How is the decay constant related to the half-life of a radioactive substance?

The decay constant is inversely proportional to the half-life of a radioactive substance.

The decay constant, denoted by λ, is a measure of the rate at which a radioactive substance decays. It is defined as the probability of decay per unit time, and has units of s^-1. The higher the decay constant, the faster the substance decays.

The half-life of a radioactive substance is the time taken for half of the original number of radioactive nuclei to decay. It is denoted by t1/2 and has units of time. The half-life is a characteristic property of a radioactive substance and is independent of the amount of substance present.

The relationship between the decay constant and the half-life can be derived using calculus. It can be shown that the decay constant is inversely proportional to the half-life, i.e. λ = ln2/t1/2. This means that substances with a shorter half-life have a higher decay constant, and vice versa.

The relationship between the decay constant and the half-life is important in many applications of radioactive decay, such as radiometric dating and nuclear power generation. It allows us to predict the rate of decay of a substance and to calculate the amount of substance remaining after a certain time.