Question

Explain qualitatively how temperature could have an effect on the appearance of the absorption spectrum of a system such as that shown, knowing that most molecules are usually in their lowest vibrational state at room temperature. Schematic potential energy diagram for ground and excited electronic singlet states of a diatomic molecule, \(\mathrm{A}-\mathrm{B}\) and \(\mathrm{A}-\mathrm{B}^{*}\), respectively. The horizontal lines represent vibrational energy levels. The wavy lines represent the arrival or departure of light quanta. It is important to recognize that the curves shown here are schematic and, for actual systems, the positions and shapes of the curves may be quite different from those shown.

Short answer

In summary, the appearance of the absorption spectrum of a diatomic molecule system is affected by temperature because it influences the distribution of molecules in various vibrational states. At higher temperatures, more molecules occupy higher vibrational energy levels, which results in a greater number of possible transitions between these levels during light absorption. Consequently, the absorption spectrum at higher temperatures appears broader and more complex than the spectrum observed at room temperature.

Step-by-step solution

In a diatomic molecule such as A-B, the electronic and vibrational energy levels are quantized, meaning that the molecule can only have discrete energy values. The ground state (A-B) has the lowest possible energy, and the excited state (A-B*) represents a higher energy level. Each electronic state also has several vibrational energy levels, with the molecule transitioning between these levels during absorption and emission of light. #Step 2: Absorption spectrum at room temperature#

At room temperature, most molecules are in their lowest vibrational state in the ground electronic state. When light is absorbed, the molecule can transition to an excited electronic state, such as A-B* state, and also to one of the available vibrational energy levels. The observed absorption spectrum at room temperature consists of vertical transitions between the ground state vibrational levels and the vibrational levels of the excited states. Each transition corresponds to a different light frequency, giving rise to a distinct pattern in the spectrum. #Step 3: Influence of temperature on molecular energy distribution#

As the temperature increases, the average molecular energy also increases. This leads to a greater number of molecules occupying higher vibrational energy levels in the ground electronic state. The distribution of molecules in various vibrational states follows the Boltzmann distribution, which dictates that as temperature rises, there are more molecules in higher energy levels. #Step 4: Absorption spectrum at higher temperatures#

With more molecules in higher vibrational states in the ground electronic state due to the increased temperature, the possible transitions between vibrational levels during absorption become more numerous. Consequently, the absorption spectrum at higher temperatures will have additional absorption bands corresponding to these new transitions, leading to a more complex and broader spectrum compared to the room temperature spectrum. To summarize, temperature has a significant effect on the appearance of the absorption spectrum of a diatomic molecule system, as the distribution of molecules in different vibrational states is highly dependent on the temperature. Higher temperatures result in a broader and more complex spectrum because more energy levels become accessible, and a larger number of transitions between vibrational states are possible during the absorption process.

Key concepts

Vibrational Energy Levels

Vibrational energy levels play a critical role in the absorption spectrum of a diatomic molecule. These levels correspond to the discrete states that a molecule can assume due to the vibratory motion of its atoms, much like the notes on a musical scale. At room temperature, molecules generally reside in the lowest vibrational state of the ground electronic state, due to their lower energy.

When light interacts with the molecule, it can be absorbed, and the molecule is excited to a higher vibrational level within an excited electronic state. This absorption process is highly specific; each jump to a higher level requires a precise quantum of energy, corresponding to a specific wavelength of light. The unique pattern of absorbed wavelengths constitutes the absorption spectrum.

The Effects of Temperature

The temperature of the system can have significant effects on these vibrational levels. As the temperature increases, the molecules gain kinetic energy and can occupy higher vibrational states. This results in more possible transitions and, therefore, a more complex absorption spectrum. Each of these additional transitions adds a new line to the absorption spectrum, thereby broadening its appearance and providing deeper insights into the molecular structure.

Electronic Singlet States

Electronic singlet states refer to the discreet energy levels associated with the electrons in a molecule. These states are characterized by pairs of electrons having opposite spins, which results in a net spin of zero. In diatomic molecules, electronic transitions occur when an electron jumps from a lower-energy singlet state to a higher-energy singlet state or vice versa.

Typically, a molecule absorbs a photon of light and transitions from the ground singlet state to an excited singlet state, which is accompanied by a change in the electron’s energy level. This can also involve a simultaneous change in the vibrational energy level, which contributes to the unique absorption lines seen in the spectrum.

Higher Energy Accessibility

As the temperature rises, more electrons are thermally excited not only to higher vibrational states but also to higher electronic states. Subsequently, the likelihood of absorption in various electronic transitions increases, leading to the presence of more features in the absorption spectra. Given these conditions, the spectrum becomes a rich source of information about both the electronic and vibrational structure of the molecule.

Boltzmann Distribution

The Boltzmann distribution is fundamental in predicting how molecules distribute themselves across different energy states at thermal equilibrium. According to this principle, the population of molecules in a given energy state is inversely proportional to the energy level's exponential function, considering the temperature and the Boltzmann constant.

At low temperatures, most molecules are found in the lowest energy states; this distribution shifts as the temperature rises, allowing higher energy states to have significant populations. Within the context of vibrational and electronic states, this means that at higher temperatures, an increasing number of molecules occupy excited states.

Understanding Temperature Impact via Boltzmann Distribution

This understanding helps us visualize why absorption spectra become more complex at elevated temperatures. With an increased number of accessible energy levels, the spectrum reflects transitions that would not occur at lower temperatures. The Boltzmann distribution thus explains the observed broadening and intensification of absorption bands as temperature increases, and helps decipher the thermal behavior of molecular systems based on their spectral data.