Question

Derive an expression for law of radioactive decay. Define one becquerel (Bq).

Hint:

Radioactive decay is a first-order process, meaning the rate depends only on the amount of "reactant" (\(N\)) you have. This always leads to an exponential decay formula, just like in chemical kinetics or capacitor discharge.

Answer 1

Derivation of the Law of Radioactive Decay: The law of radioactive decay states that the rate of disintegration of radioactive nuclei in a sample is directly proportional to the number of undecayed nuclei present at that instant.
Let \(N\) be the number of undecayed nuclei in a sample at time \(t\), and \(dN\) be the number of nuclei that decay in a small time interval \(dt\). The rate of decay is \( -dN/dt \).
According to the law: \[ -\frac{dN}{dt} \propto N \] \[ -\frac{dN}{dt} = \lambda N \] where \(\lambda\) is the decay constant, a positive constant characteristic of the radioactive substance.
Rearranging the equation to separate the variables: \[ \frac{dN}{N} = -\lambda dt \] To find the number of nuclei remaining after a time \(t\), we integrate this equation. Let \(N_0\) be the initial number of nuclei at time \(t=0\), and \(N\) be the number of nuclei at time \(t\). \[ \int_{N_0}^{N} \frac{dN}{N} = \int_{0}^{t} -\lambda dt \] \[ [\ln N]_{N_0}^{N} = -\lambda [t]_0^t \] \[ \ln N - \ln N_0 = -\lambda(t - 0) \] \[ \ln\left(\frac{N}{N_0}\right) = -\lambda t \] Taking the exponential of both sides: \[ \frac{N}{N_0} = e^{-\lambda t} \] \[ N(t) = N_0 e^{-\lambda t} \] This is the mathematical expression for the law of radioactive decay, which shows that the number of undecayed nuclei decreases exponentially with time.
Definition of Becquerel (Bq): One becquerel is the unit of activity of a radioactive sample in the SI system. It is defined as one disintegration (or decay) per second. \[ 1 \, \text{Bq} = 1 \, \text{decay/second} \]