Problem 6 - College Physics

Question

(a) Isotopes of an element must have the same (1) atomic number, (2) neutron number, (3) mass number. (b) Write two possible isotopes for gold-197.

Step-by-Step Solution

Answer

(a) Isotopes must have the same atomic number. (b) Two isotopes for gold-197 are Gold-196 and Gold-198.

Step 1: Understanding Isotopes

Isotopes of an element are atoms with the same number of protons (same atomic number) but different numbers of neutrons. This results in different mass numbers.

Step 2: Analyzing Given Options for Isotopes

Given options: (1) atomic number, (2) neutron number, (3) mass number. Isotopes must have the same atomic number, which corresponds to the number of protons in their nuclei. Neutron and mass numbers can differ among isotopes.

Step 3: Identifying Isotopes of Gold

Gold has an atomic number of 79, which means every gold atom has 79 protons. Isotopes of gold must all have 79 protons but can have different neutron counts.

Step 4: Naming Two Possible Isotopes for Gold-197

Gold-197 is an isotope with a mass number of 197. Possible isotopes of gold would also have the atomic number 79 but different mass numbers, such as Gold-198 or Gold-196, which differ by the number of neutrons compared to Gold-197.

Key Concepts

Atomic Number

The atomic number is a fundamental concept in chemistry and physics. It defines the identity of an element and is represented by the number of protons in the nucleus of an atom.
For instance, any atom with 79 protons is gold. An easy way to determine the atomic number is by looking at the periodic table, where elements are arranged according to their atomic numbers.

• The atomic number is crucial because it defines the chemical properties of an element.
• All atoms of the same element share the same atomic number, meaning they all have the same number of protons.

Since isotopes of an element have the same atomic number, isotopic variations arise from differences in neutron numbers, not protons.
In summary, understanding the atomic number is key to grasping why isotopes exist yet maintain the elemental identity through their consistent number of protons.

Neutrons

Neutrons are neutrally charged particles found within an atomic nucleus. They play a vital role in the stability of the atom and contribute to the mass number along with protons.
While protons determine the atomic number, neutrons vary among isotopes. This variance leads to different physical properties without altering the chemical identity. Consider gold isotopes like Gold-197 and Gold-198. These isotopes all have 79 protons but differ in their neutron count, which changes the total mass number.

• The number of neutrons in an atom influences the stability and radioactivity of the isotope.
• Neutrons help to offset the electrostatic repulsion between positive charges in the nucleus.

Understanding neutrons' role helps explain why isotopes might have different mass numbers, reflecting their unique neutron counts.

Mass Number

The mass number is the sum of protons and neutrons in the atomic nucleus. This number is essential for identifying and distinguishing isotopes of a single element. For example, different isotopes of gold, like Gold-197 and Gold-198, have different mass numbers due to variations in their neutron counts. A few things to note about mass numbers:

• The mass number is always a whole number.
• It is not equivalent to atomic mass, which takes into account isotope abundances and protons and neutrons' actual mass.
• When discussing isotopes, the mass number typically follows the element's name or symbol, like "Gold-197."

By understanding the mass number, students can discern differences between isotopes and calculate neutron variations.
This makes the concept pivotal in studies of atomic structure and nuclear chemistry.