The molecular structure hypothesis - that a molecule is a
collection of atoms linked by a network of bonds - was forged in the
crucible of nineteenth century experimental chemistry. It has
continued to serve as the principal means of ordering and classifying
the observations of chemistry. The difficulty with this hypothesis was
that it was not related directly to quantum mechanics, the physics
which governs the motions of the nuclei and electrons that make up
the atoms and the bonds. Indeed there was, and with some there
still is, a prevailing opinion that these fundamental concepts, while
unquestionably useful, were beyond theoretical definition. We have
in chemistry an understanding based on a classification scheme that
is both powerful and at the same time, because of its empirical
SYMBOLS AND FORMULAS
• A unique symbol is used to represent each element.
• Formulas are used to represent compounds.
• A symbol is assigned to each element. The symbol is
based on the name of the element and consists of one
capital letter or a capital letter followed by a lower case
• Some symbols are based on the Latin or German name of
• A compound formula consists of the symbols of the
elements found in the compound. Each elemental symbol
represents one atom of the element. If more than one
atom is represented, a subscript following the elemental
symbol is used.
EXAMPLES OF COMPOUND FORMULAS
• Carbon monoxide, CO (one atom of C and one atom of O
• Water, H2O (two atoms of H and one atom of O are
• Ammonia, NH3 (one atom of N and 3 atoms of H are
THE STRUCTURE OF ATOMS
• Atoms are made up of three subatomic particles, protons,
neutrons, and electrons.
• The protons and neutrons are tightly bound together to
form the central portion of an atom called the nucleus.
• The electrons are located outside of the nucleus and
thought to move very rapidly throughout a relatively large
volume of space surrounding the small but very heavy
• Protons are located in the nucleus of an atom. They carry
a +1 electrical charge and have a mass of 1 atomic mass
• Neutrons are located in the nucleus of an atom. They
carry no electrical charge and have a mass of 1 atomic
mass unit (u).
• Electrons are located outside the nucleus of an atom.
They carry a -1 electrical charge and have a mass of
1/1836 atomic mass unit (u). They move rapidly around
the heavy nucleus.
ATOMIC NUMBER OF AN ATOM
• The atomic number of an atom is equal to the number of
protons in the nucleus of the atom.
• Atomic numbers are represented by the symbol Z.
MASS NUMBER OF AN ATOM
• The mass number of an atom is equal to the sum of the
number of protons and neutrons in the nucleus of the
• Mass numbers are represented by the symbol A.
• Isotopes are atoms that have the same number of protons
in the nucleus but different numbers of neutrons. That is,
they have the same atomic number but different mass
• Because they have the same number of protons in the
nucleus, all isotopes of the same element have the same
number of electrons outside the nucleus.
SYMBOLS FOR ISOTOPES
• Isotopes are represented by the symbol
, where Z is
the atomic number, A is the mass number and E is the
• An example of an isotope symbol is 28 Ni . This symbol
represents an isotope of nickel that contains 28 protons
and 32 neutrons in the nucleus.
• Isotopes are also represented by the notation: Name-A,
where Name is the name of the element and A is the mass
number of the isotope.
• An example of this isotope notation is magnesium-26.
This represents an isotope of magnesium that has a mass
number of 26.
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• The extremely small size of atoms and molecules
makes it inconvenient to use their actual masses
for measurements or calculations. Relative
masses are used instead.
• Relative masses are comparisons of actual
masses to each other. For example if an object
had twice the mass of another object, their relative
masses would be 2 to 1.
ATOMIC MASS UNIT (u)
• An atomic mass unit is a unit used to express the
relative masses of atoms. One atomic mass unit
is equal to 1/12 the mass of a carbon-12 atom.
• A carbon-12 atom has a relative mass of 12 u.
• An atom with a mass equal to 1/12 the mass of a
carbon-12 atom would have a relative mass of
• An atom with a mass equal to twice the mass of a
carbon-12 atom would have a relative mass of
• The atomic weight of an element is the relative mass of an
average atom of the element expressed in atomic mass
• Atomic weights are the numbers given at the bottom of the
box containing the symbol of each element in the periodic
• According to the periodic table, the atomic weight of
nitrogen atoms (N) is 14.0 u, and that of silicon atoms (Si)
is 28.1 u. This means that silicon atoms are very close to
twice as massive as nitrogen atoms. Put another way, it
means that two nitrogen atoms have a total mass very
close to the mass of a single silicon atom.
