For each of the following pairs of hydrogen orbitals, indicate which is higher in energy: (a) \(1 s, 2 s ;\) (b) \(2 p\) \(3 p ;\) (c) \(3 d_{x y}, 3 d_{y c}\) (d) \(3 s, 3 d ;\) (e) \(4 f, 5 s\)

Principal quantum numbers are fundamental in understanding the structure and energy levels of electrons in atoms. Think of these numbers as the address of an electron within an atom, providing information about its energy level and distance from the nucleus.

In quantum mechanics, these are denoted by the symbol 'n' and can have values of any positive whole number. The lowest energy level starts at n=1, which is closest to the nucleus, and energy levels increase as n becomes larger.

When comparing two hydrogen orbitals, such as the pair \(1s, 2s\) from the exercise, it's the principal quantum number 'n' that determines their energy difference. In hydrogen, a \(2s\) orbital has a higher value of 'n' than a \(1s\) orbital, meaning \(2s\) is at a higher energy level. The step-by-step solution's emphasis on comparing the quantum numbers is crucial because it directly impacts the electron's energy state.

An easy way to remember this is: the larger the principal quantum number, the higher the energy level of the orbital.

Atomic orbitals are regions around the nucleus of an atom where there is a high probability of finding an electron. Each orbital is defined not just by its principal quantum number 'n', but also by other quantum numbers that describe its shape, orientation, and spin.

The shape of an orbital is indicated by the angular quantum number 'l', which can take on values from 0 to n-1. For example, when 'l' is 0, the orbital is spherical and called an 's' orbital. If 'l' is 1, the orbital has a dumbbell shape and is known as a 'p' orbital. 'd' and 'f' orbitals have more complex shapes and are labeled when 'l' is 2 and 3, respectively.

In the given exercise, when comparing \(3d_{xy}, 3d_{yz}\), both orbitals have the same energy because they have identical 'n' and 'l' values even though they are oriented differently. In hydrogen atoms, the orientation (given by magnetic quantum number 'm_l') doesn't influence energy levels, unlike in multi-electron atoms where electron repulsion comes into play.

Quantum mechanics is a fundamental theory in physics that also serves as a cornerstone in chemistry, providing insight into the behaviour of electrons in atoms and molecules. It's a theory that replaces the classical approximations of motion with quantum models that account for the discrete nature of energy levels.

In the context of chemistry, it explains why atoms form bonds, the shapes of molecules, and how chemical reactions occur at the atomic level. Quantum mechanics introduces the concept of wave-particle duality, where electrons can be treated as both waves and particles. This dual nature is vital in understanding phenomena like electron delocalization and resonance in molecules.

Applying quantum mechanics in exercises like the one provided helps students grasp that electron energies are not arbitrary but are defined by quantized levels determined by specific rules and numbers. This knowledge is vital in predicting how an atom will react chemically, as the energy of electrons affects reactivity and the types of chemical bonds that can form.