Match the statement of set of orbitals given in List-I with orbitals and energy separation given in List-II
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List-I Statements |
List-II Orbitals and energy separation |
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(A) \( t_{2g} \) set of orbitals in octahedral crystal field and energy separation from barycentre |
(I) \( d_{x^2-y^2}, d_{z^2} \) and \( 3/5\,Δ_t \) |
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(B) \( e_g \) set of orbitals in octahedral crystal field and energy separation from barycentre |
(II) \( d_{xy}, d_{xz}, d_{yz} \) and \( 2/5\,Δ_t \) |
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(C) \( t_2 \) set of orbitals in tetrahedral crystal field and energy separation from barycentre |
(III) \( d_{x^2-y^2}, d_{z^2} \) and \( 3/5\,Δ_0 \) |
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(D) \( e \) set of orbitals in tetrahedral crystal field and energy separation from barycentre |
(IV) \( d_{xy}, d_{xz}, d_{yz} \) and \( 2/5\,Δ_0 \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
In crystal field theory the five d orbitals, equal in energy in the free ion, are split by the ligands. The barycentre is their average energy, and the splitting keeps it unchanged: the energy lost by one set equals the energy gained by the other. Orbitals that point towards the ligands are repelled more and go up; the others go down.
(A) \(t_{2g}\) set of orbitals in octahedral crystal field and energy separation from barycentre — This matches (IV) \(d_{xy}, d_{xz}, d_{yz}\) and \(2/5\,Δ_0\). In an octahedron the ligands lie on the axes, so the three orbitals that lie between the axes are lowered by \( \frac{2}{5}\Delta_o \).
(B) \(e_g\) set of orbitals in octahedral crystal field and energy separation from barycentre — This matches (III) \(d_{x^2-y^2}, d_{z^2}\) and \(3/5\,Δ_0\). These two orbitals point straight at the ligands and are raised by \( \frac{3}{5}\Delta_o \); balance: \( 3 \times \frac{2}{5} = 2 \times \frac{3}{5} \).
(C) \(t_2\) set of orbitals in tetrahedral crystal field and energy separation from barycentre — This matches (II) \(d_{xy}, d_{xz}, d_{yz}\) and \(2/5\,Δ_t\). In a tetrahedron the ligands lie between the axes, so the splitting is inverted: these three orbitals are now nearer the ligands and are raised by \( \frac{2}{5}\Delta_t \).
(D) \(e\) set of orbitals in tetrahedral crystal field and energy separation from barycentre — This matches (I) \(d_{x^2-y^2}, d_{z^2}\) and \(3/5\,Δ_t\). These two orbitals are lowered by \( \frac{3}{5}\Delta_t \).
Shortcut: A set named t has three orbitals and a set named e has two. The tetrahedral \( t_2 \) set must therefore be (II), which appears in Options (1) and (2); the octahedral \( t_{2g} \) set must be (IV), which leaves Option (2).