Download Inorganic Chemistry By Dr. Khalil K. Abid

Inorganic Chemistry
By
Dr. Khalil K. Abid
Lecture 3
Atomic Structure
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Much of the experimental work on the electronic structure of atoms done prior to 1913 involved
measuring the frequencies of electromagnetic radiation (e.g. light) that are absorbed or emitted by
atoms. It was discovered that atoms absorbed or emitted only certain, sharply defined frequencies of
electromagnetic radiation. These frequencies were also found to be characteristic of each particular
element in the periodic table. And the absorption or emission spectra, i.e. the ensemble of
frequencies, were more complex for heavier elements. Before being able to understand the electronic
structures of atoms, it was natural to start studying the simplest atom of all: the hydrogen atom,
which consists of one proton and one electron.
Spectroscopic emission lines and atomic structure of hydrogen
It was experimentally observed that the frequencies of light emission from atomic hydrogen could
be classified into several series. Within each series, the frequencies become increasingly closely
spaced, until they converge to a limiting value. Rydberg proposed a mathematical fit to the observed
experimental frequencies, which was later confirmed theoretically:
1/λ = RH ( 1/n12 – 1/n22 )
where λ is the wavelength of the line, and RH is a constant known as Rydberg’s constant that has the
value 109,677 cm-1. n2 is n1 +1, Ʋ is the frequency of the light emitted, c =3 x108 m.s – 1 =3 x10 10 c.s – 1
is the velocity of light in vacuum, The energy of the electromagnetic radiation is related to its
wavelength and frequency by the following relation:
E = ɦc / λ = ɦƲ
where ɦ (=6.62617~10–34 J. s) is Planck's constant.
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The expression in Eq. shows that the emission of light from the hydrogen atom occurs at specific
discrete values of frequencies v, depending on the values of integers n1and n2 . The Lyman series of
spectral lines corresponds to n=1 for which the convergence limit is 109,678 cm– 1. The Balmer
series corresponds to n=2, and the Paschen series to n=3. These are illustrated in figure where the
energy of the light emitted from the atom of hydrogen is plotted as arrows.
Although the absorption and emission lines for most of the elements were known before the turn
of the 20th century, a suitable explanation was not available, even for the simplest case of the
hydrogen atom. Prior to 1913, the explanation for this spectroscopic data was impossible because it
contradicted the laws of nature known at the time. Indeed, very well established electrodynamics
could not explain two basic facts: that atoms could exist at all, and that discrete frequencies of light
were emitted and absorbed by atoms. For example, it was known that an accelerating charged
particle had to emit electromagnetic radiation. Therefore, in the nuclear model of an atom, an
electron moving around the nuclei has acceleration and thus has to emit light, lose energy, and fall
down to the nucleus. This meant that the stability of elements in the periodic table, which is obvious
to us, contradicted classical electrodynamics. A new approach had to be followed in order to resolve
this contradiction, which resulted in a new theory, known as quantum mechanics. Quantum
mechanics could also explain the spectroscopic data mentioned above and adequately describe
experiments in modern physics that involve electrons and atoms, and ultimately solid state device
physics.
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Although all of the developments that have been discussed are important to our understanding of
the nature of matter, there are other phenomena that provide additional insight. One of them
concerns the emission of light from a sample of hydrogen gas through which a high voltage is
placed. Balmer studied the visible light emitted from the gas by passing it through a prism that
separates the light into its components. The four lines observed are as follows.
Hα= 656 nm = 6562 Å
Hβ= 486 nm = 4861 Å
Hγ= 434 nm = 4340 Å
Hδ= 410 nm= 4101 Å
This series of spectral lines for hydrogen became known as the Balmer Series, later made more
general, as spectral lines in the ultraviolet and infrared regions of the spectrum were discovered. The
wavelengths of these four spectral lines were found to obey the relationship
ῡ=1/λ = RH ( 1/n12 – 1/n22 )
n2 is n1 +1 (n1=1 for Lyman, 2 for Balmer, 3 for Paschen and 4 for Brackett). The quantity 1/λ is
known as the wave number. From the equation it can be seen that as n assumes larger values, the
lines become more closely spaced, but when n equals infinity, there is a limit reached. That limit is
known as the series limit for the Balmer Series. Keep in mind that these spectral lines, the first to be
discovered for hydrogen, were in the visible region of the electromagnetic spectrum. Detectors for
visible light (human eyes and photographic plates) were available at an earlier time than were
detectors for other types of electromagnetic radiation.
