In the Classroom Entropy Is Simple, Qualitatively Frank L. Lambert† 2834 Lewis Drive, La Verne, CA 91750; [email protected] Qualitatively, entropy is simple. What it is, why it is so bond energies compared to water, yet they will react explo- useful in understanding the behavior of macro systems or of sively to dissipate some of that internal energy in forming molecular systems is easy to state. The key to simplicity has the lower energy water if a spark is introduced. been mentioned by chemists and physicists.1 In classical ther- In these six varied examples, and in all everyday sponta- modynamics, it is viewing entropy increase as a measure of neous physical happenings and chemical reactions, some type the dispersal of energy from localized to spread out (as in of energy flows from being localized or concentrated to be- the examples of the next section). In “molecular thermody- coming spread out to a larger space. This common language namics” it is considering the change in systems from having summary will be given precise descriptions in thermodynam- fewer accessible microstates to having many more accessible ics and in molecular behavior later. microstates (as described in later sections). Most important In most spontaneous chemical reactions the products to educators, emphasizing energy dispersal can provide a di- that are formed have lower potential energies than the start- rect relationship of molecular behavior to entropy change, ing materials. Thus, as a result of such a reaction, some of an idea that can be readily understood by beginning college the reactants’ bond energy is dispersed to cause increased students. molecular motion in the products. Less common at room This view was broached partially in previous articles that temperatures (but routine in blast furnace conditions) are showed “disorder” was outmoded and misleading as a descrip- endothermic reactions. In this kind of reaction, substances tor for entropy (1, 2). The present article substantiates the (alone or with others) react so that energy from the more- power of seeing entropy as the “spreading and sharing” of concentrated-energy surroundings is dispersed to the bonds and energy (theoretically established by Leff [3a]) in classical and motions of the molecules of the less-concentrated-energy sub- molecular thermodynamics in important equilibrium situa- stances in a system. The result is the formation of new sub- tions and chemical processes. stances with higher potential energies than their starting The preceding introduction in no way denies the subtlety materials. or the difficulty of many aspects of thermodynamics involv- Parallel with the behavior of energy in chemical phenom- ing entropy. In numerous cases the theory or calculation or ena is its dispersal in all spontaneous physical events, whether experimental determination of entropy change can be over- they are complex and exotic, or as simple and too common whelming even to graduate students and challenging to ex- as a car collision.2 perts. In complex cases, the qualitative relation of energy dispersal to entropy change can be so inextricably obscured Entropy Change: The Index of Energy Dispersal that it is moot. However, this does not weaken its explana- Clausius’ definition of entropy change, dS = dqrev/T, tory power in the common thermodynamic examples pre- could be expressed verbally as “entropy change = quantity of sented to beginning students, nor in using energy dispersal energy dispersed at a temperature T ”. Clausius’ synonym for to introduce them to spontaneous change in chemistry. entropy was Verwandlung, transformation. This definition does not imply that entropy is “disorder”, nor that it is a mea- An Overview: What Entropy Change Is sure of “disorder”, nor that entropy is a driving force. In- 2 stead, the reason for transformation in classical The Dispersal of Energy thermodynamics is energy’s dissipating, dispersing, spread- Downloaded via UNIV OF SAO PAULO on December 3, 2018 at 15:00:39 (UTC). A hot pan spontaneously disperses some of its energy to ing out from where it is confined in a small space to a larger the cooler air of a room. Conversely, even a cool room would volume whenever it is not restricted from doing so. Entropy disperse a portion of its energy to colder ice cubes placed in is the measure or the index of energy dispersal at a tempera- See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. it. When a container of nitroglycerine is merely dropped on ture T . the floor, the nitroglycerine may change into other substances explosively, because some of its internal bond energy is spread The Molecular Basis for Understanding Simple Entropy out in increasing the vigorous motions of the molecules in Change (3a, 4) the gaseous products. At a high pressure in a car tire, the com- The difference between Figure 1A and B is