Number of Stable Conformers with Non-Zero Dipole Moment

To determine the total number of stable conformers with a non-zero dipole moment, we analyze the molecular structure and stereochemistry given in the Fischer projection.

1. Determination of Stereochemistry

The given compound is 2,3-dibromo-2,3-dichlorobutane: $\text{CH}_3-\text{C}(\text{Br})(\text{Cl})-\text{C}(\text{Br})(\text{Cl})-\text{CH}_3$

Assigning priorities to the substituents at both chiral carbons ($\text{C}_2$ and $\text{C}_3$) using the Cahn-Ingold-Prelog (CIP) priority rules:

1. $-\text{Br}$ (Priority 1)
2. $-\text{Cl}$ (Priority 2)
3. $-\text{C}(\text{Br})(\text{Cl})\text{CH}_3$ (Priority 3)
4. $-\text{CH}_3$ (Priority 4)

From the given Fischer projection:

• At $\text{C}_2$: The path $1 \to 2 \to 3$ goes clockwise with the lowest priority group ($-\text{CH}_3$) on a horizontal bond, giving the $(2S)$ configuration.
• At $\text{C}_3$: The path $1 \to 2 \to 3$ goes counterclockwise with the lowest priority group ($-\text{CH}_3$) on a vertical bond, giving the $(3S)$ configuration.

Thus, the given molecule is the chiral $(2S, 3S)$-isomer (active form).

---

2. Analysis of Conformational Stability

• Stable conformers correspond to local potential energy minima, which are the 3 staggered conformations around the $\text{C}_2-\text{C}_3$ single bond.
• Eclipsed conformations are energy maxima and are thus unstable.

---

3. Dipole Moment ($\mu$) Analysis

For a conformer to have a net dipole moment of zero ($\mu = 0$), it must possess a center of inversion ($i$). In 2,3-disubstituted butane systems, a center of inversion requires identical groups on adjacent carbons to be in anti-positions relative to each other ($\text{Br}$ anti to $\text{Br}$, $\text{Cl}$ anti to $\text{Cl}$, and $\text{CH}_3$ anti to $\text{CH}_3$).

This centrosymmetric arrangement can only exist in the optically inactive meso form $(2R, 3S)$.

Since the given compound is the $(2S, 3S)$ chiral stereoisomer:

1. It lacks a center of inversion ($i$) in all of its conformations.
2. The bond dipole vectors do not cancel out completely in any of the staggered forms.
3. Therefore, all $3$ staggered (stable) conformers possess a non-zero dipole moment ($\mu \neq 0$).

---

Final Answer:

The total number of stable conformers with a non-zero dipole moment is 3.