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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?

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  • $\begingroup$ I'd rather start any answer with a reminder that BH3 does not quite exist as such; B2H6 does. $\endgroup$ – Ivan Neretin Aug 27 '19 at 6:47
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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)

  • $C_n$ (the molecule only contains a $C_n$ axis
  • $C_{n\mathrm v}$ (the molecule contains a $C_n$ axis and $n$ total $\sigma_\mathrm{v}$ planes)
  • $C_{\mathrm s}$ (the molecule only has a plane of symmetry that contains the entire molecule)

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.

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$\ce{BH3}$ is a flat triangle molecule, so it has mirror symmetry. Since the dipoles cancel out, the molecule does not have a dipole moment.

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