Whats the difference between entropy and the (dis)order of a system?

Entropy is often verbally described as the order/disorder of the thermodynamic system. However, I've been told that this description is a vague "hand-waving" attempt at describing what entropy is. For example, a messy bedroom doesn't have greater entropy than a tidy room 


My question is why is this the case? Also, what would better describe entropy verbally? 




Briefly, spontaneous processes tend to proceed from states of low probability to states of higher probability. The higher-probability states tend to be those that can be realized in many different ways.


**Entropy is a measure of the number of different ways a state with a particular energy can be realized**. Specifically, $$S=k\ln W$$ where $k$ is Boltzmann's constant and $W$ is the number of equivalent ways to distribute energy in the system. If there are many ways to realize a state with a given energy, we say it has high entropy. Often the many ways to realize a high entropy state might be described as "disorder", but the lack of order is beside the point; the state has high entropy because it can be realized in many different ways, not because it's "messy". 


Here's an analogy: if energy were money, entropy would be related to the number of different ways of counting it out. For example, there, there are only two ways of counting out two dollars with American paper money (2 1-dollar bills, or 1 two-dollar bill). But there are five ways of counting out two dollars using 50-cent or 25-cent coins (4 50-cent pieces, 3 50-cent pieces and 2 quarters, and so on). You could say that the "entropy" of a system that dealt in coins was higher than that of a system that dealt only in paper money. 


Let's look the change in entropy for a reaction $\rm A\rightarrow B$, where A molecules can take on energies that are multiples of 10 energy units, and B molecules can take on energies that are multiples of 5 units. 


Suppose that the total energy of the reacting mixture is 20 units. If we have 3 molecules of A, there are 2 ways to distribute our 20 units among energy levels with 0, 10, and 20 units:


![There are 2 ways to distribute 3 A molecules with a total energy of 20 units.](https://i.stack.imgur.com/sdTLE.png)


If we have 3 molecules of B, there are 4 ways to distribute 20 units among energy levels with 0, 5, 10, 15, and 20 units:


![enter image description here](https://i.stack.imgur.com/8i76i.png)


The entropy of B is higher than the entropy of A because there are more ways to distribute the same amount of energy in B than in A. Therefore, $\Delta S$ for the reaction $\rm A\rightarrow B$ will be positive.


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> what would better describe entropy verbally?
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We can also use "measurement of randomness" or "amount of chaos" or " energy dispersion"




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Initially in 1862, Rudolf Clausius asserted that thermodynamic process always "admits to being reduced to the **alteration** in some way or another of the arrangement of the constituent parts of the working body" and that internal work associated with these alterations is quantified energetically by a measure of **entropy** change.


But later after few years Ludwig Boltzmann translated word **alteration** from Rudolf Clausius' assertion to **order and disorder** in gas phase molecular systems.


But if you see latest books they use concept of **energy dispersion** instead of order or disorder to explain entropy. 




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Source: [Wikipedia](http://en.wikipedia.org/wiki/Entropy_(order_and_disorder))


Also have look at [physics S.E.](https://physics.stackexchange.com/a/33385/46239)


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I will take a crack at this although I admit that this topic sometimes confuses me as well. Here is how I like to think about entropy. Consider a box containing equal amounts of gases A and B. So, we have a fixed volume and fixed number of molecules. Let us also isolate the box from its surroundings so that it has a fixed energy as well. We have just created a microcanonical ensemble (constant NVE). The ensemble consists of every possible configuration of molecules having the same NVE. Now, if we were able to step back and take a broad view of the ensemble, we would observe that the vast majority of boxes contain a rather bland homogeneous mixture of gases A and B. In fact, they would be indistinguishable for all practical purposes. Let us count the number of boxes in this state and call the number $W\_1$. Continuing with the box counting, we find that there are only a handful of distinguishable boxes left, but they are quite interesting! One box, for instance, might have all of the A molecules crowded together in one corner and all of the B molecules in another corner! This one box can be labelled $W\_2$. We start to understand that some configurations of molecules will be extremely improbable because they represent such a small fraction of the total number of possible configurations. Boltzmann quantified this relationship as $S=k \log W$. If we plug $W\_1$ into this equation we get a high value for the entropy, $S$, because $W\_1$ is so large. On the other hand, if we plug in $W\_2$ the entropy we get is very low.


So, to sum up, entropy is really the measure of how likely a given system configuration is when compared to all of the possible configurations. In this sense, I would argue that you could say a messy room has a higher entropy than a clean room because there are so many more ways a room could be considered "messy" than "clean".


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