Question

One nuclide in each of these pairs is radioactive. Predict which is radioactive and which is stable: \((\mathbf{a}){ }_{20}^{40} \mathrm{Ca}\) and \({ }_{20}^{45} \mathrm{Ca},\) (b) \({ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C}\) (c) lead-206 and thorium-230. Explain your choice in each case.

Short answer

The stable isotopes are \({_{20}^{40} \mathrm{Ca}}\), \({^{12} \mathrm{C}}\), and Lead-206, while the radioactive isotopes are \({_{20}^{45} \mathrm{Ca}}\), \({^{14} \mathrm{C}}\), and Thorium-230. This is based on the magic numbers concept and the known stability of specific isotopes.

Step-by-step solution

Identifying the Basics of Isotopes and Magic Numbers

A stable isotope has a balanced ratio of protons and neutrons to maintain stability. The magic numbers provide a guideline for stability: 2, 8, 20, 28, 50, 82, and 126. Isotopes with magic number of protons and/or neutrons generally tend to be more stable.

Comparing Isotopes of Ca (Calcium)

\({_{20}^{40} \mathrm{Ca}}\) has 20 protons and 20 neutrons, and \({_{20}^{45} \mathrm{Ca}}\) has 20 protons and 25 neutrons. Both isotopes have a magic number of protons, but only \({_{20}^{40} \mathrm{Ca}}\) also has a magic number of neutrons. As a result, \({_{20}^{40} \mathrm{Ca}}\) is the stable isotope, and \({_{20}^{45} \mathrm{Ca}}\) is the radioactive isotope.

Comparing Isotopes of C (Carbon)

\({^{12} \mathrm{C}}\) has 6 protons and 6 neutrons, and \({^{14} \mathrm{C}}\) has 6 protons and 8 neutrons. While many elements tend to be more stable with an even number of both protons and neutrons, \({^{12} \mathrm{C}}\) is the stable isotope, while \({^{14} \mathrm{C}}\) is radioactive.

Comparing Lead-206 and Thorium-230

Lead-206 has 82 protons and 124 neutrons, and Thorium-230 has 90 protons and 140 neutrons. Lead-206 has a magic number of protons, while Thorium-230 does not. Moreover, it is known that isotope leads are stable elements. Thus, Lead-206 is the stable isotope, and Thorium-230 is radioactive.

Conclusion

To summarize, we've determined the stability of isotope pairs based on their position around magic numbers and other knowledge about the isotopes. - \({_{20}^{40} \mathrm{Ca}}\) is stable, and \({_{20}^{45} \mathrm{Ca}}\) is radioactive. - \({^{12} \mathrm{C}}\) is stable, and \({^{14} \mathrm{C}}\) is radioactive. - Lead-206 is stable, and Thorium-230 is radioactive.

Key concepts

Isotope Stability

Isotopes are variants of a chemical element that possess the same number of protons but different numbers of neutrons. The stability of an isotope depends primarily on the ratio of protons to neutrons within its nucleus. A balanced ratio usually contributes to stability, helping to prevent the nucleus from spontaneously breaking apart or decaying. This balance is crucial because it ensures that the attractive strong force between the nucleons (protons and neutrons) is sufficient to overcome the repulsive electromagnetic force between the positively charged protons.

There is a sweet spot where isotopes tend to be more stable, often characterized by even numbers of protons and neutrons. For instance, in the exercise, \( {_{20}^{40} \mathrm{Ca}} \) with an equal number of protons and neutrons fits this pattern. When the ratio becomes too skewed, such as in \( {_{20}^{45} \mathrm{Ca}} \), which has five extra neutrons, the isotope often becomes unstable or radioactive. This shift leads to factors like beta decay, where a neutron morphs into a proton while emitting a beta particle, in an attempt to reach a more stable state. Think of stable isotopes like balanced scales, where stability is a result of proper weight distribution.

Magic Numbers in Chemistry

Magic numbers are fascinating markers within the field of nuclear chemistry that hint at exceptional stability in isotopic nuclei. These numbers—2, 8, 20, 28, 50, 82, and 126—refer to the count of protons or neutrons in a nucleus such that, when reached, the nucleus manifests increased stability. This phenomenon is akin to the completion of valence electron shells in atoms, making magic-numbered nuclei less susceptible to radioactive decay.

Think of these numbers as milestones signifying completed energy levels within the nuclear structure. For instance, in the pair \( {_{20}^{40} \mathrm{Ca}} \) and \( {_{20}^{45} \mathrm{Ca}} \), the \( {_{20}^{40} \mathrm{Ca}} \) isotope has both its protons and neutrons equaling the magic number 20, contributing to its recognized stability. Meanwhile, \( {_{20}^{45} \mathrm{Ca}} \) diverges from this harmonic number, resulting in a propensity to decay. Similarly, Lead-206's peculiar stability is thanks to its magic number of 82 protons, setting it apart from Thorium-230, which doesn't boast any magic numbers in its structure. When students are navigating isotopes, identifying whether an isotope's nucleons add up to a magic number helps predict its likelihood of remaining unchanged over time.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and properties of atomic nuclei. At its core is the concept of radioactivity—where unstable isotopes undergo transformation to achieve stability, releasing energy in the form of radiation. This energy release occurs as the nucleus strives for a more stable configuration through alpha, beta, or gamma decay.

When considering isotopes in exercises, we delve into this transformative nature. For instance, \( {^{14} \mathrm{C}} \) is radioactive as it undergoes beta decay, transmuting into nitrogen by changing a neutron into a proton. The study of isotopes such as \( {^{14} \mathrm{C}} \), which has a longer half-life, is valuable beyond just chemistry, affecting archeology through carbon dating. Similarly, Thorium-230's radioactivity reveals its unstable quest for balance, providing insights into geological timescales through its decay to radium and eventually lead.

Understanding nuclear chemistry isn’t just about memorizing facts; it’s about comprehending how and why stability or instability occurs. Such knowledge equips us to explore practical implications ranging from energy production in nuclear reactors to medical applications in radiotherapy. In essence, nuclear chemistry offers a remarkable glimpse into the intrinsic behaviors of matter at its most elemental level.