• The relative mass of a molecule in atomic mass units is
called the molecular weight of the molecule.
• Because molecules are made up of atoms, the molecular
weight of a molecule is obtained by adding together the
atomic weights of all the atoms in the molecule.
• The formula for a molecule of water is H2O. This means
one molecule of water contains two atoms of hydrogen, H,
and one atom of oxygen, O. The molecular weight of
water is then the sum of two atomic weights of H and one
atomic weight of O:
MW = 2(at. wt. H) + 1(at. wt. O)
MW = 2(1.01 u) + 1(16.0 u) = 18.02 u
ISOTOPES AND ATOMIC WEIGHTS
• Many elements occur naturally as a mixture of several
• The atomic weight of elements that occur as mixtures of
isotopes is the average mass of the atoms in the isotope
• The average mass of a group of atoms is obtained by
dividing the total mass of the group by the number of
atoms in the group.
• A practical way of determining the average mass of a
group of isotopes is to assume the group consists of 100
atoms and use the percentage of each isotope to
represent the number of atoms of each isotope present in
• The use of percentages and the mass of each isotope
leads to the following equation for calculating atomic
weights of elements that occur naturally as a mixture of
According to this equation, the atomic weight of an element is
calculated by multiplying the percentage of each isotope in the
element by the mass of the isotope, then adding the resulting
products together and dividing the resulting mass by 100.
• A specific example of the use of the equation is shown
below for the element boron that consists of 19.78%
boron-10 with a mass of 10.01 u, and 80.22% boron-11
with a mass of 11.01u.
This calculated value is seen to agree with the value given in
the periodic table.
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THE MOLE CONCEPT APPLIED TO ELEMENTS
• The number of atoms in one mole of any element is called
Avogadro's number and is equal to 6.022x1023 .
• A one-mole sample of any element will contain the same
number of atoms as a one-mole sample of any other
• One mole of any element is a sample of the element with a
mass in grams that is numerically equal to the atomic
weight of the element.
EXAMPLES OF THE MOLE CONCEPT
• 1 mole Na = 22.99 g Na = 6.022x1023 Na atoms
• 1 mole Ca = 40.08 g Ca = 6.022x1023 Ca atoms
• 1 mole S = 32.06 g S = 6.022x1023 S atoms
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THE MOLE CONCEPT APPLIED TO COMPOUNDS
• The number of molecules in one mole of any compound is
called Avogadro's number and is numerically equal to
• A one-mole sample of any compound will contain the
same number of molecules as a one-mole sample of any
• One mole of any compound is a sample of the compound
with a mass in grams equal to the molecular weight of the
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EXAMPLES OF THE MOLE CONCEPT
• 1 mole H2O = 18.02 g H2O = 6.022x1023 H2O molecules
• 1 mole CO2 = 44.01 g CO2 = 6.022x1023 CO2 molecules
• 1 mole NH3 = 17.03 g NH3 = 6.022x1023 NH3 molecules
THE MOLE AND CHEMICAL CALCULATIONS
• The mole concept can be used to obtain factors that are
useful in chemical calculations involving both elements and
CALCULATIONS INVOLVING ELEMENTS
• The mole-based relationships given earlier as examples
for elements provide factors for solving problems.
• The relationships given earlier for calcium are:
1 mole Ca= 40.08 g Ca = 6.022x1023 Ca atoms.
• Any two of these quantities can be used to provide factors
for use in solving numerical problems.
• Examples of two of the six possible factors are:
• Calculate the number of moles of Ca contained in
a 15.84 g sample of Ca.
• The solution to the problem is:
• We see in the solution that the g Ca units in the
denominator of the factor cancel the g Ca units in
the given quantity, leaving the correct units of
mole Ca for the answer.
