Problem 27 - Chemistry: An Atoms-Focused Approach

Question

Calculate the ratio of neutrons to protons in the following stable atomic nuclei: (a) \(^{4} \mathrm{He} ;\) (b) \(^{23} \mathrm{Na} ;\) (c) \(^{59} \mathrm{Co} ;\) and (d) \(^{197}\) Au. Each of these elements exists naturally as a single isotope. What trend do you observe for the neutronto-proton ratio as \(Z\) increases?

Step-by-Step Solution

Answer

Answer: As the atomic number (Z) increases, the neutron-to-proton ratio also increases, indicating that more neutrons are required to stabilize the nucleus compared to the number of protons.

Step 1: Find Atomic Mass (A), Atomic Number (Z), and Protons (P)

To find A, Z, and P for each element, we must first look at the given nuclei notations: \((a) \ ^{4}\mathrm{He};\ (b)\ ^{23}\mathrm{Na};\ (c)\ ^{59}\mathrm{Co};\ (d)\ ^{197}\mathrm{Au}\). In each case, the superscript is the atomic mass (A), and for each element, the atomic number (Z) can be found in the periodic table. For each element, (a) \(^{4}\mathrm{He}\): A = 4, Z = 2, P = Z = 2. (b) \(^{23}\mathrm{Na}\): A = 23, Z = 11, P = Z = 11. (c) \(^{59}\mathrm{Co}\): A = 59, Z = 27, P = Z = 27. (d) \(^{197}\mathrm{Au}\): A = 197, Z = 79, P = Z = 79. Step 2: Determine the number of neutrons in each nucleus.

Step 2: Find Number of Neutrons (N)

Using the formula N = A - Z, we can find the number of neutrons for each element: (a) \(^{4}\mathrm{He}\): N = 4 - 2 = 2. (b) \(^{23}\mathrm{Na}\): N = 23 - 11 = 12. (c) \(^{59}\mathrm{Co}\): N = 59 - 27 = 32. (d) \(^{197}\mathrm{Au}\): N = 197 - 79 = 118. Step 3: Calculate the neutron-to-proton ratio for each nucleus.

Step 3: Find Neutron-to-Proton Ratio (N/P)

Now we will find the ratio of neutrons to protons for each element by dividing N by P: (a) \(^{4}\mathrm{He}\): neutron-to-proton ratio = 2 / 2 = 1. (b) \(^{23}\mathrm{Na}\): neutron-to-proton ratio = 12 / 11 ≈ 1.09. (c) \(^{59}\mathrm{Co}\): neutron-to-proton ratio = 32/ 27 ≈ 1.19. (d) \(^{197}\mathrm{Au}\): neutron-to-proton ratio = 118 / 79 ≈ 1.49. Step 4: Observe the trend of neutron-to-proton ratio as Z increases.

Step 4: Trend of Neutron-to-Proton Ratio with Increasing Z

Comparing the neutron-to-proton ratios for the four nuclei, we can observe a trend. As the atomic number (Z) increases, the neutron-to-proton ratio also increases as follows: The sequence of Z values: 2, 11, 27, and 79. The corresponding neutron-to-proton ratios: 1, 1.09, 1.19, and 1.49. This observation suggests that as atomic number (Z) increases, the number of neutrons required to stabilize the nucleus becomes more significant compared to the number of protons.

Key Concepts

Atomic Nuclei

The atomic nucleus is the very dense region at the center of an atom, composed of two types of subatomic particles: protons and neutrons.
The protons carry a positive charge, while the neutrons are neutral, bearing no charge. Together, they are known as nucleons.
The nucleus makes up nearly the entire mass of the atom, although it occupies only a small part of the atom's volume. The role of the nucleus is crucial because it holds the atom together. The protons and neutrons are bound in the nucleus by strong nuclear forces, overcoming the repulsion between the positively charged protons. In stable nuclei, these forces provide the necessary energy balance and stability. As atomic nuclei grow larger (increasing protons and neutrons), maintaining stability requires a specific balance, often involving more neutrons. This balance influences the neutron-to-proton ratio, an important concept when considering nuclear reactions and stability.

Isotopes

Isotopes are variants of a particular chemical element that have different numbers of neutrons but the same number of protons. This means they have the same atomic number but different atomic masses.
These differences can affect the stability of the isotopes, with some being stable and others being radioactive. For example:

• Helium has isotopes such as ³He and ⁴He.
• Sodium ( ²³Na) is another example where only one stable isotope is naturally abundant.

Most elements exist in nature as mixtures of isotopes. The concept of isotopes is central in fields like nuclear medicine, radiometric dating, and nuclear energy, where different isotopes can have drastically different roles and applications. Understanding isotopes provides insight into the stability of an atom and how it might interact or change.

Atomic Number

The atomic number ( \(Z\)) is a fundamental characteristic of an element and is equal to the number of protons found in the nucleus of an atom. It defines the identity of the element, as each element has a unique atomic number.
For instance:

• Helium ( \(\text{He}\)) has an atomic number of 2.
• Sodium ( \(\text{Na}\)) has an atomic number of 11.

The atomic number not only determines the place of an element on the periodic table but also its chemical properties. As elements increase in atomic number, they can exhibit changes in stability, requiring adjustments in the neutron-to-proton ratio to maintain a stable nucleus.

Neutrons

Neutrons are subatomic particles found in the nucleus of an atom that have no electric charge. Neutrons, along with protons, make up the mass of the nucleus.
The neutron number ( \(N\)) is calculated by subtracting the atomic number from the atomic mass ( \(A\)):\[N = A - Z\].The presence of neutrons is critical for the stability of the nucleus. They add attractive forces that help counterbalance the electrostatic repulsion between the positive protons.
With increasing atomic numbers, more neutrons are typically needed compared to protons to maintain stability, which is why the neutron-to-proton ratio tends to increase with larger nuclei.

Protons

Protons are positively charged subatomic particles that reside within the atomic nucleus. Each proton carries a charge of +1e and a mass very similar to that of a neutron.
The number of protons in the nucleus determines the atomic number of the element making it the same as the number of electrons in a neutral atom. Protons play a critical role in determining the identity and chemical properties of an element. The balance and interaction of protons with neutrons define much of an atom's nuclear structure and stability.
When considering nuclear stability, the presence of more protons in larger atoms prompts additional neutrons to support the nuclear forces and manage the increasing repulsive forces from multiple positive charges.