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Eventually, other series of lines were found in other regions of the electromagnetic spectrum. The
Lyman Series was observed in the ultraviolet region, whereas the Paschen, Brackett, and Pfund
Series were observed in the infrared region of the spectrum. All of these lines were observed as
they were emitted from excited atoms, so together they constitute the emission spectrum or line
spectrum of hydrogen atoms.
Energies of the light emitted from the hydrogen atom (shown by arrows). The Lyman series
corresponds to n=I , the Balmer to n=2 , the Paschen series to n=3 and Brackett to n = 4
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Following Rutherford’s experiments in 1911, Neils Bohr proposed in 1913 a dynamic model of the
hydrogen atom that was based on certain assumptions. The first of these assumptions was that there
were certain “allowed” orbits in which the electron could move without radiating electromagnetic
energy. Further, these were orbits in which the angular momentum of the electron (which for a
rotating object is expressed as mvr) is a multiple of h/2p (which is also written as ħ ),
mvr = nh/2π = nħ …………. (1)
where m is the mass of the electron, v is its velocity, r is the radius of the orbit, and n is an integer
that can take on the values 1, 2, 3, ., and ħ is h/2π . The integer n is known as a quantum number, or
more specifically, the principal quantum number.
Bohr also assumed that electromagnetic energy was emitted as the electron moved from a higher
orbital (larger n value) to a lower one and absorbed in the reverse process. This accounts for the fact
that the line spectrum of hydrogen shows only lines having certain wavelengths. In order for the
electron to move in a stable orbit, the electrostatic attraction between the electron and the proton
must be balanced by the centrifugal force that results from its circular motion. As shown in Figure
below, the forces are actually in opposite directions, so we equate only the magnitudes of the forces.
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The electrostatic force is given by the coulombic force as e2/r 2, and the centrifugal force on the
electron is mv2/r. Therefore, we can write
…
…….… (2)
From this equation we can calculate the velocity of the electron as
………… (3)
We can also solve equation 1 for v to obtain
………. (4)
Because the moving electron has only one velocity, the values for v given in Eqns 3 and 4 must
be equal.
……….. (5)
We can now solve for r to obtain
………. (6)
In Eqn 6, only r and n are variables. From the nature of this equation, we see that the value of r, the
radius of the orbit, increases as the square of n. For the orbit with n = 2, the radius is four times
that when n = 1, etc.
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…….. (7)
From Eqn (2), we see that
Multiplying both sides of the equation by ½ we obtain
…….. (8)
where the left hand side is simply the kinetic energy of the electron. The total energy of the
electron is the sum of the kinetic energy and the electrostatic potential energy, – e2/r.
……… (9)
Substituting the value for r from Eqn (6) into Eqn (9) we obtain
………… (10)
from which we see that there is an inverse relationship between the energy and the square of
the value n. The lowest value of E (and it is negative!) is for n = 1, and E = 0 when n has an
infinitely large value that corresponds to complete removal of the electron.
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By assigning various values to n, we can evaluate the corresponding energy of the electron in
the orbits of the hydrogen atom. When this is done, we find the energies of several orbits as
follows.
These energies can be used to prepare an energy level diagram such as that shown in last
figure. Note that the binding energy of the electron is lowest when n =1, and the binding energy is 0
when n = ȹ.
Although the Bohr model successfully accounted for the line spectrum of the hydrogen atom, it
could not explain the line spectrum of any other atom. It could be used to predict the wavelengths
of spectral lines of other species that had only one electron such as He+, Li2+ , Be3+, etc. Also, the
model was based on assumptions regarding the nature of the allowed orbits that had no basis in
classical physics. An additional problem is also encountered when the Heisenberg Uncertainty
Principle is considered. According to this principle, it is impossible to know exactly the position
and momentum of a particle simultaneously. Being able to describe an orbit of an electron in a
hydrogen atom is equivalent to knowing its momentum and position.
