The correct option is (c).
Explanation
An element is a pure substance consisting of atoms that all possess the same number of protons in their nuclei. This specific number is known as the atomic number ($Z$), which serves as the fundamental basis for the identification and classification of elements in the Modern Periodic Table.
Statement-wise Analysis:
- Statement 1 is Correct: Elements are fundamentally defined by the number of protons (atomic number) present in the nucleus of their atoms. For instance, any atom with exactly 6 protons is Carbon.
- Statement 2 is Incorrect: Atoms of a specific element cannot have different numbers of protons. The number of protons is constant for a given element. If the proton count changes, the atom becomes a different element. Atoms of the same element may differ in the number of neutrons (forming isotopes), but the proton count remains identical.
- Statement 3 is Correct: The atomic number is the unique characteristic identifier of a particular element. No two distinct elements share the same atomic number.
Key Takeaway:
The identity of a chemical element is determined solely by its atomic number, which equals the number of protons in its nucleus. While the number of neutrons may vary among atoms of the same element (isotopes), the number of protons is invariant.