# Rate of Radioactive Disintegration or Decay (Disintegration Rate)

The activity of a radioactive element is measured by its rate of change into the daughter element which depends upon the nature of the element and is independent of external factors like temperature, pressure or the state of chemical combination.

According to the law of radioactive decay given by Lord Rutherford, “the quantity of a radio element which disintegrates in a unit time (rate of disintegration) is directly proportional to the amount present”. Thus, radioactive disintegration may be regarded as a First-order chemical reaction.

Consider a radioactive element A which is disintegrating into another element B i.e. A ———> B.

Let ‘N0‘ be the number of atoms of A in the beginning (when t = 0) and ‘N‘ be the number of atoms at time t, suppose in a small-time dt.

The number of atoms which disintegrate is dN, then-

Rate of Disintegration = -dN/dt

According to law of Radioactive decay-

-dN/dt ∝ N or -dN/dt = λN ………………(i)

Where ‘λ‘ is constant of proportionality known as Decay constant or Radioactive constant or Disintegration constant.

[If N=1, then -dN/dt = λ, thus decay constant (λ) may be defined as the rate of radioactive disintegration of an element when it’s one atom is present. It may also be defined as the fraction of the total number of atoms which disintegrate in one second at any instant of time. The numerical value of λ depends only upon the nature of radioactive substances.]

Equation (i) may also be written as-

-dN/N = λdt; Integrating this equation, we get-

∫ -dN/N = ∫ λdt

or -lnN = λt + C ……………….(ii)

Where ‘C’ is a constant of integration.

When t = 0, N = N0 (Initial Constant)

Therefore, equation (ii) becomes-

-ln N0 = λ X 0 + C

C = -ln N0

On substituting the value of C in equation (ii), we get

-ln N = λt – ln N0

or ln N0 – ln N = λt

or λt = ln N0 – ln N

or λt = ln N0/N

or λ = 1/t ln N0/N

On changing to natural logarithm i.e. to the base 10, we get-

λ = 2.303/t log N0/N

This equation is known as an equation for the rate of radioactive disintegration.