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Atomic orbitals
Bohr's model solved the problem of the energy levels in the hydrogen atom but had several
drawbacks: it could neither explain some of the other properties of hydrogen atoms nor correctly
predict the energy levels of more complex atoms. In addition, a few years later, new experiments
pointed out that particles could behave as waves, and therefore their position could not be
determined exactly. In Bohr's model, the radius of the first Bohr orbit in the hydrogen atom was
calculated to be exactly ao=0.52917 A. This distance is a constant called the Bohr radius and is
shown in Figure(a) as a spherical surface with radius a,.
(a) The precise spherical orbit of an electron in the first Bohr orbit, for which the radius is
a0=0.5291 7 A. as calculated by Bohr's model. (b) The electron probability density pattern for the
comparable atomic orbital using a quantum mechanical model. The darker areas indicate a higher
probability of finding the electron at that location. The center cutout shows the interior of the
orbital. The outer sphere delineates the region where the electron exists 90% of the time.
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A new approach was clearly needed in order to describe matter on the atomic scale. This new
approach was elaborated during the next decade and is now called quantum mechanics. In quantum
mechanics an electron cannot be visualized as a point particle orbiting with a definite radius, but
rather as a delocalized cloud with inhomogeneous probability density around a nucleus as
illustrated in previous figure. The probability density gives the probability of finding the electron at a
particular point in space. In this picture, the Bohr radius can be interpreted as the radius a. of the
spherical surface where the maximum in the electron probability distribution occurs or, in other
words, the spherical orbit where the electron is most likely to be found.
Niels Bohr first explained the atomic absorption and emission spectra in 1913. His reasoning was
based on the following assumptions, which cannot be justified within classical electrodynamics:
(1) Stable orbits (states with energy E,) exist for an electron in an atom. While in one of these orbits,
an electron does not emit any electromagnetic radiation. An individual electron can only exist in one
of these orbits at a time and thus has an energy .
(2) The transition of an electron from an atomic orbit of energy state En1 to that of energy state En2,
corresponds to the emission (En2 >En 1) or absorption (En2<En1) of electromagnetic radiation with an
energy. The Lyman series of spectroscopic lines is observed. The other series arise when the
electron drops from higher levels to the levels with n=2 (Balmer series) and n=3 (Paschen series), as
shown in figure.
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A typical example of the interaction between an electromagnetic field and matter is a blackbody,
which is an ideal radiator of electromagnetic radiation. Using classical arguments, Rayleigh and Jeans
tried to explain the observed blackbody spectral irradiance, which is the power radiated per unit area
per unit wavelength, shown in Figure. However, as can be seen in the figure, their theoretical
predictions could only fit the data at longer wavelengths. In addition, their results also indicated that
the total irradiated energy (integral of the irradiance over all the possible (wavelengths) should
be infinite, a fact that was in clear contradiction with experiment.
In 1901, Max Planck provided a revolutionary explanation based on the hypothesis that the
interaction between atoms and the electromagnetic field could only occur in discrete packets of
energy, thus showing that the classical view that always allows a continuum of energies was incorrect.
Based on these ideas, a more sophisticated and self-consistent theory was created in 1920 and is
now called quantum mechanics.
The Probability Amplitude
In quantum mechanics, a probability amplitude is a complex number used in describing the
behavior of systems. The modulus squared of this quantity represents a probability or probability
density. Probability amplitudes provide a relationship between the wave function (or, more generally,
of a quantum state vector) of a system and the results of observations of that system, a link first
proposed by Max Born. In fact, the properties of the space of wave functions were being used to make
physical predictions (such as emissions from atoms being at certain discrete energies) before any
physical interpretation of a particular function was offered.
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uncertainty relation was derived:
Δ x Δ p > h/4π
uncertainty in position
where x denotes the position and p represents the momentum of the moving particle The more
accurately you know the position (i.e., the smaller Δx is), the less accurately you know the momentum
(i.e., the larger Δp is); and vice versa. The wavelength of a particle depends on its momentum, just
like a photon! The main difference is that matter particles have mass, and photons dont !