CALCULATIONS INVOLVING COMPOUNDS
• The mole concept applied earlier to molecules can be
applied to the individual atoms that are contained in the
• An example of this for the compound CO2 is:
1 mole CO2 molecules = 1 mole C atoms + 2 moles O atoms
44.01 g CO2 = 12.01 g C + 32.00 g O
6.022x1023 CO2 molecules = 6.022x1023 C atoms +
(2) 6.022x1023 O atoms.
• Any two of these nine quantities can be used to provide
factors for use in solving numerical problems.
• Example 1: How many moles of O atoms are contained in
11.57 g of CO2?
• Note that the factor used was obtained from two of the
nine quantities given on the previous slide.
• Example 2: How many CO2 molecules are needed to
contain 50.00 g of C?
• Note that the factor used was obtained from two of the
nine quantities given on a previous slide.
• Example 3: What is the mass percentage of C in CO 2?
• The mass percentage is calculated using the
• If a sample consisting of 1 mole of CO2 is used, the molebased relationships given earlier show that
1 mole CO2=44.01 g CO2=12.01 g C + 32.00 g O. Thus,
the mass of C in a specific mass of CO2 is known, and the
problem is solved as follows:
• Example 4: What is the mass percentage of oxygen in
• The mass percentage is calculated using the following
• Once again, a sample consisting of 1 mole of CO2 is used
to take advantage of the mole-based relationships given
1 mole CO2 = 44.01g CO2 = 12.01 g C + 32.00g O
• Thus, the mass of O in a specific mass of CO2 is
known, and the problem is solved as follows:
• We see that the % C + % O = 100% , which
should be the case because C and O are the only
elements present in CO2.
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The theory recovers the central operational concepts of the
molecular structure hypothesis, that of a functional grouping of atoms
with an additive and characteristic set of properties, together with a
definition of the bonds that link the atoms and impart the structure. Not
only does the theory thereby quantify and provide the physical
understanding of the existing concepts of chemistry, it makes possible
new applications of theory. These new applications will eventually enable
one to perform on a computer, in a manner directly paralleling
experiment, everything that can now be done in the laboratory, but more
quickly and more efficiently, by linking together the functional groups of
theory. These applications include the design and synthesis of new
molecules and new materials with specific desirable properties.
The theory of atoms in molecules enables one to take advantage of
the single most important observation of chemistry, that of a functional
group with a characteristic set of properties. This document outlines and
illustrates the topological basis of the theory and its relation to the
quantum mechanics of an open system.
What is an Atom?
Matter is composed of atoms. This is a consequence of the
manner in which the electrons are distributed throughout space in
the attractive field exerted by the nuclei. The nuclei act as point
attractors immersed in a cloud of negative charge, the electron
density (r). The electron density describes the manner in which the
electronic charge is distributed throughout real space. The electron
density is a measurable property and it determines the appearance
and form of matter. This is illustrated in the following figures. Figure
1 displays the spatial distribution of the electron density in the plane
containing the two carbon and four hydrogen nuclei of the ethane
molecule. The electron density is a maximum at the position of each
nucleus and decays rapidly away from these positions. When this
diagram is translated into three dimensions, the cloud of negative
charge is seen to be most dense at nuclear positions and to become
more diffuse as one moves away from these centers of attraction,
as illustrated in Figure 2. The presence of local maxima at the
positions of the nuclei is the general and also the dominant
topological property of (r). Figure 3 illustrates the same feature for
the 110 plane of carbon nuclei in the diamond lattice.
To determine what physics has to say about this property of the
electron density one must consider not the density itself but the field
one obtains by following the trajectories traced out by the gradient
vectors of the density. Starting at any point, one determines the
gradient of (r). This is a vector that points in the direction of
maximum increase in the density. One makes an infinitesimal step
in this direction and then recalculates the gradient to obtain the new
direction. By continued repetition of this process, one traces out a
trajectory of (r). A gradient vector map generated in this manner is
illustrated in the upper diagram of Figure 4 for the same plane of the
ethane molecule shown in Figure 1. Since the density exhibits a
maximum at the position of each nucleus, sets of trajectories
terminate at each nucleus. The nuclei are the attractors of the
gradient vector field of the electron density. Because of this
fundamental property, the space of the molecule is disjoint and
exhaustively partitioned into basins, a basin being the region of
space traversed by the trajectories terminating at a given nucleus or
attractor. Since a single attractor is associated with each basin,
an atom is defined as the union of an attractor and its basin.
