Question

Show that a molecule with a \(C_{n}\) axis cannot have a dipole moment perpendicular to the axis.

Short answer

A molecule with a Cn axis possesses n-fold rotational symmetry, meaning it looks the same when rotated by an angle of \( \frac{2\pi}{n} \) about the axis. If we assume that the molecule initially has a dipole moment perpendicular to the Cn axis, after rotation the dipole moment must also be perpendicular to the axis but in a different direction. However, due to the molecular symmetry, the dipole moment should remain unchanged. Since this contradiction arises, we conclude that it is impossible for a molecule with a Cn axis to have a dipole moment perpendicular to the axis.

Step-by-step solution

Understanding the rotational symmetry for a Cn axis

A molecule with a Cn axis possesses n-fold rotational symmetry. This means that when the molecule is rotated by an angle of \( \frac{2\pi}{n} \) (360°/n) about the axis, it will look the same as before. In other words, all charge distributions on the molecule must remain unchanged upon rotation by this angle.

Assume a dipole moment perpendicular to the axis

Let's assume that the molecule has a net dipole moment perpendicular to the Cn axis. This would mean there is a net difference in charge distribution, which can be represented as a vector pointing in a certain direction perpendicular to the axis.

Rotate the molecule about the Cn axis

Now, we rotate the molecule by an angle of \( \frac{2\pi}{n} \) about the Cn axis. Due to the n-fold rotational symmetry of the molecule, the charge distribution of the molecule must remain the same after the rotation. If the net dipole moment originally pointed in a direction perpendicular to the axis, after rotation it will point in a new direction that is also perpendicular to the axis.

Analyze the effect of rotation on the dipole moment

Since the charge distribution after rotation is the same as before, the net dipole moment vector should also be the same as before. However, we found that after rotation, the dipole moment is pointing in a new direction that is different from the original direction. This is a contradiction since the dipole moment vector should not change as a consequence of the n-fold rotational symmetry.

Conclude that a dipole moment cannot be perpendicular to the Cn axis

Based on the contradiction in step 4, we can conclude that it is impossible to have a net dipole moment perpendicular to the Cn axis in a molecule with n-fold rotational symmetry. Therefore, a molecule with a Cn axis cannot have a dipole moment perpendicular to the axis.

Key concepts

Rotational Symmetry in Molecules

Rotational symmetry in molecules is an interesting concept where an object looks the same after being rotated by a certain angle. In the case of a molecule with a Cn axis, this means it has n-fold rotational symmetry.

Imagine a round pizza cut into equal slices. As you rotate the pizza by the angle that corresponds to one slice, the pizza looks unchanged. Similarly, when a molecule with a Cn axis rotates by \( \frac{2\pi}{n} \), or 360° divided by n, it appears the same.

In molecules, this type of symmetry ensures that the spatial arrangement of atoms and the distribution of electric charges remains constant no matter how you rotate it around the axis. This stable arrangement is a fundamental characteristic.

• Rotation by \( \frac{2\pi}{n} \) results in no visual change.
• Ensures stability in molecular structures.
• Helps in understanding molecular orientation and properties.

Understanding rotational symmetry helps in predicting how the molecule behaves under different physical conditions.

Understanding Dipole Moments

A dipole moment in chemistry is like a tiny arrow pointing from one end of a molecule to the other. It represents the separation of positive and negative charges in a molecule.

When you imagine a molecule as a sports team, the dipole moment is like the team captain directing the players, indicating where there's more or less charge. This directionality is essential as it impacts how molecules interact with other molecules. A molecule with a dipole moment tends to align itself when interacting with electric fields or other dipolar molecules.

• Indicates a separation of charge within a molecule.
• Affects how molecules interact with each other and with electric fields.
• The net dipole moment is a vector, meaning direction and magnitude matter.

A key point about dipole moments is they can cancel out in symmetric structures, leading to nonpolar molecules with no net dipole moment.

Molecular Symmetry and Dipole Moments

Molecular symmetry profoundly influences whether a molecule has a dipole moment. Symmetrical molecules can often cancel out any internal dipoles, making such molecules nonpolar.

If we think of a molecule with a Cn axis, its rotational symmetry means it appears unchanged after a specific rotation. When a molecule has high symmetry, such as possessing a Cn axis, any perpendicular dipole moments would point in different directions, contradicting the symmetry.

This contradiction is precisely why a molecule with a Cn axis cannot have a dipole moment perpendicular to the axis. The alignment would break the symmetrical charge distribution given by its symmetry, leading to inconsistent molecular structure representations.

• High symmetry can cancel out dipole moments, rendering the molecule nonpolar.
• Cn axis ensures consistent orientation, supporting the lack of perpendicular dipoles.
• Symmetry maintains equilibrium in molecular charge distribution.

The relationship between molecular symmetry and dipole moments is crucial in determining the molecule's physical and chemical characteristics.