Question

The degenerate orbitals of \[{\left[ {Cr{{\left( {{H_2}O} \right)}_6}} \right]^{3 + }}\] are:
A. \[{d_{yz}}\] and \[{d_{{z^2}}}\]
B. \[{d_{{z^2}}}\] and \[{d_{xz}}\]
C. \[{d_{xz}}\] and \[{d_{yz}}\]
D. \[{d_{{x^2}}}_{ - {y^2}}\] and \[{d_{xy}}\]

Answer 1

Hint: Degenerate orbitals are known as the orbitals having the same energy levels residing in the same subshell. For the degenerate orbitals the electrons of the orbitals are not influenced by any electric field or magnetic field. The three degenerate orbitals of p orbitals having equal energies are\[{p_x}\], \[{p_y}\] and\[{p_z}\].

Complete step by step answer: The given chromium compound is a coordination compound. The complex \[{\left[ {Cr{{\left( {{H_2}O} \right)}_6}} \right]^{3 + }}\] is a cationic complex where central atom chromium is in \[ + 3\] oxidation state. The complexes with general formula \[M{L_6}\] are referred to as octahedral complexes.
In the given complex the six water molecules occupy all the hybridized orbitals of chromium. The complex is \[{d^2}s{p^3}\] hybridized. When the electrons of ligands approach the metal d orbitals, electrostatic interaction occurs. As a result in octahedral complexes, the d orbitals split into two sets of energies. The energy difference between the two sets is called crystal field splitting energy or crystal field stabilization energy.
The d orbitals consist of five orbitals. The orbitals \[{d_{{x^2}}}_{ - {y^2}}\]and \[{d_{{z^2}}}\] are oriented along the axis while the\[{d_{xy}}\], \[{d_{yz}}\] and \[{d_{xz}}\] are oriented between the axes. As in octahedral complex the ligand approach along the axes the \[{d_{{x^2}}}_{ - {y^2}}\] and \[{d_{{z^2}}}\] orbitals face more interaction than the \[{d_{xy}}\], \[{d_{yz}}\], \[{d_{xz}}\] orbitals. The energy of \[{d_{{x^2}}}_{ - {y^2}}\] and \[{d_{{z^2}}}\] orbitals become higher and the energy of \[{d_{xy}}\], \[{d_{yz}}\] and \[{d_{xz}}\] become lower.
The \[{d_{{x^2}}}_{ - {y^2}}\] and \[{d_{{z^2}}}\] orbitals are called as \[{e_g}\] set. The \[{d_{xy}}\], \[{d_{yz}}\] and \[{d_{xz}}\] orbitals are called \[{t_2}_g\] set. The energy of \[{d_{{x^2}}}_{ - {y^2}}\] and \[{d_{{z^2}}}\] are equal. Also the energy of \[{d_{xy}}\], \[{d_{yz}}\] and \[{d_{xz}}\] are equal. Hence for the given sets of orbitals option C i.e. the \[{d_{xz}}\] and \[{d_{yz}}\] are the degenerate orbitals of\[{\left[ {Cr{{\left( {{H_2}O} \right)}_6}} \right]^{3 + }}\].
So, the correct answer is “Option C”.

Note: In an octahedral complex the ligand approaches along the axes but in the tetrahedral complex the ligand approaches between the axes. Hence the \[{t_2}_g\] set of orbitals becomes higher in energy compared to the \[{e_g}\] set of orbitals.