Radioactive Decay | Chemistry for ACT PDF Download

Introduction

Reaction rates can also be evaluated by considering the time it takes for the concentration of a reactant to decrease by half of its initial value. This time period is referred to as the half-life of the reaction, denoted as t_1/2. Essentially, the half-life represents the time it takes for the concentration of the reactant ([A]) to go from [A]₀ (its initial concentration) to [A]_0/2 (half of its initial concentration). When comparing two reactions with the same order, the reaction that proceeds more rapidly will have a shorter half-life, while the reaction that proceeds more slowly will have a longer half-life.
The half-life of a first-order reaction remains constant under specific reaction conditions, unlike zeroth- and second-order reactions. In a first-order reaction, the half-life is unaffected by the reactant concentrations. This characteristic becomes apparent when we manipulate the integrated rate law for a first-order reaction, resulting in the following equation:
Substituting [A]_0/2 for [A] and t_1/2 for t (to indicate a half-life) into Equation above gives
Substituting ln2 ≈ 0.693 into the equation results in the expression for the half-life of a first-order reaction:
Thus, for a first-order reaction, each successive half-life is the same length of time, as shown in Figure 1, and is independent of [A].
Figure  1: The Half-Life of a First-Order Reaction: This plot shows the concentration of the reactant in a first-order reaction as a function of time and identifies a series of half-lives, intervals in which the reactant concentration decreases by a factor of 2. In a first-order reaction, every half-life is the same length of time.
If we know the rate constant for a first-order reaction, then we can use half-lives to predict how much time is needed for the reaction to reach a certain percent completion.
As you can see from this table, the amount of reactant left after n half-lives of a first-order reaction is (1/2)ⁿ times the initial concentration.

For a first-order reaction, the concentration of the reactant decreases by a constant with each half-life and is independent of [A].

Example 1: The anticancer drug cis-platin hydrolyzes in water with a rate constant of 1.5 × 10⁻³ min⁻¹ at pH 7.0 and 25°C. Calculate the half-life for the hydrolysis reaction under these conditions. If a freshly prepared solution of cis-platin has a concentration of 0.053 M, what will be the concentration of cis-platin after 5 half-lives? after 10 half-lives? What is the percent completion of the reaction after 5 half-lives? after 10 half-lives?
Given: rate constant, initial concentration, and number of half-lives
Asked for: half-life, final concentrations, and percent completion
Strategy:
Use Equation    to calculate the half-life of the reaction.
Multiply the initial concentration by 1/2 to the power corresponding to the number of half-lives to obtain the remaining concentrations after those half-lives.
Subtract the remaining concentration from the initial concentration. Then divide by the initial concentration, multiplying the fraction by 100 to obtain the percent completion.

A. We can calculate the half-life of the reaction using Equation   :
Thus it takes almost 8 h for half of the cis-platin to hydrolyze.
B. After 5 half-lives (about 38 h), the remaining concentration of cis-platin will be as follows:
After 10 half-lives (77 h), the remaining concentration of cis-platin will be as follows:

C. The percent completion after 5 half-lives will be as follows:
The percent completion after 10 half-lives will be as follows:
Thus a first-order chemical reaction is 97% complete after 5 half-lives and 100% complete after 10 half-lives.

Example 2: Ethyl chloride decomposes to ethylene and HCl in a first-order reaction that has a rate constant of 1.6 × 10⁻⁶ s⁻¹ at 650°C.
(a) What is the half-life for the reaction under these conditions?
(b) If a flask that originally contains 0.077 Methyl chloride is heated at 650°C, what is the concentration of ethyl chloride after 4 half-lives?

(a) 4.3 × 10⁵ s = 120 h = 5.0 days;
(b) 4.8 × 10⁻³ M

Radioactive Decay Rates

Radioactivity, also known as radioactive decay, refers to the release of particles or photons that occurs when an unstable atomic nucleus spontaneously breaks down. The rate at which radioactive decay occurs is an inherent characteristic of each radioactive isotope and is not influenced by its chemical or physical state. Additionally, this rate remains unaffected by temperature changes. In the following section, we will explore radioactive decay rates and how half-lives can be utilized to track radioactive decay processes.
In any sample of a given radioactive substance, the number of atoms of the radioactive isotope must decrease with time as their nuclei decay to nuclei of a more stable isotope. Using N to represent the number of atoms of the radioactive isotope, we can define the rate of decay of the sample, which is also called its activity (A) as the decrease in the number of the radioisotope’s nuclei per unit time:
Activity is usually measured in disintegrations per second (dps) or disintegrations per minute (dpm).
The activity of a sample is directly proportional to the number of atoms of the radioactive isotope in the sample:

Here, the symbol k is the radioactive decay constant, which has units of inverse time (e.g., s−1, yr−1) and a characteristic value for each radioactive isotope. If we combine Equation    and Equation  , we obtain the relationship between the number of decays per unit time and the number of atoms of the isotope in a sample:

Equation  is the same as the equation for the reaction rate of a first-order reaction, except that it uses numbers of atoms instead of concentrations. In fact, radioactive decay is a first-order process and can be described in terms of either the differential rate law (Equation ) or the integrated rate law:
Because radioactive decay is a first-order process, the time required for half of the nuclei in any sample of a radioactive isotope to decay is a constant, called the half-life of the isotope. The half-life tells us how radioactive an isotope is (the number of decays per unit time); thus it is the most commonly cited property of any radioisotope. For a given number of atoms, isotopes with shorter half-lives decay more rapidly, undergoing a greater number of radioactive decays per unit time than do isotopes with longer half-lives. The half-lives of several isotopes are listed in Table, along with some of their applications.
*The m denotes metastable, where an excited state nucleus decays to the ground state of the same isotope.

