The quantum mechanical model of the atom assumes that this is formed by a central nucleus made up of protons and neutrons. 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 the electrons..
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 one 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 the protons and neutrons are 2,000 times heavier than the electrons that surround them..
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One of the concepts that had the most influence on the development of the atomic model was that of duality wave - particle: the discovery that every material object has a matter wave 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 the following:
λ = h / (m v)
Where h is Planck's constant, m the mass and v speed.
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 preeminent than the particle behavior 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 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..
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, 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 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 an integer 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 It precisely comes from the fact that the electron can only take on some discrete energy values within the atom. This coincides with Planck's finding, which consisted in the discovery that a radiation of frequency F interacts with matter in packets of energy E = h f, where h is Planck's constant.
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 AND. Its mathematical expression is:
Equality in the Schrodinger equation holds only for some values of the total energy AND, giving rise to the quantization of energy. The wave function of the electrons subjected to the potential of the nucleus is obtained from the solution of the Schrodinger equation.
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 orbital, which is defined as the diffuse region occupied by the electron with 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:
The solution of the radial equation produces the discrete energy values, which depend on an integer number 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 the angular momentum and its z component, giving rise to the quantum numbers l Y ml.
The quantum number of angular momentum l goes from 0 to n-1. Quantum number ml is called the magnetic quantum number and goes from -l until +l. For example, yes l were 2 the magnetic quantum number would take the values -2, -1, 0, 1, 2.
The radial reach of the orbital is determined by the radi wave functionto the. 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.
But the shape of the orbitals is determined by the value of the angular momentum quantum number. If l = 0 we have a spherical orbital called s, if l = 1 we 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 pack 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 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.
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