Animated Solution for Chemistry - Haloalkanes & Haloarenes: The total number(s) of stable conformers with non-zero dipole moment for the following compound is (are)

Imagine you are handed a complex halogenated molecule locked in a Fischer projection, and your mission is to determine how many of its stable conformers possess a non-zero dipole moment. This isn't just a test of drawing structures; it is a profound test of symmetry and spatial reasoning.

First, we must establish the ground rules. In the dynamic world of conformational analysis, the stable conformers are the staggered ones, where steric hindrance and torsional strain are minimized. For a conformer to have a perfectly zero dipole moment ($\mu =0$), its individual bond dipoles must cancel out flawlessly. Geometrically, this almost always requires the conformer to possess a center of inversion ($i$).

The Fischer projection provided in the problem is slightly unconventional. The main carbon chain (the two methyl groups) is not fully aligned on the vertical axis. To make our symmetry analysis foolproof, we need to convert it into a standard Fischer projection.

We achieve this by performing two successive interchanges on the top chiral center. Remember, performing two interchanges on a single chiral center retains its absolute configuration.

First, we swap the top chlorine atom ($\text{Cl}$) with the right-side methyl group (${\text{CH}}_3$). Second, we swap the newly positioned chlorine with the left-side bromine atom ($\text{Br}$). Suddenly, the molecule's true backbone is revealed, with both methyl groups sitting proudly on the vertical axis.

Now comes the moment of truth. We look at our standardized Fischer projection and ask: Does this molecule have a plane of symmetry? Does the top half perfectly mirror the bottom half?

The answer is a resounding no. The arrangement of the halogens breaks the symmetry. Because the molecule lacks a plane of symmetry and a center of inversion, it is inherently chiral.

This is a massive revelation! Why? Because a chiral molecule is fundamentally asymmetric. It is a geometric impossibility for a chiral molecule to suddenly adopt a conformation that possesses a center of inversion.

Armed with the knowledge that our molecule is chiral, we can confidently draw its three stable, staggered Newman projections. We do this by holding the front carbon stationary and rotating the back carbon in $120^{\circ }$ increments.

As we cycle through the three staggered conformers—the (Me-Me) gauche, the (Br-Me) gauche, and the (Cl-Me) gauche—we observe the intricate dance of the bond dipoles. Because the parent molecule is chiral, none of these staggered states can achieve the perfect anti-alignment of all identical groups required for a center of inversion.

Since none of the three staggered conformers possess a center of inversion, their bond dipoles will never perfectly cancel. Therefore, every single one of them has a non-zero dipole moment ($\mu eq 0$).

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