Question

\({ }^{15} \mathrm{O}\) decays by positron emission with a half-life of 124 s. (a) Give the proper symbol of the product of the decay. (b) How much of a \(750 \mathrm{mg}\) sample of \({ }^{15} \mathrm{O}\) remains after 5.0 min of decay? (Contributed by Prof. W. J. Wysochansky, Georgia Southern University.)

Short answer

The product of the decay is \({ }^{15}\mathrm{N}\). After 5.0 minutes (300 seconds), approximately 91.6 mg of the original \(750\mathrm{mg}\) sample remains.

Step-by-step solution

Understand Positron Emission

Consider the positron emission process. In positron emission, a proton in the nucleus is transformed into a neutron and a positron is emitted along with a neutrino. The positron is the antimatter equivalent of an electron. The atomic number decreases by one but the mass number stays the same. The equation for the decay is \(_{Z}^{A}X \rightarrow _{Z-1}^{A}Y + _{1}^{0}e^{+} + u_{e}\).

Identify the Product of Decay

Using the decay equation from step 1, write the symbol of the product of the \({ }^{15}\mathrm{O}\) decay. Since oxygen (\(O\)) has an atomic number of 8 and positron emission decreases it by one, the atomic number of the product will be 7, which is nitrogen (\(N\)). Therefore, the decay product is \({ }^{15}\mathrm{N}\).

Calculate the Number of Half-Lives

Determine the number of half-lives passed in the given time. The half-life of \({ }^{15}\mathrm{O}\) is 124 seconds. Five minutes is 300 seconds. Calculate the number of half-lives by dividing the total decay time by the half-life: Number of half-lives = Total time / Half-life = 300 s / 124 s.

Calculate Amount Remaining

Use the number of half-lives to find the fraction remaining and then calculate the final mass. Every half-life reduces the existing sample to half its previous amount. After n half-lives, the fraction remaining is \((1/2)^{n}\). Multiply the fraction remaining by the initial mass to find the mass remaining: Mass remaining = Initial mass \times (Fraction remaining).

Key concepts

Understanding Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting particles or electromagnetic waves. There are several types of radioactive decay, one of them is positron emission, which is relevant to our exercise involving oxygen-15 ({ }^{15}O). In this type of decay, a proton inside the nucleus is converted into a neutron, releasing a positron—the antimatter counterpart of the electron—and a neutrino.

During this decay, while the atomic number decreases by one, indicating that the element changes, the mass number remains the same. This is because a proton and a neutron have approximately the same mass, and the emitted positron has negligible mass compared to them. In the case of { }^{15}O, it decays into nitrogen-15 ({ }^{15}N) as its atomic number decreases by one, from 8 (oxygen) to 7 (nitrogen).

Half-Life Calculation Demystified

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. This is a pivotal concept in nuclear chemistry because it helps us determine how quickly a substance loses its radioactivity. To perform a half-life calculation, as we did for oxygen-15, you simply divide the elapsed time by the half-life of the substance.

The formula used is: Number of half-lives = Total time / Half-life. For our exercise, the half-life is 124 seconds, and the elapsed time is 5 minutes, which converts to 300 seconds. Thus, the number of half-lives that have passed is 2.42, approximately. Understanding the half-life allows us to predict the decay pattern of a radioactive material and calculate the amount of material remaining after a certain period, which leads us to see that only about 9.45% of the oxygen-15 sample is left after 5 minutes.

Nuclear Chemistry Essentials

Nuclear chemistry is a fascinating field that studies the changes in atomic nuclei. It has significant applications ranging from medicine, where radioactive isotopes are used in diagnosis and therapy, to energy production in nuclear reactors.

In terms of our exercise, nuclear chemistry principles are applied to understand the decay process of { }^{15}O. This process, governed by quantum mechanics, involves understanding forces that operate at the nuclear level. The stability of a nucleus is a delicate balance between the nuclear strong force and electrostatic repulsion between protons. When an imbalance occurs, it results in radioactive decay, such as the positron emission we've explored.

The calculations done are based on knowledge of half-lives and radioactive decay processes. Nuclear chemists use their understanding of these processes to analyze radioactive substances, predict their behavior over time, and calculate potential risks and benefits associated with their use.