Problem 110 - Chemistry The Molecular Science

Question

Gaseous molecules in the ground vibrational state have a vibrational energy given by the equation \(E=\frac{1}{2} h v\) where \(h\) is Planck's constant and \(v\) is the frequency of the vibration. Consider a water vapor molecule, \(\mathrm{HOH}\), in which one of the lighter isotope hydrogen atoms, \(\mathrm{H},\) is replaced by the heavier isotope deuterium atom, \(\mathrm{D},\) to form \(\mathrm{HOD}\). (a) Will an HOD molecule have a higher or lower ground vibrational state energy than an HOH molecule? Explain your answer. (b) Which bond is stronger: \(\mathrm{O}-\mathrm{H}\) or \(\mathrm{O}-\mathrm{D}\) ? Explain your answer.

Step-by-Step Solution

Answer

(a) HOD has lower vibrational energy than HOH. (b) O-H bond is stronger than O-D bond.

Step 1: Understanding Vibrational Energy Equation

The vibrational energy of a molecule in the ground state is given by the equation \( E = \frac{1}{2} h v \) where \( h \) is Planck's constant and \( v \) is the frequency of the vibration. This tells us that the vibrational energy depends directly on the frequency \( v \).

Step 2: Comparing Frequencies for HOH and HOD

The frequency \( v \) of a vibrational mode is related to the reduced mass of the molecule and the force constant of the bond. The reduced mass \( \mu \) for a diatomic molecule \( AB \) is given by \( \mu = \frac{m_A m_B}{m_A + m_B} \). When an \( \mathrm{H} \) atom is replaced by \( \mathrm{D} \), the reduced mass \( \mu \) increases because deuterium is heavier. This results in a lower vibrational frequency \( v \) for \( \mathrm{O-D} \) compared to \( \mathrm{O-H} \).

Step 3: Calculating Effect on Ground Vibrational Energy

Since \( E = \frac{1}{2} h v \), a lower frequency \( v \) will result in a lower vibrational energy \( E \). Hence, an \( \mathrm{HOD} \) molecule has a lower ground vibrational state energy compared to an \( \mathrm{HOH} \) molecule.

Step 4: Exploring Bond Strengths O-H vs. O-D

Bond strength is generally associated with the force constant \( k \) of the bond. Typically, higher frequency vibrations are associated with stronger bonds. Since \( \mathrm{O-H} \) has a higher vibrational frequency than \( \mathrm{O-D} \) due to the lighter mass of hydrogen, the \( \mathrm{O-H} \) bond is considered to be stronger than the \( \mathrm{O-D} \) bond.

Key Concepts

Reduced Mass

The concept of reduced mass is crucial when considering the vibrational properties of molecules. Reduced mass, denoted as \( \mu \), is used to simplify the motion analysis of two interacting particles, particularly in diatomic molecules. It is defined by the formula:

• \( \mu = \frac{m_A m_B}{m_A + m_B} \)

Here, \( m_A \) and \( m_B \) are the masses of the two atoms involved in a bond.
This concept allows us to treat the two-body problem as a single-body problem, making calculations more manageable. When one of the hydrogen atoms in a water molecule is replaced by deuterium, the reduced mass increases due to the higher mass of deuterium. This change in mass directly affects the vibrational frequency of the molecule, leading to differences in its physical properties.

Isotopic Substitution

Isotopic substitution involves replacing an atom in a molecule with another isotope of that same element. This process is particularly significant in the study of vibrational energy because it changes the reduced mass of the molecule, thereby affecting its vibrational frequency. For instance, replacing an \( \mathrm{H} \) atom with a \( \mathrm{D} \) atom in water forms \( \mathrm{HOD} \) from \( \mathrm{HOH} \).
The deuterium atom increases the reduced mass, resulting in a lower vibrational frequency compared to the \( \mathrm{O-H} \) bond in ordinary water. As a result, \( \mathrm{HOD} \) has a lower ground vibrational state energy than \( \mathrm{HOH} \). Such substitutions are useful for studying various molecular properties and understanding vibrational dynamics.

Bond Strength

Bond strength is fundamentally linked to the force constant, which influences the vibrational frequency of a bond. The higher the force constant, the stronger the bond. Typically, bond strength is determined by a combination of factors, including atomic masses and the nature of the bond between atoms.
In the case of water isotopologues like \( \mathrm{HOH} \) and \( \mathrm{HOD} \), the \( \mathrm{O-H} \) bond has a higher vibrational frequency compared to the \( \mathrm{O-D} \) bond because hydrogen is lighter than deuterium. As higher frequencies are usually associated with stronger bonds, the \( \mathrm{O-H} \) bond is stronger than the \( \mathrm{O-D} \) bond.

Vibrational Frequency

Vibrational frequency is an essential aspect of understanding molecular vibrations and energy. The frequency of a vibration in a molecule depends on both the bond strength (force constant) and the reduced mass of the bonding atoms.
For molecules like \( \mathrm{HOH} \) (water), isotopic substitution plays a pivotal role. By replacing hydrogen with deuterium to form \( \mathrm{HOD} \), the frequency of vibration decreases due to increased reduced mass. This reflects in a lower vibrational energy level for \( \mathrm{HOD} \) compared to \( \mathrm{HOH} \).
Understanding these changes in vibrational frequency is crucial for interpreting molecular behaviors, especially in spectroscopy, where frequency shifts can indicate isotopic substitutions or changes in molecular structure.