The answer is due to lanthanide contration the size for Zr and Hf is same. Lanthanide contraction can be explained on the basis of shielding effect.In multi-electron atoms the electrons are added in outer shells.

The lanthanide contraction refers to the decrease in atomic and ionic radii that occurs as you move across the f-block of the periodic table from left to right. This contraction is primarily attributed to the poor shielding effect of the 4f electrons, which are not very effective at shielding the increasing nuclear charge experienced by the outermost electrons. As a result, the outermost electrons are pulled closer to the nucleus, leading to a decrease in atomic size.



Contrary to option 'a', the lanthanide contraction does not result in all f-block ions having equal size. In fact, the lanthanide contraction causes a gradual decrease in the size of f-block ions as you move across the series. This is because the effective nuclear charge increases across the series, pulling the electrons closer to the nucleus and decreasing the ionic radii.



Isoelectronic ions are ions that have the same number of electrons. The lanthanide contraction does not affect all isoelectronic ions equally. Therefore, option 'b' is incorrect. The decrease in size due to the lanthanide contraction is most pronounced for ions with a higher atomic number and a greater number of inner 4f electrons. Thus, isoelectronic ions with higher atomic numbers will experience a greater decrease in size.



Now, let's look at option 'c' which states that Zr and Hf have equal size. Zirconium (Zr) and hafnium (Hf) are located in the d-block of the periodic table, just below the f-block. While both elements have the same number of electrons and a similar electron configuration, the lanthanide contraction affects Hf more than Zr. This is because hafnium has a greater number of inner 4f electrons, which are less effective at shielding the increasing nuclear charge. As a result, hafnium experiences a greater decrease in atomic size compared to zirconium. Thus, option 'c' is incorrect.



Lastly, let's consider option 'd' which suggests that iron (Fe), cobalt (Co), and nickel (Ni) have equal size due to the lanthanide contraction. These elements are located in the transition metal d-block and are not directly affected by the lanthanide contraction. The lanthanide contraction primarily affects the f-block elements, specifically the 4f electrons. Therefore, option 'd' is also incorrect.

In conclusion, the correct answer is option 'C', which states that Zr and Hf have equal size. This is because the lanthanide contraction affects hafnium more than zirconium, leading to a greater decrease in atomic size for hafnium.