Question

The \(\mathrm{O}-\mathrm{H}\) bond lengths in the water molecule \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) are \(96 \mathrm{pm}\), 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? \(\mathrm{In}\) 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 oxvgen?

Short answer

The bond dipoles of the O-H bonds in a water molecule point from the hydrogen atom to the oxygen atom due to the higher electronegativity of oxygen. The dipole moment vector of the water molecule bisects the H-O-H angle and points towards the oxygen atom. Using the law of cosines and vector addition, the magnitude of the bond dipole of the O-H bonds is calculated to be 1.79 D. Comparing this bond dipole with the dipole moments of hydrogen halides, we find that it is larger except for hydrogen fluoride (HF). This result is in accord with the relative electronegativity of oxygen, as it is a more electronegative element than the halogens except for fluorine.

Step-by-step solution

Part (a) - Directions of Bond Dipoles and Dipole Moment Vector

For the water molecule, the bond dipoles of the O-H bonds point from the hydrogen atom to the oxygen atom since oxygen is more electronegative than hydrogen. The dipole moment vector of the water molecule lies in the plane of the molecule, and points in the direction of the resultant of the vectors of the two O-H bond dipoles. Since the resulting dipole moment vector points from the side with a partial positive charge to the side with a partial negative charge, it will point towards the oxygen atom and bisect the H-O-H angle.

Part (b) - Calculating the Magnitude of Bond Dipole

To find the magnitude of the bond dipole of the O-H bonds, we need to use vector addition. Let's consider the bond dipoles as vectors. We can write the bond dipole of one O-H bond as P1 and the bond dipole of the other as P2. The bond angle between these two dipoles is given as 104.5°. The net dipole moment (P_net) of the water molecule can be found by vector addition of P1 and P2. From the given information, we have P_net = 1.85 D (debyes). Now, we can use the law of cosines to find the net dipole moment: P_net^2 = P1^2 + P2^2 - 2 * P1 * P2 * cos(θ) Since both O-H bond dipoles are identical, we have P1 = P2 = P. The given bond angle θ = 104.5°. 1.85^2 = P^2 + P^2 - 2 * P * P * cos(104.5°) Substituting the value of the bond angle and solving for P, we get: P = 1.79 D.

Part (c) - Comparing Bond Dipole with Hydrogen Halides

To compare the bond dipole of oxygen in water with that of hydrogen halides, we can refer to Table 8.3 in the book. The dipole moments of H-X (where X is a halogen) increase as we go down the group from hydrogen fluoride (HF) to hydrogen iodide (HI) due to the increasing electronegativity difference between hydrogen and the halogen. The bond dipole moment of water (1.79 D) is found to be larger than that of hydrogen halides except for HF (1.91 D). This indicates that the O-H bond in water is highly polar, resulting in a more significant tendency for electron density to be concentrated near the more electronegative oxygen atom. This is also in accord with the relative electronegativity values as oxygen is a more electronegative element than the halogens except for fluorine.

Key concepts

O-H Bond

The O-H bond is a chemical bond between the oxygen (O) and hydrogen (H) atoms in a molecule. Water (H₂O) is a well-known example where each molecule features two O-H bonds. These bonds are polar, which means they have a separation of charge. This arises because of the difference in electronegativity between the oxygen and hydrogen atoms. An understanding of the O-H bond is crucial because it contributes significantly to the properties of water, such as its high boiling point and surface tension.

Oxygen, which is more electronegative than hydrogen, attracts the shared electrons in the bond more strongly. This results in a partial negative charge on the oxygen and a partial positive charge on each hydrogen. In a water molecule, the O-H bonds are angled at 104.5°, leading to a bent shape. This bent shape is key, as it ensures the molecule isn't canceled out by opposing charges. It is this arrangement that allows the water molecule to have a dipole moment, where the net charge distribution is uneven, contributing to the molecule's polarity.

Electronegativity

Electronegativity is the measure of an atom's ability to attract and hold onto electrons in a chemical bond. This property is crucial for determining how bonds form and how polar they might be. The higher the electronegativity of an atom, the stronger its attraction for electrons.

Oxygen has a higher electronegativity compared to hydrogen. This difference is what gives the O-H bond its polar nature. In a water molecule, the oxygen atom attracts the shared electrons within the O-H bonds more strongly than the hydrogen atoms do. As a result, the electrons spend more time closer to the oxygen, thus giving it a partial negative charge, and leaving the hydrogen atoms with a partial positive charge.

• Oxygen is one of the most electronegative elements, following only after fluorine in the periodic table.
• In water, this significant electronegativity difference between oxygen and hydrogen is what leads to the molecule's highly polar character.

This characteristic of water, derived from electronegativity differences, affects many of its properties, including its ability to dissolve other substances, leading to its label as a "universal solvent."

Vector Addition

Vector addition is a mathematical operation used to determine the resultant of two or more vectors. This concept is widely used in physics to solve problems involving direction and magnitude. In the context of the water molecule, vector addition helps in finding the overall dipole moment.

In a water molecule, each O-H bond can be treated as a dipole with a magnitude and a direction. Because the molecule is bent, these dipoles do not point in exactly opposite directions and can't simply cancel each other out. Instead, they have to be added as vectors to find the net dipole moment of the molecule.

• Given the bond angle between the O-H bonds, we use vector addition, considering both the direction and magnitude of each bond's dipole.
• The angle between these bond dipoles plays a crucial role in determining the net dipole moment. Since the H-O-H angle is 104.5°, calculations must account for this using trigonometric methods like the cosine rule.

The resultant dipole moment in water points towards the oxygen atom. This resultant is why water is so effective at forming hydrogen bonds with other molecules, significantly affecting its liquid state properties. Understanding vector addition provides deeper insight into why and how molecular polarity arises.