Entropy – a Further Look

We can consider the universe as being made of two parts:

• the system – the part of the universe we are directly interested in (in chemistry it will be the chemicals in the chemical reaction).  The entropy of the system is S_system
• the surroundings – everything else in the universe.  The entropy of the surroundings is S_surroundings

Everything in the universe is either a part of the system or surroundings, so

S_universe = S_system + S_surroundings

In any change the entropy of the universe increases, so S_universe > 0.   It should be pretty obvious that the entropy of the universe could increase if either (or both) of the entropies of the system or surroundings increased enough.  Individually it is possible for either S_system or S_surroundings to be increasing or decreasing, but together the entropy of the universe will increase, if a change is taking place.

Water does spontaneously fall downwards, and when it does the S_universe must increase.  As the water falls, its potential energy changes into kinetic energy.  When the water hits the ground, this kinetic energy will cause some heat – not a lot, of course, but it will warm up the surroundings a bit.  This heat energy is dispersed out into all the molecules of the surroundings, so that the S_surroundings increases.  The net result is that S_universe also increases, so the change takes place and water falls spontaneously down hill.  Even the idea that systems tend towards minimum energy is really just another consequence of the second law of thermodynamics.

The second law of thermodynamics tells us what happens in a spontaneous change, but like all scientific laws, doesn't explain why it does so.  The explanation for why there must be an increase in entropy came from Ludwig Boltzmann.  He showed that entropy is really the consequence of probability.   If you have molecules moving about at random, then it is more likely that they will become mixed than stay separate.  This is why you will sometimes hear entropy described as "randomness" or "disorder".  As entropy increases, the distribution of energies that can exist in molecules become more "mixed up".

So where does the energy get dispersed or spread-out into?  It goes into the atoms and molecules of which the substance is made.  Molecules have many different ways of dispersing the energy into all the different kinds of motion that they can do – rotation, translation, and vibration.  An increase in entropy is an increase in the number of ways in which heat energy is distributed amongst molecules.

Some important things to remember about entropy are:

• entropy always increases as the temperature increases (because there is more heat energy to be dispersed)
• entropy always increases as you change state from solid liquid gas

Sometimes it is possible to just look at a change and identify immediately if entropy is increasing.  For example:

• H₂O (l) H₂O (g) : Entropy of a liquid is less than that of a gas, so entropy of the system is increasing when water evaporates.
• O₂ (g) + N₂ (g) Air (g) : a physical mixture is by definition more "mixed", so air has higher entropy than either pure oxygen or nitrogen alone.  Gases will always mix, because entropy always increases when they do.
• Fe (l) Fe (s) : Entropy of a liquid is greater than entropy of a solid, so entropy of the system is decreasing when molten iron solidifies.

However, for other reactions it is not at all obvious.  For example, consider the following reaction, the combustion of methane

CH₄ (g) + 2O₂ (g) CO₂ (g) + 2H₂O (g)

Here everything is gas, and we have mixtures of both elements and compounds.   While we might guess that the entropy of the two product compounds is greater than the one reactant compound and one reactant element, there is no sound reason to do this.   However, there is a way to calculate the entropy change in any reaction, at least at standard conditions of 25 ^oC and 1 bar (atmospheric pressure).  The standard molar entropies, S^o, of many compounds and elements have been recorded in tables of thermodynamic data.  These can be combined using the following relationship to find the entropy change for any reaction under standard conditions:

In other words, sum up all the molar entropies of all the product molecules and subtract from this the sum of all the molar entropies of all the reactant molecules (this is the standard definition for change: final state - initial state).

Remember that what you have calculated is at the standard state.  The entropy will be different under different pressure and temperature conditions.