Question

Consider a molecule with formula \(\mathrm{AX}_{2}\). Supposing the \(\mathrm{A}-\mathrm{X}\) bond is polar, how would you expect the dipole moment of the \(\mathrm{AX}_{2}\) molecule to change as the \(\mathrm{X}-\mathrm{A}-\mathrm{X}\) bond angle decreases from \(180^{\circ}\) to \(100^{\circ} ?\)

Short answer

When the X-A-X bond angle in the AX₂ molecule decreases from 180° to 100°, the overall dipole moment of the molecule changes from zero to nonzero. This is because, in the linear 180° geometry, the two A-X bond dipoles cancel each other out, while in the bent 100° geometry, they don't. The new dipole moment can be calculated using the cosine rule and the magnitudes of the individual A-X bond dipoles: \(\mu_{total}^2 = 2\mu^2 (1 - \cos100^\circ)\).

Step-by-step solution

Understanding molecular dipole moment and bond polarity

A molecular dipole moment is a vector quantity that depends on the bond polarities and the geometrical arrangement of the bonds in the molecule. If a bond between two atoms A and X is polar, it means that the electron density is unevenly distributed between them, generating a partial positive charge on one atom and a partial negative charge on the other.

Calculating the dipole moment in AX₂

The dipole moment (\(\mu\)) of the molecule can be calculated using the formula: \[ \mu = q \cdot d \] where \(q\) is the magnitude of the charges (partial charges) and \(d\) is the distance between the charges (bond length). However, for a molecule like AX₂, we need to consider the geometry of the molecule and the bond angles. When the X-A-X angle is 180°, the molecule is linear. The dipole moments of the two A-X bonds will be opposite in direction and equal in magnitude, canceling each other out. Therefore, the overall dipole moment of the molecule when the bond angle is 180° will be zero.

Changing the bond angle and analyzing its effect

When the X-A-X bond angle decreases from 180° to 100°, the geometry of the AX₂ molecule becomes bent. In this case, the two A-X bond dipole moments will no longer cancel out due to their angle difference, creating a net molecular dipole moment that is nonzero. To quantitatively analyze the change, we can use vector addition to determine the overall molecular dipole moment of the molecule in this new geometry. Consider the two A-X dipoles as vectors \(\vec{\mu_1}\) and \(\vec{\mu_2}\). We can find the vector sum of these two dipoles as follows: \[ \vec{\mu}_{total} = \vec{\mu_1} + \vec{\mu_2} \]

Calculating the new dipole moment

Due to the symmetry of the AX₂ molecule, both \(\vec{\mu_1}\) and \(\vec{\mu_2}\) have the same magnitude, i.e., \(\mu_1 = \mu_2 = \mu\). Let's call the angle between them \(\theta\), which is our X-A-X bond angle (100°). We can use the cosine rule to calculate the magnitude of the net dipole moment: \[ \mu_{total}^2 = \mu_1^2 + \mu_2^2 - 2\mu_1\mu_2\cos\theta \] After substituting \(\mu_1 = \mu_2 = \mu\), we get: \[ \mu_{total}^2 = 2\mu^2 (1 - \cos\theta) \] Now we can plug in the values. Since we are only analyzing the change in the dipole moment, we do not need the actual values of the charges and the bond length. Let \(k = \mu^2\), and \(\theta = 100^\circ\): \[ \mu_{total}^2 = 2k \left(1 - \cos 100^\circ \right) \] Hence, the dipole moment of the AX₂ molecule will be nonzero when the X-A-X bond angle is reduced to 100° and can be calculated by taking the square root of the above expression.

Key concepts

Bond Polarity

To understand bond polarity, imagine two atoms connected by a bond. If the atoms are different, they might not share electrons equally. This unequal sharing is due to differences in electronegativity, a property that describes how strongly an atom attracts electrons. When one atom pulls the electrons toward itself more than the other, the bond becomes polar. The atom with the stronger pull gets a partial negative charge, and the other gets a partial positive charge.

A polar bond is much like a tiny magnet with a direction, represented as a vector. This vector points from the positive to the negative end, describing the bond's dipole moment. The strength of the polarity depends on the difference in electronegativity between the two atoms and the distance of the bond.

• Uneven electron sharing creates polar bonds.
• Electronegativity differences drive polarity.
• Polar bonds have a dipole moment vector.

Molecular Geometry

The shape of a molecule is crucial in determining its overall dipole moment. Molecular geometry is about how atoms are arranged in space. In an ext{AX}_2 molecule, the central atom ext{A} is bonded to two ext{X} atoms. The common shapes for ext{AX}_2 are linear and bent.

When the molecule is linear (with a bond angle of 180°), the dipoles of the ext{A-X} bonds point in exactly opposite directions. This alignment causes them to cancel out, resulting in a net dipole moment of zero.

As the ext{X-A-X} bond angle decreases, the molecule adopts a bent shape. The dipole moments no longer perfectly oppose each other. Instead, they create a resultant or net dipole moment, making the molecule polar.

• Linear shapes (180°) lead to no net dipole.
• Bent shapes result in a nonzero net dipole.
• Molecular shape affects polarity.

Vector Addition

Vector addition helps us understand how individual bond dipoles in a molecule contribute to the overall dipole moment. Think of each polar bond as a small arrow (a vector), with both magnitude and direction. When you have multiple arrows, vector addition allows you to "add" them to see the overall effect.

In an ext{AX}_2 molecule with a decreasing bond angle, the two dipole vectors no longer cancel out perfectly, as they did in a linear geometry. Instead, they are at an angle to each other. This requires using vector addition to determine the resulting dipole. This is where the cosine rule can help to mathematically find the magnitude of this resultant vector, by combining the individual dipole vectors.

• Bond dipoles are vectors with direction and magnitude.
• Vector addition shows the total effect of these dipoles.
• Molecule shape changes result in different vector sums.