Introduction:
The Modern Periodic Law states that the physical and chemical properties of elements are periodic functions of their atomic numbers. This means that as the atomic number of an element increases, its properties repeat in a periodic manner. The recognition of the Modern Periodic Law is primarily based on the periodic variation in electronic configurations of elements.

Explanation:
The electronic configuration of an element refers to the arrangement of electrons in its atomic or molecular orbitals. The number and distribution of electrons in these orbitals directly influence the chemical behavior and properties of an element. The Modern Periodic Law recognizes that as the atomic number of an element increases, the electronic configuration changes in a predictable and periodic manner.

Periodic Variation:
The periodic variation in electronic configurations is a consequence of the arrangement of electrons in energy levels and sublevels. The electronic configuration of an element determines its chemical reactivity, electronegativity, ionization energy, and other properties. As the atomic number increases, additional electrons occupy higher energy levels and sublevels, leading to changes in these properties.

Trends in Electronic Configurations:
The periodic table provides a systematic arrangement of elements based on their atomic numbers and electronic configurations. Elements in the same group or column have similar electronic configurations, resulting in similar chemical properties. For example, the elements in Group 1 (alkali metals) all have one valence electron in their outermost energy level, giving them similar reactivity and the ability to form +1 ions.

Periodic Table and Electronic Configurations:
The periodic table is divided into blocks based on the filling of different sublevels. The s-block elements have their valence electrons in s orbitals, the p-block elements have their valence electrons in p orbitals, and so on. This arrangement reflects the periodic variation in electronic configurations and helps explain the trends in chemical properties observed across the table.

Conclusion:
In conclusion, the recognition of the Modern Periodic Law is primarily based on the periodic variation in electronic configurations of elements. The arrangement of electrons in energy levels and sublevels determines the chemical properties of elements, and as the atomic number increases, these electronic configurations change in a predictable and periodic manner. This understanding is fundamental to the organization and interpretation of the elements in the periodic table.