Relative to the average enerage in the spherical crystal field the `t_(2g)` orbitals in tetrahedral field is .

Options

A. Reised `(2//5)Delta_(t)`
B. Lowered by `(2//5)Delta_(t)`
C. Reised `(3//5)Delta_(t)`
D. Lowered by `(1//5)Delta_(t)`

Explanation

To solve the question regarding the relative energy of the \( t_{2g} \) orbitals in a tetrahedral crystal field compared to the average energy in a spherical crystal field, we can follow these steps:

### Step 1: Understand Crystal Field Theory
Crystal Field Theory explains how the energies of d-orbitals are affected by the presence of ligands around a central metal ion. In a tetrahedral field, the d-orbitals split into two sets: \( t_{2g} \) and \( e_g \).

**Hint:** Remember that in tetrahedral complexes, the splitting of d-orbitals is different from octahedral complexes.

### Step 2: Identify the Splitting Energy
In a tetrahedral field, the splitting energy is denoted as \( \Delta_t \). The \( t_{2g} \) orbitals are higher in energy compared to the \( e_g \) orbitals.

**Hint:** Recall that \( t_{2g} \) orbitals are lower in energy in octahedral fields, but in tetrahedral fields, they are higher.

### Step 3: Assign Energies to the Orbitals
For tetrahedral complexes:
- The \( t_{2g} \) orbitals are assigned an energy of \( +\frac{2}{5} \Delta_t \).
- The \( e_g \) orbitals are assigned an energy of \( -\frac{3}{5} \Delta_t \).

**Hint:** Keep in mind that the energies are relative to the average energy in a spherical field, which is taken as 0.

### Step 4: Calculate the Average Energy
To find the average energy of the d-orbitals in a tetrahedral field:
- The average energy can be calculated as:
\[ \text{Average Energy} = \frac{3 \times \left( +\frac{2}{5} \Delta_t \right) + 2 \times \left( -\frac{3}{5} \Delta_t \right)}{5} \]
- Simplifying this gives:
\[ = \frac{6/5 \Delta_t - 6/5 \Delta_t}{5} = 0 \]

**Hint:** The average energy of the orbitals in a tetrahedral field is zero, which is a reference point for comparison.

### Step 5: Conclusion
Thus, the \( t_{2g} \) orbitals in a tetrahedral field are raised by \( +\frac{2}{5} \Delta_t \) relative to the average energy in a spherical crystal field.

**Final Answer:** The \( t_{2g} \) orbitals in a tetrahedral field are at an energy of \( +\frac{2}{5} \Delta_t \).