Revise Energy levels in atoms and line spectra for Cambridge 9702 (22.4)

Electrons orbiting an atom's nucleus are not free to possess any amount of energy. Instead, they are restricted to specific, fixed energy states. This fundamental concept is known as energy quantisation. These allowed states are typically visualised using an energy-level diagram, where the lowest and most stable energy state is called the ground state. Higher, less stable states are known as excited states.

Energy level diagrams are a standard way to visualise quantised energy states. By convention, the energy required to completely remove an electron from the atom (ionisation) is defined as 0 eV. Since energy must be supplied to remove a bound electron, all energy levels within the atom are negative. The ground state is the most negative value, representing the most tightly bound state. Transitions are shown as arrows: an upward arrow for absorption/excitation and a downward arrow for emission/de-excitation. The length of the arrow is proportional to the energy of the photon involved.

Electrons can move between these discrete energy levels by absorbing or emitting energy. This process is called an electron transition.

An atom can be excited, promoting an electron to a higher energy level, in two primary ways:

1. Photon Absorption: The atom absorbs an incident photon. This only occurs if the photon's energy is exactly equal to the energy difference between the electron's initial state and a higher energy level ($E_{photon} = \Delta E$).
2. Collisional Excitation: The atom collides with another particle, typically a fast-moving electron (e.g., in a fluorescent tube). The atomic electron is excited if it receives at least the required energy ($Delta E$) from the kinetic energy of the colliding particle. Any excess kinetic energy is retained by the colliding particle.

An electron in an excited state is unstable and will spontaneously transition to a lower energy level, a process called de-excitation. This transition releases energy in the form of a single photon. The energy of this photon is precisely equal to the energy difference between the initial and final levels. An electron can return to the ground state in a single jump or via a series of smaller jumps, known as a cascade. A cascade results in the emission of multiple photons, each with a different energy (and thus different frequency/wavelength).

$E_{photon} = hf = E_{initial} - E_{final}$

Here, $h$ is Planck's constant ($6.63 \times 10^{-34} \text{ Js}$), $f$ is the frequency of the emitted photon, $E_{initial}$ is the higher energy level, and $E_{final}$ is the lower energy level. Crucially, the energy of the photon is exactly the difference between the initial and final electron energy levels.

When dealing with atomic energies, Joules (J) are often inconveniently small. Physicists frequently use the electronvolt (eV), a more practical unit. One electronvolt is defined as the kinetic energy gained by a single electron when it is accelerated through an electrical potential difference of one volt.

$1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$

There are three main types of spectra:

1. Continuous Spectrum: Produced by a hot, dense source (like the filament of an incandescent light bulb). It contains all wavelengths of light, appearing as a continuous rainbow.
2. Emission Spectrum: Produced by a hot, low-density gas (like in a neon sign or a star's atmosphere). The gas emits light only at specific wavelengths corresponding to its electron transitions, appearing as a series of bright lines on a dark background.
3. Absorption Spectrum: Produced when light from a continuous source passes through a cooler, low-density gas. The gas absorbs light at the specific wavelengths that match its excitation energies, resulting in a continuous spectrum with dark lines superimposed.

Because every element has a unique set of energy levels, the pattern of spectral lines it produces is also unique. This allows an emission or absorption spectrum to be used as an 'atomic fingerprint' to identify the elements present in a substance, such as the atmosphere of a distant star.