symbolic of a pressed air tends to blow out and dissipate its more concen- transition of a molecule from a lower translational (t), or ro- trated energy to the lower pressure atmosphere. At any tational (r), or vibrational (v) energy level to a considerably pressure, ideal gases will spontaneously flow into an evacu- higher level of the same kind of motion. ated chamber, spreading the energy of their molecular mo- The energy differences between levels in a truly accu- tions over the final larger volume. Hydrogen and oxygen in rate depiction would be quantized from specific spectral lines. a closed chamber will remain unchanged for years and prob- Yet, Figure 1 is even more profoundly inaccurate than its ar- ably for millennia, despite their greater combined internal bitrary line spacing. Modern descriptions of molecular ener- getics come from quantum mechanics and the Schrödinger equation. Wave functions of the Schrödinger equation for †Professor Emeritus, Occidental College, Department of Chem- molecules that are quantified in microstates inseparably com- istry, Los Angeles, CA 90041. bine two disparate qualities: energy, and the probability of po- JChemEd.chem.wisc.edu • Vol. 79 No. 10 October 2002 • Journal of Chemical Education 1241 In the Classroom 5 4 Energy Energy 3 2 1 Smaller Larger AB Volume Volume Figure 1. Microstate (single-particle) at (A) lower temperature and Figure 2. Microstate (multi-particle) in a smaller volume compared (B) higher temperature. to an equal energy microstate of the same system in a larger vol- ume. The original energy becomes more dispersed. sition in space of the molecules. Thus, although Figure 1 could ticular macrostate. Because the number of microstates deter- be said to represent microstates, it is not an adequate sche- mines the number of ways that the total energy of a matic because it shows only energy relationships; it omits the macrostate can be spread out, any change that leads to a absolutely non-separable space occupation that is integral in the greater number of microstates means that there is greater dis- mathematics. persion of the energy in a system, that is, an increase in its Therefore, Figure 1A and B, an attempt to show the en- entropy. ergy change between two microstates, is really only symbolic Therefore, when a substance is heated, its entropy in- of microstates because the likelihood of locations in space of creases because the energy acquired and that previously within the molecule is not shown. (The location is tacitly unchanged it can be far more dispersed than before among the many between A and B.) With this omission understood in any newly accessible microstates associated with the molecules of such diagram of a microstate, Figure 1A can be considered the substance. A simplified statement would be that there as the microstate of a molecule at a moderate temperature are many more microstates available for energy dispersal. (The with t, r, and v total energies corresponding to a quantum common comment “heating causes or favors molecular dis- energy level, arbitrarily numbered 1. The molecule cannot order” is an anthropomorphic labeling of molecular behav- “access” the higher quantum levels arbitrarily numbered 2 to ior that has more flaws than utility) (1). 5 (i.e., they are not “available” to it) because the molecule’s When the volume of a gas is increased by isothermal ex- energy simply is not great enough to correspond to those lev- pansion into a vacuum, an entropy increase occurs, although els. However, if the single-molecule system of 1A is heated, not for the same reason as when a gas or other substance is the additional energy—dependent on the quanta involved— heated. There is no energy change in such an expansion of allows the molecule’s energy to be on any one of the several the gas; dq is zero. Instead, there is an increase in the density higher energy quantum levels 2 to 5, including that shown of the microstates in the gas. As symbolically shown in Fig- in Figure 1B. With the addition of energy, those variations ure 2, when a gas expands into a vacuum, the number of of Figure 1B are then said to be available or accessible or oc- microstates increases in any small energy range of the origi- cupiable microstates. nal volume when a larger volume is available. For those mol- Although it is symbolic only of a single particle’s behav- ecules whose energy is associated with that particular energy ior, the example below can serve as a guide to what happens range, this results in an increased dispersion of their original in the enormous group of molecules in a macro system. When energy because it can now be spread out to a greater number a mole of a substance is heated, the 1023 molecules (~N) are of microstates than before the gas expanded. Because of such able to access many microstates that they could not before.