Copenhagen Interpretation of Quantum Mechanics
• A system is completely described by a wave function ψ, representing an observer's subjective
knowledge of the system.
• The description of nature is essentially probabilistic, with the probability of an event related to the
square of the amplitude of the wave function related to it.
• It is not possible to know the value of all the properties of the system at the same time; those
properties that are not known with precision must be described by probabilities. (Heisenberg's
uncertainty principle)
• Matter exhibits a wave–particle duality. An experiment can show the particle like properties of
matter, or the wave-like properties; in some experiments both of these complementary viewpoints
must be invoked to explain the results.
• Measuring devices are essentially classical devices, and measure only classical properties such as
position and momentum.
• The quantum mechanical description of large systems will closely approximate the classical
description.
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Schrödinger (1926) proposed an equation that in principle allows the calculation of the amplitude,
ψ, of the wave associated with the particle as a function of position (x, y, z). This time-independent
Schrödinger (wave) equation is used for describing the properties of systems in stationary states:
the time dependency of the wave amplitude is not considered and one thereby does not study
changes in the atomic state during a transition (e.g. electron excitation) but concentrates on the
states before and after the transition. This equation is a second-order partial differential equation.
Exact solutions have been obtained only for one-electron systems. The realm of quantum mechanics
is thus mainly concerned with developing approximate methods for carrying out approximate
calculations for many (more than one) particle (electron) systems.
The form of the Schrödinger equation depends on the physical situation (see below for special
cases). The most general form is the time dependent Schrödinger equation, which gives a
description of a system evolving with time:
where i is the imaginary unit , ħ is the Plank constant divided by 2π (which is known as
the reduced Planck constant), the symbol ∂/∂t indicates a partial derivative with respect to
time t, Ψ (the Greek letter Psi) is the wave function of the quantum system, and Ĥ is the Hamilton
operator (which characterizes the total energy of any given wave function and takes different forms
depending on the situation). As before, the most famous manifestation is the non – relativistic
Schrödinger equation for a single particle moving in an electric field
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Time-independent Schrödinger equation
where μ is the particle's " reduced mass", V is its potential energy, ∇2 is the Laplacian, and Ψ is the
wave function (more precisely, in this context, it is called the "position-space wave function"). In plain
language, it means "total energy equals kinetic energy plus potential energy"
The amplitude of the wave function has no physical meaning: there is no undulating medium
through which the wave propagates. So, for example, we cannot speak of the amplitude of the wave
as a measure of displacement experienced by that medium, as compared to an average level (as is
possible for a wave propagating through a liquid, where the displacement of the surface of the liquid
at a certain location is given by the local amplitude of the wave). The square of the wave amplitude at
the position (x, y, z) does have a physical meaning: it represents the probability density for finding
the particle concerned in the volume element xyz at the position (x, y, z). This can be compared with
the calculation of the intensity of light which is given by the square of the amplitude of the
electromagnetic wave that represents the light (there is also no undulating medium (“ether”) through
which the light wave propagates. Against this background the amplitude of the wave function is also
called “probability amplitude”. According to Heisenberg’s uncertainty principle, it is impossible to
know the electron’s velocity and its position simultaneously. The exact position of the electron at any
given time cannot be known. Therefore, it is impossible to obtain a photographic picture of the atom
like we could of a busy street.
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Electrons are more like fast-moving mosquitoes in a swarm that cannot be photographed
without appearing blurred. The uncertainty about their position persists even in the photograph.
An alternative picture of the swarm can be obtained by describing the area where the mosquitoes
tend to be concentrated and the factors that determine their preference for certain locations, and
that’s the best we can do.
The quantum numbers provide us with a picture of the electronic arrangement in the atom
relative to the nucleus. This arrangement is not given in terms of exact positions, like the
photograph of a street, but rather in terms of probability distributions and potential energy levels,
much like the mosquito swarm. The potential energy levels are described by the main quantum
number n and by the secondary quantum number l. The probability distributions are given by the
secondary quantum number l and by the magnetic quantum number ml.
