The lanthanide contraction is responsible for the fact that:

Zr and Hf have about the same atomic radius

The correct answer is option 3. Zr and Hf have about the same atomic radius.

Lanthanide contraction refers to the progressive decrease in atomic and ionic radii of the elements in the lanthanide series (elements 57 through 71) as their atomic numbers increase, despite the addition of more electron shells.

Reasons for Lanthanide Contraction:

Poor Shielding by 4f Electrons:

The lanthanides have electrons added to the 4f orbitals. These 4f electrons do not shield the increasing nuclear charge (positive charge) very effectively. As a result, the effective nuclear charge felt by the outer electrons increases more significantly, pulling them closer to the nucleus.

Increased Nuclear Charge:

As you move through the lanthanide series, the number of protons in the nucleus increases. The increase in nuclear charge is not fully offset by the poor shielding of the 4f electrons. This leads to a stronger pull on the electrons, causing a decrease in atomic and ionic radii.

Impact on Elements:

Zirconium (Zr) and Hafnium (Hf):

Zirconium (Zr) is in the 4d series, and Hafnium (Hf) is in the 5d series. Due to the lanthanide contraction, the 5d electrons in Hafnium experience a similar effective nuclear charge to the 4d electrons in Zirconium. This means that the atomic radius of Hafnium (Hf) is very similar to that of Zirconium (Zr) despite Hf being one period below Zr.

Comparison with Other Elements:

Zr and Y: Yttrium (Y) is in the 4d series like Zr, but due to the different electron configurations and periodic trends, Y has a larger atomic radius compared to Zr. This comparison is not directly related to the lanthanide contraction.

Zr and Zn: Zinc (Zn) and Zirconium (Zr) have different oxidation states and are not directly comparable in terms of atomic radius due to the different electron configurations and periodic trends.

Zr and Nb: Niobium (Nb) and Zirconium (Zr) have different oxidation states (+5 for Nb and +4 for Zr), and the similarity in oxidation states is not a consequence of lanthanide contraction.

Summary:

The lanthanide contraction causes the atomic radii of elements in the 5d series (like Hafnium) to be similar to those in the 4d series (like Zirconium). This contraction happens because the 4f electrons in the lanthanide series do not shield the nuclear charge effectively, leading to a stronger pull on the electrons in the 5d series. Hence, Zr and Hf have about the same atomic radius due to this phenomenon.