What is a microstate?

There are two alternative ways of describing a microstate in this section. Either is adequate, depending on the reader's preference.

A, “A Description of Microstates” is exactly that, a verbal outline without illustration, at a moderate level of difficulty.

B, “Microstates: In Terms of Molecular Energy Levels”, is somewhat more advanced. It is a part of the article on this Web site for instructors, “Entropy Is Simple, Qualitatively”.

Dictionaries define “macro” as large and “micro” as very small but a macrostate and a microstate in thermodynamics aren't just definitions of big and little sizes of chemical systems. Instead, they are two very different ways of looking at a system. (Admittedly, a macrostate always has to involve an amount of matter large enough for us to measure its volume or pressure or temperature, i.e. in “bulk”. But in thermodynamics, a microstate isn't just about a smaller amount of matter', it is a detailed look at the energy that molecules or other particles have.) A microstate is one of the huge number of different accessible arrangements of the molecules' motional energy* for a particular macrostate.

*Motional energy includes the translational, rotational, and vibrational modes of molecular motion. (If these words are not familiar, see Figure 1 about half-way through 2ndlaw.com/entropy.html.) In calculations involving entropy, the ΔH of any phase change in a substance (“phase change energy”) is added to motional energy, but it is unaltered in ordinary entropy change (of heating, expansion, reaction, etc.) unless the phase itself is changed.

A *macrostate *is the thermodynamic state of any system that is exactly
characterized by measurement of the system's properties such as P, V, T,
H and number of moles of each constituent. Thus, a macrostate does not change
over time if its observable properties do not change.

In contrast, a *microstate* for a system is all about time and the energy of
the molecules in that system. "In a system its energy is constantly
being redistributed among its particles. In liquids and gases, the particles
themselves are constantly redistributing in location as well as changing
in the quanta (the individual amount of energy that each molecule has) due
to their incessantly colliding, bouncing off each other with (usually) a
different amount of energy for each molecule after the collision.. Each specific
way, each arrangement of the energy of each molecule in the whole system
at one instant is called a * microstate*."

One microstate then is something like a theoretical "absolutely *instantaneous* photo" of the location and momentum of *each* molecule and atom in the whole macrostate. (This is talking in ‘classical mechanics’ language where molecules are assumed to have location and momentum. In quantum mechanics the behavior of molecules is only described in terms of their energies on particular energy levels. That is a more modern view that we will use.) In the next instant the system immediately changes to another microstate. (A molecule moving at an average speed of around a thousand miles an hour collides
with others about seven times in a billionth of a second. Considering a mole
of molecules (6 x 10^{23}) traveling at a very large number of different
speeds, the collisions occur — and thus changes in energy of trillions
of molecules occurs — in far less than a trillionth of a second. That's
why it is wise to talk in terms of “an instant”!) To take a photo like that
may seem impossible and it is.

In the next instant — and that really means in an *extremely * short
time — at least a couple of moving molecules out of the 6 x 10 23 will
hit one another.. But if only one molecule moves a bit slower because it
had hit another and made that other one move an exactly equal amount faster — then
that would be a different microstate. (The total energy hasn't changed when
molecular movement changes one microstate into another. Every microstate
for a particular system has exactly the total energy of the macrostate because
a microstate is just an instantaneous quantum energy-photo of the whole system.)
That's why, in an instant for any particular *macro*state, its motional
energy* has been rearranged as to what molecule has what amount of energy.
In other words, the system — the macrostate — rapidly and successively
changes to be in a gigantic number of different microstates out of the “gazillions” of
*accessible* microstates, (In solids, the location of the particles
is almost the same from instant to instant, but not exactly, because the
particles are vibrating a tiny amount from a fixed point at enormous speeds.)

N_{2} and O_{2} molecules are at 298
K are gases, of course, and have a very wide range of speeds, from zero to
more than two thousand miles an hour with an average of roughly a thousand
miles an hour. They go only about 200 times their diameter before
colliding violently with another molecule and losing or gaining energy.
Occasionally, two molecules colliding head on at exactly the same speed would
stop completely before being hit by another molecule and regaining some speed.)
In liquids, the distance between collisions is very small, but the speeds are
about the same as in a gas at the same temperature.

Now we know what a microstate is, but what good
is something that we can just imagine as an impossible fast camera shot?
The answer is loud and clear. We can calculate the numbers for a given macrostate
and we find that microstates give us answers about the relation between molecular
motion and entropy — i.e., between molecules (or atoms or ions) constantly
energetically speeding, colliding with each other, moving distances in space
(or, just vibrating rapidly in solids) and what we measure in a macrostate
as its entropy. As you have read elsewhere, entropy is a (macro) measure
of the spontaneous dispersal of energy, how widely spread out it becomes
(at a specific temperature). Then, because the number of microstates that
are *accessible* for a system indicates all the different ways that
energy can be arranged in that system, the larger the number of microstates
accessible, the greater is a system's entropy at a given temperature.

It is *not* that the energy of a system is smeared
or spread out over a greater number of microstates that it is more dispersed.
That can't occur because all the energy of the macrostate is always in only
one microstate at one instant. The macrostate's energy is more "spread
out" when
there are larger numbers of microstates for a system because at any instant
all the energy that is in one microstate can be in any one of the now-larger
total of microstates, a greatly increased number of choices, far less chance
of being “localized” — i.e.,
just being able to jump around from one to only a dozen other microstates or'only'
a few millions or so! More possibilities mean more chances for the system to be
in one of MANY more different microstates — that is what is meant by "the
system's total energy can be more dispersed or spread out”: more choices/chances.

That might be fine, but how can we find out how
many microstates are accessible for a macrostate? (Remember, a macrostate is
just any system whose thermodynamic qualities of P, V, T, H, etc. have been
measured so the system is exactly defined.) Fortunately, Ludwig Boltzmann
gives us the answer in S = k_{B} ln W, where S is the value of entropy
in joules/mole at T, k_{B} is Boltzmann's constant of 1.4 x 10^{-23} J/K and
W is the number of microstates. Thus, if we look in “Standard State Tables” listing
the entropy of a substance that has been determined experimentally by heating
it from 0 K to 298 K, we find that ice at 273 K has been calculated to have
an S^{o} of 41.3 J/K mol.
Inserting that value in the Boltzmann equation gives us a result that should
boggle one's mind because it is among the largest numbers in science.
(The estimated number of atoms in our entire galaxy is around 10^{70} while the number for the whole universe may be about 10^{80}. A very large number in math is 10^{100} and called "a googol" — *not* Google!) Crystalline ice at 273 K has 10^{1,299,000,000,000,000,000,000,000} accessible
microstates.
(Writing 5,000 zeroes per page, it would take not just reams of paper, not just reams
piled miles high, but light years high of reams of paper to list all those microstates!)

Entropy and entropy change are concerned with the energy
dispersed in a system and its temperature, q_{rev}/T. Thus, entropy is measured
by the number of accessible microstates, in any one of which the system's total
energy might be at one instant, not by the orderly patterns of the molecules
aligned in a crystal. Anyone
who discusses entropy and calls "orderly" the energy distribution
among those humanly incomprehensible numbers of different microstates for a
crystalline solid — such as we have just seen for ice — is looking at the wrong thing.

Liquid water at the same temperature of ice, 273 K has an S^{o} of 63.3
J/K . Therefore, there are 10^{1,991,000,000,000,000,000,000,000} accessible
microstates for water.

Please see the section titled ** The Molecular Basis for Understanding
Simple Entropy Change** in this article.

entropy.lambert@gmail.com

Last revised and updated: November 2005

Last revised and updated: November 2005