The Quantum Mechanical Model
The quantum mechanical model is
based on quantum theory, which says matter also has properties associated
with waves. According to quantum theory, it’s impossible to know the exact
position and momentum of an electron at the same time. This is known as the Uncertainty
Principle.
The quantum mechanical model of the
atom uses complex shapes of orbitals (sometimes called electron
clouds), volumes of space in which there is likely to be an
electron. So, this model is based on probability rather than certainty.
It is a model that explains how electrons exist in atoms and how those electrons determine the chemical and physical properties of elements.
The Quantum Mechanical Model of the atom presents a more accurate model of the atom. It is a more sophisticated model based on complex mathematical calculations and interpretations.
The Quantum Mechanical Model of the atom presents a more accurate model of the atom. It is a more sophisticated model based on complex mathematical calculations and interpretations.
Main Energy Levels or Shells, Sublevels or Subshells and Orbitals
a.) Main Energy Levels / Principal Energy Levels, n
· n = 1,2,3,4,5,6,7
· Generally, energy increases with increasing n.
· Distance of the electron from the nucleus increases with increasing n.
Shells –
The electrons with the same principal quantum number n are said to form a shell. Proceeding from the nucleus outwards the shells are called K, L, etc. Thus if n = 1, 2, 3, 4, the shell is the K, L, M or N shell, respectively. This nomenclature had its origin in X-ray spectroscopy, in which a K-series line is due to an electron transition from an outer to the K shell.
· n = 1,2,3,4,5,6,7
· Generally, energy increases with increasing n.
· Distance of the electron from the nucleus increases with increasing n.
Shells –
The electrons with the same principal quantum number n are said to form a shell. Proceeding from the nucleus outwards the shells are called K, L, etc. Thus if n = 1, 2, 3, 4, the shell is the K, L, M or N shell, respectively. This nomenclature had its origin in X-ray spectroscopy, in which a K-series line is due to an electron transition from an outer to the K shell.
b.) Sub-shells / Sub-levels –
The electrons in each shell are arranged in sub-shells (often also called shells), specified by the value of l (l = 0,1 ,..., n − 1). In Bohr's theory, n and l determine the size and shape of the orbit. The nl sub-shell is complete or closed when it contains 2(2l +1) electrons.
The subshells are usually referred to by letters, rather than by the corresponding value of the orbital quantum number. The letters s, p, d, f, g, and h stand for values of 0, 1, 2, 3, 4, and 5, respectively. Using these letters allows us to use a shorthand to denote how many electrons are in a subshell; this is useful for specifying the ground state (lowest energy state) of a particular atom.
The electrons in each shell are arranged in sub-shells (often also called shells), specified by the value of l (l = 0,1 ,..., n − 1). In Bohr's theory, n and l determine the size and shape of the orbit. The nl sub-shell is complete or closed when it contains 2(2l +1) electrons.
The subshells are usually referred to by letters, rather than by the corresponding value of the orbital quantum number. The letters s, p, d, f, g, and h stand for values of 0, 1, 2, 3, 4, and 5, respectively. Using these letters allows us to use a shorthand to denote how many electrons are in a subshell; this is useful for specifying the ground state (lowest energy state) of a particular atom.
c.) Orbitals –
There are multiple electron orbitals within an atom. Each has their own energy level associated to them, and their own properties. Because each orbital is different, we assign them specific quantum numbers: 1s, 2s, 2p 3s, 3p,4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The numbers, (n=1,2,3, etc.) are called principal quantum numbers and can only be positive numbers. The letters (s,p,d,f) are representative of the orbital angular momentum quantum number (ℓ) and the orbital angular momentum quantum number may be 0 or a positive number but can never be greater than n-1. Each letter is paired with a specific ℓ value:
S subshell = 0
P subshell = 1
D subshell = 2
F subshell = 3
Orbitals are also described by their magnetic quantum number (mℓ). The magnetic quantum number can range from –ℓ to +ℓ. This number tells us how many orbitals there are and thus how many electrons can reside in each orbital.
Orbitals that have the same or identical energy level are known as Degenerate. An example is the 2p orbital: 2px has the same energy level to 2py. This concept becomes more important in Molecular Orbitals. The Pauli exclusion principle states that no two electrons can have the same exact orbital configuration; in other words, the same exact quantum numbers. However, the electron can exist with spin up (ms = +1/2) or with spin down (ms = -1/2). This means that the s orbital can contain up to two electrons, the p orbital can contain up to six electrons, the d orbital can contain up to 10 electrons, and the f orbital can contain up to 14 electrons.
