Problem 23 - Chemistry: The Central Science in SI Units, Global Edition

Question

Which of the following nuclides have magic numbers of both protons and neutrons: \((\mathbf{a})\) beryllium- \(10,(\mathbf{b})\) silicon- 28 , (c) chromium-52, (d) nickel-56, (e) krypton-84?

Step-by-Step Solution

Answer

The nuclide with magic numbers of both protons and neutrons is (d) Nickel-56 (Ni-56) with Z=28 and N=28.

Step 1: Identify Protons and Neutrons

Determine the number of protons and neutrons in each of the given nuclides. The atomic number (Z) of the element gives the number of protons, while the mass number (A) minus the atomic number gives the number of neutrons (N). a) Beryllium-10 (Be-10) Z = 4 (Atomic number of Be) A = 10 N = A - Z = 10 - 4 = 6 b) Silicon-28 (Si-28) Z = 14 (Atomic number of Si) A = 28 N = A - Z = 28 - 14 = 14 c) Chromium-52 (Cr-52) Z = 24 (Atomic number of Cr) A = 52 N = A - Z = 52 - 24 = 28 d) Nickel-56 (Ni-56) Z = 28 (Atomic number of Ni) A = 56 N = A - Z = 56 - 28 = 28 e) Krypton-84 (Kr-84) Z = 36 (Atomic number of Kr) A = 84 N = A - Z = 84 - 36 = 48

Step 2: Check for Magic Numbers

Now, compare each of the determined values of Z and N with the magic numbers (2, 8, 20, 28, 50, 82, and 126) to see if they have magic numbers of both protons and neutrons. a) Be-10 Z = 4 (Not a magic number) N = 6 (Not a magic number) b) Si-28 Z = 14 (Not a magic number) N = 14 (Not a magic number) c) Cr-52 Z = 24 (Not a magic number) N = 28 (A magic number) d) Ni-56 Z = 28 (A magic number) N = 28 (A magic number) e) Kr-84 Z = 36 (Not a magic number) N = 48 (Not a magic number)

Step 3: Provide the Answer

Based on the comparison above, only Nickel-56 (Ni-56) has both magic numbers of protons (Z=28) and neutrons (N=28). So, the answer is (d) Nickel-56.

Key Concepts

Protons and Neutrons

Protons and neutrons are the core components of an atom's nucleus. Protons are positively charged particles, while neutrons carry no charge. Together, they are known as nucleons. Each element is defined by its number of protons, which is known as its atomic number. Neutrons vary among isotopes of a given element, influencing the atom's stability.

• Protons determine the identity of an element.
• Neutrons contribute to the mass and stability of a nucleus.

The presence of different numbers of neutrons creates isotopes of an element. For example, both hydrogen-1 and hydrogen-2 have one proton, but hydrogen-2 has an extra neutron. The stability and properties of an atom depend on the balance between protons and neutrons, which sometimes results in magic numbers leading to particularly stable configurations.

Atomic Number

The atomic number, often represented by the symbol \( Z \), is fundamental to an element's identity. It denotes the number of protons in an atom's nucleus, directly relating to its position on the periodic table.
For instance, Beryllium has an atomic number of 4, meaning it has 4 protons. This atomic number helps in identifying and distinguishing one element from another. Thus, when chemists describe elements or compounds, they often refer back to \( Z \) to ensure accuracy and specificity in chemical reactions.

• Atomic number is unique to each element.
• It dictates the chemical properties of the element.

Atomic number is crucial not only in identifying elements but also in determining the element's interactions with other atoms as it defines the element's electron configuration.

Mass Number

Mass number, symbolized as \( A \), represents the total number of protons and neutrons in an atom's nucleus. Unlike the atomic number, \( A \) is not unique to each element as it varies among isotopes.
For example, Nickel-56 has a mass number of 56, indicating the sum of 28 protons (defined by its atomic number) and 28 neutrons.

• Mass number = Protons + Neutrons.
• Varies with different isotopes of the same element.
• Generally, greater mass numbers hint at greater atomic mass.

Understanding mass number is essential for calculations involving atomic mass and isotopic composition, as it provides insight into the atom's stability and relative abundance in nature.

Stability of Nuclei

The stability of nuclei is closely linked with the balance between protons and neutrons inside the atom's core. Certain configurations of protons and neutrons lead to particularly stable nuclei, these are referred to as having magic numbers. Magic numbers are 2, 8, 20, 28, 50, 82, and 126, representing filled quantum shells within the nucleus.
Nickel-56 is known for its particularly stable nuclear configuration because both its proton and neutron numbers align with a magic number. Such arrangements result in lower energy states and enhanced stability, hence why they play a crucial part in nuclear physics.

• Atoms with magic numbers are more stable.
• Magic numbers signify filled nuclear shells.
• Stable nuclei are less likely to undergo radioactive decay.

Recognizing these patterns helps scientists predict an isotope's stability and its likelihood of existing naturally. This understanding is essential in fields such as nuclear energy and particle physics.