When irradiated with light of wavelength \(471.7 \mathrm{nm}\) the chlorine molecule dissociates into chlorine atoms. One \(\mathrm{Cl}\) atom is formed in its ground electronic state while the other is in an excited state that is \(10.5 \mathrm{~kJ} / \mathrm{mol}\) above the ground state. What is the bond enthalpy of the \(\mathrm{Cl}_{2}\), molecule?

Planck's equation, named after Max Planck, is a fundamental principle in quantum mechanics that describes the relationship between energy and the frequency of radiation. The equation is written as \( E = h u \) or \( E = \frac{hc}{\lambda} \), where \(E\) stands for energy, \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \text{J}\cdot\text{s}\)), \(u\) is the frequency of the radiation, and \lambda\ is the wavelength of the radiation.

When light is absorbed by an atom or a molecule, it can cause the atom or molecule to move to a higher energy level. The energy of the incoming light must match the energy required for this transition; this is where Planck's equation comes into use. For the textbook exercise, this equation allows us to calculate the energy in joules of a single photon of light with a specific wavelength, which is the first crucial step in determining what happens to a chlorine molecule when it is exposed to light of a certain wavelength.

Avogadro’s number, \(6.022 \times 10^{23} mol^{-1}\), is a constant that represents the number of elementary entities (usually atoms or molecules) in one mole of a substance. It's named after the Italian scientist Amedeo Avogadro and is key to converting between the number of particles and the amount of substance in moles.

This constant is especially useful when working with the microscopic scale of atoms and molecules to relate the macroscopic world to the quantum world. For instance, in the given exercise, once you've found the energy of light in joules (using Planck's equation), you multiply this by Avogadro's number to translate this energy into terms that are relevant on a molar scale—kilojoules per mole. This is an essential step for chemists who deal with quantities of substances in moles to understand reactions at the molecular level.

Molecular dissociation is the process by which a molecule splits into smaller particles, usually smaller molecules, atoms, or ions. It often occurs when a molecule absorbs energy, such as heat or light, which overcomes the energy holding the bonds together.

In the context of our exercise, light with a specific wavelength is used to dissociate chlorine molecules. Each chlorine molecule absorbs a photon of light and then dissociates into two chlorine atoms. However, not all of the absorbed energy goes into breaking the bond; some energy is also used to propel one chlorine atom into an excited state. The amount of energy required for this dissociation is tied closely to the bond enthalpy of the molecule, which reflects the strength of the bond between the atoms within the molecule.

The excited state of an atom or a molecule is a higher energy level than the ground state (the lowest energy state). When an atom or molecule absorbs energy, its electrons can move to a higher energy orbital, leading to an excited state.

Eventually, the system will seek to return to the ground state, usually releasing energy in the form of light (photon emission) or heat. In our exercise, the absorption of energy from light causes one of the chlorine atoms to be in an excited state. Identifying the energy associated with this excited state is critical to determining the bond enthalpy; we can subtract this energy from the total energy provided by the light to find the energy that was used to break the molecular bond. Thus, understanding the excited state is key to solving problems related to molecular dissociation and bond enthalpies.