Question

Which isotope in each of the following pairs should be more stable? (a) \({ }_{14}^{28} \mathrm{Si}\) or \({ }_{14}^{29} \mathrm{Si}\) (b) \({ }_{3}^{6} \mathrm{Li}\) or \({ }_{3}^{8} \mathrm{Li}\) (c) \({ }_{11}^{23} \mathrm{Na}\) or \({ }_{11}^{20} \mathrm{Na}\)

Short answer

a) \({ }_{14}^{28} \mathrm{Si}\) or \({ }_{14}^{29} \mathrm{Si}\)? b) \({ }_{3}^{6} \mathrm{Li}\) or \({ }_{3}^{8} \mathrm{Li}\)? c) \({ }_{11}^{23} \mathrm{Na}\) or \({ }_{11}^{20} \mathrm{Na}\)? Answer: a) \({ }_{14}^{28} \mathrm{Si}\) is more stable. b) \({ }_{3}^{6} \mathrm{Li}\) is more stable. c) \({ }_{11}^{20} \mathrm{Na}\) is more stable.

Step-by-step solution

Calculate the neutron number for each isotope

For each isotope, calculate the neutron number by subtracting the atomic number (number of protons) from the mass number (total number of protons and neutrons). The atomic number is denoted by the bottom number (Z) and the mass number by the top number (A). (a) For \({ }_{14}^{28} \mathrm{Si}\): Neutron number (N) = A - Z = 28 - 14 = 14 For \({ }_{14}^{29} \mathrm{Si}\): Neutron number (N) = A - Z = 29 - 14 = 15 (b) For \({ }_{3}^{6} \mathrm{Li}\): Neutron number (N) = A - Z = 6 - 3 = 3 For \({ }_{3}^{8} \mathrm{Li}\): Neutron number (N) = A - Z = 8 - 3 = 5 (c) For \({ }_{11}^{23} \mathrm{Na}\): Neutron number (N) = A - Z = 23 - 11 = 12 For \({ }_{11}^{20} \mathrm{Na}\): Neutron number (N) = A - Z = 20 - 11 = 9

Calculate the N/Z ratios for each isotope

Divide the neutron number (N) by the atomic number (Z) to calculate the N/Z ratios for each isotope. (a) For \({ }_{14}^{28} \mathrm{Si}\): N/Z ratio = 14/14 = 1 For \({ }_{14}^{29} \mathrm{Si}\): N/Z ratio = 15/14 ≈ 1.07 (b) For \({ }_{3}^{6} \mathrm{Li}\): N/Z ratio = 3/3 = 1 For \({ }_{3}^{8} \mathrm{Li}\): N/Z ratio = 5/3 ≈ 1.67 (c) For \({ }_{11}^{23} \mathrm{Na}\): N/Z ratio = 12/11 ≈ 1.09 For \({ }_{11}^{20} \mathrm{Na}\): N/Z ratio = 9/11 ≈ 0.82

Compare N/Z ratios to determine stability

Compare the N/Z ratios to determine which isotope in each pair should be more stable. (a) For silicon, \({ }_{14}^{28} \mathrm{Si}\) has an N/Z ratio of 1, which is closer to 1 than the N/Z ratio of \({ }_{14}^{29} \mathrm{Si}\) (1.07). Therefore, \({ }_{14}^{28} \mathrm{Si}\) should be more stable. (b) For lithium, \({ }_{3}^{6} \mathrm{Li}\) has an N/Z ratio of 1, which is closer to 1 than the N/Z ratio of \({ }_{3}^{8} \mathrm{Li}\) (1.67). Therefore, \({ }_{3}^{6} \mathrm{Li}\) should be more stable. (c) For sodium, \({ }_{11}^{20} \mathrm{Na}\) has an N/Z ratio of 0.82, which is closer to 1 than the N/Z ratio of \({ }_{11}^{23} \mathrm{Na}\) (1.09). Therefore, \({ }_{11}^{20} \mathrm{Na}\) should be more stable.

Key concepts

Neutron Number

Understanding the neutron number in an isotope is essential to grasp the concept of nuclear stability. The neutron number, simply put, is the count of neutrons in an atom's nucleus. It is denoted by the letter 'N.' Since the atomic number 'Z' represents the number of protons, the neutron number can be calculated by subtracting the atomic number from the mass number 'A' (the sum of protons and neutrons). For instance, in the isotope \( { }_{14}^{28} \mathrm{Si} \), the neutron number is \( 28 - 14 = 14 \).

The neutron number plays a crucial role in determining an isotope's stability. Atoms seek a balance between protons and neutrons to achieve stability. This balance depends on several factors, including the size of the nucleus. Lighter elements typically have an equal number of protons to neutrons, while heavier elements may require more neutrons than protons to remain stable. Therefore, finding the neutron number helps predict stability as it is part of the N/Z ratio which provides an indication of nuclear stability.

N/Z Ratio

The N/Z ratio, or neutron-to-proton ratio, is a key indicator of an isotope's stability. After calculating the neutron number as described earlier, we gauge stability by dividing that number by the atomic number, which gives us the N/Z ratio. For example, for \( { }_{3}^{6} \mathrm{Li} \), with 3 protons and 3 neutrons, the N/Z ratio is \( 3/3 = 1 \).

A stable isotope often has an N/Z ratio close to 1, especially for lighter elements. As elements get heavier, the N/Z ratio for stability tends to increase slightly since additional neutrons are needed to offset the greater repulsion between the numerous protons. In the examples given, comparing the calculated N/Z ratios can help us identify the more stable isotope between pairs. This concept is fundamental in discussing nuclear reactions and decay, as isotopes with less ideal N/Z ratios are more prone to such processes in pursuit of stability.

Atomic Number

The atomic number, represented by 'Z,' is the number of protons found in the nucleus of an atom, and it defines the chemical element. Each element on the periodic table has a unique atomic number; for instance, silicon (Si) has an atomic number of 14. This number is crucial because it not only dictates the element's identity but also its position in the periodic table. Moreover, the atomic number determines the element's chemical properties, as it is directly tied to the structure of its electrons. In the context of isotope stability, the atomic number remains constant within a particular element's isotopes and is used in combination with mass number to calculate the neutron number and the resultant N/Z ratio, which are indicative of an isotope's stability.

Mass Number

The mass number, denoted as 'A,' is the total number of protons and neutrons in an atom's nucleus. It is used to differentiate between isotopes of the same element, which have the same atomic number but varying numbers of neutrons. For instance, \( { }_{14}^{28} \mathrm{Si} \) has a mass number of 28, indicating a total of 28 protons and neutrons. Isotopes of silicon can have different mass numbers, such as 29 for \( { }_{14}^{29} \mathrm{Si} \), which means it has one more neutron than \( { }_{14}^{28} \mathrm{Si} \).

The mass number is important for calculating the neutron number and for understanding the concept of isotopes. While the atomic number gives us the identity of the element, the mass number tells us about the specific isotope's mass and allows us to infer its stability by considering the neutron-proton ratio. A stable isotope typically has a balanced ratio congruent with its position in the periodic table.