Rescaling Temperature and Entropy - American Chemical Society

In the Classroom

Rescaling Temperature and Entropy John Olmsted III Professor Emeritus of Chemistry, Department of Chemistry & Biochemistry, California State University, Fullerton, Fullerton, California 92834-6866 [email protected]

Temperature, T, and entropy, S, are fundamental thermodynamic properties whose origins lie in classic studies of heat as a macroscopic phenomenon. Both also can be reinterpreted at the molecular scale, yet the most common interpretations of these quantities remain the macroscopic ones. Chemists routinely work with temperature and frequently need to calculate and interpret entropy changes, yet trying to pin down what these two quantities are and how they relate to energy is anything but routine. Neither quantity has a clear and intuitive definition that matches the definition of energy as the ability to do work. Furthermore, the units traditionally used for entropy, J/K or J/(mol K), convey little sense of what entropy is. The aim of this article is to show that a reconsideration of the units associated with these quantities helps to clarify their natures. Temperature and energy are closely related. Energy flows naturally from an object at higher temperature to one at lower temperature, indicating that a higher temperature means a greater intensity of energy. The usual units used for temperature obscure this relationship. Even though temperature is one of the seven base units in the SI system, the relationship between temperature and energy suggests that one of these properties can be expressed in terms of the other. In physics, such considerations are part of a larger exploration associated with “natural” units,1 the adoption of which can transform the values of many fundamental constants to unity. These efforts at dimensional analysis were initiated by Stoney in 1881 (1), but the most prevalent set of natural units are those first proposed by Max Planck in 1899 (2, 3). The apparent advantages of having unit values for constants are outweighed by the inconvenient magnitudes of natural units; consequently, natural units have not supplanted SI units in the realm of chemical thermodynamics. Nevertheless, some eminent chemists have recognized the advantages of expressing temperature and energy in the same units. In his revision of the classic thermodynamics text by Lewis and Randall, Pitzer observed, “...there are advantages in dividing energies by R, whereupon the values have the dimensions temperature...” and adds “we believe that an energy in kelvins is more easily interpreted than a value in joules ” (4). Whereas Pitzer advocated expressing energies in kelvins, Kalanin and Kononogov have more recently proposed that the best choice for thermodynamic purposes is the Planck temperature, Θ = kBT, expressed in joules (5). If energy is expressed in kelvins or temperature is expressed in joules, entropy becomes rescaled and is a dimensionless quantity, as was pointed out by Pitzer and Brewer a number of years ago (6). This is consistent with entropy being a measure of probability or number of accessible microstates of a system. It is also consistent with the Shannon formulation of information entropy (7).

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This article explores the insights that can be gleaned from rescaling temperature and entropy, defining rescaled temperature as T0 = kBT and rescaled entropy as S0 = S/kB. Leff has developed a similar treatment, albeit from a thermal physics perspective (8). Temperature and Energy The qualitative notion of temperature, in the sense of hot or cold, long predates the development of thermodynamics. Nevertheless, a clear definition of temperature proves to be elusive (see ref 9 for a detailed discussion). Early scientific explorations made use of temperature scales defined by the expansion properties of liquids (red wine, mercury, alcohol) and arbitrary values for fixed temperature points (10). Being based on arbitrary choices, the resulting temperature values could not be related to other fundamental properties. As thermodynamics matured, temperature could be defined at the macroscopic level as the property that determines the direction of heat flow between two objects placed in thermal contact. Although operational, this definition does not tell us what temperature is. A definition at the molecular level provides more insight into the nature of temperature. One such definition relates temperature to energy of motion: “A convenient operational definition of temperature is that it is a measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules” (11).2 The temperature-energy link is simple and direct for a monatomic ideal gas with widely spaced electronic energy levels (e.g., neon). Each atom in this system has three translational degrees of freedom, and each of these modes has average energy mode and molar internal energy Um that depend linearly on the temperature: mode ¼ ð1=2ÞkB T

and

Um ¼ ð3=2ÞNA kB T ¼ ð3=2ÞRT where NA is the Avogadro constant. Macroscopically, two samples of an ideal monatomic gas are at the same temperature when there is no net transfer of heat (energy) between them if they are placed in thermal contact. Microscopically, they have the same temperature when each sample has the same average translational kinetic energy per atom. The macroscopic statement follows from the microscopic when we analyze what will happen when a system with higher average translational kinetic energy per atom is placed in contact with one with lower average translational kinetic energy per atom. In collisions between particles from the two systems, the higher-energy set will, on average, lose energy more often than the lower-energy set; that is, heat will be transferred from the higher to the lower-energy set.

