Basic principles of shell filling
To comprehensively describe the state of each electron, four quantum numbers are used. The distribution of particles among orbitals and the probability of finding them in certain states are dictated by two fundamental rules:
- Aufbau principle (principle of minimum energy). Any physical system tends toward a minimum of potential energy (by analogy with a falling body or straightening spring). The ground state of an atom is the configuration with the minimum total energy reserve of electrons, and it is the most stable. Upon receiving excess energy, the system transitions into an excited state. It is extremely unstable and lasts only about $10^{-8}$ seconds, after which the atom returns to the baseline phase, releasing energy in the form of quantum radiation.
- Pauli exclusion principle. This is the first fundamental principle. It has absolute priority in electron distribution, is strictly enforced, and is never violated.
Filling algorithm and zone overlapping
The first energy level ($n=1$) is arranged as simply as possible and does not split into sublevels. However, as the distance from the nucleus increases, the structure becomes more complex. Starting with the principal quantum number $n=4$, partial overlapping of energy zones is observed. Because of this, orbitals are not filled strictly in numerical order of levels: for example, the $4s$ sublevel is energetically more favorable (located lower) than the $3d$ sublevel.
The actual filling sequence is determined using a matrix, where movement proceeds along diagonals from bottom to top:
- $1s$
- $2s$
- $2p \rightarrow 3s$
- $3p \rightarrow 4s$
- $3d \rightarrow 4p \rightarrow 5s$
- $4d \rightarrow 5p \rightarrow 6s$
- $4f \rightarrow 5d \rightarrow 6p \rightarrow 7s$
- $5f \rightarrow 6d \rightarrow 7p$
Magnetic quantum number
The spatial orientation of an electron orbital is determined by the magnetic quantum number ($m_l$). Its values are strictly tied to the orbital quantum number ($l$) and include the entire series of integers from $-l$ to $+l$, including zero.
This indicator got its name due to the effect of splitting of atomic spectral lines. Under normal conditions, electron states with the same energy are indistinguishable (they are degenerate). However, when exposed to a magnetic field, these states acquire different energy. Instrumentally, this is recorded precisely as the splitting of lines in the emission spectrum.
Capacity of levels and sublevels
The capacity of any energy sublevel is always equal to twice the number of its orbitals. The set of electrons occupying a single sublevel is called an atomic subshell.
The maximum number of electrons in a sublevel is calculated by the general formula $2(2l + 1)$:
- $s$ ($l=0$): 1 orbital $\rightarrow$ maximum of 2 electrons.
- $p$ ($l=1$): 3 orbitals $\rightarrow$ maximum of 6 electrons.
- $d$ ($l=2$): 5 orbitals $\rightarrow$ maximum of 10 electrons.
- $f$ ($l=3$): 7 orbitals $\rightarrow$ maximum of 14 electrons.
The sum of the capacities of all sublevels forms the atomic shell (the set of electrons of a single energy level). The maximum capacity of a level is calculated by the formula $2n^2$. Thus, the first level (K) holds only 2 electrons, the second (L) — 8, the third (M) — 18, and the fourth (N) — 32 electrons.
Methods of visualizing electronic structure
Two main formats are used to display shell structure:
- Electronic formulas: Linear notation of populated sublevels in order of increasing energy. The number of electrons is indicated as a superscript.
- Energy diagrams: A graphical representation of a portion of the overall sequence, where populated orbitals are represented graphically and electrons by arrows.
Examples of first-period elements: For hydrogen ($Z=1$), the formula is $1s^1$, and on the diagram, this is displayed as a single arrow at the $1s$ level. Helium ($Z=2$) is described by the formula $1s^2$, which corresponds to two oppositely directed arrows in the same $1s$ orbital.
Briefly: answers to topical questions
What does the Pauli exclusion principle entail during electron distribution?
The Pauli exclusion principle states that all electrons in an atom are in different states and are characterized by different sets of four quantum numbers.
This principle has priority over the Aufbau principle and is strictly observed. It follows from this that when choosing a specific orbital (where three quantum numbers are fixed), only the spin quantum number can change. Thus, a single orbital holds no more than two electrons, which form an electron pair and have opposite spins.
What is Hund's rule for orbital filling?
Hund's rule states that the ground state of an atom corresponds to the filling by electrons of the maximum number of energetically equivalent orbitals.
In this case, the spin quantum numbers (spins) of these electrons must be the same, i.e., parallel. Placing electrons in different spatial orbitals instead of pairing them in one increases the average distance between them. This minimizes interelectron repulsion and leads to a decrease in the overall energy of the system. As a result, the configuration with the maximum number of unpaired electrons is energetically the most favorable.
What four quantum numbers are used to fully describe the state of an electron?
To fully describe the state of an electron, the principal, orbital, magnetic, and spin quantum numbers are used.
- Principal quantum number ($n$) — characterizes the energy level.
- Orbital quantum number ($l$) — determines the shape of the atomic orbital; takes values from 0 to $n-1$.
- Magnetic quantum number ($m_l$) — characterizes the spatial orientation of the electron orbital; takes all integer values from $-l$ to $+l$.
- Spin quantum number ($m_s$) — characterizes the spin projection; takes values of $+1/2$ and $-1/2$.
How are electrons and orbitals graphically designated on energy diagrams?
On energy diagrams, electrons are graphically designated by arrows, and populated sublevels are depicted as energy diagrams.
The direction of the arrow corresponds to the value of the electron's spin quantum number:
- $m_s = +1/2$ is denoted by an upward arrow.
- $m_s = -1/2$ is denoted by a downward arrow.
For example, the electronic structure of hydrogen is depicted as a single arrow at the $1s$ level, while helium is shown as two oppositely directed arrows at the same level.
To write the orbital filling order without errors, create a matrix: list the sublevels in columns by levels ($1s$; $2s$, $2p$; $3s$, $3p$, $3d$, etc.) and read them along diagonal lines running from bottom to top.
Class and exam questions
Why does an atom transition to an excited state and how long does it stay in it?
An atom is excited upon receiving excess energy, but this state is extremely unstable as the system tends toward minimum potential energy. In about $10^{-8}$ seconds, the atom returns to the ground state, releasing the excess energy as radiation.
What is the difference between an atomic subshell and a shell?
A subshell unites electrons located in the same energy sublevel (for example, all electrons of the $3p$ sublevel). A shell, however, includes all electrons of an entire energy level (the sum of all $s$, $p$, $d$, etc. sublevels for a given principal quantum number $n$).
Why is the $4s$ sublevel filled before $3d$?
Starting from the principal quantum number $n=4$, the energy bands of the atom begin to partially overlap. It turns out that the $4s$ sublevel is energetically lower (more favorable) than $3d$, so electrons occupy it first according to the Aufbau principle.
What else to study on this topic
- Splitting of energy levels in magnetic fields
- Relationship between orbital and magnetic quantum numbers
- Degenerate electronic states of the atom
- Practical construction of energy diagrams using the Pauli exclusion principle
- Calculation of higher-order electron shell capacity
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