A transition of particular importance in \(\mathrm{O}_{2}\) gives rise to the 'SchumannRunge band' in the ultraviolet region.The wavenumbers (in \(\mathrm{cm}^{-1}\) ) of transitions from the ground state to the vibrational levels of the first excited state \({ }^{\left(\Sigma_{0}\right)}\) are \(50062.6,50725.4,51369.0,51988.6,52579.0,53143.4,53\) \(679.6,54177.0,54641.8,55078.2,55460.0,55803.1,56107.3,56360.3,56\) \(570.6 .\) What is the dissociation energy of the upper electronic state? (Use a Birge-Sponer plot.) The same excited state is known to dissociate into one ground-state \(\mathrm{O}\) atom and one excited-state atom with an energy \(190 \mathrm{kj} \mathrm{mol}^{-1}\) above the ground state. (This excited atom is responsible for a great deal of photochemical mischief in the atmosphere.) Ground-state \(\mathrm{O}_{2}\) dissociates into two ground-state atoms. Use this information to Calculate the dissociation energy of ground-state \(\mathrm{O}_{2}\) from the Schumann-Runge data.

Step by step solution

01

Plot a Birge-Sponer Plot

Create a Birge-Sponer plot by graphing the change in wavenumber \((\Delta \u)\) between consecutive vibrational levels against the vibrational quantum number \((u ')\).

02

Calculate the slope of Birge-Sponer Plot

Determine the slope of the linear portion of the Birge-Sponer plot. The slope is equal to twice the anharmonicity constant \((-2x_e \u_e)\), and the intercept on the y-axis gives \((\u_e - x_e \u_e)\).

03

Extrapolate to zero wavenumber change

Extrapolate the linear part of the plot to the point where the change in wavenumber is zero. This intercept point on the x-axis \( (\u '_{max}) \) represents the vibrational level at which dissociation occurs.

04

Calculate the dissociation energy for the upper electronic state

Use the equation \( D_0' = \u_e (\u '_{max} + 1/2) - \frac{x_e \u_e}{4} (\u '_{max} + 1/2)^2 \) to calculate the dissociation energy \( (D_0') \) of the upper electronic state \( { }^{(\Sigma_{0})} \).

05

Convert the dissociation energy into kJ/mol

Convert the calculated dissociation energy from wave numbers \( (\mathrm{cm}^{-1}) \) to \( \mathrm{kJ/mol} \) using the conversion factor \(1 \mathrm{cm}^{-1} = 1.986 \times 10^{-23} \mathrm{J}\), and Avogadro's number \( (6.022 \times 10^{23} \mathrm{mol}^{-1}) \).

06

Calculate the dissociation energy of ground-state \(\mathrm{O}_2\)

The dissociation energy of the ground-state \( \mathrm{O}_2 \) can be obtained by adding the energy difference between the ground state atoms and one atom in its ground state and another in the excited state (which is \(190 \mathrm{kJ/mol}\)) to the dissociation energy of the upper state calculated in Step 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dissociation Energy

Dissociation energy is a fundamental concept in chemistry, particularly in the study of molecular bonds and reactions. It refers to the amount of energy that must be supplied to a molecule to break it apart into its constituent atoms. In the context of our exercise, which examines the Schumann-Runge bands of molecular oxygen, the dissociation energy provides insights into the strength of the bond in the upper electronic state of oxygen.

By conducting a Birge-Sponer plot, one can extrapolate to find the energy required to dissociate oxygen in its excited state and thus learn about the stability of that state. Essentially, as you move to higher vibrational quantum numbers along the Birge-Sponer plot, you approach the energy at which the molecule dissociates. This information is vital in understanding molecular behavior under various energy conditions, and it also has practical implications for processes such as combustion and photochemical reactions.

Vibrational Quantum Number

The vibrational quantum number, often denoted as \(v\) or \(v'\) in spectroscopy, is integral to understanding molecular vibrations. It quantifies the vibrational state of a molecule and is used to describe the energy levels within the vibrational spectrum. Molecules can be excited to higher vibrational states by absorbing energy, and each successive level correlates with a specific quantum number.

In our Schumann-Runge bands scenario, transitions from the ground state to various excited vibrational levels involve increments in the vibrational quantum number. By examining the wavenumbers associated with these transitions, we can extract valuable information about both the vibrational energy levels of oxygen and the energy required for dissociation into atoms, which directly ties into the concept of dissociation energy.

Anharmonicity Constant

The anharmonicity constant, usually represented as \(x_e\), adjusts for the non-ideal behavior of molecular vibrations that deviate from the simple harmonic oscillator model. As vibrational energy increases, the force constant changes, and the molecule behaves less like a perfect harmonic oscillator. This non-linearity is captured by the anharmonicity constant.

In the provided solution, the slope of the Birge-Sponer plot corresponds to \( -2x_e u_e \) and is a clear indicator of anharmonic behavior in molecular vibrations. Accounting for anharmonicity is crucial for accurate calculations of dissociation energies, as it fine-tunes the results for real-world applications where molecular vibrations are rarely perfectly harmonic.

Schumann-Runge Bands

The Schumann-Runge bands refer to a series of absorption bands in the ultraviolet region of the electromagnetic spectrum, which result from electronic transitions in diatomic oxygen \(O_2\). These bands are associated with the movement of \(O_2\) from its ground state to an excited electronic state, leading to various vibrational levels within that state. Named after the scientists who discovered them, these bands are crucial for understanding the absorption behavior of oxygen in the Earth's atmosphere and contribute significantly to the photochemistry of the upper atmosphere.

The exercise requiring the analysis of Schumann-Runge bands data to determine the dissociation energy exemplifies the application of spectroscopic techniques in deducing molecular properties and the energetics of atmospheric compounds.

Photochemical Reactions

Photochemical reactions are chemical reactions initiated by light, ranging from visible to ultraviolet. They are fundamental to many processes in the atmosphere, biological systems, and technology. One product of such reactions can be an excited-state atom or molecule that has absorbed energy from light, gaining enough energy to undergo various subsequent reactions, which can sometimes be detrimental, or 'mischief', as stated in the exercise.

The study of the oxygen molecule's dissociation through the Schumann-Runge absorption data provides not only insights into the fundamental properties of oxygen but also into the potential energy pathways for photochemical reactions. Knowing the dissociation energy of oxygen's ground state enables scientists to predict and understand the behavior of oxygen under solar radiation, which is central to many environmental and atmospheric processes.