A Molecular Interpretation of Entropy

The laws of thermodynamics do not concern themselves with the molecular structure of matter. But the concept and meaning of entropy can become somewhat clearer when we consider this structure of matter, as suggested by the ensuing discussion (Denbigh). 

According to Denbigh (pp.48-52), once we recognize that matter consists of molecules in a continuous motion, then we can view - in a broad sense - every large scale process as essentially a mixing one. 

To arrive at some qualitative relationship between the concept of entropy and the consequences of this mixing process, let us consider two crystals composed of atoms A and B respectively. 

The crystids are in contact with each other and are enclosed within an adiabatic wall. For simplicity, we will assume that the two crystals are sufficiently alike in their lattice structure, so that the atoms can be interchanged without any change in the energy states of the crystals. Under these conditions we are only concerned with mixing over positions in space (spacial), but not with mixing over energy states. 

We will demonstrate next that this mixing of the two crystals leads to a loss of information about them. For ease of the calculations we will assume that each crystal contains only four atoms. 

---

Thus, a system at equilibrium remains in the same macroscopic state, even though its microscopic state is changing rapidly. There are an enormous number of microscopic states consistent with any given macroscopic state. This concept leads us at once to a molecular interpretation of entropy entropy is q measure of how many different nncroscqpii stqte arje, agivenmqavscopic state. 

Equation 8.1 is a statistical definition of entropy. Defining entropy in terms of probability provides a molecular interpretation of entropy changes as well as allowing for the cglpw iqj enfpja plangi fip qip H such as that of an ideal gas. In... 

Equation (16-2) allows the calculations of changes in the entropy of a substance, specifically by measuring the heat capacities at different temperatures and the enthalpies of phase changes. If the absolute value of the entropy were known at any one temperature, the measurements of changes in entropy in going from that temperature to another temperature would allow the determination of the absolute value of the entropy at the other temperature. The third law of thermodynamics provides the basis for establishing absolute entropies. The law states that the entropy of any perfect crystal is zero (0) at the temperature of absolute zero (OK or -273.15°C). This is understandable in terms of the molecular interpretation of entropy. In a perfect crystal, every atom is fixed in position, and, at absolute zero, every form of internal energy (such as atomic vibrations) has its lowest possible value. 

THE MOLECULAR INTERPRETATION OF ENTROPY AND THETHIRD LAW OFTHERMODYNAMICS On the molecular level, we learn that the entropy of a system is related to the number of accessible microstates. The entropy of the system increases as the randomness of the system increases. The third law of thermodynamics states that, at 0 K, the entropy of a perfect crystaiiine soiid is zero. 

Equation (18.1) provides a useful molecular interpretation of entropy, but is normally not used to calculate the entropy of a system because it is difficult to determine the number of microstates for a macroscopic system containing many molecules. Instead, entropy is obtained by calorimetric methods. In fact, as we will see shortly, it is possible to determine the absolute value of entropy of a substance, called absolute entropy, something we cannot do for energy or enthalpy. Standard entropy is the absolute entropy of a substance at 1 atm and 25°C. (Recall that the standard state refers only to 1 atm. The reason for specifying 25°C is that many processes are carried out at room temperature.) Table 18.1 lists standard entropies of a few elements and compounds Appendix 3 provides a more extensive listing. The units of entropy are J/K or J/K mol for 1 mole of the substance. We use joules rather than kilojoules because entropy values are typically quite small. Entropies of elements and compounds are all positive (that is, S° > 0). By contrast, the standard enthalpy of formation (A//f) for elements in their stable form is arbitrarily set equal to zero, and for compounds, it may be positive or negative. 

Equation (18.1) provides a useful molecular interpretation of entropy, but is normally not used to f-nlenlate the entropy of a system because it is difficult to determine the... 

The concept of entropy was developed so chemists could understand the concept of spontaneity in a chemical system. Entropy is a thermodynamic property that is often associated with the extent of randomness or disorder in a chemical system. In general if a system becomes more spread out, or more random, the system s entropy increases. This is a simplistic view, and a deeper understanding of entropy is derived from Ludwig Boltzmann s molecular interpretation of entropy. He used statistical thermodynamics (which uses statistics and probability) to link the microscopic world (individual particles) and the macroscopic world (bulk samples of particles). The connection between the number of microstates (arrangements) and its entropy is expressed in the Boltzmann equation, S = kin W, where W is the number of microstates and k is the Boltzmann constant, 1.38 x 10 JK L... 

