What if entropy were dimensionless? Harvey S. Leffa) Physics Department, California State Polytechnic University, Pomona, 3801 West Temple Avenue, Pomona, California 91768 ͑Received 23 July 1998; accepted 2 March 1999͒ One of entropy’s puzzling aspects is its dimensions of energy/temperature. A review of thermodynamics and statistical mechanics leads to six conclusions: ͑1͒ Entropy’s dimensions are linked to the definition of the Kelvin temperature scale. ͑2͒ Entropy can be defined to be dimensionless when temperature T is defined as an energy ͑dubbed tempergy͒. ͑3͒ Dimensionless entropy per particle typically is between 0 and ϳ80. Its value facilitates comparisons among materials and estimates of the number of accessible states. ͑4͒ Using dimensionless entropy and tempergy, Boltzmann’s constant k is unnecessary. ͑5͒ Tempergy, kT, does not generally represent a stored system energy. ͑6͒ When the ͑extensive͒ heat capacity Cӷk, tempergy is the energy transfer required to increase the dimensionless entropy by unity. © 1999 American Association of Physics Teachers. I. INTRODUCTION thus the energy spectrum region where the system spends most time executing its ‘‘entropy dance.’’ Section VI con- Entropy has many faces. It has been interpreted as a mea- tains a recapitulation and main conclusions. sure of disorder,1 the unavailability of work,2 the degree of energy spreading,3 and the number of accessible microstates4 in macroscopic physical systems. While each of these inter- II. DEFINITIONS OF ENTROPY AND THE pretations has a logical basis, it is unclear why any of them ‘‘ENTROPY DANCE’’ should have the dimensions of energy per unit temperature. Indeed the common entropy unit, J KϪ1, is as mysterious as Clausius defined the entropy change dS of a system at temperature T, in terms of an infinitesimal reversible heat entropy itself. The goal here is to answer the following ques- ␦ ͑ ͒ process during which the system gains energy Qrev . His tions: i Why does entropy have the dimensions of energy 11 per unit temperature? ͑ii͒ Are these dimensions necessary famous entropy algorithm is ͑ ͒ ␦ and/or physically significant? iii How would thermal phys- Qrev ics differ if entropy were defined to be dimensionless and dSϭ . ͑1͒ T temperature were defined as an energy? ͑iv͒ What range of values does dimensionless entropy per particle have? ͑v͒ If Extending this process along a finite path with a finite energy temperature is defined as an energy, called tempergy, does it transfer, integration of ͑1͒ leads to an entropy difference Ϫ have a meaningful physical interpretation? An examination Sfinal Sinitial . This procedure is commonly used as a calcu- of thermal physics books5–10 shows that although some au- lation tool to obtain path-independent entropy differences. It thors define entropy as a dimensionless quantity and some is evident from Eq. ͑1͒ that S has the dimensions of energy define temperature as an energy, no in-depth discussion of per unit temperature ͑common units are J/K͒. questions ͑i͒–͑v͒ seems to exist. What follows is in part a Notably, Clausius used the Kelvin temperature T in ͑1͒. review and synthesis of ideas that are scattered throughout This can be traced to the definition of the Kelvin temperature textbooks, with the goal of filling this gap. scale and the following property of dilute gases. If an N In Sec. II, we review common definitions of entropy in particle dilute gas has pressure p and volume V, then thermodynamics, statistical mechanics, and information pV theory. In Sec. III we observe that in each case one can ϭ␶ϭfunction of temperature. ͑2͒ replace S by a dimensionless entropy Sd and that, in a sense, N entropy is inherently dimensionless. We show how hand- The Kelvin temperature T is defined to be proportional to ␶ book entropy values can be converted to dimensionless en- and to have the value T ϭ273.16 K at the triple point of tropies per particle, assess the range and meaning of the lat- 3 ter numerical values, and use them to estimate the number of H2O. It follows that accessible states. Defining entropy as a dimensionless quan- ␶ϭkT, ͑3͒ tity leads naturally to a definition of temperature as an en- ϭ ergy ͑tempergy͒. In Sec. IV, we argue that although in very where the constant of proportionality, k 1.3807 Ϫ23 Ϫ1 special cases, tempergy is a good indicator of the internal ϫ10 JK , is Boltzmann’s constant. Extrapolation of ͑2͒ energy per particle, per degree of freedom, or per elementary to low temperatures shows that Tϭtϩ273.15, where t is the excitation, no such property holds in general. Thus tempergy Celsius temperature. The above-mentioned specification of cannot be interpreted generally as an identifiable stored en- T3 assures that the Kelvin and Celsius scales have equal ergy. In Sec. V we show that under typical laboratory con- degree sizes. An interpretation of Eq. ͑3͒ is that k is a con- ditions, tempergy is the energy transfer needed to increase a version factor that links temperature and energy. The product system’s dimensionless entropy by unity. In addition, tem- kT, which is ubiquitous in thermal physics, enables correct pergy shares with temperature the property of determining dimensions whenever an energy, heat capacity, or pressure is the internal energy ͑quantum statistical average energy͒ and expressed as a function of T in Kelvins. 