Entropy

Entropy definition

In the introduction to this article, we gave you one definition of entropy.

However, we can also describe entropy differently.

The two definitions seem very different. However, when you break them down, they start to make a little more sense.

Let's revisit the Rubik’s cube. It starts off ordered - each face contains just one colour. The first time you twist it, you disrupt the order. The second time you twist it, you might undo your first move and restore the cube to its original, perfectly solved arrangement. But it is more likely that you will rotate a different side and disrupt the order even more. Each time you randomly twist the cube, you increase the number of possible configurations that your cube could take, decrease the chance of landing upon that perfectly solved arrangement, and get more and more disordered.

Fig. 1: Randomly turning a Rubik's cube. With each side you twist, the cube tends towards greater disorder.StudySmarter Originals

Now, imagine a 3x3 Rubik's Cube. This complex cube has many more moving parts than the first, and so has more possible permutations. If you shut your eyes and twist the sides around blindly once more, the odds of chancing upon a solved cube when you open them again are even slimmer - it is extremely unlikely that your cube will have anything but a totally random, disordered configuration. A larger cube with more individual pieces has a greater tendency to become disordered, simply because there are so many more ways that it can be arranged. For example, a simple 2x2 Rubik's cube has over 3.5 million possible permutations. A standard 3x3 cube has 45 quintillion combinations - that's the number 45 followed by 18 zeros! However, a 4x4 cube trumps them all with a mind-blowing 7.4 quattuordecillion combinations¹. Ever heard of a number that large before? It is 74 followed by 44 zeros! But for all of those cubes, there is only one solved arrangement, and so the odds of randomly stumbling across that perfect combination decrease.

Notice something? As time goes on, the cube goes from solved to randomly arranged, from a state of order to disorder. In addition, as the number of moving pieces increases, the tendency to become more disordered increases because the cube has a larger number of possible arrangements.

Let’s now relate this to entropy. Imagine that each sticker represents a certain particle and amount of energy. The energy starts off neatly arranged and ordered, but quickly becomes randomly arranged and disordered. The larger cube has more stickers, and so has more particles and units of energy. As a result, there are more possible configurations of stickers and more possible arrangements of particles and their energy. In fact, it is a lot easier for the particles to move away from that perfectly ordered arrangement. With each move away from the starting configuration, the particles and their energy become more and more randomly dispersed, and more and more disordered. This fits with our two definitions of entropy:

• The larger cube has a higher number of possible arrangements of particles and their energy than the smaller cube, and so has a greater entropy.

• The larger cube tends to be more disordered than the smaller cube, and so has a greater entropy.

Properties of entropy

Now that we have a bit of an understanding of entropy, let’s look at some of its properties:

• Systems with a higher number of particles or more units of energy have a greater entropy because they have more possible distributions.

• Gases have a greater entropy than solids because the particles can move around much more freely and so have more possible ways of being arranged.

• Increasing the temperature of a system increases its entropy because you supply the particles with more energy.

• More complex species tend to have a higher entropy than simple species because they have more energy.

• Isolated systems tend towards a greater entropy. This is given to us by the second law of thermodynamics.

• Increasing entropy increases the energetic stability of a system because the energy is more evenly distributed.

Units of entropy

What do you think the units of entropy are? We can work them out by considering what entropy depends on. We know that it is a measure of energy, and is affected by temperature and the number of particles. Therefore, entropy takes the units J·K^-1·mol^-1.

Standard entropy

To compare entropy values, we often use entropy under standard conditions. These conditions are the same as the ones used for standard enthalpies:

• A temperature of 298K.

• A pressure of 100kPa.

• All species in their standard states.

Standard entropy is represented by the symbol S°.

Entropy changes: definition and formula

Entropy cannot be measured directly. However, we can measure the change in entropy (ΔS). We typically do this using standard entropy values, which have already been calculated and verified by scientists.

Each reaction firstly causes an entropy change within the system - that is, within the reacting particles themselves. For example, a solid might turn into two gases, which increases the total entropy. If the system is completely isolated, this is the only entropy change that takes place. However, isolated systems don't exist in nature; they are purely hypothetical. Instead, reactions also affect the entropy of their surroundings. For example, a reaction might be exothermic and release energy, which increases the entropy of the surroundings.

We’ll start by looking at the formula for the entropy change within a system (commonly simply known as the entropy change of a reaction, or just entropy change), before taking a deep dive into the entropy change of the surroundings and the total entropy change.

Entropy change of reaction

The entropy change of a reaction (which, you’ll remember, is also called the entropy change of the system) measures the difference in entropy between the products and the reactants in a reaction. For example, imagine your reactant is the perfectly solved Rubik’s cube, and your product is a randomly arranged cube. The product has a much higher entropy than the reactant, and so there is a positive entropy change.

We work out the standard entropy change of reaction, represented by ΔS°_system or just ΔS°, using the following equation:

$$\Delta S^\circ = {\Delta S^\circ}_{products}-{\Delta S^\circ}_{reactants}$$

Predicting entropy changes of reaction

Let’s now see how we can use what we know about entropy to predict the possible entropy change of a reaction. This is a quick way to estimate entropy changes without doing any calculations. We predict the entropy change of a reaction by looking at its equation:

• A positive entropy change of reaction means the entropy of the system increases and the products have a higher entropy than the reactants. This could be caused by:

  • A change of state from solid to liquid or liquid to gas.

  • An increase in the number of molecules. In particular, we look at the number of gaseous molecules.

  • An endothermic reaction that takes in heat.

• A negative entropy change of reaction means that the entropy of the system decreases, and the products have a lower entropy than the reactants. This could be caused by:

  • A change of state from gas to liquid or liquid to solid.

  • A decrease in the number of molecules. Once again, we look closely at the number of gaseous molecules.

  • An exothermic reaction that releases heat.

Entropy and feasible reactions

We learned earlier that, according to the second law of thermodynamics, isolated systems tend towards a greater entropy. We can therefore predict that reactions with a positive entropy change happen on their own accord; we call such reactions feasible.

But many feasible day-to-day reactions don’t have a positive entropy change. For example, both rusting and photosynthesis have negative entropy changes, and yet they are everyday occurrences! How can we explain this?

Well, like we explained above, it is because natural chemical systems aren’t isolated. Instead, they interact with the world around them and so have some sort of effect on the entropy of their surroundings. For example, exothermic reactions release heat energy, which increases their surrounding environment’s entropy, whilst endothermic reactions absorb heat energy, which decreases their surrounding environment’s entropy. Whilst total entropy always increases, the entropy of the system doesn’t necessarily increase, provided the entropy change of the surroundings makes up for it.

So, reactions with a positive total energy change are feasible. From looking at how a reaction affects the entropy of its surroundings, we can see that feasibility depends on a few different factors:

• The entropy change of the reaction, ΔS° (also known as the entropy change of the system, or just entropy change).

• The enthalpy change of the reaction, ΔH°.

• The temperature at which the reaction takes place, in K.

The three variables combine to make something called the change in Gibbs free energy.

Here’s the formula for the change in standard Gibbs free energy:

$$\Delta G^\circ={\Delta H^\circ}-T\Delta S^{\circ}$$

Like enthalpy, it takes the units kJ·mol^-1.

The change Gibbs free energy explains why many reactions with negative entropy changes are spontaneous. An extremely exothermic reaction with a negative entropy change can be feasible, provided ΔH is large enough and TΔS is small enough. This is why reactions such as rusting and photosynthesis take place.