There are multiple electron orbitals within an atom. Each has their own energy level associated to them, and their own properties. Because each orbital is different, we assign them specific quantum numbers: 1s, 2s, 2p 3s, 3p,4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The numbers, (n=1,2,3, etc.) are called principal quantum numbers and can only be positive numbers. The letters (s,p,d,f) are representative of the orbital angular momentum quantum number (ℓ) and the orbital angular momentum quantum number may be 0 or a positive number but can never be greater than n-1. Each letter is paired with a specific ℓ value:
S subshell = 0
P subshell = 1
D subshell = 2
F subshell = 3
Orbitals are also described by their magnetic quantum number (mℓ). The magnetic quantum number can range from –ℓ to +ℓ. This number tells us how many orbitals there are and thus how many electrons can reside in each orbital.
Orbitals that have the same or identical energy level are known as Degenerate. An example is the 2p orbital: 2px has the same energy level to 2py. This concept becomes more important in Molecular Orbitals. The Pauli exclusion principle states that no two electrons can have the same exact orbital configuration; in other words, the same exact quantum numbers. However, the electron can exist with spin up (ms = +1/2) or with spin down (ms = -1/2). This means that the s orbital can contain up to two electrons, the p orbital can contain up to six electrons, the d orbital can contain up to 10 electrons, and the f orbital can contain up to 14 electrons.
Quantum Numbers
Four numbers, called quantum numbers, were introduced to describe the characteristics of electrons and their orbitals:
a.) The principal quantum number
The principal quantum number n describes the average distance of the orbital from the nucleus — and the energy of the electron in an atom. It can have positive integer (whole number) values: 1, 2, 3, 4, and so on. The larger the value of n, the higher the energy and the larger the orbital. Chemists sometimes call the orbitals electron shells.
b.) The angular momentum quantum number
The angular momentum quantum number l describes the shape of the orbital, and the shape is limited by the principal quantum number n: The angular momentum quantum number l can have positive integer values from 0 to n–1. For example, if the n value is 3, three values are allowed for l: 0, 1, and 2.
The value of l defines the shape of the orbital, and the value of n defines the size.
Orbitals that have the same value of n but different values of l are called subshells. These subshells are given different letters to help chemists distinguish them from each other. The following table shows the letters corresponding to the different values of l.
- Principal quantum number: n
- Angular momentum quantum number: l
- Magnetic quantum number: ml
- Spin quantum number: ms
a.) The principal quantum number
The principal quantum number n describes the average distance of the orbital from the nucleus — and the energy of the electron in an atom. It can have positive integer (whole number) values: 1, 2, 3, 4, and so on. The larger the value of n, the higher the energy and the larger the orbital. Chemists sometimes call the orbitals electron shells.
b.) The angular momentum quantum number
The angular momentum quantum number l describes the shape of the orbital, and the shape is limited by the principal quantum number n: The angular momentum quantum number l can have positive integer values from 0 to n–1. For example, if the n value is 3, three values are allowed for l: 0, 1, and 2.
The value of l defines the shape of the orbital, and the value of n defines the size.
Orbitals that have the same value of n but different values of l are called subshells. These subshells are given different letters to help chemists distinguish them from each other. The following table shows the letters corresponding to the different values of l.
When chemists describe one particular subshell in an atom, they can use both the n value and the subshell letter — 2p, 3d, and so on. Normally, a subshell value of 4 is the largest needed to describe a particular subshell. If chemists ever need a larger value, they can create subshell numbers and letters.
Or Azimuthal quantum number l specifies the angular momentum lh/2π of the orbital motion, and is connected with the kind of symmetry possessed by the wave function (l = 0 implies spherical symmetry). Code letters for the value of l are:
Or Azimuthal quantum number l specifies the angular momentum lh/2π of the orbital motion, and is connected with the kind of symmetry possessed by the wave function (l = 0 implies spherical symmetry). Code letters for the value of l are:
The following figure shows the shapes of the s, p, and d orbitals.
As shown in the top row of the figure (a), there are two s orbitals — one for energy level 1 (1s) and the other for energy level 2 (2s). The s orbitals are spherical with the nucleus at the center. Notice that the 2s orbital is larger in diameter than the 1s orbital. In large atoms, the 1s orbital is nestled inside the 2s, just like the 2p is nestled inside the 3p.
The second row of the figure (b) shows the shapes of the p orbitals, and the last two rows (c) show the shapes of the d orbitals. Notice that the shapes get progressively more complex.
c.) The magnetic quantum number
The magnetic quantum number is designated as: ml
This number describes how the various orbitals are oriented in space. The value of this number depends on the value of l. The values allowed are integers from –l to 0 to +l. For example, if the value of l = 1 (p orbital), you can write three values for this number: –1, 0, and +1. This means that there are three different p subshells for a particular orbital. The subshells have the same energy but different orientations in space.
The second row (b) of the figure shows how the p orbitals are oriented in space. Notice that the three p orbitals correspond to magnetic quantum number values of –1, 0, and +1, oriented along the x, y, and z axes.
d.) The spin quantum number
The fourth and final quantum number is the spin quantum number, designated as: ms
This number describes the direction the electron is spinning in a magnetic field — either clockwise or counterclockwise. Only two values are allowed: +1/2 or –1/2. For each subshell, there can be only two electrons, one with a spin of +1/2 and another with a spin of –1/2.
