Problem 28 - Chemistry: An Atoms-Focused Approach

Question

Calculate the ratio of neutrons to protons in the following group 15 nuclei: (a) \(^{14} \mathrm{N} ;\) (b) \(^{31} \mathrm{P} ;\) (c) \(^{75} \mathrm{As} ;\) (d) \(^{121} \mathrm{Sb} ;\) and (e) \(^{123}\) Sb. How does the ratio change with increasing atomic number?

Step-by-Step Solution

Answer

Question: Calculate the neutron to proton ratio for each of the following isotopes and observe the changes with increasing atomic number: (a) \(^{14}\mathrm{N}\), (b) \(^{31}\mathrm{P}\), (c) \(^{75}\mathrm{As}\), (d) \(^{121}\mathrm{Sb}\), (e) \(^{123}\mathrm{Sb}\). Answer: The neutron to proton ratios for each isotope are as follows: (a) 1, (b) ≈ 1.07, (c) ≈ 1.27, (d) ≈ 1.37, and (e) ≈ 1.41. As the atomic number increases, the neutron to proton ratio also increases, indicating a need for more neutrons to balance the increased electrostatic repulsion between protons and maintain stability in heavier nuclei.

Step 1: Find the atomic number of each element.

We will start by finding the atomic number for each element, which is the number of protons. We can look up these values in the periodic table. (a) Nitrogen (N) has atomic number 7. (b) Phosphorus (P) has atomic number 15. (c) Arsenic (As) has atomic number 33. (d) Antimony (Sb) has atomic number 51. (e) Antimony (Sb) also has atomic number 51.

Step 2: Find the number of neutrons in each isotope.

Now, we need to find the number of neutrons in each isotope by subtracting the atomic number from the mass number (given by the superscript in each example). (a) In \(^{14}\mathrm{N}\), there are 14 - 7 = 7 neutrons. (b) In \(^{31}\mathrm{P}\), there are 31 - 15 = 16 neutrons. (c) In \(^{75}\mathrm{As}\), there are 75 - 33 = 42 neutrons. (d) In \(^{121}\mathrm{Sb}\), there are 121 - 51 = 70 neutrons. (e) In \(^{123}\mathrm{Sb}\), there are 123 - 51 = 72 neutrons.

Step 3: Calculate the neutron to proton ratio for each isotope.

Now, we will calculate the neutron to proton ratio for each isotope by dividing the number of neutrons by the number of protons. (a) For \(^{14}\mathrm{N}\), the ratio is 7/7 = 1. (b) For \(^{31}\mathrm{P}\), the ratio is 16/15 ≈ 1.07. (c) For \(^{75}\mathrm{As}\), the ratio is 42/33 ≈ 1.27. (d) For \(^{121}\mathrm{Sb}\), the ratio is 70/51 ≈ 1.37. (e) For \(^{123}\mathrm{Sb}\), the ratio is 72/51 ≈ 1.41.

Step 4: Analyze how the neutron to proton ratio changes with increasing atomic number.

We can now observe the pattern between the neutron to proton ratio and the increasing atomic number. As the atomic number increases, the neutron to proton ratio also increases. This pattern is a result of the need for more neutrons to balance the increased electrostatic repulsion between protons as the atomic number increases. As a result, heavier nuclei generally have a greater number of neutrons compared to protons to ensure stability.

Key Concepts

Atomic Number

The atomic number of an element is fundamental in understanding how elements are organized. It represents the number of protons found in the nucleus of an atom. Each element on the periodic table has a unique atomic number, which defines the type of atom. For example:

• Nitrogen (N) has an atomic number of 7, meaning it has 7 protons.
• Phosphorus (P) has an atomic number of 15.
• Arsenic (As) carries an atomic number of 33.
• Antimony (Sb) has an atomic number of 51, noted in two isotopes as discussed later.

Thus, the atomic number is crucial as it provides not only the identity of the element but also helps in determining the arrangement of electrons around the nucleus in a neutral atom. The position of elements on the periodic table is based on their atomic number, paving the way for a better understanding of chemical reactions and bonding.

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but a different number of neutrons. This means, isotopes have the same atomic number but different mass numbers. Here's how isotopes are important:

• They help in understanding how atoms of the same element can differ.
• Each isotope of an element can have different properties, especially when it comes to stability and nuclear reactions.

For instance, Antimony can be found in different isotopes, such as:

• In the case of the isotopes discussed, both ¹²¹Sb and ¹²³Sb have an atomic number of 51, yet different mass numbers (121 and 123 respectively).
• Understanding isotopes is crucial in studying nuclear stability and reactions as they influence the neutron to proton ratio significantly.

The study of isotopes also extends to real-world applications, such as radiocarbon dating and medical imaging.

Periodic Table

The periodic table is a powerful tool in chemistry, functioning as a display of all known chemical elements organized by increasing atomic number, electron configurations, and recurring chemical properties. Here's what you need to know:

• Elements are arranged in rows called periods and columns called groups.
• Elements in the same group often share similar chemical and physical properties.
• The periodic table helps predict how elements will react chemically, showcasing trends such as electronegativity, ionization energy, and atomic radius.

Understanding its layout can enhance the learning of how elements interact, which is visible in the neutron to proton ratios seen as atomic numbers increase:

• The discussed isotopes of Nitrogen, Phosphorus, Arsenic, and Antimony are displayed, allowing us to see increasing neutron to proton ratios with higher atomic numbers as you move across periods and down groups.

In essence, the periodic table is a reflection of the atomic structure and behaviors of the elements, making it indispensable in the study of chemistry.

Nuclear Stability

Nuclear stability is a crucial concept when analyzing the ratio of neutrons to protons within a nucleus. An atom's stability depends greatly on this balance, and slight imbalances can lead to radioactive decay. Key points about nuclear stability include:

• Stable nuclei often have neutron to proton ratios near 1 for light elements.
• For heavier elements, this ratio increases, requiring more neutrons to counterbalance the repulsive forces among protons in larger nuclei.
• As seen in the original exercise, heavier nuclei like those found in Antimony need a greater neutron count to maintain stability, shown by their higher neutron to proton ratios.

The increase in neutron to proton ratios with atomic number illustrates how nature balances forces within the nucleus. Too few or too many neutrons can make a nucleus unstable, leading to different isotopes having varying stability:

• This understanding extends into various scientific fields, including nuclear physics and chemistry, providing the framework for technological advancements like nuclear energy and medicine.

By understanding nuclear stability, scientists can better predict which isotopes are stable and which are likely to undergo decay.