"The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation."

Sir Arthur Stanley Eddington, 'The Nature of the Physical World', 1915

**From Coins to Molecules: Why Energy Spreads**

Disorder is more probable than order because there are so many more ways of achieving it. Thus coins and cards tend to assume random configurations when tossed or shuffled, and socks and books tend to become more scattered about a teenager’s room during the course of daily living. But there are some important differences between these large-scale mechanical, or macro systems, and the collections of sub-microscopic particles that constitute the stuff of chemistry.

In systems of chemical interest we are dealing with huge numbers of particles.

This is important because statistical predictions are always more accurate for larger samples. Thus although for four coin tosses there is a good chance (62%) that the H/T ratio will fall outside the range of 0.45 - 0.55, this probability becomes almost zero for 1000 tosses. To express this in a different way, the chances that 1000 gas molecules moving about randomly in a container would at any instant be distributed in a sufficiently non-uniform manner to produce a detectable pressure difference between any two halves of the space will be extremely small. If we increase the number of molecules to a chemically significant number (around 1020, say), then the same probability becomes indistinguishable from zero.

Once the change begins, it proceeds spontaneously. That is, no external agent (a tosser, shuffler, or teen-ager) is needed to keep the process going. As long as the temperature is high enough for sufficiently energetic collisions to occur between the reacting molecules in a gas, the reaction will proceed to completion on its own once the reactants have been brought together.

Thermal energy is continually being exchanged between the particles of the system, and between the system and the surroundings. Collisions between molecules result in exchanges of momentum (and thus of kinetic energy) amongst the particles of the system, and (through collisions with the walls of a container, for example) with the surroundings.

Thermal energy spreads rapidly and randomly throughout the various energetically accessible microstates of the system. The degree to which the thermal energy is dispersed amongst these microstates is known as the entropy of the system.

**Energy-Spreading Changes the World**

Energy is conserved; if you lift a book off the table, and let it fall, the total amount of energy in the world remains unchanged. All you have done is transferred it from the form in which it was stored within the glucose in your body to your muscles, and then to the book (that is, you did work on the book by moving it up against the earth’s gravitational field.) After the book has fallen, this same quantity of energy exists as thermal energy (heat) in the book and table top.

What has changed, however, is the availability of this energy. Once the energy has spread into the huge number of thermal microstates in the warmed objects, the probability of its spontaneously (that is, by chance) becoming un-dispersed is essentially zero. Thus although the energy is still “there”, it is forever beyond utilization or recovery.

The profundity of this conclusion was recognized around 1900, when it was first described at the “heat death” of the world. This refers to the fact that every spontaneous process (essentially every change that occurs) is accompanied by the “dilution” of energy. The obvious implication is that all of the molecular-level kinetic energy will be spread out completely, and nothing more will ever happen. Not a happy thought!

**Why Do Gases Tend to Expand But Never Contract?**

Everybody knows that a gas, if left to itself, will tend to expand and fill the volume within which it is confined completely and uniformly. What “drives” this expansion? At the simplest level it is clear that with more space available, random motions of the individual molecules will inevitably disperse them throughout the space. But as we mentioned above, the allowed energy states that molecules can occupy are spaced more closely in a larger volume than in a smaller one. The larger the volume available to the gas, the greater the number of microstates its thermal energy can occupy. Since all such states within the thermally accessible range of energies are equally probable, the expansion of the gas can be viewed as a consequence of the tendency of thermal energy to be spread and shared as widely as possible. Once this has happened, the probability that this sharing of energy will reverse itself (that is, that the gas will spontaneously contract) is so minute as to be unthinkable.

Imagine a gas initially confined to one half of a box. We then remove the barrier so that it can expand into the full volume of the container. In its expanded, lower-pressure state, the allowed translational microstates of the gas are more closely spaced. Because more microstates are energetically accessible to the gas molecules in full volume than in half, the additional microstates are quickly populated.

In other words, the configuration of the system corresponding to the expanded state of the gas is so massively more probable than the initial state (because there are so many more ways of realizing it) that probability of the reverse process is vanishingly small.

**Why Heat Flows From Hot to Cold**

Just as gases spontaneously change their volumes from “smaller-to-larger”, the flow of heat from a warmer body to a cooler one always operates in the direction “warmer-to-cooler” because this allows thermal energy to populate a larger number of energy microstates as new ones are made available by bringing the cooler body into contact with the warmer one; in effect, the thermal energy becomes more “diluted”. In this simplified schematic diagram, the "cold" and "hot" bodies differ in the numbers of translational microstates that are occupied, as indicated by the shading. When they are brought into thermal contact, a hugely greater number of microstates are created, as is indicated by their closer spacing in the rightmost section of the diagram, which represents the combined bodies in thermal equilibrium. The thermal energy in the initial two bodies fills these new microstates to a level (and thus, temperature) that is somewhere between those of the two original bodies.

Note that this explanation applies equally well to the case of two solids brought into thermal contact, or two the mixing of two fluids having different temperatures.

As you might expect, the increase in the amount of energy spreading and sharing, and thus the entropy, is proportional to the amount of heat transferred q, but there is one other factor involved, and that is the temperature at which the transfer occurs. When a quantity of heat q passes into a system at temperature T, the degree of dilution of the thermal energy is given by: q /T

To understand why we have to divide by the temperature, consider the effect of very large and very small values of T in the denominator. If the body receiving the heat is initially at a very low temperature, relatively few thermal energy states are initially occupied, so the amount of energy spreading into vacant states can be very great. Conversely, if the temperature is initially large, more thermal energy is already spread around within it, and absorption of the additional energy will have a relatively small effect on the degree of thermal disorder within the body.

###### Ponder this

Can entropy be reversed without any inputs of energy?

In the far future of a possible Big Crunch, can we say that entropy is reversed?

###### Discuss

What will happen to the world in the future as entropy keeps on increasing? What will happen to its oceans, forests, mountains?

###### Further readings

"Entropy Is Simple...If You Avoid The Briar Patches!", a 'simple' explanation to entropy by Prof Frank Lambert. Your mileage may vary...

Entropy Law, a more readable explanation. Variations to mileage is smaller...

Entropy and life, where the laws of thermodynamics meets biology