ATOMS AND MOLECULES
• One of the most powerful developments in the history of
science is the atomic/molecular model of matter, which can be
used to explain and predict a large variety of phenomena.
Ideas about elements, atoms, and their combination in
molecules or large arrays develop from two notions: that
matter is made of invisibly tiny pieces and that the enormous
variety of materials in the world is the result of different
combinations of a relatively small number of basic ingredients.
• In high school, these ideas extend to benchmarks about the
relation of a material's properties to its atomic or molecular
make-up, the structure of the atom itself, and the existence of
isotopes and radioactivity. The sections Understanding
Fire and Splitting the Atom in Science for All Americans and
Benchmarks Chapter 10: HISTORICAL PERSPECTIVES
could provide context in instruction for the study of
The atoms in molecules or atoms-in-molecules or quantum
theory of atoms in molecules (Qtaim) approach is a quantum
chemical model that characterizes the chemical bonding of a
system based on the topology of the quantum charge density. In
addition to bonding, AIM allows the calculation of certain physical
properties on a per-atom basis, by dividing space up into atomic
volumes containing exactly one nucleus. Developed by Professor
Richard Bader since the early 1960s, during the past decades
QTAIM has gradually become a theory for addressing possible
questions regarding chemical systems, in a variety of situations
hardly handled before by any other model or theory in Chemistry .
In QTAIM an atom is defined as a proper open system, i.e. a
system that can share energyand electron density, which is
localized in the 3D space. Each atom acts as a local attractor of the
electron density, and therefore it can be defined in terms of the
local curvatures of the electron density. The mathematical study of
these features is usually referred in the literature as charge
density topology. Nevertheless, the term topology is used in a
different sense in Mathematics
In chemistry and physics, atomic theory is a theory of the nature of matter,
which states that matter is composed of discrete units called atoms, as opposed
to the obsolete notion that matter could be divided into any arbitrarily small
quantity. It began as a philosophical concept in ancient Greece and India and
entered the scientific mainstream in the early 19th century when discoveries in
the field of chemistry showed that matter did indeed behave as if it were made
up of particles.
The word "atom" (from the ancient Greek adjective atoms', 'undivisible') was
applied to the basic particle that constituted a chemical element, because the
chemists of the era believed that these were the fundamental particles of matter.
However, around the turn of the 20th century, through various experiments
with electromagnetism and radioactivity, physicists discovered that the so-called
"indivisible atom" was actually a conglomerate of various subatomic particles (
chiefly, electrons, protons and neutrons) which can exist separately from each
other. In fact, in certain extreme environments such as neutron stars, extreme
temperature and pressure prevents atoms from existing at all. Since atoms were
found to be actually divisible, physicists later invented the term "
elementary particles" to describe indivisible particles. The field of science which
studies subatomic particles is particle physics, and it is in this field that
physicists hope to discover the true fundamental nature of matter.
Earliest empirical evidence
Near the end of the 18th century, two laws about chemical reactions
emerged without referring to the notion of an atomic theory. The first was
the law of conservation of mass, formulated by Antoine Lavoisier in 1789,
which states that the total mass in a chemical reaction remains constant
(that is, the reactants have the same mass as the products). The second
was the law of definite proportions. First proven by the French chemist
Joseph Louis Proust in 1799, this law states that if a compound is broken
down into its constituent elements, then the masses of the constituents will
always have the same proportions, regardless of the quantity or source of
the original substance.
