Which bond angle, `theta` would result in the maximum dipole moment for the triatomic molecule `XY_2` ?

Options

A. `theta =90^(@)`
B. `theta = 120^(@)`
C. `theta =150^(@)`
D. `theta = 180^(@)`

Answer

To determine which bond angle \( \theta \) would result in the maximum dipole moment for the triatomic molecule \( XY_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of \( XY_2 \)**:
- The molecule \( XY_2 \) consists of one central atom \( X \) and two identical atoms \( Y \) bonded to it. The arrangement of these atoms can affect the dipole moment.

2. **Defining the Dipole Moment**:
- The dipole moment \( \mu \) of a molecule can be expressed in terms of the bond lengths and the angle between the bonds. For the molecule \( XY_2 \), the dipole moment can be represented as:
\[ \mu = \sqrt{\mu_1^2 + \mu_2^2 + 2 \mu_1 \mu_2 \cos(\theta)} \]
- Here, \( \mu_1 \) and \( \mu_2 \) are the dipole moments associated with each \( Y \) atom, and \( \theta \) is the bond angle between the two \( Y \) atoms.

3. **Analyzing the Effect of the Bond Angle**:
- As the bond angle \( \theta \) varies, the cosine of the angle \( \cos(\theta) \) will also change. Specifically:
- At \( \theta = 90^\circ \), \( \cos(90^\circ) = 0 \).
- As \( \theta \) increases from \( 90^\circ \) to \( 180^\circ \), \( \cos(\theta) \) becomes negative.

4. **Finding the Maximum Dipole Moment**:
- The dipole moment \( \mu \) is maximized when the contributions from the two dipole moments are as additive as possible. This occurs when \( \theta = 90^\circ \) because at this angle, the two dipole moments are perpendicular to each other, leading to maximum vector addition.

5. **Conclusion**:
- Therefore, the bond angle \( \theta \) that results in the maximum dipole moment for the triatomic molecule \( XY_2 \) is:
\[ \theta = 90^\circ \]

### Final Answer:
The bond angle \( \theta \) that would result in the maximum dipole moment for the triatomic molecule \( XY_2 \) is \( 90^\circ \).

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