How Can I Interpret Entropy Changes?

An increase of entropy is often stated to be equivalent to an increase of disorder or randomness or mixed-upness or probability when these are simply short- hand for the number of accessible eigenstates (energy) for an isolated system. Th e number of eigenstates is directly related to the concept of “mixed- upness” in two special cases which serve to illustrate the conceptual relationship. First we suppose we have two noninteracting gases each in an isolated container and separated from each other by an impermeable membrane. When the mem- brane is broken, an increased volume is available for each molecule and in addi- tion the number of available combinations of translational energy eigenvalues increase, which we have seen make up its energy. Second, we consider crystals at temperatures close to zero where the geometrical orientations of the mole- cules on the lattice sites may be regular or irregular and may be ordered or dis- ordered. In both cases, the number of accessible eigenstates is simply related to the purely geometrical or spatial “disorder” and also entropy.

However, for normal and realizable chemical processes there is no sim- ple geometrical interpretation of the entropy change and it is not possible to extrapolate statistical-mechanical conclusions for systems of noninteracting particles or crystals at T → 0 (discussed in Chapter 3 with Nernst’s heat theo- rem) to beakers of liquids at T = 293 K.

Entropy is a state variable with the same status as temperature and pres- sure and is measurable (or at least diff erences are). Let us accept this simple and refreshing statement as a fact and, after introducing the second law in Chapter 3, review again the misconception of mixed-upness and better still ask another question: How would you measure the entropy change that accompa- nies the mixing of two gases?

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101

2

_{Law of Thermodynamics}

3.1 INTRODUCTION

In Chapter 1 of this book we argue that thermodynamics is an experimental science consisting of a collection of axioms, derivable from statistical mechan- ics and in many circumstances from Boltzmann’s distribution. So far we have introduced the 0th _{law and the 1}st _{law of thermodynamics that interrelate phys-} ical quantities some of which are far more easily measured than others. We are also armed with two types of “thermodynamic-meter”: (1) a thermometer to measure temperature and (2) a calorimeter used to measure diff erences in energy and enthalpy. Both of these will be put to good use in this chapter, which considers the 2nd _{law of thermodynamics. We will also introduce a third “meter”:} a chemical potentiometer used to measure diff erences in chemical potential.

Clausius provided the fi rst broad statements of the 2nd _{law of thermodynamics} (1850a, 1850b, and 1851) and these were refi ned by Th omson,* and those readers interested in the history of the formulation of the laws of thermodynamics should consider consulting the work of Atkins (2007) and Rowlinson (2003 and 2005).