Question

Consider a molecule with formula \(\mathrm{AX}_{3}\) . Supposing the \(\mathrm{A}-\mathrm{X}\) bond is polar, how would you expect the dipole moment of the \(\mathrm{AX}_{3}\) molecule to change as the \(\mathrm{X}-\mathrm{A}-\mathrm{X}\) bond angle increases from \(100^{\circ}\) to \(120^{\circ}\)

Short answer

As the X-A-X bond angle in an AX3 molecule with trigonal planar geometry increases from 100° to 120°, the net dipole moment's magnitude changes. At the ideal 120° bond angle, the net dipole moment becomes zero, as the individual A-X bond dipole moments are evenly distributed and cancel each other out in the plane of the molecule.

Step-by-step solution

Consider the molecular geometry of AX3

In general, the molecular geometry of an AX3 molecule can be trigonal planar, without a lone pair on the central atom A, or trigonal pyramidal, with a lone pair on the central atom A. For this exercise, we'll consider the trigonal planar geometry.

Calculate the dipole moment vector components

Since the A-X bond is polar, there will be a net dipole moment in each A-X bond. We can consider each bond's dipole moment as a vector. The net dipole moment of the AX3 molecule will be the vector sum of these three individual bond dipole moment vectors. To find the net dipole moment, let's split each dipole moment vector's components into their x and y-components, considering the molecule in a two-dimensional plane. If the bond angle X-A-X increases, the molecule remains in the plane.

Determine how the net dipole moment changes as the bond angle increases

Now, let's investigate the net dipole moment as the bond angle X-A-X increases from 100° to 120°. In the XY plane, each A-X bond's dipole moment contributes to both the x- and y-components. When the bond angle is 100°, the molecule is less symmetrical, leading to a non-zero net dipole moment. As the bond angle increases to 120°, the AX3 molecule becomes more symmetrical (the ideal bond angle in a trigonal planar molecule is 120°). In this case, the dipole moment vectors become evenly distributed and cancel each other out, resulting in a net dipole moment of zero.

Conclusion

As the X-A-X bond angle increases from 100° to 120° in an AX3 molecule with a trigonal planar geometry, the net dipole moment's magnitude changes. At the ideal 120° bond angle, the net dipole moment becomes zero, as the individual A-X bond dipole moments are evenly distributed and cancel each other out in the plane of the molecule.

Key concepts

Trigonal Planar Geometry

Trigonal planar geometry is a molecular shape that arises when a central atom, like "A" in \( \text{AX}_3 \) molecules, is bonded to three other atoms without any lone pairs affecting the geometry. In this setup, all constituent atoms lie on the same plane forming a triangular shape.

The key feature of a trigonal planar arrangement is that the bond angles are each ideally 120 degrees. This creates symmetry on the molecular level. When considering the polar bonds in such molecules, it's the symmetry established by these angles that becomes vital in assessing the overall molecular dipole moment.

• Geometry: Flat, triangular appearance
• Bond angle: Ideally 120°
• Effect of symmetry: Can lead to dipole cancellation.

When the bond angles in the molecules are altered, such as increasing from 100° to 120°, the molecular symmetry increases, leading potentially to changes in the dipole moment.

Polar Bonds

Polar bonds occur in molecules when there is an electronegativity difference between the bonded atoms. In the molecule \( \text{AX}_3 \), the A-X bond being polar implies that electron density is unequally shared between atom "A" and atom "X".

Because each bond acts as a dipole (imagine a tiny arrow pointing from the less electronegative atom to the more electronegative one), the question becomes how these bond dipoles interact to create a molecular dipole. In a perfectly trigonal planar \( \text{AX}_3 \) molecule, these polar bonds arrange themselves such that their effects can cancel each other out if the angles are perfect.

• Electronegativity difference causes bond polarity
• Each polar bond is a dipole with direction and magnitude
• Symmetrical arrangements can lead to zero net molecular dipole

When the bond angle changes from 100° to 120°, this re-distribution can lead to a balanced state in the molecule, thus neutralizing the overall dipole as observed.

Vector Components

In understanding why and how dipoles cancel, it becomes essential to view each bond's dipole as a vector—a quantity that has direction and magnitude.

In a plane, like the one formed by the trigonal planar structure, these vectors have components in both x and y directions. What you do is break down each bond dipole into these components and see how they combine.
The vector nature of each bond helps explain how individual bond polarities add up, or rather, cancel out in the case of certain symmetries.

• Vectors have both magnitude (how strong) and direction (which way)
• Components: Usually broken into x and y for simplification
• Symmetrical vectors: Add up to zero net result in ideal geometry

Thus, as the \( \text{X-A-X} \) bond angle expands from 100° to the more symmetric 120°, the directionality of these vectors aligns in such a way that they offset, resulting in zero net dipole moment.