John Dalton studied and expanded upon this previous work and developed
the law of multiple proportions: if two elements can together form more than
one compound, then the ratios of the masses of the second element which
combine with a fixed mass of the first element will be ratios of small integers
. For instance, Proust had studied tin oxides and found that their masses
were either 88.1% tin and 11.9% oxygen or 78.7% tin and 21.3% oxygen
(these were tin(II) oxide and tin dioxide respectively). Dalton noted from
these percentages that 100g of tin will combine either with 13.5g or 27g of
oxygen; 13.5 and 27 form a ratio of 1:2. Dalton found an atomic theory of
matter could elegantly explain this common pattern in chemistry - in the
case of Proust's tin oxides, one tin atom will combine with either one or two
Dalton also believed atomic theory could explain why water
absorbed different gases in different proportions: for example, he
found that water absorbed carbon dioxide far better than it
absorbed nitrogen. Dalton hypothesized this was due to the
differences in mass and complexity of the gases' respective
particles. Indeed, carbon dioxide molecules (CO2) are heavier and
larger than nitrogen molecules (N2).
Dalton proposed that each chemical element is composed of atoms
of a single, unique type, and though they cannot be altered or
destroyed by chemical means, they can combine to form more
complex structures (chemical compounds). This marked the first
truly scientific theory of the atom, since Dalton reached his
conclusions by experimentation and examination of the results in an
Various atoms and molecules as depicted in John Dalton's A New
System of Chemical Philosophy (1808).
In 1803 Dalton orally presented his first list of relative atomic
weights for a number of substances. This paper was published in
1805, but he did not discuss there exactly how he obtained these
figures. The method was first revealed in 1807 by his acquaintance
Thomas Thomson, in the third edition of Thomson's textbook, A
System of Chemistry. Finally, Dalton published a full account in his
own textbook, A New System of Chemical Philosophy, 1808 and
Dalton estimated the atomic weights according to the mass ratios in which they
combined, with hydrogen being the basic unit. However, Dalton did not conceive
that with some elements atoms exist in molecules — e.g. pure oxygen exists as
O2. He also mistakenly believed that the simplest compound between any two
elements is always one atom of each (so he thought water was HO, not H 2O).
This, in addition to the crudity of his equipment, resulted in his table being
highly flawed. For instance, he believed oxygen atoms were 5.5 times heavier
than hydrogen atoms, because in water he measured 5.5 grams of oxygen for
every 1 gram of hydrogen and believed the formula for water was HO (an
oxygen atom is actually 16 times heavier than a hydrogen atom).
The flaw in Dalton's theory was corrected in 1811 by Amedeo Avogadro.
Avogadro had proposed that equal volumes of any two gases, at equal
temperature and pressure, contain equal numbers of molecules (in other words,
the mass of a gas's particles does not affect its volume). Avogadro's law
allowed him to deduce the diatomic nature of numerous gases by studying the
volumes at which they reacted. For instance: since two liters of hydrogen will
react with just one liter of oxygen to produce two liters of water vapor (at
constant pressure and temperature), it meant a single oxygen molecule splits in
two in order to form two particles of water. Thus, Avogadro was able to offer
more accurate estimates of the atomic mass of oxygen and various other
elements, and firmly established the distinction between molecules and atoms.
In 1827, the British botanist Robert Brown observed that pollen particles floating
in water constantly jiggled about for no apparent reason. In 1905,
Albert Einstein theorized that this Brownian motion was caused by the water
molecules continuously knocking the grains about, and developed a hypothetical
mathematical model to describe it. This model was validated experimentally in
1908 by French physicist Jean Perrin, thus providing additional validation for
particle theory (and by extension atomic theory).
Discovery of subatomic particles
Atoms were thought to be the smallest possible division of matter until 1897
when J.J. Thomson discovered the electron through his work on
cathode rays. A Crookes tube is a sealed glass container in which two
electrodes are separated by a vacuum. When a voltage is applied across
the electrodes, cathode rays are generated, creating a glowing patch where
they strike the glass at the opposite end of the tube. Through
experimentation, Thomson discovered that the rays could be deflected by
an electric field (in addition to magnetic fields, which was already known).
He concluded that these rays, rather than being a form of light, were
composed of very light negatively charged particles he called "corpuscles"
(they would later be renamed electrons by other scientists).