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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 11 November 2010 10.1021/ed900008z Published on Web 08/31/2010

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Journal of Chemical Education

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In the Classroom

If temperature is a measure of average energy, it can be expressed in joules ( J) rather than in kelvins (K). Two ubiquitous constants can be viewed as conversion factors between these two units. At the molecular level, the conversion factor is the Boltzmann constant, kB = 1.380650310-23 J/K; at the macroscopic level, the conversion factor is the gas constant, R = 8.314472 J/(K mol). These two conversion factors are related through the Avogadro constant NA: R = NAkB. In recognition of this linkage, the International Committee for Weights and Measures is currently considering redefining the temperature unit in terms of energy and the Boltzmann constant: “The kelvin, unit of thermodynamic temperature, is such that the Boltzmann constant is exactly kB = 1.38065xx 10-23 J/K.” Equivalently, “The kelvin is the thermodynamic temperature at which the mean translational kinetic energy of atoms in an ideal gas at equilibrium is exactly (3/2)1.38065xx 10-23 J” (12). To express temperature in joules, define a rescaled temperature. This can be done in either of two ways T 0 ¼ kB T

or T 00 ¼ RT

where T0 relates temperature to average molecular thermal energy and T00 relates temperature to average molar energy. Inasmuch as R = NAkB, these choices differ “only” by the Avogadro constant: NAT0 = T00 . Because kBT appears in many exponential expressions used in statistical thermodynamics, we prefer T0 , but note that Pitzer (4) chose to rescale using R rather than kB. Using rescaled temperature, the energy expressions for an ideal monatomic gas are mode ¼ ð1=2ÞT 0

and

Um ¼ ð3=2ÞNA T 0

The equation for Um is easily interpreted. There are NA atoms per mol and 3 translational modes. Multiplying these two factors by the average energy per mode, (1/2)T0 , gives the molar energy. Energy is an extensive quantity, but temperature, average energy, and molar energy are intensive properties. Just as the careful specification of units for molar energy includes “per mole” (units of Um are J/mol), a careful specification of the units associated with T0 includes “per mode” (units of T0 are J/mode). The total internal energy of a sample of a monatomic ideal gas is obtained by multiplying the appropriate intensive quantity by an appropriate amount: U ðjouleÞ ¼ 3ðmode=atomÞ nðmolÞ NA ðatom=molÞ ð1=2 T 0 Þðjoule=modeÞ ¼ ð3=2ÞnNA T 0

Several small diatomic molecular gases at moderate temperatures also have internal energies that are linear functions of temperature: Um ¼ ð5=2ÞRT ¼ ð5=2ÞNA T 0


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Vol. 87 No. 11 November 2010

