Does a messier room actually have higher entropy?

When my chemistry teacher was explaining entropy, she used the analogy of a clean vs. messy room. What I'm wondering is whether a messy room really does have higher entropy. Does this still work on a larger scale? If not, can I trust the analogy?

• Rooms don't have entropy. Other than that, the analogy is good. – Ivan Neretin Aug 25 '16 at 20:16
• Calculating the entropy of a small number of objects is not very meaningful, neither is calculating the entropy of an assortment of macroscopic objects.You cannot predict anything sensible from the result. – Karl Aug 25 '16 at 20:38
• Fun fact: Every letter in a common sentence contains roundabout one bit of information. Is the rest noise (=entropy), or is it structure, predetermined by the information? – Karl Aug 25 '16 at 20:50

A room can be a complicated system, (at least mine can be) so I shall assume a simplified version: an enclosed space bound by 4 walls, a ceiling and a floor and the only piece of furniture inside the room is a desk, and two books on it -- book 1 and book 2.

In the this scenario, I would define a clean room as one with both the books neatly stacked on the desk, and in a messy room you can find one of the books on the floor.

Now, if I were to try enumerate the number of ways of organising the books in a clean and a messy room, I would find that there is exactly one way in which I can achieve a "clean room configuration", and that is by having both books on the desk. On the other hand if you wish to achieve a "messy room configuration", you can do so in two ways-- either have book 1 on the floor and book 2 on the desk, or have book 2 on the floor and book 1 on the desk.

The no. of ways in which you can achieve a configuration is called its weight and is defined as:

$$W = \frac{N!}{N_0!N_1!N_2!....}$$

In the first case, $W=1$ (clean) and in the second $W=2$ (messy)

The boltzmann formula for entropy is: $S = k_b\ln W$

Plugging in $W$, we get $$S_\text{clean} = 0$$ and $$S_\text{messy} = k_b\ln2$$

Thus, the messier room has a higher entropy. This somewhat "silly" argument can be reproduced for a room with more furniture, and more items in it. Feel free to toss in some clothes, papers, chairs, book cases, toys, shoes, pets...go nuts.

So, I believe the analogy holds good. But again, what I did here is a silly exercise to illustrate a point, and it isn't meaningful as such.

If you wish to learn more, watch this lecture by Leonard Susskind

This is a tricky question. The answer is no.

From Wikipedia

In thermodynamics, entropy (usual symbol S) is a measure of the number of microscopic configurations Ω that correspond to a thermodynamic system in a state specified by certain macroscopic variables.

It can not be defined the messy and clean states thermodynamically. (Which would be the reasonable thermodynamic variables?) If it would be possible, after further assumptions, the problem is reduced to the @getafix answer. Well, it is also needed that all the book remains stacked in the desk (all of them with the same propertied but distinguishable).

It is more important to note that the non technical concept of messy is not related to entropy.

EDIT:

I have decided to make it explicit (although at first I though it was) as the content of the answer was qualified as arbitrary and it was down-voted.

The OP said:

What I'm wondering is whether a messy room really does have higher entropy

The other questions turns to be meaning less if we can not assert that the messy room has higher entropy.

1. It is clear from the context ("When my chemistry teacher was explaining entropy...") that we are talking about entropy in thermodynamics.

2. It is also clear from the context that we are talking at an introductory level.

3. In order to compare the entropy of two systems, it is required that the states of both systems can be defined in terms of state variables.

4. The system state variables can not be established for messy nor clean rooms.

5. As the system states can not be specified, then it is not true that a messy room has higher entropy.

The usage of analogies is not just matter of taste. At first, there is a goal when an analogy is used while teaching a concept $X$, clearly it should be to help the students to learn $X$. In order to describe $X$, a well known $Y$ is used. $X$ and $Y$ must share a common set of properties. Aided by $Y$, some features of $X$ are inferred by inductive reasoning.

It is not evident which are the common features that links entropy with the level of cleanliness of a room, so a valid analogy can not be built. Of course there could be established some analogy when a lot of considerations are taking into account. But, if the student have reached a conceptual level to understand "why" they are made, then, the analogy will be of little use. If they did not, take for sure that they won't understand the concept (because they won't know when they can trust the analogy).

Does this still work on a larger scale? If not, can I trust the analogy?

I think I was no clear when I said "no". I really should have said "sometimes, entropy has nothing to do with cleanliness".

• It's your wilful decision to drive the analogy to the point where it breaks. I don't like it very much either. But saying it is just wrong is, well, arbitrary. – Karl Sep 4 '16 at 19:10
• @Karl , I expanded my answer in order to address the points of our comments. Thanks for sharing your thoughts. It help to improve the contents and clarity of the answers. – user1420303 Sep 4 '16 at 22:31

"whether a messy room really does have higher entropy".

1. One messy room left untouched does not have a higher entropy than a clean one. Both represent one way of arranging the things in the room (a microstate), so the entropy is zero for both.

2. If you have a messy person living in the room then, yes, over time the messy room will have a high entropy. Things will move around but never be in the same place (many microstates), so you will have many different arrangements of things that always will be characterised as "messy" (macrostate).

3. A collection of several messy rooms, e.g. identically furnished but unmade rooms in a hotel, has a higher entropy than the same collection of rooms after house keeping has been to them all and arranged the things in exactly the same way.

• +1 You touched an interesting point where an analogy can be drawn, if the internal structure of macroscopic objects is not considered. I would like to read "ergodic" scattered somewhere in point 3, to turn the statement a little more accurate. – user1420303 Sep 19 '16 at 15:20