By solving the Schrödinger equation (Hy = Ey), we obtain a set of mathematical equations,
called wave functions (y), which describe the probability of finding electrons at certain energy
levels within an atom. A wave function for an electron in an atom is called an atomic orbital; this
atomic orbital describes a region of space in which there is a high probability of finding the
electron. Energy changes within an atom are the result of an electron changing from a wave
pattern with one energy to a wave pattern with a different energy (usually accompanied by the
absorption or emission of a photon of light). Each electron in an atom is described by four
different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and
the fourth (ms) specifies how many electrons can occupy that orbital.
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In the case of the electrons in the atom, those at lower levels of potential energy (lower shells,
or lower n) are more stable and less easily disrupted than those at higher levels of potential energy.
Chemical reactions are fundamentally electron transfers between atoms. In a chemical reaction, it is
the electrons in the outermost shell that react, that is to say, get transferred from one atom to
another. That’s because they are the most easily disrupted, or the most available for reactions. The
outermost shell is the marketplace where all electron trade takes place. Accordingly, it has a special
name. It is called the valence shell. Now, the solar system model of the atom is outmoded because
it does not accurately depict the electronic distribution in the atom. Electrons do not revolve around
the nucleus following elliptical, planar paths. They reside in 3-D regions of space of various shapes
called orbitals.
An orbital is a region in 3-D space where there is a high probability of finding the electron.
An orbital is, so to speak, a house where the electron resides. Only two electrons can occupy
an orbital, and they must do so with opposite spin quantum numbers ms . In other words, they must
be paired. The type and shape of orbital is given by the secondary quantum number l. As we know,
this number has values that depend on n such that l = 0, 1, ... n-1. Furthermore, orbitals are not
referred to by their numerical l values, but rather by small case letters associated with those values.
Thus, when l = 0 we talk about s orbitals. When l = 1 we talk about p orbitals. When l = 2 we talk
about d orbitals, and so on. In organic chemistry, we are mostly concerned with the elements of the
second row and therefore will seldom refer to l values greater than 1. We’ll be talking mostly about
s and p orbitals, and occasionally about d orbitals in reference to third row elements.
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The potential energy
nucleus
+
n =1
increases as the distance
from the nucleus increases
n =2
n =3
n=4
THE RELATIONSHIP BETWEEN POTENTIAL ENERGY AND STABILITY IS INVERSE
As the potential energy of a system increases, the system’s stability is more easily disrupted.
As an example consider the objects on the earth. Objects that are positioned at ground level have
lower potential energy than objects placed at high altitudes. The object that’s placed at high altitude,
be it a plane or a rock at the top of a mountain, has a higher “potential” to fall (lower stability) than
the object that’s placed at ground level. Systems tend towards lower levels of potential energy, thus
the tendency of the plane or the rock to fall. Conversely, an object placed in a hole on the ground
does not have a tendency to “climb out” because its potential energy is even lower than the object
placed at ground level. Systems do not naturally tend towards states of higher potential energy.
Another way of saying the same thing is to say that systems tend towards states of higher stability.
Energy levels of electrons in an atom are quantized (experimental evidence from spectroscopy).
We use equations derived from quantum mechanics to describe both the energy of an electron, and
the probability of finding that electron in a region of space.
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Heisenbergʼs Uncertainty Principle tells us that we canʼt know both the energy and the position of
an electron Particles at the atomic-molecular level have wave-like properties (De Broglie). We call
these regions of high probability for finding electrons and each orbital can be described by a set of
quantum numbers - that are derived from quantum mechanical Calculations. There are four types
of quantum numbers n, l, ml, and ms.
They allow us to understand the arrangement of electrons in atoms and the arrangement of the
periodic table. Important consequences. (why do we need to know this?)
Understanding the idea that electrons can be described by orbitals of different shapes and
definite energies – allows us to understand how elements bond and react, and the arrangement of
the periodic table.
Some questions to be answered
1 – Niels Bohr is credited with what scientific contribution?
