If the experimentally measured dipole moment of a standard chlorobenzene molecule is represented as — Haloalkanes and Haloarenes Chemistry Question
Question
If the experimentally measured dipole moment of a standard chlorobenzene molecule is represented as the variable $\mu$, the net resultant dipole moment of $m$ -dichlorobenzene can be perfectly described by the mathematical expression $x\mu$ (assuming an idealized regular hexagonal geometry). What is the exact numerical value of $x$?
💡 Solution & Explanation
In $m$ -dichlorobenzene, the two $C-Cl$ bond dipole vectors (each of magnitude $\mu$) originate from the geometric center and are separated by a $120^\circ$ angle. The resultant vector magnitude ($\mu_R$) for two identical vectors separated by angle $\theta$ is determined by the formula: $\mu_R = \sqrt{\mu^2 + \mu^2 + 2\mu(\mu)\cos\theta}$. Plugging in $\theta = 120^\circ$: $\mu_R = \sqrt{2\mu^2 + 2\mu^2(-0.5)} = \sqrt{2\mu^2 - \mu^2} = \sqrt{\mu^2} = \mu$. Since the final dipole moment is exactly $1 \times \mu$, the value of the coefficient $x$ is 1.