As you work through this section, if you find that you need a bit more background material to help you understand the topics at hand, you can consult “Chemistry: The Molecular Science” (5th ed. Moore and Stanitski) Chapter 18-3 through 18-9, and/or Chapter 9.13-9.15 in the Additional Reading Materials section.

D20.1 Half-Life of a Reaction

The half-life (t_1/2) of a reaction is the time required for the  concentration of a reactant to be reduced to half its initial value. In each succeeding half-life, the remaining concentration of the reactant is halved.

First-Order Reactions

An equation for the half-life of a first-order reaction can be derived from the integrated rate law:

If we set the time t equal to the half-life, t_1/2, then [latex][A]_{t_{1/2}} = \frac{1}{2}[A]_0[/latex].  Therefore:

Thus:

The half-life of a first-order reaction is inversely proportional to the rate constant k, and, conveniently, is independent of the concentration of the reactant. A fast reaction (shorter half-life) has a larger k; a slow reaction (longer half-life) has a smaller k.

Example 1

Calculation of a First-order Rate Constant using Half-Life

Some dyes can be decomposed by sunlight. Use the data in the figure to determine whether the reaction is first order and to calculate the rate constant for decomposition of a green dye dissolved in water and left outdoors in sunlight at 40 °C.

Solution

It takes 6 hours for the 1.000-M solution to decrease to 0.500 M; it takes another 6 hours for the concentration to reach 0.250 M, and each successive halving also takes 6 h. Therefore the half-life for the decomposition is 6 h or 2.16 × 10⁴ s. Because the half-life is independent of concentration, it is reasonable to assume the reaction is first order. (Subsequent sections in today’s work show that for other reaction orders, half-lives vary with concentration.) To find the rate constant at 40 °C:

[latex]\begin{array}{rcl} t_{1/2} & = & \frac{0.693}{k} \\[0.5em] k & = & \frac{0.693}{t_{1/2}} = \frac{0.693}{2.16\;\times\;10^4\text{s}} = 3.21\;\times\;10^{-5}\text{s}^{-1} \end{array}[/latex]

Check Your Learning

The first-order radioactive decay of iodine-131 exhibits a rate constant of 0.138 d⁻¹. What is the half-life for this decay?

Answer:

5.02 d.

Second-Order Reactions

The equation for half-life of a second-order reaction can be derived this way:

When t = t_1/2 and [latex][A]_{t_{1/2}} = \frac{1}{2}[A]_0[/latex],

Thus,

For a second-order reaction, t_1/2 is inversely proportional to the concentration of the reactant, and hence is not constant throughout the reaction. It increases as the reaction proceeds due to decreasing concentration of reactant. Consequently, unlike the situation with first-order reactions, the rate constant of a second-order reaction cannot be calculated directly from the half-life unless the initial concentration is known.

Zeroth-Order Reactions

For a zeroth-order reaction,

When t = t_1/2 and [latex][A]_{t_{1/2}} = \frac{1}{2}[A]_0[/latex]:

and

The half-life of a zeroth-order reaction decreases as the initial concentration decreases.

Equations for both differential and integrated rate laws and the corresponding half-lives for zero-, first-, and second-order reactions are summarized in Table 1.

D20.2 Radioactive Half-Lives

Most radioactive decay follows first-order kinetics, and hence radioactive decay has a characteristic, constant half-life. An isotope’s half-life allows us to determine how long a sample of a useful isotope will be available, and how long a sample of an undesirable or dangerous isotope must be stored before it decays to a sufficiently low radiation level.

For example, cobalt-60, an isotope that emits gamma rays used to treat cancer, has a half-life of 5.27 years (Figure 1). In a given cobalt-60 source, since half of the [latex]_{27}^{60}\text{Co}[/latex] nuclei decay every 5.27 years, both the amount of material and the intensity of the radiation emitted is cut in half every 5.27 years. (Note that for a given substance, the intensity of radiation produced is directly proportional to the rate of decay of the substance and the quantity of the substance.)

