The shape and extent of electronic orbitals is determined by several magnitudes: the potential of the nucleus and the quantified levels of energy and angular momentum of the electrons.
According to quantum mechanics, electrons have a double wave-particle behavior and, on an atomic scale, they are diffuse and non-point. The dimensions of the atom are practically determined by the extension of the electronic orbitals that surround the positive nucleus.
The size of a helium atom is on the order of an angstrom (1 Å), ie 1 x 10^-10 m. While its core size is on the order of a femtometer (1 fm), this is 1 x 10^-15 m.
Despite being comparatively small, 99.9% of the atomic weight is concentrated in the small nucleus. This is because protons and neutrons are 2000 times heavier than the electrons that surround them.
Atomic scale and quantum behavior
One of the concepts that had the greatest influence on the development of the atomic model was the wave-particle duality : the discovery that each material object has a wave of matter associated with it.
The formula for calculating the wavelength λ associated with a material object was proposed by Louis De Broglie in 1924 and is as follows:
λ = h / (mv)
Where h is Planck’s constant, m is mass and v is velocity.
According to De Broglie’s principle, every object has a dual behavior, but depending on the scale of interactions, speed and mass, the behavior of waves may be more prominent than that of the particle or vice versa.
The electron is light, its mass is 9.1 × 10 ^ -31 kg. The typical speed of an electron is 6000 km / s (fifty times less than the speed of light). This velocity corresponds to energy values in the range of tens of electron volts.
With the above data, and using De Broglie’s formula, the wavelength of the electron can be obtained:
The electron at typical energies of atomic levels has a wavelength of the same order of magnitude as that of the atomic scale; therefore, at this scale, it has a wave and non-particle behavior.
First Quantum Models
With the idea that the atomic scale electron has wave behavior, the first atomic models based on quantum principles were developed. Among them, the Bohr atomic model stands out, which perfectly predicted the emission spectrum of hydrogen, but not that of other atoms.
The Bohr model and later the Sommerfeld model were semi-classical models. That is, the electron was treated as a particle subject to the electrostatic attraction force of the nucleus that orbited around it, governed by Newton’s second law.
In addition to classical orbits, these early models took into account that the electron had an associated material wave. Only orbits whose perimeter was an integer number of wavelengths were allowed, since those that do not meet this criterion disappear from destructive interference.
It is then that the quantization of energy in the atomic structure first appears.
The word quantum comes precisely from the fact that the electron can only receive a few discrete values of energy within the atom. This coincides with Planck’s discovery, which consisted in the discovery that a radiation of frequency f interacts with matter in energy packets E = hf , where h is Planck’s constant.
Dynamics of material waves
There was no doubt that the electron at the atomic level behaved like a material wave. The next step was to find the equation that governs their behavior. This equation is neither more nor less than Schrodinger’s equation, proposed in 1925.
This equation refers to and determines the ψ wave function associated with a particle, such as an electron, with its potential for interaction and its total energy of E . Its mathematical expression is:
The equality in the Schrodinger equation is achieved only for some values of the total energy E , resulting in the quantization of energy. The wavefunction of the electrons subject to the nucleus potential is obtained from the solution of the Schrodinger equation.
The absolute value of the squared wave function | , | ^ 2, gives the probability amplitude of finding the electron at a given position.
This leads to the concept of an orbital, which is defined as the diffuse region occupied by the electron with a probability amplitude other than zero, for discrete values of energy and angular momentum determined by the solutions of the Schrodinger equation.
Knowledge of orbitals is very important as it describes the atomic structure, chemical reactivity and possible bonds to form molecules.
The hydrogen atom is the simplest of all, having a solitary electron and is the only one that admits an exact analytical solution of the Schrodinger equation.
This simple atom has a nucleus formed by a proton, which produces a central coulomb attraction potential that depends only on the radius r , being a system with spherical symmetry.
The wave function depends on the position, given by the spherical coordinates in relation to the nucleus, since the electric potential has central symmetry.
Furthermore, the wavefunction can be written as the product of a function that depends exclusively on the radial coordinate and another that depends on the angular coordinates:
The solution of the radial equation produces the discrete values of energy, which depend on an integer n, called the principal quantum number , which can take on positive integer values 1, 2, 3, …
Discrete energy values are negative values given by the following formula:
The angular solution equation defines the quantified values of angular momentum and the z component, resulting in quantum numbers l and ml .
The angular momentum quantum number l ranges from 0 to n-1 . The quantum number ml is called the magnetic quantum number and ranges from -l to +l . For example, if I were 2, the magnetic quantum number would take on the values -2, -1, 0, 1, 2.
Shape and size of orbitals
The radial range of the orbital is determined by the em radio wave function . It is larger as the energy of the electron grows, that is, as the principal quantum number increases.
Radial distance is usually measured in Bohr radii, which for the lowest hydrogen energy is 5.3 X 10 -11 m = 0.53 Å.
But the shape of the orbitals is determined by the value of the quantum number of angular momentum. If l = 0 you have a spherical orbital called s, if l = 1 you have a lobulated orbital called p which can have three orientations according to the magnetic quantum number. The following figure shows the shape of the orbitals.
These orbitals are packed into each other according to the energy of the electrons. For example, the following figure shows the orbitals in a sodium atom.
The quantum mechanical model of the Schrödinger equation does not incorporate electron spin. But this is taken into account by the Pauli exclusion principle, which indicates that orbitals can be filled with up to two electrons with quantum spin numbers s = + ½ and s = -½.
For example, the sodium ion has 10 electrons, that is, if we refer to the previous figure, there are two electrons for each orbital.
But if it is the neutral sodium atom, there are 11 electrons, the last of which would occupy a 3s orbital (not shown in the figure and with a radius larger than the 2s). The rotation of the atom determines the magnetic characteristics of a substance.