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

The product of decay of \({ }^{36} \mathrm{Cl}\) is \({ }^{36} \mathrm{Ar}\) through beta decay. The nuclides \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\) are more stable than \({ }^{36} \mathrm{Cl}\) because they have even numbers of neutrons and more favorable N/Z ratios, whereas \({ }^{36} \mathrm{Cl}\) has an odd number of neutrons and a less favorable N/Z ratio, making it less stable and radioactive.

Step-by-step solution

Find the product of decay

\({ }^{36} \mathrm{Cl}\) goes through beta decay, which results in the emission of an electron, also known as a beta particle, and an antineutrino from the nucleus. In beta decay, one of the neutrons gets converted into a proton, with no change in the atomic mass number, but an increase of 1 in the atomic number. Therefore, our decay product should have the same atomic mass number 36 and an atomic number of 18 instead of 17. \( _{17}^{36}\textrm{Cl}\rightarrow _{18}^{36}\textrm{X} + e + \bar{\nu}_e \) Now, we can check which element has an atomic number of 18 and find the product of decay. The element with atomic number 18 is Argon, symbol Ar. Thus, \( _{17}^{36}\textrm{Cl}\rightarrow _{18}^{36}\textrm{Ar} + e + \bar{\nu}_e \)

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

Empirical rules of nuclear stability are based on the number of protons and neutrons present in the nucleus. There are a few rules to keep in mind: 1. Even numbers of protons and neutrons are generally more stable due to pair formations. 2. A stable nucleon is one with a neutron-to-proton ratio (\(\textrm{N/Z}\) ratio) that is close to 1 for light elements and increases up to 1.5 for heavy elements. 3. Elements with an atomic number larger than 83 are inherently unstable and radioactive. Now, let's analyze each nuclide: 1. \({ }^{35} \mathrm{Cl}\): The atomic number, Z, is 17 and atomic mass number, A, is 35. Number of neutrons, N = A - Z = 35 - 17 = 18, which is an even number. The N/Z ratio = 18/17 = 1.06, which is close to 1. 2. \({ }^{37} \mathrm{Cl}\): The atomic number is still 17, and the atomic mass number is 37. Number of neutrons, N = 37 - 17 = 20, which is an even number. The N/Z ratio = 20/17 = 1.18, which is also close to 1. Comparing to \({ }^{36} \mathrm{Cl}\): 3. \({ }^{36} \mathrm{Cl}\): The atomic number is 17, and the atomic mass number is 36. Number of neutrons, N = 36 - 17 = 19, which is an odd number. The N/Z ratio is 19/17 = 1.12, which is close to 1 but the number of neutrons is odd. Based on the empirical rules of nuclear stability, both \({ }^{35} \mathrm{Cl}\) and \({ }^{37} \mathrm{Cl}\) have even numbers of neutrons, which generally make them more stable. In contrast, \({ }^{36} \mathrm{Cl}\) has an odd number of neutrons and a less favorable N/Z ratio, making it less stable than the other two nuclides and resulting in its radioactive nature.

Key concepts

Chlorine Isotopes

Chlorine is an interesting element with three notable isotopes: \ \[^{35} \textrm{Cl}, \, ^{36} \textrm{Cl}, \text{ and } \, ^{37} \textrm{Cl}\]\ . Each of these isotopes has its unique characteristics, especially when it comes to stability. For starters, both \ \[^{{35}} \textrm{Cl} \text{ and } ^{37} \textrm{Cl} \]\ are stable isotopes found naturally on Earth. Their stability makes them non-radioactive, allowing them to exist without spontaneously transforming into other elements.
On the other hand, \ \[^{36} \textrm{Cl} \]\ is a radioactive isotope, meaning it is unstable and will eventually decay over time. This behavior arises from its unique nuclear structure which creates pressure within the nucleus, leading to radiation and transformation. Despite being less common than its stable counterparts, \ \[^{36} \textrm{Cl} \]\ can still be found naturally and in certain scientific applications.
Knowing the different isotopes of chlorine and their properties is essential for understanding how elements transform and contribute to various chemical and nuclear processes.

Beta Emission

Beta emission is a fascinating form of radioactive decay. It occurs when an unstable nucleus attempts to reach stability by releasing particles. Specifically, beta emission involves the conversion of a neutron into a proton while emitting a beta particle (an electron) and an antineutrino. This transformation increases the atomic number by one but keeps the mass number constant. For example, when \ \[^{36} \textrm{Cl} \]\ undergoes beta emission, it transforms into \ \[^{36} \textrm{Ar} \]\ . Here’s the process: \ - A neutron in the \ \[^{36} \textrm{Cl} \]\ nucleus is turned into a proton. - A beta particle and an antineutrino are ejected from the nucleus. - The atomic number increases from 17 to 18, becoming argon.This process doesn't alter the atomic mass of the isotope, only the atomic number. Understanding beta emission is crucial in nuclear chemistry and physics, as it explains how certain elements transition into others while releasing radiation.

Nuclear Stability

Nuclear stability is an essential concept in understanding why certain isotopes are stable while others are not. This stability relates closely to the structure of an atomic nucleus, particularly the ratio of neutrons to protons. Several key rules help us determine nuclear stability:

• An even number of protons and neutrons generally leads to a more stable nucleus due to a pairing effect.
• The neutron-to-proton (\( \text{N/Z} \) ratio) should ideally be close to 1 for lighter elements and up to 1.5 for heavier elements to maintain balance.
• Isotopes of elements with atomic numbers higher than 83 are typically unstable and radioactive.

When examining \ \[^{35} \textrm{Cl}, \, ^{36} \textrm{Cl}, \text{ and } \, ^{37} \textrm{Cl} \]\ , the stability differs due to these factors.
- \ \[^{35} \textrm{Cl} \]\ has 18 neutrons (an even number), resulting in greater stability with an \( \text{N/Z} \) ratio of 1.06.- \ \[^{37} \textrm{Cl} \]\ also boasts an even number of 20 neutrons and an \( \text{N/Z} \) ratio of 1.18.- \ \[^{36} \textrm{Cl} \]\ , however, has 19 neutrons, an odd number, giving it an \( \text{N/Z} \) ratio of 1.12 and a lower stability compared to the other two isotopes.
Understanding nuclear stability helps explain why some isotopes naturally occur in abundance while others are rare and prone to radioactive decay.