Entropy

Imagine if you had a bed and you slept on it every day, but you never had anyone, including yourself, tidy it and clean it. You would just sleep on it and do what you need to every night, then wake up and not bat an eye towards it until you went to go back to sleep. Over time, you'd realize that this bed will become very disordered and dirty over time with no one to clean it up. When you spontaneously let your bed be, it becomes naturally dirty over time. No matter what happens, every possibility of your room's state will always have a messy bed if no one ever does anything to clean it.

This right here is the concept of entropy. Entropy isn't exactly the easiest thing to explain, even at a higher level. However, it is usually associated with the idea of disorder. The second law of thermodynamics states that the entropy of the universe can never decrease over time, only stay the same or increase. Essentially, any process that is left unchecked(spontaneous) will always proceed towards a state of disorder. By the way, spontaneous processes are just processes that don't need external energy to proceed. They just happen by themselves.

But, why do processes move towards disorder? Why does dropping food coloring dye in water cause it to disperse throughout the fluid? Why do bedrooms get messier over time when people don't make them? Why does ice melt into water but not the other way around?

This can be internalized through the concept of probability. Imagine these four molecules:

Let's see how many microstates we can make in these two flasks with these 4 molecules. A microstate is another word for arrangement. The arrangement shown above is one with all four moleccules on the left side.

We can display the possible molecular arrangements as below:

There are 6 possible arrangements with 2 of the molecules in one flask and the other 2 in the other.

There are four arrangements where 3 of the molecules are in the left flask and 1 is in the right flask.

There's one arrangement where all the molecules are in the right flask.

So, there's a total of 16 possible arrangements of these four molecules. The most likely is of course that there's two molecules in each flask, as there's 6 possible arrangements for this. This means there's a 38% chance of getting this, meaning a system with 4 molecules and two flasks is most likely to proceed towards disorder, with the molecules spread apart and not concentrated. Since entropy is associated with disorder, if the disorder increases, the entropy increases, which is consistent with the second law.

Now, 38% may not seem like a lot. If you were given flasks like these, you may not even see there be 2 molecules in each flask because other arrangements aren't too unlikely. However, let's imagine these flasks but instead of just 4 molecules, we now have over a mole(6.022 * 10²³) of them. The probabilities of configurations other than a half-half disordered arrangements become very, very, very, very small. For example, the probability that all the molecules are in the right flask is 1/16. If we take a mole of molecules, that probability becomes the reciprocal of 2 to the power of Avogadro's constant, which is pretty much 0. This means that the system will occupy the state which is most probable, being one where half the molecules are in each flask.

So, now that we've explained the idea behind entropy, and how it is related to disorder, let's go over some rules.

The first rule is that liquids have higher entropies than solids, and gases have a lot higher entropies than liquids. This is because solids are molecularly tight in pretty much one arrangement. Liquids, however, consist of more dispersed molecules floating about. Gases have molecules zipping around at intense speeds so they're the most disordered out of the 3. Solutions have higher entropies than solids but not higher than liquids. Thus, the progression goes:

S_solid < S_solution < S_liquid << S_gas

Something to note here is that the letter S stands for entropy.

Lastly, 2 moles of a substance have more entropy than one mole of the same substance at the same phase of matter.

This helps us analyze the standard change in entropy ΔS° of a reaction. Let's take the reaction below for example:

4NH₃(g) + 5O₂(g) → 4NO₂(g) + 6H₂O(g)

On the reactant side, there are 9 gaseous molecules. On the products, there are 10 gaseous molecules. There's more on the product side, meaning that as the reaction proceeds, there will be more molecules zipping around. This means the entropy increases as you go from left to right. If this reaction was reversed, the entropy would decrease as it proceeds.