Entropy - "Wiggle"?

Question

The title is not a reference to a Jason Derulo song.

In any case:

1) How is change in entropy measured, experimentally? I've Googled this for a bit and I've found all sorts of mathematical equations such as delta S = q/T as a way to get a grasp on entropy. However, I want to know how entropy is actually determined in the lab (as opposed to on paper). I understand that the units on entropy are in terms of energy/temperature. So if I had to guess I would suppose that some sort of calorimetry is used.

2) What is the best way to teach entropy (at an introductory level)? I know that textbooks like to say it's a "measure of disorder." On the other hand I have lots of websites telling me that entropy is best NOT described at a measure of disorder. So what is it really, and what do you think about the term "disorder" as a synonym? My professor refers to it as the degrees of freedom in a system; he also notes that longer carbon chains tend to have higher entropy values due to a greater ability to "wiggle" along their carbon chains. What do you think of this?

Discarding the archaic idea of "disorder" in regard to entropy is essential. It just doesn't make scientific sense in the 21st century and its apparent convenience often is flat-out misleading. As of November 2005, fifteen first-year college texts have deleted “entropy is disorder” although a few still retain references to energy “becoming disorderly”. (This latter description is meaningless

Answer 1

You can't measure entropy directly, any more than you can measure interatomic distances. You measure other quantities -- for instance often you can measure energy gain/loss and temperature, and then you integrate $dS=dE/T$.

How to explain it? One of the best expositions I know is The Second Law by Henry A. Bent. It is full of insightful examples, lays the ground carefully and avoids woolly talk, unlike many other thermodynamics books.

Some elementary but valid comments about the link between information-theoretical entropy and thermodynamical entropy can be found here: http://www.madsci.org/posts/archives/1998-10/909712896.Ph.r.html

Edit: Greg reminds us that energy and enthalpy are not directly measurable either. That's why I was careful to write energy gain/loss, which are somewhat more accessible to measurement; but of course even when doing a calorimetric experiment you're not measuring heat per se but other quantities: how much gas you burned or how intense the current was and how long it was left on. And even some of these quantities again are indirectly measured.

I agree 100% that "entropy is not more abstract than energy" -- they are all abstractions. Ultimately, however, I do think the measurement of thermodynamic entropy is one step more indirect than that of energy, if only because it involves the notion of temperature, which (if you think about it carefully) is subtle indeed.

Answer 2

As an interesting bit of history, Boltzmann was the first one to describe entropy as a "measure of disorder" of a system. It's worth noting that he didn't know this was an oversimplification. In reality, entropy is best described as a measure of the number of ways that energy can be distributed in energy levels between or within particles.

From this, it's clear how the number of possible particle arrangements correlate with entropy: a greater number of particle arrangements means that more energy distributions are possible. I would avoid conflating particle disorder with entropy/microstates though; they are correlated, but not one and the same.

As for how entropy changes are measured experimentally, I would think the easiest way to do that would be dividing the heat lost by the system by the temperature of the surroundings. This essentially tells you how much energy has dispersed.

Answer 3

Here are two common ways of measuring the entropy change in a reaction:

1. Measure the equilibrium constant $K$ at multiple temperatures. This gives you the Gibbs energy (via $\Delta_r G^\circ = - R T \ln{K}$) and, via the van't Hoff relationship, the enthalpy. You can calculate the entropy from those two.
2. Use microcalorimetry in a titrating mode. You get the enthalpy directly, and the equilibrium constant by fitting the trace (which tells you the extent of reaction at different concentrations). From those two, you obtain the entropy.

The entropy change is concentration-dependent, so once you have the standard entropy of reaction, you might have to correct for concentrations that are different from those at standard state.

2) What is the best way to teach entropy (at an introductory level)?

It depends. However, my favorite analogy to use when guessing which state has higher energy is "freedom of atoms to move". This works for solid vs. gas, for bound vs. in solution, and for constrained conformations vs. rotatable bonds. It also hand-wavingly makes the connection to the number of microstates that Martin references in a comment. The other term you sometimes find is "dispersion of energy", which nicely explains how entropy increases when a rock (that fell off a cliff) hits the ground.