Radioactive decay occurs in atomic nuclei and therefore does not depend on chemical bonding or collisions of molecules. Each nuclide has a particular probability that a nucleus will decay in a given time. The more nuclei there are, the greater the number of nuclear decays during that time, so the rate of radioactive decay is proportional to the number of atomic nuclei present, N. The rate constant, k, for nuclear decay is called the decay constant. Hence, the rate of radioactive decay is:

The decay rate in nuclear decays per second is called the activity of the radioactive sample. The decay constant is related to the half-life by,

The first-order equations relating amount, N, and time are:

where N₀ is the initial number of nuclei or moles of the isotope, and N_t is the number of nuclei/moles remaining at time t.

D20.3 Radiometric Dating

Several radioisotopes have half-lives and other properties that make them useful for radiometric dating, the process of determining the date of origin of objects such as geological formations, archaeological artifacts, or formerly living organisms. Radiometric dating  has been responsible for many breakthrough scientific discoveries about the geological history of the earth, the evolution of life, and the history of human civilizations. We will explore some of the most common types of radiometric dating and how the particular isotopes work for each type.

Radiometric Dating Using Carbon-14

Radiocarbon dating or carbon-14 dating can provide reasonably accurate dating of carbon-containing substances up to  about 50,000 years old.

Naturally occurring carbon consists of three isotopes: [latex]_{\;\;6}^{12}\text{C}[/latex], which constitutes 99% of the carbon on earth; [latex]_{\;6}^{13}\text{C}[/latex], about 1% of the total; and trace amounts of [latex]_{\;\;6}^{14}\text{C}[/latex]. Carbon-14 is formed in the upper atmosphere by the reaction of nitrogen atoms with neutrons:

All isotopes of carbon react with oxygen to produce CO₂. In the atmosphere, the ratio of ¹⁴CO₂ to ¹²CO₂ depends on the ratio of ¹⁴C to ¹²C, and the natural abundance of ¹⁴C is approximately 1 part per trillion. Analysis of carbon-containing gas samples trapped in Greenland ice sheets provides data on the ¹⁴C:¹²C ratio in the past, and the ratio has been nearly constant.

The incorporation of ¹⁴CO₂ and ¹²CO₂ into plants is a regular part of photosynthesis, which means that the ¹⁴C:¹²C ratio found in a living plant is the same as the ¹⁴C:¹²C ratio in the atmosphere. But when the plant dies, it no longer traps additional carbon through photosynthesis. Because ¹²C is a stable isotope and does not undergo radioactive decay, its concentration in the plant does not change. However, ¹⁴C decays by β emission with a half-life of 5730 years:

Thus, the ¹⁴C:¹²C ratio gradually decreases after the plant dies. The decrease in the ratio over time provides a measure of the time that has elapsed since the death of the plant. Throughout their lifetimes organisms that feed on plants also take up ¹⁴C with abundance the same as the atmosphere; when they die the ¹⁴C decays and the ¹⁴C:¹²C ratio gradually decreases. Figure 2 depicts this process.

For example, if the ¹⁴C:¹²C ratio in a wooden object found in an archaeological dig is half what it is in a living tree, this indicates that 5730 years have elapsed since the death of the tree from which the wooden object was made. Highly accurate determinations of ¹⁴C:¹²C ratios can be obtained from very small samples (as little as a milligram) by the use of a mass spectrometer.

Example 3

Radiocarbon Dating

The first radiocarbon dating of the Dead Sea Scrolls was in 1950. A tiny piece of paper (produced from formerly living plant matter) taken from the Dead Sea Scrolls was found to have an activity of 10.8 disintegrations per minute per gram of carbon. If the initial ¹⁴C activity was 13.6 disintegrations/min/g of C, estimate the age of the Dead Sea Scrolls.  (The half-life of ¹⁴C is 5730 years)

Solution

The activity, or rate of decay, is proportional to the amount of radioactive ¹⁴C left in the paper (decay rate = kN), so we can substitute the decay rates for the amount of ¹⁴C, N, in the relationship:

[latex]t = -\frac{1}{k}\text{ln}(\frac{N_t}{N_0})\;{\longrightarrow}\;t = -\frac{1}{k}\text{ln}(\frac{\text{Rate}_t}{\text{Rate}_0})[/latex]

where the subscript 0 represents the time when the plants were cut to make the paper, and the subscript t represents the current time.

