Define the term “half-life” and derive a relationship between the half-life and λ

The term "half-life" is commonly used in nuclear physics and chemistry, and it refers to the time required for a quantity to reduce to half of its initial value. The term is commonly used in the context of nuclear decay, where it refers to the time required for half of the radioactive atoms in a sample to decay.

The decay of radioactive atoms is a random process, but it occurs at a rate that is characteristic of the isotope. The decay rate is often described in terms of the half-life of the isotope.

The decay rate is mathematically expressed by the first-order decay equation:

N = N0 * e^(-λt)

where:

• N is the quantity of the substance at time t,
• N0 is the initial quantity of the substance,
• λ (lambda) is the decay constant,
• t is the time.

The half-life (T) is related to the decay constant by the equation:

T = ln(2) / λ

This equation is derived from the first-order decay equation by setting N = N0/2 (since half of the substance has decayed) and solving for t (which we call T when N = N0/2).