To understand the correct explanation of the modern periodic law of elements, let's examine the statements provided and determine which aligns with the scientific understanding of the periodic law:

1. Atomic number has no effect on chemical properties: This statement is incorrect. The periodic law is fundamentally based on the atomic number of elements, which directly influences their chemical properties.
2. Properties repeat periodically with increasing atomic number values: This is the correct statement. The modern periodic law states that the physical and chemical properties of elements are a periodic function of their atomic numbers. This means that when elements are arranged in order of increasing atomic number, elements with similar properties occur at regular intervals.
3. Properties depend only on atomic mass values of elements: This was the basis of Mendeleev's original periodic law but was later corrected. The modern periodic table is based on atomic numbers, not atomic masses.
4. Properties vary randomly without any periodic trend observed: This is incorrect as well. The periodic table is a systematic arrangement where elements show periodic trends in properties.

Based on this analysis, the statement "Properties repeat periodically with increasing atomic number values" best describes the modern periodic law of elements. This periodic repetition reflects the recurring pattern of electron configurations as the atomic number increases, which in turn affects chemical reactivity and other properties.

To summarize, modern periodic law emphasizes the periodicity of elemental properties when arranged by atomic number, ensuring elements with similar properties are aligned in groups, which aids in predicting chemical behavior.