Question

A measurement of the vibrational energy levels of \(^{12} \mathrm{C}^{16} \mathrm{O}\) gives the relationship \\[ \widetilde{\nu}(n)=2170.21\left(n+\frac{1}{2}\right) \mathrm{cm}^{-1}-13.461\left(n+\frac{1}{2}\right)^{2} \mathrm{cm}^{-1} \\] where \(n\) is the vibrational quantum number. The fundamental vibrational frequency is \(\tilde{\nu}_{0}=2170.21 \mathrm{cm}^{-1} .\) From these data, calculate the depth \(D_{e}\) of the Morse potential for \(^{12} \mathrm{C}^{16} \mathrm{O}\) Calculate the bond energy of the molecule.

Short answer

The depth of the Morse potential, \(D_e\), for \(^{12}C^{16}O\) is \(34982.64 cm^{-1}\), and the bond energy of the \(^{12}C^{16}O\) molecule is \(34447.53 cm^{-1}\).

Step-by-step solution

1. Identify the Frequency Parameter and Anharmonicity Constant

Compare the given vibrational energy levels expression to the energy levels in the Morse potential: Given relationship: \(\widetilde{\nu}(n) = 2170.21 (n+\frac{1}{2}) cm^{-1} - 13.461 (n+\frac{1}{2})^{2} cm^{-1}\) Morse potential expression: \(E_{n} = \hbar \omega_e (n + \frac{1}{2}) - \hbar \omega_e x_e (n + \frac{1}{2})^2\) It can be easily observed that: \[ \omega_e = 2170.21 cm^{-1} \] \[ x_e = 13.461 cm^{-1} \]

2. Calculate the Depth of Morse Potential (\(D_e\))

We can find the depth of the Morse potential using the expression: \[ D_e = \frac{\hbar^2 \omega_e^2}{4e^2 x_e} \] Insert the identified values of \(\omega_e\) and \(x_e\) into the equation and calculate: \[ D_e = \frac{\hbar^2 (2170.21)^2}{4e^2 (13.461)} = \frac{(2170.21)^2}{4\cdot (13.461)} cm^{-1} = 34982.64 cm^{-1} \] The depth of the Morse potential, \(D_e\), for \(^{12}C^{16}O\) is \(34982.64 cm^{-1}\).

3. Calculate the Bond Energy

The bond energy is equal to the dissociation energy, which is the energy needed to break the bond. In the Morse potential, the dissociation energy is given by: \[ D_0 = D_e - \frac{1}{2}\hbar \omega_e \] Insert the values of \(D_e\) and \(\omega_e\) and calculate: \[ D_0 = 34982.64 - \frac{1}{2}(2170.21) = 34447.53 cm^{-1} \] The bond energy of the \(^{12}C^{16}O\) molecule is \(34447.53 cm^{-1}\).

Key concepts

Morse Potential

The Morse potential is an important model to represent the potential energy of a diatomic molecule. Unlike the simpler harmonic oscillator, the Morse potential provides a more realistic depiction of bond interactions because it accounts for bond breaking and non-linear behavior.
The Morse potential is defined by the mathematical expression:

• \(E_{n} = \hbar \omega_e \left(n + \frac{1}{2}\right) - \hbar \omega_e x_e \left(n + \frac{1}{2}\right)^2\)

In this equation:

• \(\omega_e\) is the vibrational frequency,
• \(x_e\) is the anharmonicity constant,
• \(n\) is the vibrational quantum number.

Key features of the Morse potential include predicting the energy levels of vibrating molecules and accurately describing bond energy. It accounts for the weakening of bonds as they stretch, which the harmonic oscillator model fails to consider.
Overall, the Morse potential effectively handles bond dissociation and reveals more realistic vibrational states.

Vibrational Quantum Number

In the study of molecular vibrations, the vibrational quantum number \(n\) plays a significant role. It indicates the quantized levels of vibrational energy a diatomic molecule can possess in its ground and excited states.
The vibrational quantum number can take non-negative integer values starting from 0.

• The 0th level (\(n = 0\)) represents the lowest energy state, also known as the ground state.
• Higher values of \(n\) correspond to higher energy levels, denoting excited vibrational states.

The energy associated with these vibrational levels in the Morse potential is calculated using the given mathematical formulas.
The vibrational quantum number is crucial because it defines each level's energy, affecting the molecule's vibrational spectrum and contributing to spectroscopic analysis.

Bond Energy

Bond energy refers to the amount of energy needed to dissociate a chemical bond within a molecule. In the context of the Morse potential, bond energy is obtained by calculating the energy required to break the molecule from its equilibrium state to free atoms.
The bond energy relates directly to dissociation energy, which is crucial in understanding chemical reactions, stability, and molecular interactions.

• The dissociation energy is denoted as \(D_0\) and calculated from the Morse potential depth \(D_e\) subtracted by half the energy of the zero-point vibrational frequency.
• This can be mathematically expressed as \(D_0 = D_e - \frac{1}{2}\hbar \omega_e\).

Accurate calculation of bond energy helps predict reaction behavior and provides insights into molecular properties.
In molecular spectroscopy, analyzing vibrational spectra aids in determining bond energies and thus understanding molecular structure and dynamics.

Anharmonicity Constant

The anharmonicity constant \(x_e\) is an essential parameter in the Morse potential model. It measures the degree of deviation from the simple harmonic oscillator model’s assumptions.
In nature, molecular bonds do not oscillate perfectly; they undergo anharmonic vibrations, especially at higher energy levels.

• The constant \(x_e\) quantifies how much the potential energy deviates from an ideal harmonic behavior as the bond stretches.
• In practical terms, \(x_e\) allows the Morse potential to predict changes in vibrational energy levels more accurately as the molecule approaches bond dissociation.

The presence of \(x_e\) in energy calculations also influences the spacing of the higher vibrational levels, which typically become closer as energy increases.
Understanding and calculating the anharmonicity constant is vital for more accurate vibrational spectroscopy analysis, providing a clearer picture of molecular dynamics.