All isotopes of the same chemical element have which of the following in common?

Introduction / Context:
This question deals with the fundamental definitions of isotopes, which are variants of the same chemical element. Understanding isotopes is crucial for topics such as radioactivity, nuclear chemistry and applications like radiocarbon dating. The key is to know which subatomic particles are the same and which differ among isotopes of a given element, and how this relates to atomic number and atomic mass.

Given Data / Assumptions:

• Isotopes are being compared within the same element.
• Atomic number is defined as the number of protons in the nucleus.
• Atomic mass (or mass number) is the total number of protons and neutrons.
• Options describe different possible relationships between atomic number and atomic mass for isotopes.
• We assume a basic understanding of protons, neutrons and electrons.

Concept / Approach:
By definition, isotopes of an element have the same number of protons in the nucleus, and therefore the same atomic number, but differ in the number of neutrons. Because the atomic mass (mass number) equals protons plus neutrons, different neutron counts lead to different mass numbers. The number of electrons in a neutral atom equals the number of protons, but isotopes can be neutral or form ions. The constant property across isotopes of the same element is therefore atomic number, while atomic mass can vary.

Step-by-Step Solution:
Step 1: Recall that the atomic number (Z) is the number of protons in an atom and uniquely identifies the element. Step 2: For isotopes of the same element, the number of protons is the same, so the atomic number is the same. Step 3: The mass number (A) is the sum of protons and neutrons: A = Z + number of neutrons. Step 4: Isotopes differ in their number of neutrons, so they have different mass numbers while sharing the same atomic number. Step 5: Therefore, isotopes of a single element have the same atomic number but different atomic masses. Step 6: This matches option C.

Verification / Alternative check:
Consider carbon isotopes as a familiar example. Carbon-12 has 6 protons and 6 neutrons, mass number 12. Carbon-14 has 6 protons and 8 neutrons, mass number 14. Both are carbon because they each have 6 protons, so their atomic number is 6. The difference is in the neutron count and therefore the mass. Similar patterns hold for hydrogen isotopes: protium (1H) has 1 proton and 0 neutrons, deuterium (2H) has 1 proton and 1 neutron and tritium (3H) has 1 proton and 2 neutrons. These examples clearly show constant atomic number and varying mass number among isotopes.

Why Other Options Are Wrong:
Option A suggests that isotopes have different atomic numbers and different masses, which would make them entirely different elements, not isotopes of the same element. Option B suggests different atomic numbers with the same mass, which again would indicate two different elements accidentally having the same mass number, not isotopes. Option D claims the same atomic number and the same atomic mass, which would describe identical atoms, not isotopes. Option E mentions different numbers of protons and electrons, which could describe ions but does not capture the defining feature of isotopes. Thus, only option C matches the precise definition of isotopes.

Common Pitfalls:
A common misunderstanding is to confuse mass number with atomic weight or to think that isotopes differ in both protons and neutrons. Remember that changing the number of protons changes the element itself, so isotopes must keep the proton count constant. Another pitfall is ignoring neutrons altogether and focusing only on protons and electrons. To avoid this, always remember the simple rule: isotopes are same element (same atomic number), different mass number due to different neutrons.

Final Answer:
All isotopes of the same element have the same atomic number but different atomic masses.