Question

a box contains two halves separated by a partition. Initially, there are 3 ideal gas molecules in the left half, and a vacuum in the right half. The partiition is then removed so the gas goes through a "free expansion" so that the speeds of the molecules are unaffected. In the final state, each molecule has probability of 1/2 of being in the left half of the box and probability of 1/2 being in the right half od the box. Because each of the 3 molecules now has twice as many possible positions than before, the number of microstates of the system has increased by a factor of 2x2x2=2^3. Determine the change in entropy of the system as a result of removing the partition. (Note: We can only calculate the change in entropy of the system but not the initial and final entropy values without knowing more information) What is the probability that all three molecules will simultanesouly be in the left half of the box?

Answer 1

Second law of thermodynamics: In a natural thermodynamic process, the sum of the entropies of the participating thermodynamic systems increases. Equivalently, perpetual motion machines of the second kind are impossible.

You are discussing six molecules. Thermodynamics is a theory formulated to fit observations for large numbers of particles , of the order of the Avogadro number ~10^23. It is a very successful theory used in many aspects of engineering, but that does not mean that it scales down to small numbers . So there is no problem with your example.

The statement you quote

"A small probability exists that all the molecules of a system of confined gas might appear for an instant in just one corner of the container. This is called energy fluctuation."- Boltzmann]

Is a visualized extension from small numbers, adding more and more molecules will make the probabilities smaller and smaller, and because of the 10^23 extremely improbable.

This is the study of thermodynamics as it emerges from statistical mechanics, where it is shown that entropy is not a "law" in the same sense as energy conservation is a law, but can be derived from thestatistics of many bodies.

In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate,

Each individual microstate is equally improbable as getting all the molecules on one side of the box, it is the sheer numbers of the average mix that define entropy at the thermodynamic framework, giving probabilities of order of 10^-12. The link may he

• The probability that the first particle is somewhere in the left half of the box is one-half.
• The probability that the second particle is somewhere in the left half of the box is one-half.
• �

This tells that the probability of having all six particles in the left half of the box is ?1.2%, as you say: that is only one configuration out of the 26=64 possibilities. By comparison there are "six choose three," or

(3c3)=1

possible configurations where you have three particles on the left side. (3 could be the first one over, two remain to be the second, 3 for the third, but there are 3!=6 ways to order them, and we don't want to over-count.) That means that you'll find exactly three particles on the left side about 20/64?31% of the time, as you say. In fact you'll have 3�3? particles on the left side somewhere between 70% and 97% of the time, depending on whether you count the 5/1 state or not. Here's a little histogram of how much time your box will be in each state:

hat for these two I haven't copied over most of the rows where the probability is too low to print a dot. This is honestly hidden data: those zeros really are zeros. The shape of the curve is called thebinomial distribution and it becomes quite similar to the better-known bell curve for many particles. If you have 2N particles in your box, then

• about 85% of the time you'll find N�N??? on the left side;
• about 99.5% of the time you'll find N�2N??? on the left side;
• about 99.998% of the time you'll find N�3N??? on the left side.

If you had only 1000 particles in your box, there are only about five configurations out of every 1010 that a split between the two sides more extreme than 400/600; if your box took a millisecond to cross, like a typical meter-sized box of gas at room temperature, you'd see two-to-one density difference between the sides about once a month. For a millisecond.

The thing that breaks your intuition about statistical mechanics is the sheer size of the numbers involved. I gave you some examples for up to 1000 particles because I happen to be using computer software which tolerates numbers up to 21024. But a liter of gas at STP contains something like 1023particles, which have 21023 ways to partition themselves between the two halves of a box. The natural density fluctuations due to particles randomly moving between the two sides of the box are at the level of 1023?????=1011.5 particles � changes which begin in the eleventh digit of the number. For example, if I expected to find 100 000 000 000 000 000 000 000 atoms on the left side of the box on average, about 7.5% of the time I'd randomly find more than 100 000 000 000 030 000 000 000 atoms, and about 0.001% of the time I'd find more than 100 000 000 000 100 000 000 000 atoms. Totally unmeasurable differences. The fluctuations are made small by the statistical size of the ensemble.

It's not impossible for a mole of gas to briefly, randomly find itself in one half of its container. But it's a safe bet that no such extreme fluctuation has occurred for any such volume in the observable universe since the Big Bang. Your intuition has failed you because 6?1023, not even for extremely large values of six.