Question

Chlorine has two stable nuclides, \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\). In contrast, \({ }^{36} \mathrm{Cl}\) is a radioactive nuclide that decays by beta emission. (a) What is the product of decay of \({ }^{36} \mathrm{Cl}\) ? (b) Based on the empirical rules about nuclear stability, explain why the nucleus of \({ }^{36} \mathrm{Cl}\) is less stable than either \({ }^{35} \mathrm{Cl}\) or \({ }^{37} \mathrm{Cl}\).

Short answer

(a) The product of decay of \({ }^{36}\mathrm{Cl}\) is \({ }^{36}\mathrm{Ar}\). (b) \({ }^{36}\mathrm{Cl}\) is less stable than \({ }^{35}\mathrm{Cl}\) and \({ }^{37}\mathrm{Cl}\) primarily due to its odd N and odd Z combination, which is associated with increased instability in isotopes with odd atomic numbers.

Step-by-step solution

(a) Identify decay product of \({ }^{36}\mathrm{Cl}\) through beta emission

When a nuclide undergoes beta emission (β⁻ decay), a neutron in the nucleus is transformed into a proton. This increases the atomic number (number of protons) by 1, while the mass number (total of protons and neutrons) remains unchanged. The equation for the beta decay can be written as: \[n \rightarrow p^{+} + e^{-} + \overline{\nu}_e\] Understanding this process, let's find the decay product of \({ }^{36}\mathrm{Cl}\). The mass number will still be 36, and the atomic number will increase by 1. This atomic number of 18 corresponds to Argon (Ar). So, the product of the decay of \({ }^{36}\mathrm{Cl}\) is \({ }^{36}\mathrm{Ar}\).

(b) Compare the stability of \({ }^{36}\mathrm{Cl}\) to \({ }^{35}\mathrm{Cl}\) and \({ }^{37}\mathrm{Cl}\)

We will examine the neutron-to-proton ratios (N/Z) of \({ }^{35}\mathrm{Cl}\), \({ }^{36}\mathrm{Cl}\), and \({ }^{37}\mathrm{Cl}\), as well as whether they have any magic numbers contributing to their stability. 1. \({ }^{35}\mathrm{Cl}\): - Protons (Z) = 17 - Neutrons (N) = 35 - 17 = 18 - N/Z = 18/17 = 1.06 2. \({ }^{36}\mathrm{Cl}\): - Protons (Z) = 17 - Neutrons (N) = 36 - 17 = 19 - N/Z = 19/17 = 1.12 3. \({ }^{37}\mathrm{Cl}\): - Protons (Z) = 17 - Neutrons (N) = 37 - 17 = 20 - N/Z = 20/17 = 1.18 As we can see, \({ }^{36}\mathrm{Cl}\) has a lower N/Z ratio than \({ }^{37}\mathrm{Cl}\) but higher than \({ }^{35}\mathrm{Cl}\). However, while having stable isotopes with odd atomic numbers, nuclides with an odd N and odd Z are generally unstable, making \({ }^{36}\mathrm{Cl}\) less stable than \({ }^{35}\mathrm{Cl}\) and \({ }^{37}\mathrm{Cl}\). Examining for magic numbers, none of these nuclides have neutron or proton numbers associated with recognized magic numbers that organize stable shells. Thus, magic numbers don't play a role in the stability contrast between these Chlorine isotopes. In conclusion, the lower stability of \({ }^{36}\mathrm{Cl}\) as compared to \({ }^{35}\mathrm{Cl}\) and \({ }^{37}\mathrm{Cl}\) relies primarily on its odd N and odd Z combination, which is associated with increased instability in isotopes with odd atomic numbers.

Key concepts

Beta Decay

Beta decay is a fascinating process in nuclear physics that involves the transformation of a neutron into a proton. During beta decay, specifically beta-minus decay (\(\beta^-\)), a neutron (\(n\)) turns into a proton (\(p^+\)), releasing an electron (\(e^-\)) and an antineutrino (\(\overline{u}_e\)) in the process.
The key takeaway here is that beta decay changes the identity of the atom because the atomic number increases by one. Meanwhile, the mass number, which totals protons and neutrons, remains unchanged.

• Neutron converts into proton
• Electron and antineutrino are emitted
• Atomic number increases by 1
• Mass number remains the same

In the case of \(^{36}\mathrm{Cl}\), when it undergoes beta decay, it transforms into \(^{36}\mathrm{Ar}\) (argon) by increasing its atomic number from 17 (chlorine) to 18 (argon), while the mass number stays 36. This transformation is central to understanding why some isotopes are naturally unstable.

Isotopes

Isotopes are different forms of the same element. They have the same number of protons but a different number of neutrons. This difference in neutron count leads to variations in their nuclear properties. For instance, while \(^{35}\mathrm{Cl}\) and \(^{37}\mathrm{Cl}\) are stable isotopes of chlorine, \(^{36}\mathrm{Cl}\) is radioactive due to its unique balance of neutrons and protons.
Key points about isotopes include:

• Same atomic number; different mass numbers
• Exist in both stable and unstable forms
• The variation in nuclear stability can lead to radioactivity

Isotopes play a crucial role in nuclear stability. The stability often depends on neutron-to-proton (N/Z) ratios. Stable isotopes like \(^{35}\mathrm{Cl}\) and \(^{37}\mathrm{Cl}\) typically have ratios that conform to empirical stability rules, whereas unstable ones, like \(^{36}\mathrm{Cl}\), do not.

Neutron-to-Proton Ratio

The neutron-to-proton ratio (\(N/Z\)) is a fundamental factor in determining nuclear stability. It represents the number of neutrons (\(N\)) divided by the number of protons (\(Z\)) in a nucleus. A proper balance in this ratio helps stabilize the nucleus, reducing radioactive decay events.
For chlorine isotopes:

• \(^{35}\mathrm{Cl}: N/Z = \frac{18}{17} = 1.06\)
• \(^{36}\mathrm{Cl}: N/Z = \frac{19}{17} = 1.12\)
• \(^{37}\mathrm{Cl}: N/Z = \frac{20}{17} = 1.18\)

When examining these ratios, we see that a slight imbalance in \(^{36}\mathrm{Cl}\) compared to \(^{35}\mathrm{Cl}\) and \(^{37}\mathrm{Cl}\) can lead to decreased nuclear stability. Generally, isotopes with ratios too high or too low compared to their stable counterparts develop nuclear instability, often resulting in decay processes. Thus, the neutron-to-proton ratio is a crucial indicator of why certain isotopes are radioactive.