Um ¼ 3NA kB T ¼ 3NA T 0 The relationship between internal energy and temperature would be straightforward if our world were composed entirely of such simple systems, but it is not. The linear relationships that hold for these simple systems break down whenever energy is not partitioned equally among all degrees of freedom. Linking atoms together into molecules constrains the motion of the atoms, such that some translational degrees of freedom are replaced by rotational and vibrational ones. The energy distributions among these multiple degrees of freedom depend on the spacing of energy levels associated with each degree of freedom. Whenever the energy level spacing is comparable to T0 , the number of modes available to accept energy increases with T0 , with the result that Um is not a linear function of T0 . As stated on the Hyperphysics Web site “The concept of temperature is complicated by internal degrees of freedom like molecular rotation and vibration and by the existence of internal interactions in solid materials which can include collective modes” (13). Despite the complicated relationship between total internal energy U and T0 for any real system, rescaled temperature is linearly related to the translational kinetic energy (14). To see this, consider two systems in thermal contact: any real fluid and a monatomic ideal gas. If the atoms in the gas have a different average translational kinetic energy than the molecules in the fluid, the systems will approach thermal equilibrium by exchanging heat. At thermal equilibrium, the two systems must have the same average molecular or atomic translational kinetic energy. Otherwise, heat will continue to be transferred from the system with higher average to the one with lower average translational kinetic energy. At thermal equilibrium, the two systems have the same temperature, and because T0 = kBT, they also have the same rescaled temperature. Thus, for any fluid, temperature measures average translational kinetic energy of its constituent particles. Solids can be approached using similar reasoning. Replace the real fluid with any real solid. Instead of translational kinetic energy of the particles in the fluid, this system has vibrational kinetic energy of its particles locked, yet vibrating with some average energy, in the solid lattice. In thermal contact with a monatomic ideal gas, the vibrational kinetic energies of the particles in the solid will equilibrate with the translational kinetic energies of the particles in the gas. At thermal equilibrium, the average vibrational kinetic energy for particles in the solid will equal the average translational energy for particles in the gas. Again, at thermal equilibrium, the two systems have the same temperature, and because T0 = kBT, they also have the same rescaled temperature. Thus, for any solid, temperature measures average vibrational kinetic energy of its constituent particles. Heat Capacities

For these gases, there are two rotational degrees of freedom in addition to the three of translational motion, for a total of 5 degrees of freedom, each contributing mode = (1/2)T0 . A third simple system for which energy and temperature are linearly related is an Einstein solid at sufficiently high temperature. Each atom in such a solid vibrates independently in three dimensions, and each vibrational mode has both potential energy and kinetic energy. Each of these degrees of freedom contributes 1196

(1/2)T0 to the internal energy:

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Using rescaled temperature eliminates both R and kB from other thermodynamic expressions. For example, the constantvolume molar heat capacity CV,m is defined to be CV , m ¼ ðDUm =DT ÞV Heat capacity traditionally has units of J/K (extensive) or J/(mol K) (molar). A monatomic ideal gas has CV,m = (3/2)R, and solids

pubs.acs.org/jchemeduc

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r 2010 American Chemical Society and Division of Chemical Education, Inc.

In the Classroom

at sufficiently high temperature have CV,m = 3R (The latter relationship is the Dulong and Petit law). Rescaling T to T0 leaves the definition unchanged: CV , m 0 ¼ ðDUm =DT 0 ÞV Upon rescaling,3 however, heat capacities become dimensionless, and the corresponding values are CV , m 0 ¼ ð3=2ÞNA ðmonatomic ideal gasÞ

and

0

CV , m ¼ 3NA ðhigh-T solidÞ To interpret these numbers, note that CV,m0 = (∂Um/∂T0 )V is the rate of change of total energy with average energy per mode. When we pump energy into a classical system (monatomic gas or Einstein solid), the added energy is distributed evenly among all the available modes. Total energy of the system increases by a factor that is the number of particles (NA) times 3 modes of motion per particle times the average energy per mode mode = (1/2)T0 . Comparing the heat capacities of a monatomic and a diatomic gas, under conditions where both behave ideally, helps to show how “internal” interactions create complications. Consider 1 mol of N2 versus 2 mol of Ne, each of which contains 2 mol of atoms, as summarized in Table 1. Both samples have the same total number of degrees of freedom, but diatomic molecules have constraints on the freedom of individual atoms. This has consequences for how energy is distributed among the degrees of freedom because of the quantization of energy in rotational and vibrational modes. As the last row in the table shows, the experimental heat capacity of Ne matches the classical prediction. The value for nitrogen is smaller by 0.5NA, because for N2 at room temperature, the vibrational mode of motion is not able to accept energy. The reason for degrees of freedom not being available to accept heat energy is well-known: the energy levels of any system are quantized, and higher energy levels are not accessible to incoming heat transfers if the spacing between energy levels is large relative to the average energy of each degree of freedom. At room temperature, the average energy per mode of translational motion is 0.8 kJ/mol, much larger than the rotational energy level spacing for N2 (about 0.01 kJ/mol) but much smaller than the vibrational energy level spacing of 28 kJ/mol. Thus, molecular nitrogen at room temperature behaves like a classical system with five degrees of freedom. Energy added to this system flows equally into the three translational and two rotational modes of motion, so total energy increases at 5 times the rate of increase of average energy per accessible mode of motion. When the energy level spacing is comparable to the average energy per classical degree of freedom, the methods of statistical Table 1. Degrees of Freedom of Ne and N2 System

