Think about this. The visible universe consists
mostly of less than 100 naturally occurring
elements. Yet there is immense diversity and
complexity reflected in not only the myriad
of substances that can be found in the universe,
but also substances that can be found within
your own body.
How is it that we can have a near boundless
number of chemical substances with only about 100
relatively simple building blocks, called
atoms, to work with? This is because the
naturally occurring atoms are rarely found
alone.
They are mostly found in combinations with
other atoms, through the process of chemical
bonding. The combinations of chemicals and
material that can be built with these various
combinations is so vast, that it can give
rise not only to the variety of phenomena
that we can observe in the universe, but also
the complexity required to create life, fuel
for the energy we need, consciousness that
we enjoy, and every other macro scale process
that affects us?
Why is the universe
not full of just about 100 different types
of atoms floating around in a sea of dull
monotony?
Lucky us, that these atoms bond together to
form rich chemicals that have properties fundamentally
different than the atoms comprising them.
Why does this happen? The answer to this important
question lies in understanding the role that
energy plays in the formation of molecules,
and its roots in where else, but quantum mechanics.
That explanation is coming up right now…
Some of the most important chemicals to us
like the air we breathe consists not of individual
atoms, but molecules. Oxygen for example,
is not found in the air as individual oxygen
atoms, but as O2 or two oxygen atoms bound
together. Nitrogen is also found as N2, two
nitrogen atoms bound together. Water, without
which we could not survive, is H2O – two
atoms of hydrogen combined with one oxygen atom.
Why are atoms prevalent mostly as molecules?
The key to understanding why atoms link together
is energy. All natural systems tend to adopt
a state of lowest energy.
A marble at the top of a hill has high potential
energy due to gravity. If given the opportunity,
it will roll naturally to the bottom of the
hill where it will have a lower potential
energy. This is the same reason a river flows
in one direction, from high ground to lower
ground.
So how does energy play a role in the formation
of molecules? Let’s look at the simplest
atom, hydrogen, which consists of only one
proton and one electron. It is found on earth
usually not as individual atoms of hydrogen,
but in pairs, H2, or hydrogen gas.
By understanding the role of quantum mechanics
in the formation of the hydrogen gas molecule,
we can begin to understand why other atoms
also form molecules in order to adopt a lower
energy state.
A hydrogen atom consists of just one electron
and one proton. As we saw in an earlier video,
the electron forms a cloud around the nucleus.
The shape of this cloud is determined by the
Schrodinger equation, which contains a wave
function. The wave function of the hydrogen
atom, simply put, represents the probabilities
of the possible results if we were to measure
its properties.
The hydrogen atom by itself will be in its
lowest energy state, called the ground state.
But when a second hydrogen atom is introduced
into this system, some interesting things
begin to happen. Not much happens if the atoms
are very far apart. Both atoms are in their
respective ground state.
But as they come closer to each other, a few
things happen as the same time. First, the
electrons, since they are both negatively charged
repel each other. But the electron of hydrogen
atom 1 also starts to get affected by the
positive charge of the proton in hydrogen
atom 2. Similarly the electron of atom 2 starts
to get attracted to the proton of atom 1.
So the electrons of each of the 2 atoms tend
to get pulled slightly to the other one’s
proton. And if they get close enough, the
cloud begins to spread to the space between
the two atoms.
Now, if the atoms get too close, then the protons
begin to repel each other and push each other
apart. So there is a sweet spot distance that
the two protons prefer to be in, such that
the electrons are happier being shared, and
the protons are not too repelled to each other.
You might ask why do these two atoms get attracted
in the first place because shouldn’t the
electron clouds be repelling each other, and
not allow them to get anywhere near each other?
This is an excellent question. What you have
to understand is the repulsion of the electrons
is not the only interaction taking place here.
There are a multitude of interactions happening.
And what happens to the entire system is determined
by the total energy of the system.
To calculate the lowest energy of this two
atom system, or molecule of hydrogen, we have
to take into account the following: the kinetic
energy each atom, the potential energy between
the two protons, the potential energy between
the two electrons, and the potential energy
between each electron and each proton.
The sum of the possible outcomes of kinetic
and potential energy of this entire system
in quantum mechanics is referred to the Hamiltonian,
represented by capital H.
The Hamiltonian for our 2 atom hydrogen system
looks something like this, when you do all
the calculations.
To be clear, the Hamiltonian is an operator
corresponding to the energy of the system,
and once you plug it into the time-independent
Schrodinger equation, written here, you can
solve to get possible values for energy. Now,
as you might imagine, this is not a trivial
equation to solve.
But it can be represented for simplicity by
the following graph.
And as we move the from right to left, you
can see what happens to the energy when the
two atoms go from being far apart to being
closer together. The dip in the graph represents
the lowest energy state of the two atom system.