A) The discovery of the electron, B)The discovery of the photon, C)The observation that matter
emits light , D)The discovery of the equation E = hv, E)The theory that electrons in atoms are
arranged in shells with discrete energies
2 – Using Bohr's equation for the energy levels of the electron in the hydrogen atom, determine the
energy (J) of an electron in the n = 4 level.
A) -5.45 x 10-19
B) -1.84 x 10-29
D) +1.84 x 10-29
E) -7.34 x 1018
3 – How many protons, neutrons and electrons are in
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54
C) -1.36 x 10-19
Cr ?
4 – What the A. N. of an element had the following quantum number: n=3, l=2, ml= - 2 , ms=+1/2
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THE FOUR QUANTUM NUMBERS
The quantum numbers are parameters that describe the distribution of electrons in the atom, and
therefore its fundamental nature. They are:
1 – PRINCIPAL QUANTUM NUMBER (n) - Represents the main energy level, or shell, occupied by an
electron. It is always a positive integer, that is n = 1, 2, 3 ...
2 – SECONDARY QUANTUM NUMBER (l ) - Represents the energy sublevel, or type of orbital,
occupied by the electron. The value of l depends on the value of n such that l = 0, 1, ... n-1. This
number is sometimes also called azimuthal, or subsidiary.
3 – MAGNETIC QUANTUM NUMBER (ml ) - Represents the number of possible orientations in 3-D
space for each type of orbital. Since the type of orbital is determined by l, the value of ml ranges
between -l and +l such that ml = -l, ...0, ...+l.
4 – SPIN QUANTUM NUMBER (mS ) - Represents the two possible orientations that an electron can
have in the presence of a magnetic field, or in relation to another electron occupying the same
orbital. Only two electrons can occupy the same orbital, and they must have opposite spins. When
this happens, the electrons are said to be paired. The allowed values for the spin quantum number
ms are +1/2 and -1/2.
Since the value of l depends on the value of n, only certain types of orbitals are possible for
each n, as follows (only the highest energy level is shown for each row of elements):
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FIRST ROW ELEMENTS: n = 1
l=0
SECOND ROW ELEMENTS: n = 2
only s orbitals are possible, denoted as 1s orbitals.
l=0
s orbitals are possible, denoted as 2s orbitals, l = 1 p
orbitals are possible, denoted as 2p orbitals.
THIRD ROW ELEMENTS: n = 3
l=0
s orbitals are possible, denoted as 3s orbitals, l = 1 p
orbitals are possible, denoted as 3p orbitals, l = 2
and d orbitals are possible, denoted as 3d
orbitals.
The shapes associated with s and p orbitals are shown below. The red dot represents
the nucleus
p orbital
s orbital
Spherical, or
Dumbbell, or
Finally, the orientations of each orbital in 3-D space are given by the magnetic quantum number ml.
This number depends on the value of l such that ml = -l, ...0, ...+l. Thus, when l = 0, ml = 0. There is only
one value, or only one possible orientation in 3-D space for s-orbitals. That stands to reason, since they
are spherical. In the case of p-orbitals l = 1, so ml = -1, 0, and +1. Therefore, there are three possible
orientations in 3-D space for p-orbitals, namely along the x, y, and z axes of the Cartesian coordinate
system. More specifically, those orbitals are designated as px, py, and pz respectively.
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The distribution of electrons among the orbitals of an atom is called the electron configuration.
The electrons are filled in according to a scheme known as the Aufbau principle (“building-up”),
which corresponds (for the most part) to increasing energy of the subshells:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f
It is not necessary to memorize this listing, because the order in which the electrons are filled in can
be read from the periodic table in the following fashion:
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
5s
5p
5d
5f
6s
6p
6d
6f
7s
7p
7d
7p
Another way to indicate the placement of electrons is an orbital diagram, in which each orbital is
represented by a square (or circle), and the electrons as arrows pointing up or down (indicating the
electron spin). When electrons are placed in a set of orbitals of equal energy, they are spread out as
much as possible to give as few paired electrons as possible (Hund’s rule). In a ground state
configuration, all of the electrons are in as low an energy level as it is possible for them to be. When an
electron absorbs energy, it occupies a higher energy orbital, and is said to be in an excited state.
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