Note: Radioactive decay is a first-order process.

Radioisotope Dating Techniques

In our previous conversation, we utilized the concept of half-life in first-order reactions to determine the duration of the reaction. Similarly, in nuclear decay reactions, which adhere to first-order kinetics and possess a rate constant unaffected by temperature or the surrounding conditions, we can employ the half-lives of isotopes to estimate the ages of geological and archaeological objects. These methods, referred to as radioisotope dating techniques, have been developed specifically for this purpose.
The primary method used for determining the age of ancient objects is carbon-14 dating. Carbon-14, a radioactive isotope, is continuously formed in the upper regions of Earth's atmosphere and combines with atmospheric oxygen or ozone to create ¹⁴CO₂. Consequently, the CO₂ utilized by plants as a carbon source for synthesizing organic compounds contains a certain proportion of ¹⁴CO₂ molecules alongside nonradioactive ¹²CO₂ and ¹³CO₂. When animals consume plants, they acquire a mixture of organic compounds that reflects the same isotopic proportions as those present in the atmosphere. Upon the death of an organism, the carbon-14 nuclei in its tissues undergo radioactive decay, transforming into nitrogen-14 nuclei through a process called beta decay. This decay emits low-energy electrons (β particles) that can be detected and measured.
The half-life for this reaction is 5700 ± 30 yr.
The ratio of ¹⁴C to ¹²C in living organisms is 1.3 × 10⁻¹², and its decay rate is 15 disintegrations per minute per gram of carbon (dpm/g) (Figure 2). By comparing the disintegrations per minute per gram of carbon in an archaeological sample with that of a recently living sample, scientists can estimate the age of the artifact, as demonstrated. However, this method assumes that the ratio of ¹⁴CO₂ to ¹²CO₂ in the atmosphere remains constant, which is not entirely accurate. To account for minor changes in the ¹⁴CO₂/1²CO₂ ratio over time, other techniques such as tree-ring dating have been employed to calibrate radiocarbon dates. Consequently, all reported radiocarbon dates are now adjusted for these variations.

Figure  2: Radiocarbon Dating: A plot of the specific activity of ¹⁴C versus age for a number of archaeological samples shows an inverse linear relationship between ¹⁴C content (a log scale) and age (a linear scale).
Example 3:  In 1990, the remains of an apparently prehistoric man were found in a melting glacier in the Italian Alps. Analysis of the ¹⁴C content of samples of wood from his tools gave a decay rate of 8.0 dpm/g carbon. How long ago did the man die?
Given: isotope and final activity
Asked for: elapsed time
Strategy:
A. Use Equation    to calculate N₀/N. Then substitute the value for the half-life of ¹⁴C into Equation    to find the rate constant for the reaction.
B. Using the values obtained for N₀/N and the rate constant, solve Equation    to obtain the elapsed time.

We know the initial activity from the isotope’s identity (15 dpm/g), the final activity (8.0 dpm/g), and the half-life, so we can use the integrated rate law for a first-order nuclear reaction (Equation  ) to calculate the elapsed time (the amount of time elapsed since the wood for the tools was cut and began to decay).

A. From Equation , we know that A = kN. We can therefore use the initial and final activities (A₀ = 15 dpm and A = 8.0 dpm) to calculate N₀/N:

Now we need only calculate the rate constant for the reaction from its half-life (5730 yr) using Equation :
This equation can be rearranged as follows:
B. Substituting into the equation for t,
From our calculations, the man died 5200 yr ago.

Example 4: It is believed that humans first arrived in the Western Hemisphere during the last Ice Age, presumably by traveling over an exposed land bridge between Siberia and Alaska. Archaeologists have estimated that this occurred about 11,000 yr ago, but some argue that recent discoveries in several sites in North and South America suggest a much earlier arrival. Analysis of a sample of charcoal from a fire in one such site gave a 14C decay rate of 0.4 dpm/g of carbon. What is the approximate age of the sample?

30,000 yr

Summary

• The half-life of a first-order reaction is independent of the concentration of the reactants.
• The half-lives of radioactive isotopes can be used to date objects.

The half-life of a reaction corresponds to the duration needed for the concentration of a reactant to decrease by half of its initial value. For first-order reactions, the half-life remains constant and can be determined using the equation: t_1/2 = 0.693/k, where k represents the rate constant. Radioactive decay reactions adhere to first-order kinetics. The activity or rate of decay of a radioactive substance refers to the decline in the number of radioactive nuclei per unit time.