In compound `PCl_(x)F_(5-x)`, possible values of x are 0 to 5, then calculate value of `x_(1)+x_(2)+x_(3)` (where `x_(1), x_(2) and x_(3)`… are possible values of x, with zero dipole moment for given compound).

Explanation

To solve the problem of finding the possible values of \( x \) in the compound \( PCl_xF_{5-x} \) that result in a zero dipole moment, we will analyze the molecular geometry and the arrangement of the atoms.

### Step-by-step Solution:

1. **Understanding the Compound**:
The compound is given as \( PCl_xF_{5-x} \), where \( x \) can take values from 0 to 5. The total number of atoms of phosphorus (P) is 1, and the total number of halogen atoms (Cl and F) is 5.

2. **Molecular Geometry**:
The phosphorus atom in this compound adopts a trigonal bipyramidal geometry due to the presence of five bonding pairs of electrons. In this geometry, there are two axial positions and three equatorial positions.

3. **Identifying Possible Values of \( x \)**:
We need to find values of \( x \) that lead to a non-polar molecule (zero dipole moment). The possible values of \( x \) are 0, 1, 2, 3, 4, and 5. We will analyze each case:

- **Case \( x = 0 \)**: The compound is \( PF_5 \).
- Structure: All fluorine atoms are in the equatorial positions.
- Dipole Moment: The dipoles cancel out, resulting in zero dipole moment.
- **Conclusion**: \( x_1 = 0 \)

- **Case \( x = 1 \)**: The compound is \( PClF_4 \).
- Structure: One chlorine in an equatorial position and four fluorines in the axial positions.
- Dipole Moment: The dipole from fluorine does not cancel out due to the presence of chlorine.
- **Conclusion**: Not zero dipole moment.

- **Case \( x = 2 \)**: The compound is \( PCl_2F_3 \).
- Structure: Two chlorines in equatorial positions and three fluorines in axial positions.
- Dipole Moment: The dipoles do not cancel out completely.
- **Conclusion**: Not zero dipole moment.

- **Case \( x = 3 \)**: The compound is \( PCl_3F_2 \).
- Structure: Three chlorines in equatorial positions and two fluorines in axial positions.
- Dipole Moment: The dipoles from fluorine dominate, but the structure can be balanced.
- **Conclusion**: Zero dipole moment.
- **Conclusion**: \( x_2 = 3 \)

- **Case \( x = 4 \)**: The compound is \( PCl_4F \).
- Structure: Four chlorines in equatorial positions and one fluorine in axial position.
- Dipole Moment: The dipoles do not cancel out completely.
- **Conclusion**: Not zero dipole moment.

- **Case \( x = 5 \)**: The compound is \( PCl_5 \).
- Structure: All chlorines, symmetrical arrangement.
- Dipole Moment: The dipoles cancel out completely.
- **Conclusion**: \( x_3 = 5 \)

4. **Summing the Values**:
The possible values of \( x \) that result in zero dipole moment are \( x_1 = 0 \), \( x_2 = 3 \), and \( x_3 = 5 \). Therefore, we calculate:
\[ x_1 + x_2 + x_3 = 0 + 3 + 5 = 8 \]

### Final Answer:
The value of \( x_1 + x_2 + x_3 \) is **8**.