2 mol Ne

1 mol N2

Translational modes

6 NA

3 NA

Rotational modes

0

2 NA

Vibrational modes

0

NA

Experimental heat capacities, nCV,m 0 a

3.00 NA

2.50 NA

Accessible modes

6 NA

5 NA

mechanics must be applied to determine how energy distributes among the various degrees of freedom. Although the details are complicated, the qualitative result is straightforward: heat capacity values are intermediate between those predicted for the inaccessible and classical situations. Thus, for example, CV,m0 = 2.47NA for H2 at room temperature, reflecting the fact that the vibrational mode is inaccessible and that the rotational modes are approaching, but have not quite reached, the classical limit (CV,m0 = 2.50NA). To obtain further insight into what heat capacities are, consider how the heat capacity relates changes in internal energy to temperature changes: dU ¼ CV 0 dT 0

Entropy When temperature is rescaled, entropy also becomes rescaled. This can be shown using the classical defining equation for entropy change, dS = δqrev/T. Substituting T = T0 /kB into this statement and rearranging gives dS/kB = δqrev/T0 , and defining S0 =S/kB yields dS 0 ¼ δqrev =T 0 Inasmuch as both q and T0 have energy units, rescaled entropy is dimensionless, as is also apparent from its definition (units of S are J/K, units of kB are J/K). Upon rescaling, the Boltzmann statement of entropy, S = kB ln W, (where W is the number of states accessible to the system) becomes S 0 ¼ ln W This equation is a special case of a probability relationship for the condition where the probability of finding the system in any of the W states is the same. The more general statement, covering conditions where states have different probabilities, is X X pi ln pi and S 0 ¼ pi ln pi S ¼ - kB i

i

When all i states have the same probability, pi = 1/W for all i, and the requirement that the total probability be 1 leads to the Boltzmann statement. By examining rescaled entropy, we find first that rescaled entropy is dimensionless: it measures a number rather than an amount. This statement echoes the description given by Rock (16): “Entropy is not a thing. There are no entropy meters. Entropy is not a directly measurable quantity...” Divorcing entropy from the apparent units, J/K, helps clarify this feature. Second, rescaled entropy is a number that converts energy intensity, T0 , into energy, as illustrated by the following thermodynamic equalities:

The experimental heat capacities are from ref 15; Cp values converted to CV,m 0 using CV,m 0 = NA[(CP/R) - 1].

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dT 0 ¼ dU =CV 0

Viewing from the perspective of increasing internal energy, heat capacity tallies the number of “slots” into which added energy can be placed: the greater the number of slots (magnitude of CV0 ), the more total energy (U) grows per unit change in average energy (T0 ). Conversely, the greater the number of slots (magnitude of CV0 ), the more slowly the average energy (T0 ) grows with increase in internal energy (U).