A molecular interpretation of the fact that rubber-like elasticity is primarily entropic in origin had to await H. Staudinger s much more recent demonstration, in the 1920s, that polymers were covalently bonded molecules, rather than being some type of association complex best studied by the colloid chemists [1]. In 1932, W. Kuhn used this observed constancy in volume to point out that the changes in entropy must therefore involve changes in orientations or spatial configurations of the network chains. These basic qualitative ideas are shown in the sketch in Fig. 1.5 [9], where the arrows represent some typical end-to-end vectors of the network chains. 

It is often claimed, with some justification, that statistical theory explains the basis of the second law and provides a mechanistic interpretation of thermodynamic quantities. For example, the Boltzmann expression for entropy is S = ks In W, where W is the number of ways the total energy is distributed among the molecules. Thus entropy in statistical theory is connected to the availability of microstates for the system. In classical theory, on the other hand, there are no pictures associated with entropy. Hence the molecular interpretation of entropy changes depends on statistical theory. 

Application of this criterion requires the determination of entropy changes, which are considered next, followed by a series of Examples that demonstrate how these entropy changes are used in applications of the second law, developing thus our ability to use it. (To provide for some understanding of the concept of entropy, however, a molecular interpretation of it is presented before the Examples). 

But first, to develop some sense about the physical meaning of entropy, we consider a molecular interpretation of it. 

In Chapter 7 (Thermochemistry), we have updated the notation to ensure that we are using, for the most part, symbols that are recommended by the lUPAC. For example, standard enthalpies of reaction are represented by the symbol (not AH°) and are expressed in kj mol (not kj). We have added a molecular interpretation of specific heat capacities (in Section 7-2) and an introduction to entropy (in Section 7-10). 

As we have seen, the third law of thermodynamics is closely tied to a statistical view of entropy. It is hard to discuss its implications from the exclusively macroscopic view of classical themiodynamics, but the problems become almost trivial when the molecular view of statistical themiodynamics is introduced. Guggenlieim (1949) has noted that the usefiihiess of a molecular view is not unique to the situation of substances at low temperatures, that there are other limiting situations where molecular ideas are helpfid in interpreting general experimental results ... 

This is a law about the equilibrium state, when macroscopic change has ceased it is the state, according to the law, of maximum entropy. It is not really a law about nonequilibrium per se, not in any quantitative sense, although the law does introduce the notion of a nonequilibrium state constrained with respect to structure. By implication, entropy is perfectly well defined in such a nonequilibrium macrostate (otherwise, how could it increase ), and this constrained entropy is less than the equilibrium entropy. Entropy itself is left undefined by the Second Law, and it was only later that Boltzmann provided the physical interpretation of entropy as the number of molecular configurations in a macrostate. This gave birth to his probability distribution and hence to equilibrium statistical mechanics. 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

The lateness of this development is perhaps surprising, yet a mere catalogue of entropies would have been useless before data became available to interpret the entropies in terms of molecular structure and reaction mechanisms. Data of this sort are now being obtained by means of new tools which probe into the molecular structure of adsorbed species. 

Over a long period of time experimental results on amphiphilic monolayers were limited to surface pressure-area ( r-A) isotherms only. As described in sections 3.3 and 4, from tc[A) Isotherms, measured under various conditions, it is possible to obtain 2D-compressibilities, dilation moduli, thermal expansivities, and several thermodynamic characteristics, like the Gibbs and Helmholtz energy, the energy cmd entropy per unit area. In addition, from breaks in the r(A) curves phase transitions can in principle be localized. All this information has a phenomenological nature. For Instance, notions as common as liquid-expanded or liquid-condensed cannot be given a molecular Interpretation. To penetrate further into understanding monolayers at the molecular level a variety of additional experimental techniques is now available. We will discuss these in this section. 

From Equation 13.9 we see that the entropy of a gas increases during an isothermal expansion (V2 > Vi) and decreases during a compression (V2 < Vt). Boltzmann s relation (see Eq. 13.1) provides the molecular interpretation of these results. The number of microstates available to the system, H, increases as the volume of the system increases and decreases as volume decreases, and the entropy of the system increases or decreases accordingly. 

The doctrine of molecular chaos, leading to the interpretation of entropy as probability, is in a somewhat different case again. It is based, though not upon direct experiment, upon the primary hypothesis of aU chemistry, that of the existence of molecules, and upon the assumption, common to most of physics, that these particles are in motion. It is related very closely to such facts of common observation as diffusion and evaporation, and it takes its place among the major theories about the nature of things. In scope and significance it is of a different order from rather colourless assertions about the geometry of lines and surfaces constructed with the variables of state. 

This section is devoted to a detailed elaboration of thermodynamic quantities employed in the literature to deal with dilute solutions. These are the so-called standard free energies, entropies, enthalpies, etc. associated with the transfer of a solute from one phase to another. There are actually many quantities which are referred to as standard quantities. We deal essentially with one of these, the one which is most directly amenable to a molecular interpretation. 

---