KEY2CHEM

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.