It is a well-established experimental fact that the internal energies of a given atom or a molecule — Physical Chemistry — Kinetics Chemistry Question
Allowed energy levels and requirements for absorption of light
It is a well-established experimental fact that the internal energies of a given atom or a molecule M are confined to discrete values, so called “quantized energies”. “Internal” energy is the total energy of M excluding its translational energy. Translational energy of M is equal to the kinetic energy of a free particle that has the mass of M, which is moving along a straight line with a constant speed. It is not quantized, and it does not play a role in absorption of light by M. Internal energy of an atom is the energy associated with the motion of its electrons around the nucleus. In molecules, there are additional contributions from rotational and vibrational motions. The “allowed” internal energies of M can be numbered as E1, E2, E3, ... in increasing order of energy. These are called the “energy levels” of M. The lowest energy level, E1, is called the “ground” level, and when M has this lowest possible energy, M is said to be in its ground state. All the other, higher energy levels are referred to as “excited” levels of M, and if M is in a state with one of these higher energies it is said to be in an excited state. There is one and only one ground level whereas there are infinitely many excited states of M. Each atom or molecule has its own characteristic set of energy levels.
When a sample of M molecules is exposed to a beam of monochromatic light with wavelength λ there may be an energy exchange between the light and the M molecules. In its interaction with M, the monochromatic light beam is considered to consist of identical “photons”, with all photons moving in parallel and along the direction of the beam with the speed of light. Each photon carries an energy given by Ephoton = hν, where h is Planck’s constant and ν is the frequency of the light, related to its wavelength by: ν = c/λ. Since c is a constant, a given monochromatic light may be characterized by stating either λ or ν.
A molecule M may take energy from an external source such as light, thereby changing its initial energy level Einitial to a final level Efinal. For example, consider a case, where M is initially in its ground state with the lowest energy E1. Its final energy (Efinal) can only be one of E2, E3, E4, ... As a consequence, the amount of energy that M can accept from the external source is restricted to the values: ∆E = En – E1, where n = 2, 3, ... Conservation of total energy requires that if M gains energy equal to one of these allowed ∆E values, the external source must provide precisely the same amount of energy.
When light is used as the energy source, a photon from the light beam may or may not be absorbed by an M molecule in the sample, depending on the frequency ν of light used. Only when the photon energy is exactly equal to one of the allowed ∆E values of M, the energy of the photon may be accepted by M. The fundamental condition for absorption of light by M is expressed as hν = ∆E. This is a minimum requirement for absorption of light. Depending on whether M is an atom or molecule, and the nature of the energy levels involved in the transition, additional conditions called “selection rules” may have to be concurrently satisfied.
A closed test tube containing gaseous H atoms is irradiated by monochromatic light. Six experiments are done, differing from each other only by the wavelength, λ, of light employed.
Experiment No. 1 2 3 4 5 6
λ (nm) 129.6 121.6 101.6 97.25 94.97 94.11
Find out the experiments in which light will be absorbed by the H atoms in the sample, and describe the transitions involved.
Additional Data:
The allowed energy levels for the electron in a hydrogen atom are given (in SI units) by
[VISUAL]
where RH = 2.1787×10–18 J is a constant; i.e. RH is same for all values of the “n” quantum number. Assume that initially all of the hydrogen atoms in the sample are in their ground electronic states. Conservation of total energy is the only requirement for absorption of a photon by a H atom; i.e. there are no extra selection rules regarding the “n” quantum number.
Model Answer
For the H-atom, the allowed ∆E values from the ground state (n = 1) are:
∆E = En – E1= RH (1 – 1 / n^2), n = 2,3, ...
n: 2, 3, 4, 5, 6
∆E (10^–18 J): 1.6340, 1.9366, 2.0425, 2.0916, 2.1182
λ (nm): 121.57, 102.57, 97.255, 94.972, 93.780
RH = 2.1787 ⋅ 10^–18 J, h = 6.6261 ⋅ 10^–34 J s, and c = 2.9979 ⋅ 10^8 m s^–1
For light of frequency ν to be absorbed, the fundamental condition is:
hν = ∆E or λ = hc / ∆E
λ values corresponding to some of the allowed ∆E values are shown in the last row of the table above. Comparison of these λ values with those in the experiments shows that in Experiments 2, 4, and 5 the light will be absorbed by H-atoms in n = 1 level.
- In Experiment 2 (λ = 121.6 nm), the light is absorbed, corresponding to the electronic transition from n = 1 to n = 2 (theoretical wavelength 121.57 nm).
- In Experiment 4 (λ = 97.25 nm), the light is absorbed, corresponding to the electronic transition from n = 1 to n = 4 (theoretical wavelength 97.255 nm).
- In Experiment 5 (λ = 94.97 nm), the light is absorbed, corresponding to the electronic transition from n = 1 to n = 5 (theoretical wavelength 94.972 nm).