Complex `[ML_(5)]` can exhibit trigonal bipyramidal and square pyramidal geometry. Determine total number of `180^@, 90^@ & 120^@` L-M-L bond angles.

Answer

To determine the total number of `180^@`, `90^@`, and `120^@` L-M-L bond angles in the complex `[ML_(5)]`, which can exhibit both trigonal bipyramidal and square pyramidal geometries, we will analyze each geometry step by step.

### Step 1: Analyze Trigonal Bipyramidal Geometry

1. **Identify the Geometry**: In trigonal bipyramidal geometry, there are 5 ligands around the central metal atom. The arrangement consists of 3 ligands in a plane (equatorial) and 2 ligands above and below this plane (axial).

2. **Bond Angles**:
- **120° Angles**: The equatorial ligands form bond angles of `120°` with each other. There are 3 pairs of equatorial ligands, contributing to 3 bond angles of `120°`.
- **90° Angles**: Each axial ligand forms a `90°` angle with each of the equatorial ligands. Since there are 3 equatorial ligands, each axial ligand contributes `3` angles of `90°`. Since there are 2 axial ligands, the total number of `90°` angles is `3 (from one axial) + 3 (from the other axial) = 6`.
- **180° Angles**: There is only one `180°` angle formed between the two axial ligands. Thus, there is `1` angle of `180°`.

3. **Total for Trigonal Bipyramidal**:
- `120°`: 3 angles
- `90°`: 6 angles
- `180°`: 1 angle

### Step 2: Analyze Square Pyramidal Geometry

1. **Identify the Geometry**: In square pyramidal geometry, there is a square base formed by 4 ligands and one ligand at the apex.

2. **Bond Angles**:
- **90° Angles**: All four ligands in the square base form `90°` angles with each other. This contributes `4` angles. Additionally, the apex ligand forms `90°` angles with each of the 4 base ligands, contributing another `4` angles. Thus, the total number of `90°` angles is `4 (base) + 4 (apex) = 8`.
- **120° Angles**: There are no `120°` angles in square pyramidal geometry. Thus, the count is `0`.
- **180° Angles**: There are two `180°` angles formed between the ligands across the square base. Thus, there are `2` angles of `180°`.

3. **Total for Square Pyramidal**:
- `90°`: 8 angles
- `120°`: 0 angles
- `180°`: 2 angles

### Step 3: Combine Results

Now, we sum the angles from both geometries:

- **Trigonal Bipyramidal**:
- `120°`: 3
- `90°`: 6
- `180°`: 1
- **Total**: 3 + 6 + 1 = 10 angles

- **Square Pyramidal**:
- `90°`: 8
- `120°`: 0
- `180°`: 2
- **Total**: 8 + 0 + 2 = 10 angles

### Final Calculation

Adding the totals from both geometries gives us:

- Total angles = 10 (from trigonal bipyramidal) + 10 (from square pyramidal) = **20 angles**.

### Conclusion

The total number of `180°`, `90°`, and `120°` bond angles in the complex `[ML_(5)]` is **20**.

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