Question

One of the nuclides in each of the following pairs is radioactive. Predict which is radioactive and which is stable: \((\mathbf{a})_{19}^{39} \mathrm{K}\) and \(_{19}^{40} \mathrm{K},\) \((\mathbf{b})^{209} \mathrm{Bi}\) and \(^{208} \mathrm{Bi}\) \((\mathbf{c})\) nickel-58 and nickel-65.

Short answer

The radioactive nuclides in each pair are: \(\mathbf{a)}\) \(_{19}^{40} \mathrm{K}\), \(\mathbf{b)}\) \(^{208}\mathrm{Bi}\), and \(\mathbf{c)}\) Nickel-65. This is based on their neutron-proton ratios, which help determine their stability.

Step-by-step solution

Pair A: \(_{19}^{39} \mathrm{K}\) and \(_{19}^{40} \mathrm{K}\)

First, let's find the n/p ratios for both nuclides. For \(_{19}^{39} \mathrm{K}\): - Number of protons (Z) = 19 - Number of neutrons (N) = Mass number (A) - Protons (Z) = 39 - 19 = 20 - n/p ratio = N/Z = 20/19 ≈ 1.05 For \(_{19}^{40} \mathrm{K}\): - Number of protons (Z) = 19 - Number of neutrons (N) = Mass number (A) - Protons (Z) = 40 - 19 = 21 - n/p ratio = N/Z = 21/19 ≈ 1.11 Both nuclides have a similar n/p ratio, but since both are lighter elements (Z ≤ 20), \(_{19}^{39} \mathrm{K}\) with a n/p ratio closer to 1 is more stable. Thus, \(_{19}^{40} \mathrm{K}\) is radioactive.

Pair B: \(^{209} \mathrm{Bi}\) and \(^{208}\mathrm{Bi}\)

Let's find the n/p ratios for both nuclides. For \(^{209} \mathrm{Bi}\): - Number of protons (Z) = 83 - Number of neutrons (N) = Mass number (A) - Protons (Z) = 209 - 83 = 126 - n/p ratio = N/Z = 126/83 ≈ 1.52 For \(^{208} \mathrm{Bi}\): - Number of protons (Z) = 83 - Number of neutrons (N) = Mass number (A) - Protons (Z) = 208 - 83 = 125 - n/p ratio = N/Z = 125/83 ≈ 1.51 Both nuclides are heavier elements (Z>20) with a similar n/p ratio. However, \(^{209} \mathrm{Bi}\) is slightly more stable, given its ratio is closer to 1.5; thus, \(^{208}\mathrm{Bi}\) is radioactive.

Pair C: Nickel-58 and Nickel-65

Let's find the n/p ratios for both nuclides. For Nickel-58: - Number of protons (Z) = 28 (Nickel atomic number) - Number of neutrons (N) = Mass number (A) - Protons (Z) = 58 - 28 = 30 - n/p ratio = N/Z = 30/28 ≈ 1.07 For Nickel-65: - Number of protons (Z) = 28 (Nickel atomic number) - Number of neutrons (N) = Mass number (A) - Protons (Z) = 65 - 28 = 37 - n/p ratio = N/Z = 37/28 ≈ 1.32 Nickel is a lighter element (Z ≤ 20), so a n/p ratio close to 1 indicates stability. Therefore, Nickel-58 is stable, and Nickel-65 is radioactive.

Key concepts

n/p ratio

The neutron to proton (n/p) ratio is a fundamental aspect in determining the stability of an isotope. It compares the number of neutrons (N) to the number of protons (Z) in an atom's nucleus. The stability of atomic nuclei is closely related to this ratio, with different rules applying based on whether the element is light (with a lower atomic number, Z ≤ 20) or heavy (with a higher atomic number, Z>20).

Stable light elements tend to have n/p ratios close to 1, which means they have nearly equal numbers of neutrons and protons. As elements get heavier, a greater number of neutrons is needed to offset the increasing electrostatic repulsion between the positively charged protons. Therefore, heavier stable isotopes generally have n/p ratios that are increasingly greater than 1, often approaching or exceeding a value around 1.5.

When the n/p ratio deviates significantly from the expected stable range, the nucleide tends to be radioactive, as the forces within the nucleus are not balanced, leading to various modes of radioactive decay to reach a more stable state.

stable isotopes

Stable isotopes are versions of elements that do not undergo radioactive decay over time. They remain unchanged indefinitely. Each element can have one or several stable isotopes, and the key characteristic of these stable isotopes is their n/p ratio, which falls within a certain range that allows the forces within their nuclei to exist in equilibrium.

Stable isotopes are incredibly useful in a variety of scientific applications. For instance, they serve as tracers in biological and geological studies because their non-radioactive nature does not pose a danger to living organisms or the environment. Carbon-12 and Carbon-13 are examples of stable isotopes of carbon, with Carbon-12 being the standard against which atomic weights are measured.

radioactive isotopes

Radioactive isotopes, or radioisotopes, are the unstable forms of elements that emit radiation as they decay into more stable forms. Unlike stable isotopes, radioactive isotopes have n/p ratios that do not align with stability, leading to various modes of decay such as alpha, beta, and gamma decay, each accompanied by the emission of particles or energy.

Radioisotopes have many practical applications, ranging from medical diagnostic and treatment procedures, such as in cancer therapy with isotopes like Technetium-99m, to archaeological dating using radiocarbon dating with Carbon-14. These isotopes are time-sensitive tracers due to their half-lives, which provide scientists with a clock to measure the duration of processes and age of objects. However, the same radioactivity that makes them useful also requires careful handling to prevent unwanted exposure to radiation.