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 \({ }^{36} \mathrm{Cl}\) by beta emission is \({ }^{36} \mathrm{Ar}\). (b) The nucleus of \({ }^{36} \mathrm{Cl}\) is less stable than either \({ }^{35} \mathrm{Cl}\) or \({ }^{37} \mathrm{Cl}\) because its neutron-to-proton (N/P) ratio is less optimal (1.12) compared to the other isotopes (1.06 and 1.18), and it does not have magic numbers of protons or neutrons, unlike \({ }^{37} \mathrm{Cl}\) which has a magic number of neutrons (20).

Step-by-step solution

Determine the product of decay

The radioisotope \({ }^{36} \mathrm{Cl}\) decays by beta emission (β- decay). In beta decay, a neutron in the nucleus is transformed into a proton, and a beta particle (an electron) is emitted. So, we consider the following transformation: \[n \rightarrow p + e^-\] In the case of \({ }^{36} \mathrm{Cl}\), the atomic number (Z) is 17 and the mass number (A) is 36. According to the beta decay, the number of protons increases by 1 while the mass number remains constant. So, we get a new element with an atomic number of 18 and mass number 36. The element with an atomic number 18 is Argon (Ar). Hence, the product after decay is \({ }^{36} \mathrm{Ar}\). (b)

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

To analyze the stability of these isotopes, we'll consider two empirical rules about nuclear stability: 1. The neutron-to-proton ratio (N/P): Stable nuclides have a neutron-to-proton ratio close to 1 for lighter elements and gradually increase for heavier elements. Comparing the neutron-to-proton ratio of our three isotopes: • \({ }^{35} \mathrm{Cl}\) : (18 neutrons, 17 protons) → N/P = 18/17 ≈ 1.06 • \({ }^{36} \mathrm{Cl}\) : (19 neutrons, 17 protons) → N/P = 19/17 ≈ 1.12 • \({ }^{37} \mathrm{Cl}\) : (20 neutrons, 17 protons) → N/P = 20/17 ≈ 1.18 Comparing these ratios, we can see that \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\) are closer to the ideal ratio, making them more stable compared to \({ }^{36} \mathrm{Cl}\). 2. The magic number rule: Nuclei are more stable if they have "magic numbers" of protons or neutrons (2, 8, 20, 28, 50, 82, 126). These magic numbers correspond to a completely filled nuclear shell. \({ }^{35} \mathrm{Cl}\) has 18 neutrons, which is two less than the magic number 20. \({ }^{37} \mathrm{Cl}\) has 20 neutrons and therefore exhibits increased stability due to completely filled nuclear shells. However, \({ }^{36} \mathrm{Cl}\) does not have a magic number of protons or neutrons, making it less stable compared to \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\). Thus, considering these empirical rules about nuclear stability, the nucleus of \({ }^{36} \mathrm{Cl}\) is less stable than either \({ }^{35} \mathrm{Cl}\) or \({ }^{37} \mathrm{Cl}\).

Key concepts

Beta Decay

Let's dive into the intriguing world of beta decay, which is a type of radioactive decay crucial for understanding nuclear stability. At its core, beta decay involves a neutron inside an atom's nucleus transforming into a proton. During this process, an electron, referred to as a beta particle, is emitted alongside an antineutrino. The emission of the beta particle is nature's way of trying to balance out the number of protons and neutrons within the nucleus.

When a nucleus has too many neutrons, beta decay helps to increase the number of protons, maintaining equilibrium within the atom. This change can lead to the transformation from one element to another due to the increase in atomic number, which defines an element's identity in the periodic table. The case of Chlorine-36 becoming Argon-36 after beta decay is a perfect example of this transmutation. The process can be represented by the equation:

\[n \rightarrow p^+ + e^- + \bar{u}_e\]
Where n is a neutron, p^+ is a proton, e^- is the beta particle (electron), and \( \bar{u}_e \) is the antineutrino. This transformation sheds light on the intricate balance within atomic nuclei and their quest for stability through beta decay.

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a fundamental concept that governs the intricate balance within an atomic nucleus. This ratio is pivotal in determining whether a nucleus is stable or prone to radioactive decay. For lighter elements, a ratio close to 1:1 between neutrons and protons generally indicates a stable nucleus. However, as we progress to heavier elements, the optimal ratio gradually increases to accommodate the growing electrical repulsion amongst the densely packed protons.

Why does this ratio matter? Simply put, a mismatch in this delicate balance can render an atom unstable. An excess of neutrons, for instance, can lead to beta decay, where a neutron is converted into a proton to stabilize the nucleus. This is the case for Chlorine-36, where the presence of one extra neutron compared to its stable counterparts leads to beta decay. Through evaluating the neutron-to-proton ratios:

• Chlorine-35: 18/17 ≈ 1.06
• Chlorine-36: 19/17 ≈ 1.12
• Chlorine-37: 20/17 ≈ 1.18

we observe that Chlorine-36 exceeds the stable zone, whereas Chlorine-35 and Chlorine-37 sit closer to the ideal ratio, underpinning their stability.

Magic Numbers in Nuclei

Peering into the nucleus, we encounter the concept of magic numbers in nuclei, which are akin to winning numbers in nature's lottery of stability. These numbers – 2, 8, 20, 28, 50, 82, and 126 – signify the count of protons or neutrons that lead to exceptionally stable configurations known as 'closed shells'.

Why are these numbers 'magic'? They reflect the structure of the nucleus where protons and neutrons arrange themselves in layers, much like electrons around the nucleus. A full layer, or shell, equates to enhanced stability, much like the noble gases which have complete electron shells and are notably nonreactive.

Let's put this in context with our chlorine isotopes:

• Chlorine-35 with 18 neutrons approaches the magic number 20, suggesting a proximity to greater stability.
• Chlorine-37, holding 20 neutrons, hits the magic number squarely, delivering a stable nucleus.
• Yet, Chlorine-36 lacks a magic number of protons or neutrons, teetering on the edge of instability and therefore undergoes beta decay to reach a more stable state.

This concept plays a pivotal role in nuclear physics, informing us why certain isotopes are more stable than others and driving the behavior of radioactive substances across the periodic table.