Thomson believed that the corpuscles emerged from the molecules of gas
around the cathode. He thus concluded that atoms were divisible, and that
the corpuscles were their building blocks. To explain the overall neutral
charge of the atom, he proposed that the corpuscles were distributed in a
uniform sea of positive charge; this was the plum pudding model as the
electrons were embedded in the positive charge like plums in a plum
pudding (although in Thomson's model they were not stationary)
First steps toward a quantum physical
model of the atom
The planetary model of the atom had two significant shortcomings. The first is that,
unlike planets orbiting a sun, electrons are charged particles. An accelerating
electric charge is known to emit electromagnetic waves according to the
Larmor formula in classical electromagnetism; an orbiting charge should steadily lose
energy and spiral toward the nucleus, colliding with it in a small fraction of a second.
The second problem was that the planetary model could not explain the highly
peaked emission and absorption spectra of atoms that were observed.
The Bohr model of the atom
Quantum theory revolutionized physics at the beginning of the 20th century, when
Max Planck and Albert Einstein postulated that light energy is emitted or absorbed in
discrete amounts known as quanta (singular, quantum). In 1913, Niels Bohr
incorporated this idea into his Bohr model of the atom, in which an electron could
only orbit the nucleus in particular circular orbits with fixed angular momentum and
energy, its distance from the nucleus (i.e., their radii) being proportional to its energy.
Under this model an electron could not spiral into the nucleus because it could not
lose energy in a continuous manner; instead, it could only make instantaneous "
quantum leaps" between the fixedenergy levels. When this occurred, light was
emitted or absorbed at a frequency proportional to the change in energy (hence the
absorption and emission of light in discrete spectra).
Bohr's model was not perfect. It could only predict the spectral lines of hydrogen; it
couldn't predict those of multielectron atoms. Worse still, as spectrographic
technology improved, additional spectral lines in hydrogen were observed which
Bohr's model couldn't explain. In 1916, Arnold Sommerfeld added elliptical orbits to
the Bohr model to explain the extra emission lines, but this made the model very
difficult to use, and it still couldn't explain more complex atoms.
Discovery of isotopes
• While experimenting with the products of radioactive decay, in
1913 radio chemist Frederick Soddy discovered that there
appeared to be more than one element at each position on
the periodic table. The term isotope was coined by Margaret
Todd as a suitable name for these elements.
• That same year, J.J. Thomson conducted an experiment in
which he channeled a stream of neon ions through magnetic
and electric fields, striking a photographic plate at the other
end. He observed two glowing patches on the plate, which
suggested two different deflection trajectories. Thomson
concluded this was because some of the neon ions had a
different mass. The nature of this differing mass would later
be explained by the discovery of neutrons in 1940.
Quantum physical models of the
n 1924, Louis de Broglie proposed that all moving particles — particularly
subatomic particles such as electrons — exhibit a degree of wave-like
behavior. Erwin Schrödinger, fascinated by this idea, explored whether or
not the movement of an electron in an atom could be better explained as a
wave rather than as a particle. Schrödinger's equation, published in 1926,
describes an electron as a wavefunction instead of as a point particle.
This approach elegantly predicted many of the spectral phenomena that
Bohr's model failed to explain. Although this concept was mathematically
convenient, it was difficult to visualize, and faced opposition. One of its
critics, Max Born, proposed instead that Schrödinger's wave function
described not the electron but rather all its possible states, and thus could
be used to calculate the probability of finding an electron at any given
location around the nucleus.This reconciled the two opposing theories of
particle versus wave electrons and the idea of wave-particle duality was
introduced. This theory stated that the electron may exhibit the properties of
both a wave and a particle. For example, it can be refracted like a wave, and
has mass like a particle.
• A consequence of describing electrons as
waveforms is that it is mathematically impossible to
simultaneously derive the position and momentum
of an electron; this became known as the
Heisenberg uncertainty principle after the theoretical
physicist Walter Heisenberg, who first described it.
This invalidated Bohr's model, with its neat, clearly
defined circular orbits. The modern model of the
atom describes the positions of electrons in an atom
in terms of probabilities. An electron can potentially
be found at any distance from the nucleus, but,
depending on its energy level, tends to exist more
frequently in certain regions around the nucleus
than others; this pattern is referred to as its atomic
orbital. The orbital's come in a variety of shapes,
manifesting from a simple sphere of the full helium
orbital, to the dumbbell shape of the full neon orbital,
with the nucleus in the middle.