a


and


δqrev ¼ T 0 dS 0

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and A ¼ U - T 0 S 0

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In these and other thermodynamic expressions, an intensive quantity, T0 , must be multiplied by S0 to obtain an extensive energy that can be used in combination with other extensive energy measures (A, U, δq). Loosely speaking, S0 can be considered to be a count of the number of “slots” among which energy is distributed. Specifying the sense of “slots” more explicitly requires the machinery of statistical mechanics. The mathematical details, which, as Leff observes (17), “can distract students from the physics”, are elaborate, but the result can be stated in words as “the number of accessible states” (8), the “available microscopic energy storage modes” (17), or “the number of microstates associated with a macrostate” (18). Other Properties Even though rescaling affects both S and T, it leaves other thermodynamic state functions unchanged. From the two rescaled definitions, T0 = kBT and S0 =S/kB, it follows that any function containing the product TS (including SdT and TdS) remains unchanged under these new definitions. Thus, for example, G = H - TS = H - T0 S0 is independent of rescaling. Rescaling changes the units associated with equalities involving various derivatives but does not otherwise affect these equalities. For instance, the Maxwell relation (∂S/∂V)T = (∂P/∂T)V becomes (∂S0 /∂V)T = (∂P/∂T0 )V, and the derivative relation (∂U/∂S)V = T becomes (∂U/∂S0 )V = T0 . In other words, using rescaled temperature and entropy alters none of the mathematical “machinery” of thermodynamics. Similarly, rescaling does not affect the mathematical development of statistical mechanics. One needs to look only at the central quantity of statistical mechanics, the canonical partition function, Z, which is unaltered by rescaling: X X 0 e -Ej =kB T ¼ e -Ej =T Z ¼ j

j

Inasmuch as every thermodynamic property can be expressed as a mathematical formula in terms of Z, the only effect of rescaling is that kBT is replaced by T0 in all those formulas. Rescaling emphasizes that each exponential term Ej/T0 is an energy ratio, the energy of level j divided by the energy intensity of the system. Besides clarifying temperature and entropy, rescaled temperature expressed in energy units provides different perspectives for other properties that involve either kB or R. As discussed above, heat capacities can be better understood as enumeration of the number of degrees of freedom among which added energy is distributed. “Characteristic temperatures” are restated as “characteristic energies”. The characteristic rotational temperature, Θrot = Bhc/kB, where B is the rotational constant, is replaced by Θ0 rot = kBΘrot = Bhc, equal to the rotational energy level spacing. Likewise, the characteristic vibrational temperature, Θvib = hυ/kB is replaced by Θ0 vib = kBΘvib = hυ, equal to the vibrational energy level spacing. Rescaling also puts into perspective the significance of the gas constant. As Pitzer observed (19), “The gas constant R is probably the most important numerical constant of thermodynamics. It appears in many equations that are otherwise unrelated to the ideal gas.” Two such equations involve relationships between energy and temperature: ΔG o ¼ - RT ln Keq k ¼ Ae - Ea =RT

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ΔG o ¼ - T 0 ln Keq

k ¼ Ae - Ea =T


0

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ðequilibriumÞ

ðkinetic rate constantÞ


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Rescaling eliminates the gas constant from these equations, because that constant is just a unit conversion factor between kelvins and joules. Expressing temperature in energy units eliminates the need for such conversions. The ideal gas equation less obviously relates energy to temperature: PV ¼ nRT

PV ¼ nNA T 0

Here, the product PV, although not explicitly an energy term, nevertheless has energy units. Alternatively, we can view this equation as one describing pressure: P ¼ nRT =V

P ¼ nNA T 0 =V

From this perspective, it is similar to the equation for osmotic pressure: Π ¼ MRT

Π ¼ MT 0

In both these equations, pressure must have units of force/area = energy/volume; hence the need for R, to convert temperature units into energy units. Conclusion Rescaling of temperature and entropy allows these quantities to be viewed from a different perspective. Temperature can be understood as energy/mode or energy intensity, and entropy becomes dimensionless. Further, from this perspective, the gas constant and the Boltzmann constant are unit conversion factors that must be applied whenever a temperature is to be related to an energy. Presenting this perspective to students in chemistry courses has several advantages. For example, “hotness is high energy intensity” is a statement that may help students get a better feel for temperature. Emphasizing that the gas constant is a unit conversion factor helps to explain why this constant appears in equations that are unrelated to the behavior of gases. Showing that entropy is dimensionless helps to emphasize that entropy is a number, not a physical property. Rescaled temperature and entropy can provide insights into how to interpret these important properties. Nevertheless, the Kelvin scale is too firmly established as the preferred temperature scale for it to be readily replaced. Moreover, even recognizing that temperature is energy intensity, a specific unit for temperature is useful, just as the joule is useful as an energy unit even though energy can be expressed in the fundamental units of mass (length)2(time)-2. In addition, using joules for total internal energy and temperature for energy intensity helps remind us that these two quantities generally are not linearly related. Notes 1. Natural unit systems specify five fundamental properties: length, mass, time, electrical charge, and temperature. The units of other properties, including energy, are expressed in terms of the units associated with these five properties. The unit sizes for these properties are selected so that the numerical values of five fundamental constants (typically, speed of light, gravitational constant, reduced Planck constant, Coulomb constant, and Boltzmann's constant) are unity. These unit sizes are either very large (temperature unit = 1.416 1032 K) or very small (length unit = 1.616 10-35 m). For details see ref 3.