*Pauli’s Exclusion Principle states that no two electrons can have all their four quantum numbers the same.
The second row of the figure (b) shows the shapes of the p orbitals, and the last two rows (c) show the shapes of the d orbitals. Notice that the shapes get progressively more complex.
c.) The magnetic quantum number
The magnetic quantum number is designated as: ml
This number describes how the various orbitals are oriented in space. The value of this number depends on the value of l. The values allowed are integers from –l to 0 to +l. For example, if the value of l = 1 (p orbital), you can write three values for this number: –1, 0, and +1. This means that there are three different p subshells for a particular orbital. The subshells have the same energy but different orientations in space.
The second row (b) of the figure shows how the p orbitals are oriented in space. Notice that the three p orbitals correspond to magnetic quantum number values of –1, 0, and +1, oriented along the x, y, and z axes.
d.) The spin quantum number
The fourth and final quantum number is the spin quantum number, designated as: ms
This number describes the direction the electron is spinning in a magnetic field — either clockwise or counterclockwise. Only two values are allowed: +1/2 or –1/2. For each subshell, there can be only two electrons, one with a spin of +1/2 and another with a spin of –1/2.
*Pauli’s Exclusion Principle states that no two electrons can have all their four quantum numbers the same.
Electronic Configuration
In atomic physics and quantum
chemistry, the electron configuration
is the distribution of electrons of an atom or molecule (or other physical
structure) in atomic or molecular orbitals. For example, the electron
configuration of the neon atom is 1s2 2s2 2p6.
Electronic configurations describe electrons as each moving independently in an orbital, in an average field created by all other orbitals. Mathematically, configurations are described by Slater determinants or configuration state functions.
According to the laws of quantum mechanics, for systems with only one electron, an energy is associated with each electron configuration and, upon certain conditions, electrons are able to move from one configuration to another by the emission or absorption of a quantum of energy, in the form of a photon.
Electronic configurations describe electrons as each moving independently in an orbital, in an average field created by all other orbitals. Mathematically, configurations are described by Slater determinants or configuration state functions.
According to the laws of quantum mechanics, for systems with only one electron, an energy is associated with each electron configuration and, upon certain conditions, electrons are able to move from one configuration to another by the emission or absorption of a quantum of energy, in the form of a photon.
Transition of Electrons
a.) Atomic electron transition is a change of an electron from one
quantum state to another within an atom or artificial atom. It appears
discontinuous as the electron "jumps" from one energy level to
another in a few nanoseconds or less. It is also known as atomic transition,
quantum jump, or quantum leap.
Electron transitions cause the emission or absorption of electromagnetic radiation in the form of quantized units called photons. Their statistics are Poissonian, and the damping of statistic values of time between jumps is exponential on average. The damping time constant (which ranges from nanoseconds to a few seconds) relates to the natural, pressure, and field broadening of spectral lines. The farther the electron jumps, the shorter the wavelength of the photon emitted, meaning they emit different colors based on how far they jump.
Although changes of quantum state occur on the submicroscopic level, in popular discourse, the term "quantum leap" refers to a large increase.
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. The ground state of a quantum field theory is usually called the vacuum state or the vacuum.
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state.
In quantum mechanics an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). The temperature of a group of particles is indicative of the level of excitation (with the notable exception of systems that exhibit Negative temperature).
b.) Molecular electronic transitions take place when electrons in a molecule are excited from one energy level to a higher energy level. The energy change associated with this transition provides information on the structure of a molecule and determines many molecular properties such as color. The relationship between the energy involved in the electronic transition and the frequency of radiation is given by Planck's relation.
Electron transitions cause the emission or absorption of electromagnetic radiation in the form of quantized units called photons. Their statistics are Poissonian, and the damping of statistic values of time between jumps is exponential on average. The damping time constant (which ranges from nanoseconds to a few seconds) relates to the natural, pressure, and field broadening of spectral lines. The farther the electron jumps, the shorter the wavelength of the photon emitted, meaning they emit different colors based on how far they jump.
Although changes of quantum state occur on the submicroscopic level, in popular discourse, the term "quantum leap" refers to a large increase.
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. The ground state of a quantum field theory is usually called the vacuum state or the vacuum.
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state.
In quantum mechanics an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). The temperature of a group of particles is indicative of the level of excitation (with the notable exception of systems that exhibit Negative temperature).
b.) Molecular electronic transitions take place when electrons in a molecule are excited from one energy level to a higher energy level. The energy change associated with this transition provides information on the structure of a molecule and determines many molecular properties such as color. The relationship between the energy involved in the electronic transition and the frequency of radiation is given by Planck's relation.