Why is there a big space in the periodic table between the 2nd & 3rd periods? The activity of the electrons is related somehow.

3d level starts in the fourth period. There is no d filling in the second or first period

1s
2s...2P
3s...3p
4s3d4p

I think you mean big spaces IN the 2nd and 3rd periods. There are no elements in columns #3 to #12 because there is no "2d" subshell of electrons. Also because the energy of the 3d subshsell is higher than that of the "4s" subshell.

The first row of elements that contains "d" subshell electrons is the 4th row, atomic numbers 11 to 20. In that range the 3d subshell gradually fills up as we go from Sc to Zn. There are no "d" subshells below 3d and no "d-block" elements in the first three rows.

The big space between the 2nd and 3rd periods in the periodic table is called the d-block or transition metals block. This gap exists because it represents the series of elements where electrons fill up the 3d orbitals in their atomic structure.

To understand why this gap exists, it's important to know how the electrons are organized in an atom. Electrons occupy different energy levels or shells around the nucleus. The first shell can hold up to 2 electrons, the second shell can hold up to 8 electrons, and the third shell can hold up to 18 electrons.

In the second period of the periodic table, the atoms of elements like lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon have their outermost or valence shell completely filled with electrons after having 2 or 8 electrons (except for helium, which has only 2 electrons in total). These elements have properties that are quite different from the elements that follow them in the third period.

In the third period, the electrons first start filling the 3d orbitals in addition to the 3s and 3p orbitals. This additional set of orbitals, the 3d orbitals, lies at a higher energy level than the 2p orbitals but lower than the 4s orbitals (which will be filled in the following periods).

The filling order of these orbitals is based on the Aufbau principle, which states that electrons occupy the lowest energy orbitals available first. So, after the 2p orbitals are filled, the 3s orbitals are filled, and then the 3d orbitals are gradually filled.

The filling of the d-orbitals creates the large gap in the periodic table between the 2nd and 3rd periods. As a result, elements with electrons filling the 3d orbitals have distinct properties compared to the elements before and after them. These elements are generally categorized as transition metals and tend to exhibit similar chemical behavior due to their electron configurations.

In summary, the big gap between the 2nd and 3rd periods of the periodic table exists because of the additional 3d orbitals being filled with electrons. This leads to the emergence of transition metals with unique properties compared to other elements.