**Half-life **

The rate constant of a reaction is specific for a particular reaction at a given temperature. The rate constant is also related to the half-life, which is the time required for the concentration of the reactant to decrease to exactly one half of its initial value. For a first-order process, the half-life is independent of concentration and is expressed by the equation \(t_{1/2} = \frac{ln \;2}{k}\). Radioactive decay is an example of a first-order process, and the half-life of decay processes for radioactive isotopes can be very important for their safe disposal.

**Example 1.**

The rate constant for a first order process is \(0.00320\text{ s}^{-1}\). What is the half-life for this process?

A. \(217\text{ s}\)

B. \(320\text{ s}\)

C. \(0.00223\text{ s}\)

*Solution*

A. \(217\text{ s}\)

\(t_{1/2} = \frac{ln \;2}{k} = \frac{ln\; 2}{0.00320\text{ s}^{-1}} = 217\text{ s}\)

**Example 2.**

The half-life of the decay of the \(^{14}\text{C}\)isotope (used in carbon dating) is \(5730\text{ years}\). If the decay process follows first order kinetics, what is the rate constant for \(^{14}\text{C}\) decay?

A. \(1.75\times 10^{-3}\text{ yr}^{-1}\)

B. \(1.21\times 10^{-4}\text{ yr}^{-1}\)

C. \(5.25\times 10^{-5}\text{ yr}^{-1}\)

*Solution*

B. \(1.21\times 10^{-4}\text{ yr}^{-1}\)

\(\begin{align} t_{1/2} &= \frac{ln \;2}{k}\\ k &= \frac{ln \;2}{t_{1/2}}\\ k &=\frac{ln \; 2}{5730\text{ years}} = 1.21\times 10 ^{-4}\text{ yr}^{-1}\end{align}\)

**Example 3.**

Which statement about the half-life of a first order reaction is true?

A. The half-life is not related to the rate constant.

B. The half-life does not depend on the reactant concentration.

C. The half-life does not depend on the reaction temperature.

*Solution*

B. The half-life does not depend on the reactant concentration.

The half-life of a first order process (\(t_{1/2} = \frac{ln \;2}{k}\)) does not depend on the reactant concentration. Because the half-life is related to the rate constant, and the rate constant is dependent on reaction temperature, the half-life also depends on the reaction temperature.