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In the Classroom

2. In order for the average, and hence the temperature, to be welldefined, the system must contain a large number of atoms and molecules and be, at least locally, at equilibrium. 3. Using the alternate rescaling, T00 = RT, would eliminate the Avogadro constant from these and other expressions for molar heat capacities: CV0 = 3/2 (monatomic ideal gas) and CV0 = 3 (high-T solid).

Literature Cited 1. Stoney, G. Philos. Mag. 1881, 11, 381–391. 2. Planck, M. Sitzungsberichte der Ko.niglich Preussischen Akademie der Wissenschaften zu Berlin 1899, 5, 440–480. 3. Planck or Natural Units. http://en.wikipedia.org/wiki/Planck_ units (accessed Jul 2010). 4. Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill, Inc.: New York, 1995; p 50. 5. Kalinin, M.; Kononogov., S. Meas. Tech. 2005, 48, 632–636 (English translation of Izmeritel'naya Tekhnika 2005, 7, 5-8). 6. Pitzer, K. S.; Brewer, L. J. Phys. Chem. Ref. Data 1979, 8, 917–919. Pitzer, K. S.; Brewer, L. High Temp. Sci. 1979, 11, 49 (cited in ref 4). 7. Shannon, C. E. Bell Syst. Tech. J. 1948, 27, 379-423, 623-656. 8. Leff, H. S. Am. J. Phys. 1999, 67, 1114–1122. 9. Temperatures.com Home Page. http://www.temperatures.com/ wit.html (accessed Jul 2010).


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10. A Brief History of Temperature Measurement Home Page. http:// home.comcast.net/∼igpl/Temperature.html (accessed Jul 2010); Zytemp Web site. The History of Temperature and Thermometry, http://www.zytemp.com/infrared/thermometry_history.asp (accessed February 2010). 11. Hyperphysics Home Page. http://hyperphysics.phy-astr.gsu.edu/ hbase/hframe.html /Heat and Thermodynamics/Temperature (accessed November 2009). 12. Fischer, J.; Gerasimov, S.; Hill, K. D.; Machin, G.; Moldover, M. R.; Pitre, L.; Steur, P.; Stock, M.; Tamura, O.; Ugur, H.; White, D. R.; Yang, I.; Zhang, J. Int. J. Thermophys. 2007, 28, 1753–1765. 13. Hyperphysics Web site. http://hyperphysics.phy-astr.gsu.edu/ hbase/hframe.html /Heat and Thermodynamics/Temperature/A More General View of Temperature (accessed November 2009). 14. Levine, I. N. Physical Chemistry, 4th ed.; McGraw-Hill: New York, 1995; p 425. 15. Handbook of Chemistry and Physics, 83rd ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2002. 16. Rock, P. A. Chemical Thermodynamics; University Science Books: Mill Valley, CA, 1983; p 78. 17. Leff, H. S. Am. J. Phys. 1996, 64, 1261–1271. 18. Baierlein, R. Am. J. Phys. 1994, 62, 15–26. 19. Pitzer, K. S. Thermodynamics, 3rd ed.; McGraw-Hill, Inc.: New York, 1995; p 23.


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Rescaling Temperature and Entropy - American Chemical Society

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