While studying dipole moment in organic chemistry, I got stuck at a particular statement written in the book :
Symmetrical molecules without lone pairs of electrons will have $\mu = 0.$
The main problem was, why would they have $\mu =0$ ?
What type of Symmetrical molecules are possible?
The book gave the example of $\ce{BH3}$. How is this a symmetrical molecule?
Yes, the statement in your book is very vague. Molecules are often classified by the symmetry elements they contain. For example, $\ce{BH3}$ contains a $C_3$ axis perpendicular to the plane containing the $\ce{BH3}$ molecule. "$C_3$" means that if you rotate the molecule around this axis by $360^\circ/3 = 120^\circ$, you'll get a molecule that is indistinguishable from the molecule you started with. $\ce{BH3}$ also contains 3 $\sigma_\mathrm{v}$ planes of symmetry. These planes contain the $C_3$ axis and one of the $\ce{B-H}$ bonds. $\ce{BH3}$ also contains one $\sigma_\mathrm{h}$ symmetry plane that is perpendicular to the $C_3$ axis and contains the $\ce{BH3}$ molecule. In addition to these symmetry elements, a number of other symmetry elements exist as well.
It turns out that only molecules that belong to symmetry classes (point groups)
can have a permanent dipole moment. If the molecule belongs to any other point group it cannot have a dipole moment. Since $\ce{BH3}$ contains a $\sigma_\mathrm{h}$ plane of symmetry ($\ce{BH3}$ belongs to point group $D_{3\mathrm h}$) - it therefore does not belong to groups $C_n$, $C_{n\mathrm v}$ or $C_{\mathrm s}$ - and cannot have a dipole moment.
$\ce{BH3}$ is a flat triangle molecule, so it has mirror symmetry. Since the dipoles cancel out, the molecule does not have a dipole moment.
