Question

Of the following isotopes, which one is likely to be the most stable? (a) \(\mathrm{Zn}^{63}\) (b) \(\mathrm{Zn}^{67}\) (c) \(\mathrm{Zn}^{71}\) (d) \(\mathrm{Zn}^{64}\)

Short answer

\( \mathrm{Zn}^{64} \) is likely to be the most stable due to having an even number of neutrons and a favorable neutron-to-proton ratio.

Step-by-step solution

Consider Nuclear Stability Rules

We need to consider the general rules of nuclear stability to address this question. Stable nuclei tend to have even numbers of protons and neutrons, and they also have a neutron-to-proton ratio that is close to the value for stable isotopes of that element. Some isotopes are particularly stable if they have 'magic numbers' of protons or neutrons (2, 8, 20, 28, 50, 82, or 126), but this is not applicable here since Zinc (Zn) has a fixed atomic number of 30.

Determine the Neutron Number of Each Isotope

To continue analyzing stability, we must calculate the number of neutrons in each isotope. The number of neutrons (N) is the difference between the mass number (A) and the atomic number (Z). So, for the isotopes provided: \( \mathrm{Zn}^{63} \) has 33 neutrons (63 - 30), \( \mathrm{Zn}^{67} \) has 37 neutrons, \( \mathrm{Zn}^{71} \) has 41 neutrons, and \( \mathrm{Zn}^{64} \) has 34 neutrons.

Identify the Isotope with an Even Number of Neutrons

Now, we look for the isotope(s) with even numbers of both protons and neutrons. Since zinc has 30 protons, which is an even number, we are looking for the isotopes with even neutron numbers. \( \mathrm{Zn}^{64} \) has 34 neutrons, which is even, while the others do not.

Assess Neutron-to-Proton Ratio

A stable neutron-to-proton (N/Z) ratio is typically around 1 for lighter elements (Z < 20) and increases towards 1.5 for heavier elements. Since zinc is a heavier element, we can expect a stable N/Z ratio to be slightly above 1 but not too far from it. \( \mathrm{Zn}^{64} \) has an N/Z ratio of 34/30 = 1.13, which is closer to the stable range than the other isotopes provided.

Conclusion

Considering the even number of neutrons and the appropriate neutron-to-proton ratio, isotope \( \mathrm{Zn}^{64} \) is likely to be the most stable out of the options given.

Key concepts

Neutron-to-Proton Ratio and Nuclear Stability

Understanding the neutron-to-proton ratio is crucial in assessing the stability of a nucleus. For lighter elements (those with an atomic number less than 20), the ratio tends to approximate 1:1. However, as we progress to heavier elements, this ratio slightly increases, being more towards 1.5. This is because additional neutrons are needed to counterbalance the increased proton-proton repulsion due to their positive charges. The added neutrons reduce the electrostatic forces between protons and thereby promote a more stable arrangement.

In our example, four Zinc isotopes are given. Zinc, with a fixed atomic number of 30, falls into the category where we can expect the stable neutron-to-proton ratio to be slightly above 1. When we look at \(\text{Zn}^{64}\) with 34 neutrons, the neutron-to-proton ratio is \(\frac{34}{30}\approx 1.13\), which aligns well with the expected ratio for heavier elements like Zinc. This suggests that this neutron-to-proton ratio contributes to the isotope's stability, indicating why \(\text{Zn}^{64}\) is most likely the most stable among the given options.

Magic Numbers in Nuclear Physics

While the neutron-to-proton ratio is a general guide for nuclear stability, 'magic numbers' in nuclear physics are like winning combinations providing enhanced stability. These numbers refer to the number of nucleons (either protons or neutrons) within a nucleus that constitutes 'completed shells'. When either the number of protons or neutrons is 2, 8, 20, 28, 50, 82, or 126, the nucleus tends to be extremely stable. These magic numbers are akin to closed electron shells in atomic chemistry, which result in noble gas stability.

Although the concept of magic numbers does not apply directly to our exercise since Zinc has a constant number of protons, it is important to understand this concept because it can explain certain anomalies in the pattern of stability across the periodic table. For the isotopes of Zinc discussed, none feature magic numbers of neutrons; however, the understanding of magic numbers can help explain why certain nuclei may exhibit unexpected stability based on neutron or proton count alone.

Isotope Stability

When discussing specific isotopes, such as those of Zinc, stability can be influenced by several factors, including the neutron-to-proton ratio and the presence of magic numbers. However, other general rules apply, such as the tendency of nuclei to be more stable if they have even numbers of protons and neutrons. This is known as the even-even rule, indicating that nuclei with even Z and N tend to be more stable than those that don't.

Among the isotopes listed, \(\text{Zn}^{64}\) not only has a favorable neutron-to-proton ratio but also adheres to this even-even rule, giving it an extra edge in stability. It possesses an even number of protons (30) and an even number of neutrons (34), which statistically makes it more stable when compared to its counterparts. Thus, these concepts collectively offer a rationale behind the increased stability of \(\text{Zn}^{64}\) among the Zinc isotopes presented in the exercise.