# Microstates

- Page ID
- 96628

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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. 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.

## Contributors and Attributions

- Frank L. Lambert, Professor Emeritus, Occidental College