Introduction:
The modern periodic law states that the physical and chemical properties of elements are periodic functions of their atomic numbers. This means that when elements are arranged in order of increasing atomic number, their properties repeat at regular intervals.

Explanation:
The modern periodic table is based on the concept of atomic number, which is the number of protons in the nucleus of an atom. Atomic number is a fundamental property of an element and determines its position in the periodic table.

When elements are arranged in order of increasing atomic number, certain patterns and trends in their properties emerge. These patterns occur because the arrangement of electrons in atoms, which is determined by the atomic number, directly influences the chemical behavior of elements.

Periods:
The periodic table is divided into periods, which are horizontal rows. Each period represents a different energy level or shell in which electrons are arranged. As we move from left to right across a period, the atomic number increases, and so does the number of protons and electrons.

The properties of elements within a period gradually change as we move across the table. This is because the number of valence electrons, which are the outermost electrons involved in chemical bonding, increases by one from left to right within a period. The increase in valence electrons leads to changes in properties such as atomic size, ionization energy, and electronegativity.

Groups:
The periodic table is also divided into groups, which are vertical columns. Elements within the same group have similar properties because they have the same number of valence electrons. The number of valence electrons in a group remains constant as we move down the table.

Elements in the same group tend to have similar chemical reactivity and form similar types of compounds. This is because the number of valence electrons determines an element's ability to gain, lose, or share electrons during chemical reactions.

Conclusion:
In summary, the modern periodic law states that the properties of elements repeat at regular intervals when the elements are arranged in order of increasing atomic number. This arrangement allows for the observation of patterns and trends in the properties of elements within periods and groups. Understanding these patterns is crucial for predicting and explaining the behavior of elements in various chemical reactions.