QUANTUM-MECHANICAL MODEL OF THE ATOM: BEHAVIOR, EXAMPLES - PHYSICAL - 2025

The quantum-mechanical model of the atom assumes that it is made up of a central nucleus made up of protons and neutrons. The negatively charged electrons surround the nucleus in diffuse regions known as orbitals.

The shape and extent of electronic orbitals is determined by several magnitudes: the potential of the nucleus and the quantized levels of energy and angular momentum of electrons.

Figure 1. Model of a helium atom according to quantum mechanics. It consists of the cloud of probability of the two electrons of helium that surround a positive nucleus 100 thousand times smaller. Source: Wikimedia Commons.

According to quantum mechanics, electrons have dual wave-particle behavior and at the 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.

Figure 1 shows the structure of the helium atom, which has a nucleus with two protons and two neutrons. This nucleus is surrounded by the cloud of probability of the two electrons that surround the nucleus, which is one hundred thousand times smaller. In the following image you can see the helium atom, with the protons and neutrons in the nucleus and the electrons in orbitals.

The size of a helium atom is of the order of an angstrom (1 Å), that is, 1 x 10 ^ -10 m. While the size of its nucleus is of the order of a femtometer (1 fm), that is, 1 x 10 ^ -15 m.

Despite being so comparatively small, 99.9% of the atomic weight is concentrated in the tiny nucleus. This is because protons and neutrons are 2,000 times heavier than the electrons that surround them.

Atomic scale and quantum behavior

One of the concepts that had the most influence on the development of the atomic model was that of the wave - particle duality: the discovery that each material object has an associated wave of matter.

The formula for calculating the wavelength λ associated with a material object was proposed by Louis De Broglie in 1924 and is the following:

Where h is Planck's constant, m is mass, and v is velocity.

According to the de Broglie principle, every object has a dual behavior, but depending on the scale of the interactions, the speed and the mass, the wave behavior may be more pre-eminent than 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 slower than the speed of light). This speed corresponds to energy values ​​in the range of tens of electron volts.

With the above data, and by using the de Broglie formula, the wavelength for the electron can be obtained:

λ = 6.6 x 10 ^ -34 J s / (9.1 × 10 ^ -31 kg 6 x 10 ^ 6 m / s) = 1 x 10 ^ -10 m = 1 Å

The electron at the typical energies of the atomic levels, has a wavelength of the same order of magnitude as that of the atomic scale, so that at that scale it has a wave behavior and not a particle.

First quantum models

With the idea in mind that the atomic-scale electron has wave behavior, the first atomic models based on quantum principles were developed. Among these, Bohr's 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. In other words, the electron was treated as a particle subjected to the electrostatic attractive force of the nucleus that orbited around it, governed by Newton's second law.

In addition to classical orbits, these first models took into account that the electron had an associated material wave. Only orbits whose perimeter was a whole number of wavelengths were allowed, since those that do not meet this criterion are vanished by destructive interference.

It is then that the quantization of energy appears for the first time in the atomic structure.

The word quantum precisely comes from the fact that the electron can only take on some discrete values ​​of energy within the atom. This coincides with Planck's finding, which consisted in the discovery that 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 longer any 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. That equation is neither more nor less than the Schrodinger equation, proposed in 1925.

This equation relates and determines the wave function ψ associated with a particle, such as the electron, with its interaction potential and its total energy E. Its mathematical expression is:

Equality in the Schrodinger equation holds only for some values ​​of the total energy E, leading to the quantization of the energy. The wave function of the electrons subjected to the potential of the nucleus is obtained from the solution of the Schrodinger equation.

Atomic orbitals

The absolute value of the wave function squared - ψ - ^ 2, gives the probability amplitude of finding the electron at a given position.

This leads to the concept of the orbital, which is defined as the diffuse region occupied by the electron with a non-zero probability amplitude, for the discrete values ​​of energy and angular momentum determined by the solutions of the Schrodinger equation.

The knowledge of the orbitals is very important, because it describes the atomic structure, the chemical reactivity and the possible bonds to form molecules.

The hydrogen atom is the simplest of all, because it has a solitary electron and it is the only one that admits an exact analytical solution of the Schrodinger equation.

This simple atom has a nucleus made up of a proton, which produces a central potential of Coulomb attraction that only depends on the radius r, so it is a system with spherical symmetry.

The wave function depends on the position, given by the spherical coordinates with respect to the nucleus, since the electric potential has central symmetry.

Furthermore, the wave function can be written as the product of a function that depends only on the radial coordinate, and another that depends on the angular coordinates:

Quantum numbers

The solution of the radial equation produces the discrete energy values, which depend on an integer n, called the principal quantum number, which can take positive integer values ​​1, 2, 3,…

Discrete energy values ​​are negative values ​​given by the following formula:

The angular equation solution defines the quantized values ​​of angular momentum and its z component, giving rise to the 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 l were 2, the magnetic quantum number would take the values ​​-2, -1, 0, 1, 2.

Shape and size of orbitals

The radial range of the orbital is determined by the radio wave function. It is greater as the energy of the electron increases, that is, as the principal quantum number increases.

The radial distance is usually measured in Bohr radii, which for the lowest energy of hydrogen is 5.3 X 10-11 m = 0.53 Å.

Figure 2. Bohr's radius formula. Source: F. Zapata.

But the shape of the orbitals is determined by the value of the angular momentum quantum number. 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.

Figure 3. Shape of the s, p, d, f orbitals. Source: UCDavis Chemwiki.

These orbitals pack into each other according to the energy of the electrons. For example, the following figure shows the orbitals in a sodium atom.

Figure 4. 1s, 2s, 2p orbitals of the sodium ion when it has lost an electron. Source: Wikimedia Commons.

The spin

The quantum mechanical model of the Schrödinger equation does not incorporate the spin of the electron. But it is taken into account through the Pauli exclusion principle, which indicates that orbitals can be populated with up to two electrons with spin quantum 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 greater radius than the 2s). The spin of the atom is decisive in the magnetic characteristics of a substance.

References

1. Alonso - Finn. Quantum and statistical fundamentals. Addison Wesley.
2. Eisberg - Resnick. Quantum physics. Limusa - Wiley.
3. Gasiorowicz. Quantum physics. John Wiley & Sons.
4. HSC. Physics course 2. Jacaranda plus.
5. Wikipedia. Schrodinger's atomic model. Recovered from: Wikipedia.com