1114 Am. J. Phys. 67 ͑12͒, December 1999 © 1999 American Association of Physics Teachers 1114 Modern-day values of Boltzmann’s constant are obtained a ‘‘spreading’’ function.3 This ‘‘spreading’’ can usefully be from data on the universal gas constant R envisioned as a dance over the system’s accessible states. ϭ8.3145 J KϪ1 molϪ1, evaluated from dilute gas data, and When the dance involves more states ͑or phase space vol- 12–15 ͑ ͒ ͑ ͒ ume͒ then the entropy is greater; thus we call it an entropy Avogadro’s number, NA . That is, combining 2 and 3 with the corresponding standard molar equation pVϭnRT, dance. Last, but not least, we observe that although the factor ϭ ϭ k in Eq. ͑6͒ gives S its conventional dimensions of energy/ where n is the mole number, gives k R/(N/n) R/NA .We note that k and R are two versions of the same constant using temperature, S/k contains all the essential physics. In this different unit systems. sense, entropy is inherently a dimensionless entity. Given the definition of temperature above and the fact that In information theory, the missing information, MI, is de- all reversible Carnot cycles operating between fixed reser- fined as voir temperatures T and T have efficiency 1ϪT /T , one c h c h ϭϪ ͑ ͒ finds that for any such Carnot cycle, MI c͚ Pi ln Pi , 7 i Qh Qc ϩ ϭ0, ͑4͒ where Pi is the probability of finding the system in state i, T T h c with energy eigenvalue Ei , and c is an arbitrary multiplica- tive constant.20 Equilibrium statistical mechanics at constant where (Qh ,Qc) are the energies gained by the working fluid at (T ,T ). In many textbooks, these properties are used to temperature can be obtained by maximizing the function MI h c under the constraint that the average energy is fixed.21 This obtain the Clausius inequality, ⌺ Q /T р0, for a cycle with i i i procedure leads to the conclusion that if one makes the iden- segments iϭ1,2,..., and this inequality is then used to obtain tifications SϭMI and cϭkϭBoltzmann’s constant, then the entropy algorithm, Eq. ͑1͒.16,17 consistency with classical thermodynamics is obtained, and In Callen’s modern formulation of classical thermodynamics,18 the entropy function S is postulated to ϭ Ϫ1 ͑ ␶͒ ϭ ͑ ␶͒ ͑ ͒ exist for equilibrium thermodynamic states. It is specified to Pi Z exp Ei / with Z ͚ exp Ei / . 8 i be extensive, additive over constituent subsystems, and to ϭ ץ ץ vanish in a state for which ( U/ S)V,N 0. Here U is the Z is the canonical partition function of statistical mechanics. internal energy and the derivative is taken holding particle Notably, in both Pi and Z, temperature occurs only through numbers and external parameters fixed. The dimensions of S the product ␶ϭkT. As in the microcanonical formalism, en- come from the requirement that the latter derivative be iden- tropy’s dimensions come solely from the factor k. Here the tified as the Kelvin temperature; i.e., essential physics is contained within the set ͕Pi͖. In this -S͒ ϭT, ͑5͒ sense, the canonical ensemble entropy is inherently dimenץ/Uץ͑ V,N sionless. and this gives S the dimensions energy/temperature. Again the dimensions of S are linked to the definition of tempera- III. DIMENSIONLESS ENTROPY AND TEMPERGY ture. In statistical physics, entropy is usually defined using the The arguments above suggest that the tempergy, ␶ϭkT, Boltzmann form, has significance. Nevertheless, it is the temperature T, and ␶ 22 Sϭk ln W. ͑6͒ not , that is commonly used as a thermodynamic variable. Recall that ␶ is traditionally written as kT in order to obtain The factor k is Boltzmann’s constant, which was actually an absolute temperature scale with a degree size equal to the introduced into Eq. ͑6͒ by Planck. Using radiation data, Celsius degree. Had the temperature been defined as ␶, then Planck obtained the first numerical value of k.19 In classical ͑ ͒ ␶ ␶ Th and Tc in Eq. 4 would have been replaced by h and c , statistical mechanics, W is proportional to the total phase respectively. Similarly, the Clausius algorithm, ͑1͒, would space volume accessible to the system. This volume is in the have taken the form 6N-dimensional phase space consisting of the position and ␦ ␦ Qrev Qrev momentum coordinates for the N particles.