According to the modern periodic law, the chemical and physical properties of elements are periodic functions of which fundamental atomic property?

Introduction / Context:
This question tests understanding of the modern periodic law, which underlies the arrangement of the periodic table. Earlier versions of the periodic law used atomic mass as the organising principle, but with the discovery of protons and atomic number, the modern version was adopted. Knowing exactly which property controls periodic trends is essential for reasoning about groups, periods, and periodic properties such as ionisation energy, electronegativity, and atomic radius.

Given Data / Assumptions:
- The question lists four possible controlling properties: valency, atomic mass, number of neutrons, and atomic number.
- We are asked which one appears in the wording of the modern periodic law.
- We assume standard definitions used in school level chemistry.

Concept / Approach:
The modern periodic law states that the physical and chemical properties of elements are periodic functions of their atomic numbers. Atomic number is defined as the number of protons in the nucleus, which also equals the number of electrons in a neutral atom. While Mendeleev originally arranged elements by atomic mass, anomalies such as the position of tellurium and iodine were resolved when atomic number was used instead. Valency and number of neutrons are related but do not define the primary sequence of the periodic table, and atomic mass is no longer the defining property in the modern law.

Step-by-Step Solution:
Step 1: Recall the exact wording of the modern periodic law: properties of elements are periodic functions of their atomic numbers.
Step 2: Consider valency. Although valency shows periodicity and is influenced by position in the table, it is an outcome of periodic behaviour, not the cause used to arrange elements.
Step 3: Atomic mass was used in the original Mendeleev periodic law, but this led to a few misplacements that were corrected when atomic number was adopted.
Step 4: The number of neutrons can vary among isotopes of the same element and does not uniquely define the element, so it cannot serve as the ordering principle for the periodic table.
Step 5: Atomic number, which is the count of protons in the nucleus, uniquely identifies each element and forms the basis of the modern periodic law and the present arrangement of the periodic table.

Verification / Alternative check:
If you open any modern chemistry textbook or periodic table chart, the elements are listed in increasing order of atomic number from hydrogen to oganesson. Group trends such as similar valence electron configurations and recurring properties appear at regular intervals when ordered this way. This visual pattern shows that atomic number is the fundamental ordering property, and all references to the modern periodic law explicitly mention atomic number rather than atomic mass or valency.

Why Other Options Are Wrong:
- Valency: It does show periodic behaviour but is a property that results from atomic structure and position in the periodic table, not the primary ordering principle in the modern law.
- Atomic mass: It was used in the original periodic law but was replaced by atomic number in the modern formulation because atomic mass created some inconsistencies in element placement.
- Number of neutrons: Different isotopes of the same element have different numbers of neutrons but share the same atomic number, so neutron count cannot define the periodic order of elements.

Common Pitfalls:
Students sometimes remember that Mendeleev used atomic mass and mistakenly apply that fact to the modern periodic law. Others may confuse atomic mass and atomic number because both are numbers printed in periodic table boxes. To avoid this, remember that atomic number is always the smaller whole number and defines the element, while atomic mass is usually a decimal reflecting a weighted average of isotopes. The wording modern periodic law is a clear signal that the correct property is atomic number.

Final Answer:
Atomic number is the fundamental property in the modern periodic law, and the properties of elements are periodic functions of their atomic numbers.