Question

The \(\mathrm{O}-\mathrm{H}\) bond lengths in the water molecule \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) are \(0.96 \AA\), and the \(\mathrm{H}-\mathrm{O}-\mathrm{H}\) angle is \(104.5^{\circ} .\) The dipole moment of the water molecule is \(1.85 \mathrm{D} .\) (a) In what directions do the bond dipoles of the \(\mathrm{O}-\mathrm{H}\) bonds point? In what direction does the dipole moment vector of the water molecule point? (b) Calculate the magnitude of the bond dipole of the \(\mathrm{O}-\mathrm{H}\) bonds. (Note: You will need to use vector addition to do this.) (c) Compare your answer from part (b) to the dipole moments of the hydrogen halides (Table 8.3). Is your answer in accord with the relative electronegativity of oxygen?

Short answer

(a) The bond dipoles of the O-H bonds point towards the oxygen atom, and the overall dipole moment vector of the water molecule also points towards the oxygen atom. (b) The magnitude of the O-H bond dipole is approximately \(1.56 \mathrm{D}\). (c) The O-H bond dipole moment aligns with the relative electronegativity of oxygen, as it is larger than the H-Cl bond (\(1.11 \mathrm{D}\)) and smaller than the H-F bond (\(1.91 \mathrm{D}\)).

Step-by-step solution

(a) Bond dipole and dipole moment directions

Since oxygen is more electronegative than hydrogen, the bond dipoles point towards the oxygen atom in both O-H bonds. To find the direction of the overall dipole moment of the water molecule, we have to sum the bond dipoles as vectors. Due to the V-shaped structure and similar bond dipoles, the resultant dipole moment vector also points towards the oxygen atom.

(b) Magnitude of O-H bond dipole

Let's denote the O-H bond dipoles as vectors \(\vec{P_1}\) and \(\vec{P_2}\), and the dipole moment of the water molecule as \(\vec{P}\). The magnitude of \(\vec{P}\) is given as \(1.85 \mathrm{D}\). First, we find the angle between the bond dipoles. Since the angle between O-H bonds is \(104.5^{\circ}\), the angle between bond dipoles (\(\theta\)) is also \(104.5^{\circ}\). Using the law of cosines for vector addition, we can express \(\vec{P}\) in terms of \(\vec{P_1}\) and \(\vec{P_2}\). Let \(p\) denote the magnitude of \(\vec{P_1}\), \(\vec{P_2}\), and \(\vec{P}\). \[p^2 = p_1^2 + p_2^2 - 2p_1p_2\cos{\theta}\] Since both O-H bond dipoles are equal in magnitude, we have \(p_1 = p_2\). Therefore, \[p^2 = 2p_1^2 - 2p_1^2\cos{104.5^{\circ}}\] We can now solve for \(p_1\): \[p_1^2 = \frac{p^2}{2 - 2\cos{104.5^{\circ}}}\] \[p_1 = \sqrt{\frac{(1.85 \mathrm{D})^2}{2 - 2\cos{104.5^{\circ}}}}\] \[p_1 \approx 1.56 \mathrm{D}\] Hence, the magnitude of the O-H bond dipole is approximately \(1.56 \mathrm{D}\).

(c) Comparison with hydrogen halides and electronegativity

Comparing the dipole moment of the O-H bond with the hydrogen halides (Table 8.3), we see that it lies in the expected range, considering the relative electronegativities of atom pairs. The O-H bond has a larger dipole moment than the H-Cl bond (\(1.11 \mathrm{D}\)), and smaller dipole moments than the H-F bond (\(1.91 \mathrm{D}\)). Furthermore, since oxygen is more electronegative than other elements such as nitrogen and carbon, the O-H bond dipole moment is indeed in accord with the relative electronegativity of oxygen.

Key concepts

Bond Dipoles

In a molecule, bond dipoles are directional vectors representing the separation of charge between atoms within a bond. When one atom in the bond is more electronegative, it pulls the shared electrons closer, creating a partial negative charge towards that atom and a partial positive charge towards the other atom. For instance, in a water molecule (H₂O), the oxygen atom is more electronegative than hydrogen atoms.
Thus, the bond dipoles in the O-H bonds point towards the oxygen atom, indicating the direction of the electron density shift.

• Bond dipoles depend on the difference in electronegativity between bonded atoms.
• Directionality is important as it affects the overall molecular dipole moment.

This specific direction of bond dipoles is crucial for understanding the resulting shape and polarity of a molecule.

Electronegativity

Electronegativity is a fundamental concept used to describe the tendency of an atom to attract shared electrons in a chemical bond. It essentially measures how strongly an atom can "pull" on electrons. The greater the electronegativity difference between two bonded atoms, the more polar the bond.
Oxygen, for example, is highly electronegative compared to hydrogen, which is why the O-H bonds in water are polar. As a result:

• Oxygen pulls the electrons closer, creating a partial negative charge on oxygen and partial positive on hydrogen.
• This difference sets up a stronger bond dipole when compared to a bond with less electronegative atoms.

Understanding electronegativity enables prediction of molecule polarity and the direction of dipoles within molecules.

Vector Addition

Vector addition is essential when dealing with molecular dipoles, as the overall dipole moment in a molecule like water results from the vector sum of individual bond dipoles. This involves both direction and magnitude of the bond dipoles. The V-shape of the water molecule, with an H-O-H angle of 104.5°, makes the addition non-trivial.
Using vector addition, each bond dipole (for each O-H bond) is treated as a vector quantity. The overall molecular dipole is calculated by summing these vectors:

• Consider each bond dipole as a vector pointing from hydrogen to oxygen.
• The angle between these vectors (104.5°) affects the resultant direction and magnitude.

This careful addition helps predict the overall molecular polarity, crucial for understanding molecular interactions.

Dipole Moment Calculation

The dipole moment calculation involves not only knowing the bond dipoles but also employing vector addition principles to find the total dipole for a molecule. For the water molecule, the combined dipole moment results from its structural geometry and the individual O-H bond dipoles.
The given formula helps calculate the bond dipole magnitude if the overall molecular dipole is known. For water with a dipole moment of 1.85 D:

• The law of cosines is used to find bond dipole magnitude based on given molecular geometry.
• Using the overall dipole and angle, the bond dipole is approximately 1.56 D.

This computational step is vital for comparing molecular characteristics, like with hydrogen halides, and understanding electronegativity effects on molecule polarity.