Problem 31 - Chemistry: The Central Science

Question

Each statement below refers to a comparison between two radioisotopes, \(\mathrm{A}\) and \(\mathrm{X}\). Indicate whether each of the following statements is true or false, and why. (a) If the half-life for \(\mathrm{A}\) is shorter than the half-life for \(\mathrm{X}\), A has a larger decay rate constant. (b) If \(X\) is "notradioactive," its half-life is essentially zero. (c) If A has a half-life of 10 years, and \(\mathrm{X}\) has a half-life of 10,000 years, A would be a more suitable radioisotope to measure processes occurring on the 40 -year time scale.

Step-by-Step Solution

Answer

Statement (a) is True because a shorter half-life indicates a larger decay rate constant, as they are inversely proportional. Statement (b) is False because a non-radioactive isotope does not have a half-life of essentially zero; instead, its half-life should be considered infinitely long or undefined. Statement (c) is True because isotope A, with a half-life of 10 years, is more suitable for measuring a process on a 40-year time scale compared to isotope X's half-life of 10,000 years.

Step 1: Statement (a) - Comparing Decay Rate Constants and Half-Lives

The decay rate constant λ is the probability per unit time that a nucleus will decay, while the half-life (t) denotes the time it takes for half of a sample to decay. These quantities are inversely proportional: \[ t = \frac{0.693}{\lambda} \] To determine if statement (a) is true or false, we have to assess the relationship between the decay rate constant λ and half-life t when comparing A and X. If the half-life of A is shorter than that of X, that means the time it takes for A to decay is less than X. Since half-life and decay rate constant are inversely proportional, a shorter half-life would mean a larger decay rate constant. Therefore, the statement (a) is True.

Step 2: Statement (b) - Non-Radioactive Isotope's Half-Life

A non-radioactive isotope is an isotope that does not undergo radioactive decay. Since radioactive decay is the process by which an unstable nucleus loses energy, a non-radioactive isotope's nucleus is stable and does not lose energy. Therefore, it does not have a half-life. However, saying that the half-life of a non-radioactive isotope is essentially zero is incorrect. A half-life of zero would imply that the isotope decays instantly, which is not the case for a non-radioactive isotope. Instead, a non-radioactive isotope's half-life should be considered infinitely long or undefined. Thus, the statement (b) is False.

Step 3: Statement (c) - Suitability of Radioisotopes for Measuring Processes

When measuring processes, the half-life of the radioisotope used should roughly be within the time scale of the process that's being measured. In this case, isotope A has a half-life of 10 years, and isotope X has a half-life of 10,000 years, while the process being measured has a 40-year time scale. Isotope A would be a better candidate for measuring the 40-year process because its half-life is closer to the time scale of the process. Isotope A will undergo several decay cycles within the 40 years, providing more accurate measurements. In contrast, isotope X's half-life of 10,000 years is too long to effectively measure a 40-year process, as its decay will be too slow to provide meaningful data. Therefore, the statement (c) is True.

Key Concepts

Half-life

Half-life is a term used to describe the time it takes for half of a radioactive sample to decay. It is a very important concept in understanding how quickly different radioisotopes lose their radioactivity.
For instance, if you start with 100 grams of a particular isotope, after one half-life, you'll have 50 grams remaining. After another half-life, only 25 grams will be left, and so on until it is no longer measurable.
Key points to remember about half-life:

• It is not affected by environmental factors like temperature or pressure.
• Shorter half-life means the isotope is more radioactive and decays faster.
• A knowledge of half-life helps in practical applications like dating fossils or tracing the movement of isotopes in the environment.

Knowing the half-life can help scientists and engineers choose the right isotope for dating objects or for use in medical applications.

Decay Rate Constant

The decay rate constant, often denoted by the Greek letter lambda (\( \lambda \)), is a measure of how quickly a radioactive material decays. It is directly related to the stability of a radioisotope and inversely related to its half-life.
Mathematically, decay rate constant and half-life are linked by the equation:\[t = \frac{0.693}{\lambda}\] where \( t \) is the half-life and \( \lambda \) is the decay rate constant.
More on decay rate constant:

• A bigger \( \lambda \) suggests a faster rate of decay.
• If two isotopes have different half-lives, the one with the shorter half-life has the larger decay rate constant.
• It helps in determining the "speed" of the radioactive process, which is crucial in fields such as nuclear medicine and radiometric dating.

Understanding the decay rate constant is invaluable for practical applications where knowing how long a radioactive source remains active is important.

Radioactive Decay

Radioactive decay is the process by which a nucleus of an unstable atom loses energy by emitting radiation. It can occur in different forms such as alpha decay, beta decay, or gamma decay, each involving different particles and energy emissions.
Some essential facts about radioactive decay:

• It is a random process at the level of single atoms, meaning it is impossible to predict when a particular atom will decay.
• On a larger scale, it follows a predictable pattern described by the half-life of the substance.
• This process is responsible for the heat production in the Earth's interior and plays a role in the natural background radiation in our environment.

The understanding of radioactive decay is fundamental in various applications, from energy production in nuclear power plants to radiocarbon dating in archaeology. Knowing how different isotopes decay helps predict outcomes in scientific experiments and practical applications in industry and medicine.