Problem 21 - Chemistry: An Atoms-Focused Approach

Question

If the mass number of a nuclide is more than twice the atomic number, is the neutron-to-proton ratio less than, greater than, or equal to \(1 ?\)

Step-by-Step Solution

Answer

Answer: The neutron-to-proton ratio is greater than 1.

Step 1: Express the mass number in terms of the atomic number

According to the given condition, the mass number (A) is more than twice the atomic number (Z). We can write this as: \(A > 2Z\)

Step 2: Express the neutron number in terms of the mass number and atomic number

The mass number (A) is the sum of the number of protons (Z) and the number of neutrons (N). So, we can write the neutron number (N) in terms of the mass number (A) and atomic number (Z) as: \(N = A - Z\)

Step 3: Find the neutron-to-proton ratio in terms of A and Z

We need to find the neutron-to-proton ratio, which can be expressed as \(\frac{N}{Z}\). Using the expression for N from step 2, we can calculate this ratio as: \(\frac{N}{Z} = \frac{A - Z}{Z}\)

Step 4: Analyze the inequality to find the relationship between the neutron-to-proton ratio and 1

Now, we need to determine if the neutron-to-proton ratio is less than, equal to, or greater than 1 based on the given condition that \(A > 2Z\). Replacing A in the inequality with the expression for the neutron-to-proton ratio from step 3, we have: \(\frac{A - Z}{Z} > 1\) Now, we will multiply both sides by Z to get rid of the denominator: \(A - Z > Z\) Rearrange this inequality to isolate A: \(A > 2Z\) Here, we can see that the inequality we derived is the same as the given condition, which means the neutron-to-proton ratio is indeed greater than 1. In conclusion, if the mass number of a nuclide is more than twice the atomic number, the neutron-to-proton ratio is greater than 1.

Key Concepts

Mass Number

The mass number is a fundamental concept in understanding the composition of an atomic nucleus. It is represented by the letter \(A\) and signifies the total number of protons and neutrons in the nucleus of an atom. For any given atom, subtracting the atomic number from the mass number reveals the count of neutrons. This can be summarized with the formula:

• Mass Number (A) = Number of Protons (Z) + Number of Neutrons (N)

This holistic figure is crucial in identifying isotopes, which are variations of elements that have the same number of protons but different numbers of neutrons. The significance of the mass number becomes evident when comparing the stability and types of isotopes. In simple isotopic notation, such as \(\text{^{A}_{Z}X}\), \(A\) is crucial for distinguishing between isotopes like \(\text{^{14}_6C}\) and \(\text{^{12}_6C}\), both forms of carbon. Understanding this helps us comprehend atomic behaviors and the building blocks of matter.

Atomic Number

The atomic number is a central concept in chemistry, represented by the letter \(Z\). It indicates the number of protons found in the nucleus of an atom. The atomic number is singularly responsible for defining the identity of an element. Differentiating elements from one another, the atomic number offers a unique fingerprint to each element on the periodic table.
When considering the atomic number:

• It equals the number of protons in an atom's nucleus.
• It also equals the number of electrons in a neutral atom.

The atomic number is vital for the element's chemical properties. For example, all carbon atoms have an atomic number of 6. This defines what makes carbon, carbon, and distinguishes it from other elements like oxygen or nitrogen. Stability, reactivity, and the chemical bonding of the atom are highly influenced by its atomic number, further underscoring its importance in science.

Nuclide

A nuclide is a term that refers to a specific nucleus with a particular composition of protons and neutrons. It embodies the idea of a distinct atomic form and encompasses both isotopes and individual elements. Understanding nuclides helps in appreciating how varied forms exist within the same element, due to differences in neutron numbers.
Key aspects of nuclides include:

• Nuclides are specified by their exact proton count (atomic number \(Z\)) and neutron count (\(N\)).
• For example, carbon-12 and carbon-14 are different nuclides of the same element, carbon, sharing the same atomic number but differing in mass numbers and neutron count.

In scientific notation, a nuclide is often represented as \(\text{^{A}_{Z}X}\), where \(X\) represents the chemical element, \(A\) denotes the mass number, and \(Z\) is the atomic number. Recognizing nuclides is critical for nuclear chemistry and physics, where transformations and reactions depend on these specific structural details. By exploring nuclides, we uncover the principles governing nuclear stability and radioactivity.