Why classical mechanics is unsuitable for the atom
For large macroscopic objects, wave properties are completely indistinguishable. For example, if you calculate the wavelength for the first artificial Earth satellite (mass 84 kg, speed 7900 m/s), it is a negligible $10^{-40}$ meters. This value can be safely ignored.
The situation changes dramatically when moving to microscopic objects due to the extremely small mass of the electron ($9.11 \cdot 10^{-31}$ kg). On the scale of the atom, wave properties become decisive. The electron wavelength actually coincides with the circumference of its orbit, defining the physical dimensions of the atom itself. The movement of an electron cannot be described as the flight of a tiny ball — it is more like the oscillations of a standing wave on a fixed string, where the transition between states occurs exclusively in a jump-like manner, without intermediate stages.
The electron cloud and the concept of the orbital
Due to its enormous speed and wave nature, the electron is seemingly "smeared" in space, creating the effect of a cloud. The density of this electron cloud (the probability of finding the particle in a given volume) is maximum near the nucleus and gradually decreases with distance from it.
In quantum chemistry, a strict definition is used: an orbital is the region of space around the atomic nucleus where the probability of finding an electron is 90%.
Quantum numbers: the electron's "address"
Absolute values of the physical quantities of the electron are replaced by a set of discrete parameters — quantum numbers.
- Principal quantum number ($n$)
Determines the energy level and orbital radius. Takes integer values ($1, 2, 3...$). Upon absorption of a light quantum, the principal number and radius increase (for example, in hydrogen, the radius increases from 53 pm to 212 pm upon transition from $n=1$ to $n=2$).
- Orbital quantum number ($l$)
Responsible for the shape of the orbital. It depends on the principal number and takes values from $0$ to $n-1$.
- $l=0$ ($s$-orbital): spherical shape.
- $l=1$ ($p$-orbital): dumbbell shape.
- $l=2$ ($d$-orbital): four-leaf clover shape.
- $l=3$ ($f$-orbital): complex spatial configuration.
- Magnetic quantum number ($m_l$)
Sets the spatial orientation of the orbitals. For example, for the $p$-subshell there are three orientation options (along the coordinate axes), and for the $d$-subshell — five.
- Spin quantum number ($m_s$)
Spin characterizes the intrinsic rotation of the electron around its axis. Only two jump-like states are possible: clockwise ($+1/2$, arrow up) and counterclockwise ($-1/2$, arrow down).
Fundamental laws of orbital filling
Pauli exclusion principle Ensures the uniqueness of the electronic structure of elements. According to it, there cannot be two electrons in an atom with the same set of all four quantum numbers. If three numbers coincide (electrons on the same orbital), they must have different spins. Consequence: any orbital holds no more than two electrons (an electron pair).
Hund's rule Describes the population of energetically equivalent orbitals (e.g., three $p$-orbitals). Electrons tend to occupy the maximum number of empty orbitals singly, and with parallel (identical) spins. This minimizes the mutual repulsion of negatively charged particles and brings the system into an energetically most favorable, stable state.
Briefly: answers to topical questions
What specific numerical values does the magnetic quantum number take?
The magnetic quantum number ($m_l$) takes all integer values from $-l$ to $+l$, including zero. The number of possible values is strictly related to the orbital quantum number ($l$) and determines the spatial orientation of the electron orbital:
- For $l = 0$ ($s$-subshell) — $0$.
- For $l = 1$ ($p$-subshell) — $-1, 0, +1$.
- For $l = 2$ ($d$-subshell) — $-2, -1, 0, +1, +2$.
- For $l = 3$ ($f$-subshell) — $-3, -2, -1, 0, +1, +2, +3$.
By what formula is the maximum number of electrons in an energy level calculated?
The maximum number of electrons in an energy level is calculated using the general formula $2n^2$, where $n$ is the principal quantum number. The total capacity of an energy level is the sum of the capacities of all subshells included in it.
| Level | Subshells included | Maximum number of electrons |
|---|---|---|
| $n = 1$ (K) | $s$ | 2 |
| $n = 2$ (L) | $s$, $p$ | 8 |
| $n = 3$ (M) | $s$, $p$, $d$ | 18 |
| $n = 4$ (N) | $s$, $p$, $d$, $f$ | 32 |
Class and exam questions
Why do energy levels split into sublevels?
This occurs in all multi-electron atoms due to the mutual repulsion of electrons. As a result, the energy begins to depend not only on the principal ($n$) but also on the azimuthal ($l$) quantum number. The exception is the hydrogen atom, where the energy level depends solely on $n$.
What does negative electron energy mean?
Zero energy corresponds to a free electron torn away from the atom. When an electron enters the sphere of attraction of the nucleus, its energy decreases (it becomes more stable), which is why energy values in an atom always go into the negative.
How is the shape of the orbital related to the standing wave?
The shape of the orbital is analogous to the number of half-waves of a vibrating string. A state with one half-wave gives a spherical $s$-orbital, and with two half-waves — a dumbbell-shaped $p$-orbital with a characteristic node in the middle.
What else to study on this topic
- Mathematical calculation of electron energy in a hydrogen atom
- Absorption and emission spectra during transitions between levels
- Calculation of electron shell and subshell capacity
- The influence of orbital shape on the geometry of chemical bonds in molecules
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