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 ${}^{36}\mathrm{C}$ is less stable than either ${}^{35}\mathrm{Cl}$ or ${}^{37}\mathrm{Cl}$.

Short answer

The product of decay of ${}^{36}\mathrm{Cl}$ undergoing beta emission is ${}^{36}\mathrm{Ar}$. The radioactive nuclide ${}^{36}\mathrm{Cl}$ is less stable than the stable nuclides ${}^{35}\mathrm{Cl}$ and ${}^{37}\mathrm{Cl}$ due to having an odd number of neutrons and not obeying the empirical rules for nuclear stability, such as the even-odd rule and the magic number rule.

Step-by-step solution

Determine the Product of Decay

The decay we are considering is beta emission (β- decay), which occurs when a neutron in an unstable nucleus converts into a proton, an electron (β- particle), and an electron antineutrino. The general equation for beta decay is: $n\to p+e^{-}+{\overline{\nu }}_e$ Now, applying this rule to ${}^{36}\mathrm{Cl}$: ${}_{17}^{36}\mathrm{Cl}{\to }_Z^A\mathrm{X}+e^{-}+{\overline{\nu }}_e$ Since one neutron transforms into a proton, the mass number (A) remains the same but the atomic number (Z) increases by one. So the product of decay ($\mathrm{X}$) has: atomic number (Z) = 17+1 = 18 mass number (A) = 36 By looking at the periodic table, the element with atomic number 18 is Argon ($\mathrm{Ar}$). Thus, the product of decay for ${}^{36}\mathrm{Cl}$ is: ${}_{17}^{36}\mathrm{Cl}{\to }_{18}^{36}\mathrm{Ar}+e^{-}+{\overline{\nu }}_e$

Nuclear Stability Analysis

To analyze the stability of the given nuclides, we can use the two empirical rules of nuclear stability: 1. Even-odd rule: Isotopes with even number of protons and neutrons are more stable than isotopes with odd number of protons and neutrons. 2. Magic number rule: Isotopes with a "magic number" of protons or neutrons (2, 8, 20, 28, 50, 82, or 126) tend to be more stable. For ${}^{35}\mathrm{Cl}$: - Number of protons (Z) = 17 (odd) - Number of neutrons (N) = 35 - 17 = 18 (even) For ${}^{37}\mathrm{Cl}$: - Number of protons (Z) = 17 (odd) - Number of neutrons (N) = 37 - 17 = 20 (even and magic number) For ${}^{36}\mathrm{Cl}$: - Number of protons (Z) = 17 (odd) - Number of neutrons (N) = 36 - 17 = 19 (odd) According to the even-odd rule, both stable isotopes ${}^{35}\mathrm{Cl}$ and ${}^{37}\mathrm{Cl}$ have an even number of neutrons, while the unstable isotope ${}^{36}\mathrm{Cl}$ has an odd number of neutrons. Furthermore, ${}^{37}\mathrm{Cl}$ has a magic number of neutrons (20), making it more stable. In conclusion, the product of decay of ${}^{36}\mathrm{Cl}$ undergoing beta emission is ${}^{36}\mathrm{Ar}$. The radioactive nuclide ${}^{36}\mathrm{Cl}$ is less stable than the stable nuclides ${}^{35}\mathrm{Cl}$ and ${}^{37}\mathrm{Cl}$ due to having an odd number of neutrons and not obeying the empirical rules for nuclear stability.

Key concepts

Nuclear Stability

Nuclear Stability is a fundamental concept in nuclear physics that determines whether an atomic nucleus is stable or prone to decay. Stability in the atomic nucleus comes from a delicate balance between the attractive nuclear force that binds protons and neutrons together and the repulsive electromagnetic force between protons.

For an atomic nucleus to be stable, it often follows certain rules, like the even-odd rule and magic number rule discussed in the exercise solution. Isotopes with an even number of both protons and neutrons are generally more stable than those with odd numbers. This stability arises because pairs of nucleons (protons or neutrons) form more stable configurations.

Moreover, there are certain 'magic numbers' of protons or neutrons (2, 8, 20, 28, 50, 82, or 126) that make a nucleus extraordinarily stable. Elements with magic numbers of protons and neutrons are often found in nature and have very long half-lives if they are radioactive. These rules provide a basic framework for predicting nuclear stability, albeit many exceptions exist which are explained by more complex nuclear models.

Isotopes

Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons in their nuclei. This variance in neutron number leads to differences in mass and stability among isotopes of a single element. For chlorine, as presented in the exercise, the isotopes in question are ³⁵Cl, ³⁷Cl, and ³⁶Cl.

Isotopes can be stable or unstable (radioactive), depending on the balance of forces within the nucleus. Stable isotopes, like ³⁵Cl and ³⁷Cl, do not change over time, while unstable isotopes, such as ³⁶Cl, undergo radioactive decay to become more stable. The study of isotopes is crucial in fields ranging from archaeology (carbon dating) to medicine (radiotherapy), where the specific properties of isotopes are harnessed for various applications.

Understanding isotopes also plays a key role in nuclear chemistry and the predictable patterns of radioactive decay, which can be utilized to predict the products of such processes and their implications in terms of stability and radiation.

Radioactive Decay

Radioactive decay is a process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. There are different types of radioactive decay, such as alpha decay, beta decay, and gamma decay, each involving the emission of different particles or radiation.

Beta emission, the focus of our exercise, involves the transformation of a neutron into a proton within the nucleus, which results in the emission of an electron (also known as a beta particle) and an antineutrino. This transformation causes the element to change into another element, as seen with ³⁶Cl decaying into ³⁶Ar. A good mnemonic to remember the process of beta emission is that the 'B' in beta stands for 'Bigger', meaning the atomic number gets bigger by one.

Understanding radioactive decay is important not only for physics but also for understanding the behavior of radioactive isotopes in different environments. It explains why certain materials are radioactive and how they can be managed safely in various applications, such as nuclear power generation, medical treatments, and radiometric dating techniques.