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Notes for a “Conversation About Entropy”

November 8, 2005 at California State Polytechnic University, Pomona

A clarification of the meaning of “configurational” entropy:
the  measure of   energy    dispersal
in statistical mechanics calculations.

1.       When thermodynamic entropy in chemistry is properly viewed as a combination of two factors, both the misleading term, ‘configurational’ (‘positional’) entropy, and the chasm between thermodynamic entropy and information “entropy” are clarified:

          “Energy of all types changes from being localized to becoming dispersed or spread out, if it is not hindered from doing so.  The overall process is an increase in thermodynamic entropy, enabled in chemistry by the motional energy of molecules (that, in chemical reactions, arises from bond energy change) and actualized because the process makes available a larger number of microstates, a maximal probability.”

           The two factors, energy and probability, are both necessary for thermodynamic entropy change but neither is sufficient alone.  In sharp contrast, information ‘entropy’ depends only on  -k Σ[Pi log Pi], where k is an arbitrary constant that never is required to involve energy, as does kB. (‘Sigma entropy’ in physics, σ = S/kB, depends only on the one factor of probability, ln W, where all states of W are equally probable.)

2.       Thermal and ‘configurational’ entropy both involve calculation of an actualization factor of increased microstates, but ‘configurational’ entropy calculations via statistical mechanics seem to have no enabling factor of molecular motional energy. (In actuality, those stat mech results are the macro measure of how widely the initial energy of the system has been dispersed in gas expansion and all types of mixing. )

          The processes in thermal entropy are the transfer of motional energy from hotter surroundings to a cooler system, either the irreversible transfer in heating a cooler system, or a reversible phase change.  In both, energy dispersal and entropy increase occur because the increase in energy in the system results in additional accessible microstates.  (In simply heating a system, additional higher energy levels become accessible.  In phase change from solid to liquid or liquid to vapor, translational energy levels become closer together.  Both lead to many additional accessible microstates.)

          The processes in ‘configurational or positional’ entropy are gas expansion into a vacuum, gas or liquid mixing, or dissolving of a solid solute in a solvent.  Each of those processes involves a literal spreading out of the motional energy of a substance in space — gas expansion into a vacuum being obviously so.  The mixing of two different liquids (or solute in solvent) is a more subtle spreading of each component’s energetic molecules, but clearly they become more separated from one another than when in the pure state they were colliding only with identical adjacent molecules.  (This may involve only a slight change in volume but all mixing processes result in an increase in the density of energy levels.  Consequently, there is a large increase in the number of accessible microstates for the final mixture just as in a pure gas after expansion.)

          A designation of ‘configurational’ or ‘positional’ entropy is an unfortunate artifact from poor communication by statistical mechanics experts.  They developed the mathematics of entropy change in mixing and gas expansion well but never described the details of what was occurring energy-wise behind their calculations.  The result has been that, unintentionally, they have misled generations of textbook authors and beginning students.

          When countable numbers of cells in a system are used as a model, each of those cells is a microstate containing the entire energy of the tiny countable system.  In the next step, factorial or combinatoric calculations extend the count to a system of N molecules and then the final results are in terms of V2/V1 for gas expansion and moles with mole fractions (n1 ln X1 + n2 ln X2) for mixing..Statistical mechanics writers and general chemistry text authors thereby lead beginners to focus on the increase in positions or configurations of molecules in space — the ‘actualizing’ probability factor — rather than on the enormous increase in numbers of microstates for the original energy, the “enabling” energy factor.  Entropy is an energy function and each of those combinatoric positions/configurations associated with a final ‘position’ represents a microstate —  an accessible energy distribution for that system.  Thus, the variables of volume or of mole fraction in “configurational” entropy measure how much more spread out in the final state the original motional energy has become 

          As I emphasized in “Entropy Is Simple, Qualitatively”, a greatly increased number of microstates for any system (including the mixtures that we have been discussing) means that its energy has become more dispersed — but not in the sense of the total energy of the system becoming smeared or spread out over more microstates!  Rather, it is the lesser probability that the system in one microstate at one instant would return to that microstate in the next instant (i.e., be localized) but instead would have many many more ‘septillions’ of choices of microstates to change to each moment than would the pure components.

          Texts that develop “positional” or “configurational” entropy by showing dots of molecules in a glass bulb and speaking solely about their additional probable locations if a stopcock is opened to another bulb (i.e., a final greater volume) are actually ‘smuggling in’ thermodynamic entropy.  All of those probable locations (the ‘second factor’ of thermodynamic entropy) are meaningless picture-drawing unless there are molecules that have motional energy (the ‘first factor’ of thermo entropy) to enable them to spread out to such locations.  A focus on ‘probability’ that doesn’t include motional energy makes the whole process of gas expansion appear to be an example of information “entropy” that is no different from somebody having flipped coins or tossed dice so that these static objects fell in probable arrangements.  Coins don’t spontaneously flip.   But molecules, because of their inherent motion above 0 K. are enabled to move to probable locations if that movement is allowed or ‘actualized’ by some process.  “Some process” includes not only opening a stopcock to a larger volume for a gas but also mixing of gases or liquids, and dissolving a solid in a solvent.  Each leads to an increase in entropy that is related to the initial unchanged molecular energy becoming more dispersed. (More fundamentally, of course, the final result is an increase of accessible microstates for the system of molecules in the new larger volume or solution.)

          Four chemistry texts that talk about “the dispersal of energy” occurring when substances are heated or melt or boil, unfortunately also state that “the dispersal of matter” causes molecules to move spontaneously into a larger volume and entropy to increase in mixing.  This is a hapless variation of ‘configurational’ entropy.  Molecules cannot mix or move to spread out to a greater volume without motional energy — any aspect of “the dispersal of matter” is implicit in viewing and measuring the processes as due to the dispersal of energy.

 

entropy.lambert@gmail.com
Created: November 2005

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