Isotopes are atoms of the same chemical element that have the same atomic number but different values of which quantity?

Introduction / Context:
The concept of isotopes is central to modern atomic theory and has important applications in nuclear chemistry, medical imaging, and dating of archaeological samples. Isotopes explain how atoms of the same element can have slightly different masses while retaining the same chemical properties. This question focuses on the definition of isotopes and asks which subatomic quantity differs between isotopes of the same element.

Given Data / Assumptions:

• We are discussing atoms of the same element, so they must have the same atomic number.
• Atomic number is defined as the number of protons in the nucleus.
• The options mention neutrons, electrons, protons, and atomic numbers.

Concept / Approach:
Atoms of the same element must have the same number of protons, because the atomic number defines the element. For example, all carbon atoms have 6 protons. Isotopes are variants of the same element that differ in mass number, which is the total number of protons plus neutrons. Since the protons stay the same, the difference must come from the number of neutrons. Therefore, isotopes have the same number of protons and electrons in neutral atoms, but different numbers of neutrons.

Step-by-Step Solution:
Step 1: Recall that the atomic number equals the number of protons and defines the identity of the element. Step 2: Remember that isotopes are described as atoms of the same element with different mass numbers. Step 3: Note that mass number equals number of protons plus number of neutrons. Step 4: Since the element is the same, the number of protons cannot change, so the change must be in the number of neutrons. Step 5: Conclude that isotopes differ in their numbers of neutrons, which leads to different mass numbers.

Verification / Alternative check:
Consider the common isotopes of carbon: carbon 12 and carbon 14. Both have 6 protons, because they are carbon, but carbon 12 has 6 neutrons while carbon 14 has 8 neutrons. The atomic number remains 6 in both cases, but the mass numbers are 12 and 14. Neutral atoms of both isotopes also have 6 electrons. This confirms that the defining difference between isotopes of the same element is the number of neutrons in the nucleus.

Why Other Options Are Wrong:
Option B, numbers of electrons, is not correct because ions of the same element can have different numbers of electrons, but isotopes are defined by neutron difference, not electron count. Option C, numbers of protons, is wrong because changing the number of protons would change the element itself, not create isotopes of the same element. Option D, atomic numbers, is incorrect because the atomic number is exactly the same for isotopes of one element. These alternatives do not match the precise definition of isotopes.

Common Pitfalls:
A common confusion is between isotopes and ions. Ions of the same element differ in electron numbers, while isotopes differ in neutron numbers. Another pitfall is mixing up atomic number and mass number, especially when students first learn about nuclear notation. To avoid these mistakes, remember that isotopes share atomic number but have different mass numbers because of different neutron counts. Keeping this distinction clear helps in solving many nuclear chemistry and radioactivity problems.

Final Answer:
Isotopes are atoms of the same element that differ in their numbers of neutrons while having the same atomic number.