Secular Equilibrium:
Secular equilibrium occurs in a situation where the daughter nuclide is also radioactive and has a long half-life compared to the parent nuclide. In this case, the rate of decay of the parent nuclide is much faster than the rate of decay of the daughter nuclide. As a result, the activities of both nuclides become equal and remain constant over time.

Transient Equilibrium:
Transient equilibrium occurs when the daughter nuclide is also radioactive, but has a short half-life compared to the parent nuclide. In this case, the rate of decay of the parent nuclide is much faster than the rate of decay of the daughter nuclide. As a result, initially, the activity of the daughter nuclide increases until it reaches a maximum value and then decreases as it approaches secular equilibrium.

Radioactive Decay Law:
The radioactive decay law states that the rate of decay of a radioactive substance is proportional to the number of radioactive nuclei present. Mathematically, it can be expressed as:
N(t) = N0 * e^(-λt)
where N(t) is the number of radioactive nuclei at time t, N0 is the initial number of radioactive nuclei, λ is the decay constant, and e is the base of the natural logarithm.

Half-Life:
Half-life is the time taken for half of the radioactive nuclei in a sample to decay. It is a constant characteristic of each radioactive nuclide and is used to determine the rate of decay of a substance.

Decay Constant:
The decay constant (λ) is a measure of how quickly a radioactive substance decays. It is inversely proportional to the half-life of the substance, with a larger decay constant indicating a faster rate of decay.