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, we need to determine the possible values of \( x \) in the compound \( PCl_xF_{5-x} \) for which the dipole moment is zero. The possible values of \( x \) range from 0 to 5. We will analyze each case step by step.

### Step 1: Analyze \( x = 0 \)
For \( x = 0 \), the compound is \( PF_5 \).
- **Structure**: The structure of \( PF_5 \) is trigonal bipyramidal.
- **Dipole Moment**: The dipole moment is zero because the molecule is symmetric.

### Step 2: Analyze \( x = 1 \)
For \( x = 1 \), the compound is \( PClF_4 \).
- **Structure**: The structure has one chlorine atom in an axial position and four fluorine atoms in equatorial positions.
- **Dipole Moment**: The dipole moment is not zero due to the presence of one chlorine atom, which has a different electronegativity compared to fluorine.

### Step 3: Analyze \( x = 2 \)
For \( x = 2 \), the compound is \( PCl_2F_3 \).
- **Structure**: The structure has two chlorine atoms and three fluorine atoms. Chlorine atoms will occupy equatorial positions.
- **Dipole Moment**: The dipole moment is not zero because of the unequal distribution of electronegativities.

### Step 4: Analyze \( x = 3 \)
For \( x = 3 \), the compound is \( PCl_3F_2 \).
- **Structure**: The structure has three chlorine atoms and two fluorine atoms. The chlorine atoms will occupy equatorial positions while the fluorine atoms will be in axial positions.
- **Dipole Moment**: The dipole moment is zero because the molecule is symmetric.

### Step 5: Analyze \( x = 4 \)
For \( x = 4 \), the compound is \( PCl_4F \).
- **Structure**: The structure has four chlorine atoms and one fluorine atom. The chlorine atoms will occupy equatorial positions.
- **Dipole Moment**: The dipole moment is not zero because of the presence of one fluorine atom.

### Step 6: Analyze \( x = 5 \)
For \( x = 5 \), the compound is \( PCl_5 \).
- **Structure**: The structure of \( PCl_5 \) is also trigonal bipyramidal.
- **Dipole Moment**: The dipole moment is zero because the molecule is symmetric.

### Summary of Results
The possible values of \( x \) for which the dipole moment is zero are:
- \( x_1 = 0 \) (for \( PF_5 \))
- \( x_2 = 3 \) (for \( PCl_3F_2 \))
- \( x_3 = 5 \) (for \( PCl_5 \))

Now, we calculate \( x_1 + x_2 + x_3 \):
\[ x_1 + x_2 + x_3 = 0 + 3 + 5 = 8 \]

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