Question

Each statement below 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 A is shorter than the half-life for \(\mathrm{X}\), A has a larger decay rate constant. (b) If \(\mathrm{X}\) 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.

Short answer

(a) True. The half-life for A is shorter than for X, meaning that A has a larger decay rate constant, as \(T = \frac{ln(2)}{k}\) shows an inverse relationship between half-life and decay rate constant. (b) False. For a nonradioactive isotope, it would be better to say that its half-life is essentially infinite and not zero. (c) True. Radioisotope A is more suitable for measuring processes occurring on a 40-year time scale since its half-life is closer to the 40-year duration (10 years) compared to X (10,000 years).

Step-by-step solution

(a) Comparing decay rate constants

To determine if statement (a) is true or false, let's understand the relationship between half-life and decay rate constant. The equation that relates half-life (T) to the decay rate constant (k) is: \(T = \frac{ln(2)}{k}\) If the half-life for A is shorter than the half-life for X, that would mean that the denominator k for A must be larger. Therefore, the statement is true.

(b) Half-life of non-radioactive isotopes

Now, let's discuss statement (b) about the half-life of a nonradioactive sample. If an isotope is not radioactive, that means it does not decay and does not have a half-life. Saying that the half-life is essentially zero is incorrect. In fact, it would be better to say that its half-life is essentially infinite. So, statement (b) is false.

(c) Choosing suitable radioisotope for 40-year time scale measurement

Comparing the half-life of A (10 years) and X (10,000 years), we want to determine which is more suitable for measuring processes occurring on the 40-year time scale. If we use a radioisotope with a shorter half-life, it will undergo a significant number of decays during the time scale, yielding a more detectable change indicating the progress of the process in question. In this case, radioisotope A is more appropriate for a 40-year time scale, since its half-life is closer to the 40-year duration. Therefore, statement (c) is true.

Key concepts

half-life

Half-life is a crucial concept when dealing with radioisotopes. It refers to the time required for half of the atoms in a radioactive sample to decay. This is an intrinsic property of an isotope and helps determine how long it remains active or detectable.

The formula to find the half-life (T) of a sample is given by \(T = \frac{\ln(2)}{k}\), where \(k\) is the decay rate constant. A shorter half-life means that the isotope decays faster, which can be particularly useful when measuring processes that occur within that timespan.

Understanding half-life allows scientists to select the proper radioisotope for specific tasks. For instance, in a 40-year process, a radioisotope with a half-life close to that duration will undergo a sufficient number of decays, providing valuable information about the process.

decay rate constant

The decay rate constant, denoted as \(k\), is another fundamental concept in radioactive decay. It indicates the probability per unit time that an atom will decay. A higher decay rate constant signifies a faster decay process.

From the formula \(T = \frac{\ln(2)}{k}\), it is clear that the decay rate constant is inversely proportional to the half-life. This means that if one radioisotope has a shorter half-life than another, it will have a larger decay rate constant. Such a relationship is why radioisotope A, with a shorter half-life, has a larger decay rate constant compared to radioisotope X.

Understanding the decay rate constant helps in predicting the rate of decay over time, which is crucial for both scientific studies and practical applications such as medical treatments or archaeological dating.

non-radioactive isotopes

Non-radioactive isotopes are the counterparts of radioisotopes. As the name suggests, they do not undergo radioactive decay, meaning no change in their atomic structure over time.

Because they do not decay, these isotopes technically have an infinite half-life. Thus, stating that a non-radioactive isotope has a half-life of zero is misleading. Instead, it's best understood that they remain stable indefinitely.

The study and identification of non-radioactive isotopes are essential in various scientific fields, including chemistry and geology, as they often serve as stable reference points or are used in applications where stability is needed, such as in nutritional or medical contexts.

measuring radioactive decay

Measuring radioactive decay is a vital part in many scientific areas, from determining the age of archaeological finds to diagnosing diseases using nuclear medicine. It involves monitoring the decrease in the number of radioactive atoms over time.

When selecting a radioisotope for measurement over a specific time scale, like a 40-year time period, the isotope's half-life becomes a key factor. For effective measurement, the chosen radioisotope should have undergone a significant proportion of decay within the process's timeframe.

Accurate measurements require understanding both the half-life and decay rate constant, allowing researchers to estimate the decay process's progress over time. For scientists, knowing which radioisotope to use ensures that they can obtain valuable results and insights into the processes they are studying.