The decay constant can be determined from the half-life of ¹⁴C:

[latex]{k} = \frac{\text{ln}\;2}{t_{1/2}} = \frac{0.693}{5730\;\text{y}} = 1.21\;\times\;10^{-4}\text{y}^{-1}[/latex]

Substituting and solving, we have:

[latex]t = -\frac{1}{k}\;\text{ln}\;(\frac{\text{Rate}_t}{\text{Rate}_0}) = -\frac{1}{1.21\;\times\;10^{-4}\;\text{y}^{-1}}\;\text{ln}\;(\frac{10.8\;\text{dis/min/g}}{13.6\;\text{dis/min/g}}) = 1910\;\text{y}[/latex]

Therefore, the Dead Sea Scrolls are approximately 1900 years old.

Check Your Learning

More accurate dates of the reigns of ancient Egyptian pharaohs have been determined recently using plants that were preserved in their tombs. Samples of seeds and plant matter from King Tutankhamun’s tomb have a ¹⁴C decay rate of 9.07 disintegrations/min/g of C. How long ago did King Tut’s reign come to an end?

Answer:

about 3350 years ago, or approximately 1340 BC

The accuracy of this technique depends on the ¹⁴C:¹²C ratio in a living plant being the same now as it was in an earlier era, but this is not always valid. For example, right now, due to the increasing accumulation of CO₂ (largely ¹²CO₂) in the atmosphere, caused by combustion of fossil fuels in which essentially all of the ¹⁴C has decayed, the ¹⁴C:¹²C ratio in our atmosphere is decreasing. This in turn affects the ¹⁴C:¹²C ratio in currently living organisms on the earth. Fortunately, we can use other data, such as tree dating via examination of annual growth rings, to calculate correction factors. With these correction factors, more accurate dates can be determined. In general, radioactive dating only works for about 10 half-lives; therefore, the limit for carbon-14 dating is about 57,000 years.

Radiometric Dating Using Nuclides Other than Carbon-14

Radiometric dating can also use other radioactive nuclides with longer half-lives to date older events.

For example, uranium-238 can be used for establishing the age of rocks (and the approximate age of the oldest rocks on earth). Since ²³⁸U has a half-life of 4.5 billion years, it takes that amount of time for half of the original ²³⁸U to decay into ²⁰⁶Pb. In a sample of rock that does not contain appreciable amounts of ²⁰⁸Pb, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed. Therefore, by measuring and analyzing the ratio of ²³⁸U:²⁰⁶Pb, we can determine the age of the rock. This  assumes that all of the ²⁰⁶Pb present came from the decay of ²³⁸U. If there is ²⁰⁶Pb from other sources than ²³⁸U present, which is indicated by the presence of lead isotopes with mass number other than 206 in the sample, it is necessary to make adjustments in the calculations.

Potassium-argon dating uses a similar method. ⁴⁰K decays by positron emission and electron capture to form ⁴⁰Ar with a half-life of 1.25 billion years. If a rock sample is crushed and the amount of ⁴⁰Ar gas that escapes is measured, determination of the ⁴⁰Ar:⁴⁰K ratio yields the age of the rock. Other methods, such as rubidium-strontium dating (⁸⁷Rb decays into ⁸⁷Sr with a half-life of 48.8 billion years), operate on the same principle. To estimate the lower limit for the earth’s age, scientists determine the age of various rocks and minerals, making the assumption that the earth is older than the oldest rocks and minerals in its crust. As of 2014, the oldest known rocks on earth are the Jack Hills zircons from Australia, found by uranium-lead dating to be almost 4.4 billion years old.