Question

Each statement that follows refers to a comparison between two radioisotopes, $A$ and $X$. 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 $\mathrm{X},\mathrm{A}$ has a larger decay rate constant. (b) If $X$ is "not radioactive," its half-life is essentially zero. (c) If A has a half-life of $10\mathrm{yr}$, and $\mathrm{X}$ has a half-life of $10,000\mathrm{yr}$, A would be a more suitable radioisotope to measure processes occurring on the 40 -yr time scale.

Short answer

(a) True, since half-life $t_A<t_X$ implies decay rate constant $k_A>k_X$. (b) False, the half-life for a non-radioactive isotope would be essentially infinite, not zero. (c) True, A is more suitable than X to measure processes occurring on the 40-year time scale, as approximately 1% of A remains after 40 years, while approximately 99% of X remains.

Step-by-step solution

(Statement a)

For this statement, we will first establish the relation between half-life (t) and decay rate constant (k). The formula to calculate half-life is given by: $t=\frac{ln(2)}{k}$ Now let's evaluate the given statement: If the half-life of A is shorter than that of X, we can say $t_A<t_X$ Using the relation between half-life and decay rate, we get: $\frac{ln(2)}{k_A}<\frac{ln(2)}{k_X}$ Cancelling the $ln(2)$ term from both sides, we have: $k_A>k_X$ Hence the statement (a) is True, meaning if the half-life of A is shorter than the half-life of X, A has a larger decay rate constant.

(Statement b)

The given statement says that if X is "not radioactive," its half-life is essentially zero. We know that the concept of a "non-radioactive" isotope implies that it does not decay. In other words, we can say that the decay rate constant (k) is zero. Now, let's use the relation between half-life and decay rate constant to evaluate this statement. If the decay rate constant, $k_X$, is zero: $t_X=\frac{ln(2)}{k_X}=\frac{ln(2)}{0}$ However, dividing by zero is undefined, which means the half-life would be infinite, not zero. This means that the statement (b) is False, as the half-life for a non-radioactive isotope would be essentially infinite, not zero.

(Statement c)

The given statement says that if A has a half-life of 10 years and X has a half-life of 10,000 years, then A would be a more suitable radioisotope to measure processes occurring on the 40-year time scale. To determine if A is more suitable, we need to compare the fraction of the initial quantity of each isotope remaining after 40 years. The formula to calculate the fraction of remaining isotope is: $F=\frac{N}{N_0}=e^{-k\cdot t}$ Given the half-life of A, we can find the decay rate constant ($k_A$): $k_A=\frac{ln(2)}{t_A}=\frac{ln(2)}{10}$ Given the half-life of X, we can find the decay rate constant ($k_X$): $k_X=\frac{ln(2)}{t_X}=\frac{ln(2)}{10000}$ Now, we can calculate the fraction of A and X remaining after 40 years using their respective decay rate constants: $F_A=e^{-k_A\cdot 40}=e^{-\frac{ln(2)}{10}\cdot 40}$ $F_X=e^{-k_X\cdot 40}=e^{-\frac{ln(2)}{10000}\cdot 40}$ Evaluating $F_A$, we find that approximately 1% of A remains after 40 years, whereas when evaluating $F_X$, we find that approximately 99% of X remains after 40 years. Since the goal is to measure processes occurring on a 40-year time scale, it would be more suitable to use an isotope that has a considerable amount of decay over that period. Hence, the statement (c) is True, as A is more suitable than X to measure processes occurring on the 40-year time scale.

Key concepts

Decay Rate Constant

To understand the decay rate constant, let's first define what it represents. In the world of radioisotopes, decay rate constant ($k$) is a measure of how quickly a radioactive substance undergoes decay. It is intrinsically linked to the half-life of the isotope, which is the time it takes for half of the material to decay.

The mathematical relationship between these two concepts is expressed by the formula:$$t=\frac{\ln (2)}{k}$$Here, $t$ is the half-life, and $\ln (2)$ is a constant derived from the natural logarithm of 2.

Since the equation shows that the half-life and decay rate constant are inversely proportional, a smaller half-life indicates a larger decay rate constant. This means that substances with a high decay rate constant disintegrate more rapidly. Understanding this relationship is crucial when comparing the stability and longevity of different radioisotopes.

Radioactive Decay

Radioactive decay is a natural process in which an unstable atomic nucleus loses energy by emitting radiation. This process results in the transformation of a parent isotope into a daughter isotope, often accompanied by the release of particles or electromagnetic waves.

There are different types of radioactive decay, including alpha, beta, and gamma decay. Each type involves the release of different particles:

• Alpha decay releases helium nuclei.
• Beta decay involves the conversion of a neutron into a proton, emitting an electron.
• Gamma decay emits electromagnetic radiation in the form of gamma rays.

The rate at which a substance undergoes radioactive decay can be consistently quantified using its decay rate constant, giving scientists a reliable measure of how quickly an isotope will reduce to half its initial amount, known as its half-life.

In practical terms, knowing the rate of decay helps in various fields such as archaeology through carbon dating or in medicine with radiotherapy.

Measurement of Processes Using Isotopes

The precise measurement of processes using isotopes is a powerful tool in science. Isotopes have applications that extend across various domains, from dating ancient artifacts to diagnosing medical conditions.

When considering measurements, the half-life of a radioisotope is a crucial factor. For processes occurring over thousands of years, isotopes with long half-lives like uranium-238 are preferable. Meanwhile, shorter-lived isotopes like carbon-14 are useful for dating more recent events.

To measure processes accurately, it's essential to select the appropriate isotope whose decay correlates well with the timeframe of the event or process you are studying. If an isotope like A (with a half-life of 10 years) is used to measure processes spanning decades, appreciable decay can be beneficial for tracking the progression. In contrast, isotopes like X with much longer half-lives are more stable and change little over short timescales.

This concept not only elucidates how we learn about the history and transformation of materials but also how isotopic tracers aid researchers in tracking chemical pathways and environmental changes.