All atoms of the same chemical element have the same value for which of the following quantities?

Introduction / Context:
The identity of a chemical element is defined by its atomic structure. Even though atoms of the same element can differ in mass number or charge, they remain the same element as long as one key quantity is unchanged. This question asks which property must be the same for all atoms of a given element, such as all atoms of carbon or all atoms of oxygen. Understanding this concept is fundamental to atomic theory and the structure of the periodic table.

Given Data / Assumptions:

• An atom consists of protons and neutrons in the nucleus and electrons in surrounding orbitals.
• Different isotopes of an element have different numbers of neutrons.
• Ions of an element have gained or lost electrons but retain the same nuclear structure.
• The periodic table organises elements by atomic number, which is tied to a specific nuclear property.

Concept / Approach:
All atoms of a particular element have the same atomic number. The atomic number is defined as the number of protons in the nucleus. For example, all carbon atoms have six protons, all oxygen atoms have eight protons and all sodium atoms have eleven protons. Isotopes of an element differ in neutron number, and ions differ in electron number, but the proton count remains the same. Therefore, the invariant quantity for atoms of the same element is the number of protons in the nucleus, and this is also what atomic number represents.

Step-by-Step Solution:
Step 1: Recall that the atomic number Z of an element is defined as the number of protons in the nucleus of each atom of that element. Step 2: Recognise that isotopes, such as carbon 12 and carbon 14, have the same number of protons but different numbers of neutrons, so neutron count is not fixed. Step 3: Understand that ions, such as Na+ or Cl-, differ from neutral atoms by gaining or losing electrons, so electron count is not fixed either. Step 4: Realise that mass number, the total of protons plus neutrons, varies among isotopes, so it is not the defining constant of an element. Step 5: Conclude that the only quantity that must be identical for all atoms of the same element is the number of protons in the nucleus.

Verification / Alternative check:
Look at the periodic table: each element is labeled by its atomic number, such as 1 for hydrogen, 6 for carbon and 26 for iron. Experimental measurements show that the charge on an atomic nucleus is proportional to the number of protons it contains, and this nuclear charge determines the element unique chemical identity. Changing the number of protons changes the element entirely; for example, if you could change one proton in a nitrogen nucleus into a proton plus neutron combination, you would get oxygen, not a different form of nitrogen. This confirms that proton number is the defining feature shared by all atoms of a given element.

Why Other Options Are Wrong:
- Number of neutrons in the nucleus: Neutron number can vary between isotopes, such as chlorine 35 and chlorine 37, which are both chlorine but have different neutron counts.
- Mass number: Since mass number equals protons plus neutrons, it also varies between isotopes of the same element.
- Number of electrons in the neutral atom: Neutral atoms of an element do have equal numbers of electrons and protons, but ions of that element can have different electron counts and still be the same element.
- Number of valence electrons in every ion: Valence electron count often changes when ions form, and even in neutral atoms it can differ across oxidation states in molecules.

Common Pitfalls:
Learners sometimes confuse atomic number with mass number and may think that mass number defines the element. Others focus on electrons because they determine chemical reactivity and bonding, forgetting that electrons can be lost or gained without changing the element identity. Always remember that it is the number of protons in the nucleus that fixes which element an atom belongs to, and this is the quantity that is the same for all atoms of a given element.

Final Answer:
All atoms of the same element have the same Number of protons in the nucleus.