The electron in a hydrogen atom is excited to a 4d-orbital. Calculate the energy of the photon released if the electron were then to move to each of the following orbitals: (a) \(1 \mathrm{~s}\); (b) \(2 \mathrm{p}\); (c) \(2 \mathrm{~s}\); (d) \(4 \mathrm{~s}\). (e) If the outermost electron in a potassium atom were excited to a 4d-orbital and then fell to the same orbitals, describe qualitatively how the emission spectrum would differ from that of hydrogen (do not do any calculations). Explain your answer.

The Rydberg formula is pivotal to understanding the behaviour of electrons in a hydrogen atom and similar hydrogen-like atoms. It represents the energy of an electron found at a certain principal quantum number level, denoted as 'n'. The energy associated with each orbital level is given as

\[ E_n = -R_H \left(\frac{1}{n^2}\right) \]
where \( R_H \) is the Rydberg constant and \( n \) is the principal quantum number. The negative sign indicates that the energy is lower when an electron is bound to the nucleus, and as \( n \) increases, the energy levels become less negative and closer together. When an electron transitions between orbits, it can absorb or emit a photon, and the energy of that photon, referred to as \( E_{photon} \), is the difference in energy between the two levels. By algebraically manipulating the Rydberg formula, we can calculate the energy of released photons during such transitions.

Electron orbitals represent the regions around an atom's nucleus where electrons are most likely to be found. These orbitals are characterized by quantum numbers that designate their shape, orientation, and energy level. An important aspect of electron orbitals to understand is that they come in various shapes—'s' orbitals are spherical, 'p' orbitals are dumbbell-shaped, and 'd' orbitals have more complex shapes.

For the hydrogen atom, electron transitions between these orbitals, such as from a 4d to a 2p, represent a change in the electron's energy state. The likelihood of finding an electron in a particular region of space changes based on its orbital, with 's' orbitals fully symmetrical and 'p' and 'd' orbitals more complex.

The emission spectrum is the spectrum of light released from an atom when electrons drop from higher energy levels to lower ones, emitting photons. Each element has a unique emission spectrum based on its electronic structure. For hydrogen, this spectrum consists of lines at specific frequencies corresponding to different energy transitions. These lines form what is known as the Balmer series, Lyman series, and so on, in the visible, ultraviolet, and other ranges of light. This unique fingerprint of light can be used to identify elements in lab measurements as well as astronomical observations.

Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. These include:

• The principal quantum number (\( n \)), which indicates the relative size and energy of an orbital
• The angular momentum quantum number (\( l \)), defining the shape of the orbital
• The magnetic quantum number (\( m_l \)), reflecting the orbital's orientation in space
• The spin quantum number (\( m_s \)), determining the direction of the electron's spin

These numbers are essential to paint a complete picture of an electron's position and energy. They obey specific rules and principles, like the Pauli exclusion principle, which says no two electrons can have the same set of quantum numbers in a given atom.

Hydrogen-like atoms are those with a single electron, such as He+, Li2+, etc. Their energy levels can be determined by the same principles as hydrogen's, but with adjustments for their specific atomic numbers. As a hydrogen-like atom's nuclear charge increases, the energy levels become more spread out and the emission lines in their spectrum shift accordingly. This concept helps to understand differences in the emission spectra compared to hydrogen, such as the variations that would be observed with potassium mentioned in the exercise. Factors such as the increased nuclear charge and the shielding effect of core electrons influence the energy levels and, consequently, the spectrum emitted during electron transitions.