Problem 23 - Chemistry The Science in Context

Question

If the mass number of an isotope 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 when the mass number is more than twice the atomic number.

Step 1: Expressing Neutron-to-Proton Ratio in Terms of A and Z

To express the neutron-to-proton ratio in terms of the mass number (A) and atomic number (Z), we will use the equation A = Z + N. Rearrange the equation to solve for N: N = A - Z Now find the neutron-to-proton ratio as \(\frac{N}{Z}\):

Step 2: Evaluating the Neutron-to-Proton Ratio under the Condition A > 2Z

Now, we will substitute the expression for N derived in the previous step into the neutron-to-proton ratio and evaluate it under the given condition: Neutron-to-Proton ratio = \(\frac{N}{Z}\) = \(\frac{A-Z}{Z}\) = \(\frac{A}{Z}-1\) The question states that A > 2Z, so we can rewrite the inequality as: \(\frac{A}{Z} > 2\) (Divide both sides by Z, since Z > 0 for any element) Now, subtracting 1 from both sides: \(\frac{A}{Z} - 1 > 1\) We see that: Neutron-to-Proton ratio = \(\frac{A}{Z}-1\) > 1 Thus, the neutron-to-proton ratio is greater than \(1\) when the mass number (A) is more than twice the atomic number (Z).

Key Concepts

Mass Number

The mass number of an isotope is a crucial concept in understanding the composition of an atom. It is represented by the symbol \(A\) and defines the total number of protons and neutrons in the nucleus of an atom. Protons and neutrons, together, are called nucleons. The more nucleons an atom has, the heavier its nucleus becomes.
This number is always a whole number, as it is simply the sum of discrete particles, and it helps distinguish one isotope from another of the same element. While the atomic number stays constant among isotopes of an element, their mass numbers can vary, leading to different isotopes.

• Mass Number Formula: \(A = Z + N\), where \(Z\) is the atomic number (number of protons), and \(N\) is the number of neutrons.

A very high mass number compared to the number of protons results in a high neutron-to-proton ratio, indicating nuclear stability characteristics, which can explain natural isotopic distribution and stability.

Atomic Number

The atomic number, symbolized as \(Z\), is one of the most fundamental properties of an atom. It indicates the number of protons in the nucleus of an atom. As each element has a unique number of protons, the atomic number becomes the identifier of the element within the periodic table.
The atomic number is crucial because:

• It defines the chemical properties of an atom.
• It determines the element's position in the periodic table.
• The number of electrons in a neutral atom equals the number of protons.

For example, the element with \(Z = 6\) is carbon, meaning every carbon atom has 6 protons in its nucleus. This consistent number across all atoms of the same element provides coherence to chemical reactions and bonding.
An interesting point is that while isotopes of an element differ in their mass number due to variations in neutrons, their atomic number remains unchanged, preserving the chemical nature of the element.

Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons, leading to different mass numbers. This concept plays a significant role in understanding atomic structure and nuclear chemistry.
Despite their differences in mass, isotopes of the same element are generally chemically similar because they share the same number of electrons and protons.

• Stable Isotopes: Do not undergo radioactive decay.
• Radioactive Isotopes: Are unstable and decay over time, emitting radiation.

These differences in neutron count among isotopes lead to variations in their physical properties.
Natural samples of elements typically contain a mix of isotopes, the ratio of which can be very specific and consistent across samples. These isotopic distributions are crucial in applications like radiometric dating, medical diagnostics, and nuclear power, each relying on the unique characteristics of specific isotopes.