Problem 28 - Chemistry: An Atoms-Focused Approach

Question

Nine isotopes of sulfur have mass numbers ranging from 30 to \(38 .\) Five of the nine are radioactive: \(^{30} \mathrm{S},^{31} \mathrm{S},^{35} \mathrm{S},^{37} \mathrm{S}\) and \(^{38} \mathrm{S}\). Which of these isotopes do you expect to decay by \(\beta\) decay?

Step-by-Step Solution

Answer

Answer: All five isotopes, \(^{30}S\), \(^{31}S\), \(^{35}S\), \(^{37}S\), and \(^{38}S\), will undergo beta decay.

Step 1: Determine the stable neutron-to-proton ratio

An atom of sulfur has 16 protons. In general, for isotopes with low atomic numbers (less than 20), the ratio of neutrons to protons is approximately 1:1. For heavier isotopes, the ratio tends to be slightly more than 1.

Step 2: Calculate the number of neutrons for each isotope

For each radioactive sulfur isotope, we can calculate the number of neutrons by subtracting the number of protons (16) from the mass number. This gives us the following neutron counts: - \(^{30}S\): 14 neutrons - \(^{31}S\): 15 neutrons - \(^{35}S\): 19 neutrons - \(^{37}S\): 21 neutrons - \(^{38}S\): 22 neutrons

Step 3: Determine the neutron-to-proton ratios for the radioactive isotopes

Now that we have calculated the number of neutrons for each isotope, we can determine the neutron-to-proton ratios as follows: - \(^{30}S\): 14/16 = 0.875 - \(^{31}S\): 15/16 = 0.9375 - \(^{35}S\): 19/16 = 1.1875 - \(^{37}S\): 21/16 = 1.3125 - \(^{38}S\): 22/16 = 1.375

Step 4: Compare the neutron-to-proton ratios to predict beta decay

Comparing the calculated ratios to the stable neutron-to-proton ratio, we see that \(^{30}S\) and \(^{31}S\) have a lower ratio, indicating they have too few neutrons relative to protons and will likely undergo beta-minus decay (converting a proton into a neutron). On the other hand, \(^{35}S\), \(^{37}S\), and \(^{38}S\) have a higher ratio, indicating they have too many neutrons relative to protons and will likely undergo beta-plus decay (converting a neutron into a proton) or electron capture. So, the isotopes expected to decay by beta decay are: \(^{30}S\), \(^{31}S\), \(^{35}S\), \(^{37}S\), and \(^{38}S\).

Key Concepts

Sulfur Isotopes

Isotopes are different forms of an element that have the same number of protons but vary in the number of neutrons. This variance in neutron count gives each isotope a different mass number. Sulfur has several isotopes, with mass numbers ranging from 30 to 38. In plain terms, if we're talking about sulfur isotopes, we are considering sulfur atoms which each have 16 protons, but a different count of neutrons. Each isotope has unique properties, including their stability. Some remain stable, while others might be radioactive. This difference stems from how the protons and neutrons are arranged in the nucleus. By knowing the neutron count, we can identify whether an isotope of sulfur is stable or has a tendency to undergo radioactive processes.

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a critical factor in determining the stability of an isotope's nucleus. For elements with lower atomic numbers, like sulfur, a balance close to 1:1, where the protons are equal to neutrons, often indicates stability. When analyzing sulfur isotopes, calculating this ratio reveals insights into the isotope's characteristics:

• If the ratio is less than 1, the isotope might lack enough neutrons to be stable and could undergo beta-minus decay, where a proton turns into a neutron.
• If the ratio is more than 1, there's an excess of neutrons, and the isotope might undergo beta-plus decay or electron capture, where a neutron turns into a proton.

Understanding these ratios is essential for predicting the type of beta decay an isotope might experience.

Radioactive Decay

Radioactive decay is a natural process where unstable isotopes transform into more stable forms. This transformation occurs through the emission of particles. In the case of sulfur isotopes, this decay often manifests as beta decay. Beta decay comes in two types:

• Beta-minus decay: Here, a neutron is converted into a proton, liberating an electron (a beta particle) and an antineutrino.
• Beta-plus decay or electron capture: A proton becomes a neutron, emitting a positron or capturing an electron, and releasing energy in the form of a neutrino.

This process helps isotopes reach a more stable neutron-to-proton ratio. It is a spontaneous event, meaning it happens without any external cause, driven by the intrinsic instability of the radioactive isotope.

Stable Isotopes

Stable isotopes are variants of an element that do not undergo radioactive decay under normal conditions. They maintain a balanced neutron-to-proton ratio, preventing them from spontaneously transforming into another element or isotope. For sulfur, stable isotopes maintain a healthy balance:

• Their neutron count adjusts perfectly to the number of protons, avoiding the need for decay.
• Such isotopes do not emit radiation over time, making them safe and predictable.

Understanding stable isotopes helps scientists gauge which isotopes are safe for practical applications, including in medicine, environmental science, and industrial processes.