Question

The table to the right gives the number of protons \((p)\) and neutrons \((n)\) for four isotopes. (a) Write the symbol for each of the isotopes. (b) Which of the isotopes is most likely to be unstable? (c) Which of the isotopes involves a magic number of protons and/or neutrons? (d) Which isotope will yield potassium-39 following positron emission? \begin{equation}\begin{array}{|c|c|c|c|}\hline & {\text { (i) }} & {\text { (ii) }} & {\text { (iii) }} & {\text { (iv) }} \\ \hline p & {19} & {19} & {20} & {20} \\ \hline n & {19} & {21} & {19} & {20} \\ \hline\end{array} \end{equation}

Short answer

(a) The symbols for the isotopes are: Isotope (i) - \(K^{38}\), Isotope (ii) - \(K^{40}\), Isotope (iii) - \(Ca^{39}\), and Isotope (iv) - \(Ca^{40}\). (b) Isotope (ii) with 19 protons and 21 neutrons is most likely to be unstable. (c) Isotope (iv) has magic numbers of both protons and neutrons, and Isotope (iii) has a magic number of protons. (d) Isotope (iii) \(Ca^{39}\) will yield potassium-39 following positron emission, as it results in \(K^{39}\).

Step-by-step solution

Writing the symbol for each of the isotopes.

For each isotope, determine the element based on the number of protons and add the mass number as a superscript: - Isotope (i): 19 protons → Potassium (K), K\(^{19+19}\) → \(K^{38}\) - Isotope (ii): 19 protons → Potassium (K), K\(^{19+21}\) → \(K^{40}\) - Isotope (iii): 20 protons → Calcium (Ca), Ca\(^{20+19}\) → \(Ca^{39}\) - Isotope (iv): 20 protons → Calcium (Ca), Ca\(^{20+20}\) → \(Ca^{40}\) Symbols for each isotope: - Isotope (i): \(K^{38}\) - Isotope (ii): \(K^{40}\) - Isotope (iii): \(Ca^{39}\) - Isotope (iv): \(Ca^{40}\)

Identify the most likely unstable isotope.

Determine which of these isotopes is likely to be unstable based on unusual proton-neutron ratios. Comparing the isotopes: - Isotope (i): 19 protons and 19 neutrons - Isotope (ii): 19 protons and 21 neutrons - Isotope (iii): 20 protons and 19 neutrons - Isotope (iv): 20 protons and 20 neutrons Isotope (ii) with 19 protons and 21 neutrons has a higher ratio of neutrons to protons compared to other isotopes, so it is the most likely unstable isotope.

Identify isotopes with magic numbers.

Magic numbers of protons and neutrons are 2, 8, 20, 28, 50, 82, and 126. Check if any of the isotopes have these magic numbers: - Isotope (i): 19 protons and 19 neutrons (none) - Isotope (ii): 19 protons and 21 neutrons (none) - Isotope (iii): 20 protons (magic) and 19 neutrons (none) - Isotope (iv): 20 protons (magic) and 20 neutrons (magic) Isotope (iv) has magic numbers of both protons and neutrons, and isotope (iii) only has a magic number of protons.

Identify the isotope that yields potassium-39 after positron emission.

To find the isotope that yields potassium-39 after positron emission (loses a proton and gains a neutron), look for an isotope that will have 19 protons and a mass number of 39 after this process: - Isotope (i): \(K^{38}\) → no - doesn't have the right mass number - Isotope (ii): \(K^{40}\) → no - doesn't have the right mass number - Isotope (iii): \(Ca^{39}\) → yes, after positron emission, it becomes \(K^{39}\) - Isotope (iv): \(Ca^{40}\) → no - doesn't have the right mass number Therefore, isotope (iii) will yield potassium-39 following positron emission, as it results in \(K^{39}\).