Posts Tagged ‘configurational entropy’


Tucked away at the back of Volume One of The Scientific Papers of J. Willard Gibbs, is a brief chapter headed ‘Unpublished Fragments’. It contains a list of nine subject headings for a supplement that Professor Gibbs was planning to write to his famous paper “On the Equilibrium of Heterogeneous Substances”. Sadly, he completed his notes for only two subjects before his death in April 1903, so we will never know what he had in mind to write about the sixth subject in the list: On entropy as mixed-up-ness.

Mixed-up-ness. It’s a catchy phrase, with an easy-to-grasp quality that brings entropy within the compass of minds less given to abstraction. That’s no bad thing, but without Gibbs’ guidance as to exactly what he meant by it, mixed-up-ness has taken on a life of its own and has led to entropy acquiring the derivative associations of chaos and disorder – again, easy-to-grasp ideas since they are fairly familiar occurrences in the lives of just about all of us.

Freed from connexion with more esoteric notions such as spontaneity, entropy has become very easy to recognise in the world around us as a purportedly scientific explanation of all sorts of mixed-up-ness, from unmade beds and untidy piles of paperwork to dysfunctional personal relationships, horse meat in the food chain and the ultimate breakdown of civilization as we know it.

This freely-associated understanding of entropy is now well-entrenched in popular culture and is unlikely to be modified. But in the parallel universe occupied by students of classical thermodynamics, chaotic bed linen and disordered documentation are not seen as entropy-driven manifestations. Sure, how these things come about may defy rational explanation, but they do not happen by themselves. Some external agency, human or otherwise, is always involved.

To physical chemists of the old school like myself, entropy has always been seen as the driver of spontaneously occurring thermodynamic processes, in which the combined entropy of system and surroundings increases to a maximum at equilibrium. This view of entropy partly explains why many of us had difficulty in absorbing the notion of entropy as chaos, since equilibrium always seemed to us a very calm and peaceful thing, quite the opposite of chaos.

Furthermore, we were quite sure that entropy was an extensive property, i.e. one that is dependent on the amount of substance or substances present in a system. But disorder didn’t at all have the feeling of an extensive property. If one (theoretically) divided a thermodynamically disordered system into two equal parts, would each part be half as disordered as the whole? We didn’t think so. To us, there were serious conceptual obstacles to accepting the notion of entropy as disorder.

But while our fundamental understanding of entropy was grounded in the thermal theories of Rankine and Clausius, we did give a statistical nod in the direction of Boltzmann when seeking to explain spontaneous isothermal phenomena. We accepted the notion of aggregation and dispersal as arbiters of entropy change, which we viewed (rightly or wrongly) as separate and distinct from changes in thermal entropy. We even had a name for it – configurational entropy.

Having not one but two different kinds of entropy to play with turned out to be quite useful at times. For example, it helped to explain counter-intuitive spontaneous happenings such as the following:


This is an experiment I remember well from my college days. The diagram shows a sealed Dewar flask containing a supercooled, saturated solution of sodium thiosulphate (aka thiosulfate). A tiny seeding crystal is dropped through a hole in the lid. Crystallization immediately occurs, with an apparent increase in organisation as piles of highly regular crystals form in the solution. It’s an awesome sight to behold.

The experiment provides an unequivocal demonstration that visually-assessed disorganisation and entropy cannot be regarded as synonymous, for while the former unquestionably decreases, the latter must surely increase because the process is spontaneous.

And in overall terms, indeed it does. Although the configurational entropy of the system decreases due to the aggregation of Na+ and S2O32- ions into crystals, the other kind of entropy – thermal entropy – more than compensates as the heat of crystallization causes the temperature of the system to rise. For the whole process ΔSsystem > 0, and therefore ΔSuniverse >0 since the system is isolated from its surroundings.

As I said, having two kinds of entropy to play with can be useful in explaining things that are otherwise counter-intuitive. The above experiment also serves to show that the fashion in popular culture to interpret entropy simply as mixed-up-ness can end up being more than mildly misleading.

One of the devices that college lecturers like to employ when illustrating the concept of entropy is the one pictured above. The narrative goes along these lines: When cream is added to coffee, diffusion and convection produces in the course of time a homogeneous-looking mixture. Compared to its components it is very stable, and never reverts back of its own accord to the initial state of pure cream and black coffee. The diffusion of these two miscible liquids is spontaneous, irreversible and entropy-producing, since the mixture has more microstates, and thus more entropy, than the pure components.

I have never found this illustration particularly helpful, for two reasons.

Firstly, it complicates the picture by mixing together (no pun intended) two distinct concepts. Since the coffee is assumed to be hot and the cream cold, the mixing process involves thermal entropy due to heat exchange as well as configurational entropy due to diffusion.

Secondly, it ignores the spoon.

The spoon has an interesting role, because although stirring doesn’t affect the end point at which configurational entropy attains a maximum, it certainly accelerates the process of reaching it.

In the configurational sense – the sole concern of this article from here on – the spoon represents assisted entropy. The phrase floated into my head one day and remained there, with the consequence that I have developed the habit of noting examples of assisted entropy in the world at large.

The interrelation of work and configurational entropy deserves some mention. One formulation of the second law of thermodynamics is that the direction of spontaneous change in an inanimate system is such that work can be obtained in suitable circumstances. In the coffee and cream example, it is difficult to imagine just how the work accompanying such a spontaneous process would be obtained. But it’s real enough when looked at through the equivalent converse statement, that in order to reverse this mixing process and return the system to its initial configurational state, the amount of work that would need to be provided from the surroundings would be at least the same.

If a process resulting in increased configurational entropy is not spontaneous, but is achieved by doing work on a stable system – mixing together separate piles of black and white peppercorns is an example – then the  amount of work required to return the system to its initial configurational state would again be at least the same.

It’s instructive to apply this latter example of assisted entropy to stable deposits of relatively rare minerals that are sourced, processed, manufactured into products, traded and consumed worldwide, and then discarded.

Think of a lithium ion button battery. The valuable lithium inside it originates perhaps from a concentrated brine on some salt flat in Bolivia, then after a lengthy process of extraction, transportation, processing, manufacture, trading, distribution, retailing and purchasing, it ends up in the pocket calculator of someone in New York. The battery comes to the end of its useful life and then, being a tiny worthless object, it gets tossed without much thought into the trash can. From there it is loaded onto a waste truck that drives it about 1,000 kilometres to Ohio where it is dumped along with countless other lithium ion button batteries into a big hole where it gets mixed in with thousands of tons of other NYC waste.

This is a gigantic exercise in assisted entropy, in which the initial configurational entropy of the lithium in the concentrated brine in Bolivia is vastly increased by the time it reaches the hole in Ohio. The work required simply to recover it, let alone recycle it, is astronomical, as would be the cost. It is commercially irrecoverable. The lithium is lost.

The same applies to any consumer electronics product with a limited lifetime, negligible or zero resale value, and a size small enough to be discarded into unsorted waste that will end up as landfill – mobile phones, MP3 players, pocket cameras, gadgets of that sort. Disposable objects that contain increasing amounts of rare earth elements, that will as a result become more than rare, by becoming irrecoverable.

Market forces are the big spoon of assisted entropy, and the world at large has yet to focus on its irreversible effects. But the need already exists for systems that will enable large-scale recovery of small-scale consumer products containing scarce resources, before they hit the trash can and are lost.