equation-2, we get-
ln N = − λt + ln N_o
or, ln N − ln N_o = − λt
or, ln N/N_o = − λt
or, ln N_o/N = λt
or,

If t = t_1/2 then, N = N/2
Putting the value of t and N in the above equation, we get-

Hence, the half life of radioactive substance is inversely proportional to its decay constant.

A radio element has activity of 630 and 610 counts per minute at times t₁ and t₂ differing by one hour. What is its half life?

HINTS:
Radioactive decay follows first order kinetics, hence-