Entropy Measures Amount of Choice R. M. Swanson Biochemistry & Biophysics, Texas A & M University, College Station, TX 77843 One readily applies the concept of energy in thinking about molecular behavior but seldom makes use of e n t r o ~ v in the same way. Yet entropy is just as significant as energy in characterizing the behavior of moleculev in chemical svstems. The purpose of this paper is to present a correct and simple interpretation of entropy in terms of per-molecule beha\,ior, and LO put entropy on a closer toequal footing with energy as an intuitively useful chemical concept. Even when the total energy of a chemical system is constant, the enerev of anv particular molecule in a chemical system repeated& changes, as a result of collisions and other interactions with neighbors. The "enerw Der molecule" reoresents the average kalue of this varying quantity. A fair characterization of the "entropy per molecule" is that i t describes the number of different energy states that any particular molecule in the system eventually occupies. The body of this article begins by introducing a way of characterizing the variety available in a mixture, the "opt i o n ~ " and , relates this concept to information theory and to chemical entropy. With the insight provided by this discussion, the entropy of melting ice is compared with the results of a molecular dynamics simulation of the behavior of liquid water. The Concept of an Effective Number of Chokes
Imagine a container with 10,000 peanuts in it. Imagine also a second container with 9,999 peanuts and one cashew, a third with 9.900 oeanuts and 100 cashews. and a fourth with 5,000 peanuts a i d 5,000 cashews. In the krst container the varietv available to a oerson blindlv removine a sinele nut is one. I* the last contaiher it's two. &t what agout thk second and third containers? One's intuition suggests that even though every container but the first actually holds two different kinds of nuts, the effective variety offered by some of them is smaller than the others. The container with 9,999 peanuts and one cashew seems more like the container with only peanuts than like the one with equal amounts of each kind of nut. An interesting way t o characterize the effective variety offered by the different containers would he to assign a number greater than 1 but less than 2 t o the 1:9,999 and 100:9,990 containers. An expression (based on the information theory expression for entropy) that will satisfy this requirement is
py.) Expression 1 above has the following connections to entroov. .. In information theory (I) entropy is calculated from an exoression that is the loearithm of the one above. Similarlv. -. the chemical entropy 2 mixing is given hy an analogous expression in terms of mole fractions (2).Further. the same fo;m is used for the entropy in a canonical ensemble (3). The relation between eq 1 and the statistical mechanical expression, entropy = k X i n W ( 4 ) ,can be demonstrated by straightforward, but tedious, algebra, which I will send to any interested reader on request. The two expressions yield the same value for entropy under the assumptions normally made. The main conceptual difference between them is that eq 1 transforms entropy into a number that describes the number of different molecule states a molecule mav occu~v. while the statistical mechanical expression gives ehtropy'k terms of the loearithm of the number of multinarticle states accessible to a-macroscopic numher of mole&les. The insight I want to convey is how the macroscopic, abstract, multiparticle concept is expressible in terms of a concrete, molecular description. Although the illustration above was in terms of just two kinds of things (nuts), the concept is applicable to any numher of components, with any arbitrary proportions. A more chemical example with more possible species of things is the followine. ~ o n s i d e rsymmetrically substituted dibromethane. How many "different kinds" of dibromoethane are there in a mole of kolecules? One way to think about this problem is to consider that each molecule has one hond with a 3-minimum rotational potential, and therefore three main conformational states. In addition to exhibiting these main conformations, a molecule occasionally passes through higher energy ("in between") conformations. The molecules in the main conformations are like three major kinds of nuts. Their contribution to the optiony is neatest when all three main conformations are eouallv . .occupied and is reduced by inequalities of population. For example, perhaps the "trans" conformation is more favorble than the othe; conformations-then more molecules will be found in it, fewer will be found in the other conformations, and the optiony will be lowered, just as with the excess of peanuts in the 9,999:l (optiony = 1.001) or 9,990:100 (optiony = 1.06) cases compared with the 5,000:5,000 (optiony = 2.) case. Higher energy conformations are like scarcer nuts.
Opllony fw Sample Mixtures where the quantity yjspecifies the fraction of the totalrepresented hy the ith kind of nut. Call the effective variety represented by a collection of nuts its "optiony" and calculate it as above. The table displays numbers for the containers described, as well as for some additional compositions. Evidently, the optiony of a container describes in a quantitative way an intuitive impression about the variety of its contents. O ~ t i o n vis closelv related t o e n t r o ~ v .(The relation of optiony to entropy is somewhat like therelation of hydrogen ion concentration t o pH; optiony is the exponential of entro208
Journal of Chemical Education
Species:
Fractions:
peanut
cashew
1.0000 0.9999 0.0001 0.9900 0.0100 0.5000 0.5000
almond
walnut
pecan
Optiony
-
-
-
1. 1.001
-
-
-
1.06 2~
They do add to the variety of conformations availahle, but not greatly, and the rarer they are, the less they add. The "kinds" of dibromoethane are identified in terms of their eeometric conformations for convenience. A more complete,-though still approximate, description is that each mole c d e exists in a particular quantum state with a particular energy. In each quantum state, there is aprobability density associated with the positions of the component electrons and nuclei, so that strictly speaking a molecule does not have a single rigidly defined conformation. However, only a relatively small proportion of the possible arrangements of the electrons and nuclei have a significant probability of occurrine. We can and do describe these fluid but freauentlv occurring arrangements in terms of fixed geometries. Furthercorrectionsto thisdescriptionarestill needed tocomply with our best theoretical notions of the behavior of interacting individual molecules, but i t is adequate here toacknowledge both the usefulness and the limitations of the simplified description given. I t is satisfadorv to use simolified descrivtions in talkine ahout entropy if