If the distance between the protons gets any
smaller than that, the energy goes up significantly.
And if the distance gets larger, it also goes
up, although not as steeply.
So the two atoms find a natural sweet spot
such that they are both happier being at a
lower energy state together than when they
were farther apart by themselves.
It turns out the energy of this two atom
system is less than the energy of two separate
one atom systems. This is the reason if a
bunch of hydrogen atoms are near each other,
they will naturally combine to form a molecule
of H2 rather than float around by themselves.
This sharing of electrons by two atoms of
hydrogen is called a covalent bond.
And similarly, covalent bonds formed by other
atoms work analogously.
Now, if you took high school chemistry, you
will know that not all atoms form bonds with
atoms of their own kind, nor with just any
other atom.
One of the most remarkable things about chemical
bonding of atoms is that all the substances
that you see all around you, comprised of
molecules is due to the remarkable stability
of atoms that have or share certain seemingly
magical numbers of electrons – 2, 10, 18,
36, 54, or 86 electrons in so called shells
around the nucleus of atoms.
And all this is due to the fact that certain
combinations of atoms that contain these numbers
of electrons tend to have the lowest potential
energy.
If you look at the periodic table, you will
see that these numbers correspond to the number
of electrons contained in the 6 naturally
occurring noble gases. These are inert elements,
meaning they do not react to form bonds with
other atoms under standard conditions. That’s
because they already contain the number of
electrons needed to form highly stable shells
around the nucleus.
Other elements strive to contain a full set
of electrons in their outer shell called the
valence shell. Any element with an unfilled
outer shell has a much higher chemical potential
energy than these noble gases.
So for example, if you look at the halogen
elements right next to the noble gases, fluorine,
Chlorine, etc. they are only one electron
away from being highly stable, so they are
desperate to attract one electron from any
atom that wants to get rid of one of their
electrons.
Also, if you look the alkali metals on the
far left side of the table, these are elements
that have one too many electrons, so they
are desperate to get rid of one electron.
Thus, when these alkali metals get together
with the halogen elements, guess what happens?
They form very strong bonds, and the resulting compounds have so
much lower energy that the excess energy is
released in the form of heat in an exothermic
reaction.
In fact, one very important compound results
from this type of reaction. It’s when Sodium
bonds with chlorine to form sodium chloride,
or ordinary table salt.
So atoms can either share electrons with one
or more other atoms, resulting in covalent
bonds, or they can give away or take on electrons
from other atoms resulting in ionic bonds.
There are other types of bonds as well, but
these are the only two we are discussing today.
You might ask, why are these numbers I talked
about earlier such magical numbers?
Chemists will say, well, these are the numbers
that allow atoms to fill their electron shells.
Or, in other words, all atoms strive to form
what’s called a full valence set of electrons.
And this attractive force for atoms to share
electrons in order to form a full valence
shell, is balanced by the repulsive forces
of their electron clouds and protons.
But what is the underlying reason? Why should
10 electrons have lower energy than 6 electrons
for example? Or why is the number 18 more
special than some arbitrary number like
16?
The answer lies, as many phenomena do, again
in quantum mechanics. It all has to do with
potential energy of multi atom systems. There
are two concepts in quantum mechanics that
are most important in determining the energy
of these systems. The first is the Schrodinger
equation, as I pointed out in our example of
energy related to the hydrogen molecule.
And the second is the Pauli exclusion principle.
This principle basically states that no two
fermions, (an electron is a fermion) can occupy
the same quantum state. For our purposes,
this means that in a molecule, two electrons
that are in the same orbital must have opposite
spins.
These two principles taken together, can reproduce
those exact sequence of numbers that I called
"magical." And solving the equations will show
that these are the precise numbers that result
in lowest potential energy of chemical systems.
So for example, if we wanted to look at water,
H2O, consisting of 2 hydrogen atoms and one
oxygen atom, a wave function can be written
for this structure. Energy can be calculated
by solving the equations. And bond lengths
can be determined that result in the lowest
potential energy.
But precise calculations get very mathematically
complex, so approximations called Morse potential,
represented by the equation below is usually
used. Plugging in the numbers, a graphical
representation of Morse potential for the
O-H bond in water can be shown as the following. Here U0 is the bond energy
and r0 is the bond length that can be read
in a tables like the one here.
So, as usual it all comes down to quantum mechanics.
But, what I want to impart to you is that the
quantum mechanics behind these phenomena only
allows you to see how nature behaves. It does
not explain why nature is the way that it
is. It only shows you how nature works.
The question of why things are the way 
they are, is what I think the most interesting
question. And there may be some underlying
reason, for example that lies in string theory,
or some other theory of everything.
And that’s what I hope the next generation
of scientists will keep pursuing.
Now, if you are such a scientist, whether professional,
or hobbyist who wants to pursue profound ideas
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