**Understanding Half-Life and Its Derivation:****Concept of Half-Life:**Half-life is the time required for a quantity to reduce to half its initial value. It is commonly associated with the decay of radioactive materials, but it also applies to various exponential decay processes, such as chemical reactions and pharmacokinetics.**Deriving Half-Life Formula for Zero-Order Decay:**A zero-order reaction is one where the rate of reaction is constant and independent of the concentration of the reactant. The rate law for a zero-order reaction can be expressed as:\[ \frac{d[A]}{dt} = -k \]Where:- \([A]\) is the concentration of the reactant- \(k\) is the rate constant- \(t\) is timeRearranging and integrating from \([A]_0\) to \([A]\) and from 0 to \(t\), the expression becomes:\[ [A] = [A]_0 - kt \]To find the half-life (\(t_{1/2}\)), set \([A] = \frac{[A]_0}{2}\):\[ \frac{[A]_0}{2} = [A]_0 - kt_{1/2} \]Solving for \(t_{1/2}\), we get:\[ t_{1/2} = \frac{[A]_0}{2k} \]This formula indicates that for a zero-order reaction, the half-life is directly proportional to the initial concentration \([A]_0\) and inversely proportional to the rate constant \(k\).

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 indicates the time required for half of the radioactive atoms in a sample to decay. However, it can also be used to describe any quantity which follows an exponential decay.In the context of a batch system with zero-order decay, the rate of decay is independent of the current amount. This means that the quantity decreases at a constant rate over time. The mathematical expression for this is: N(t) = N0 - kt, where N(t) is the remaining quantity at time t, N0 is the initial quantity, k is the decay constant, and t is the time.To derive the half-life formula, we set N(t) to half of the initial quantity, i.e., N0/2. Substituting this into the equation gives us: N0/2 = N0 - kt.Rearranging the equation to solve for t gives us: t = (N0 - N0/2) / k = N0 / (2k). This is the formula for the half-life in a system with zero-order decay.Half-life is the time required for a quantity to reduce to half of its initial value. In a system with zero-order decay, the half-life can be calculated using the formula t = N0 / (2k), where N0 is the initial quantity and k is the decay constant.