Absolute entropy functional

A cryogenic calorimeter measures Cp,m as a function of temperature. We have seen that with the aid of the Third Law, the Cp,m data (along with AHm for phase changes) can be integrated to give the absolute entropy... 

The entropy of any chemical substance increases as temperature increases. These changes in entropy as a function of temperature can be calculated, but the techniques require calculus. Fortunately, temperature affects the entropies of reactants and products similarly. The absolute entropy of every substance increases with temperature, but the entropy of the reactants often changes with temperature by almost the same amount as the entropy of the products. This means that the temperature effect on the entropy change for a reaction is usually small enough that we can consider A Sj-eaction he independent of temperature. 

Figure 2.8 A plot of the function describing the change in absolute entropy as a function of temperature. The discontinuities occur at phase changes...

The energy and entropy functions have been defined in terms of differential quantities, with the result that the absolute values could not be known. We have used the difference in the values of the thermodynamic functions between two states and, in determining these differences, the process of integration between limits has been used. In so doing we have avoided the use or requirement of integration constants. The many studies concerning the possible determination of these constants have culminated in the third law of thermodynamics. 

The stable phase of all substances, except helium, at sufficiently low temperatures is the solid phase. We therefore consider the solid phase as the condensed state whose entropy is zero at 0 K, and exclude helium from the discussion for the present. The absolute entropy of a pure substance in some state at a given temperature and pressure is the value of the entropy function for the given state taking the value of the entropy of the solid phase at 0 K... 

In the first discussion of equilibrium (Ch. 5) we recognized that there may be states of a system that are actually metastable with respect to other states of the system but which appear to be stable and in equilibrium over a time period. Let us consider, then, a pure substance that can exist in two crystalline states, a and p, and let the a phase be metastable with respect to the p phase at normal temperatures and pressures. We assume that, on cooling the a. phase to the lowest experimental temperature, equilibrium can be maintained within the sample, so that on extrapolation the value of the entropy function becomes zero. If, now, it is possible to cool the p phase under the conditions of maintaining equilibrium with no conversion to the a phase, such that all molecules of the phase attain the same quantum state excluding the lattice vibrations, then the value of the entropy function of the p phase also becomes zero on the extrapolation. The molar absolute entropy of the a phase and of the p phase at the equilibrium transition temperature, Tlr, for the chosen... 

Liquid helium presents an interesting case leading to further understanding of the third law. When liquid 4He, the abundant isotope of helium, is cooled at pressures of < 25 bar, a second-order transition takes place at approximately 2 K to form liquid Hell. On further cooling Hell remains liquid to the lowest observed temperature at 10 5 K. Hell does become solid at pressures greater than about 25 bar. The slope of the equilibrium line between liquid and solid helium apparently becomes zero at temperatures below approximately 1 K. Thus, dP/dT becomes zero for these temperatures and therefore AS, the difference between the molar entropies of liquid Hell and solid helium, is zero because AV remains finite. We may assume that liquid Hell remains liquid as 0 K is approached at pressures below 25 bar. Then, if the value of the entropy function for sol 4 helium becomes zero at 0 K, so must the value for liquid Hell. Liquid 3He apparently does not have the second-order transition, but like 4He it appears to remain liquid as the temperature is lowered at pressures of less than approximately 30 bar. The slope of the equilibrium line between solid and liquid 3He appears to become zero as the temperature approaches 0 K. If, then, the slope is zero at 0 K, the value of the entropy function of liquid 3He is zero at 0 K if we assume that the entropy of solid 3He is zero at 0 K. Helium is the only known substance that apparently remains liquid as absolute zero is approached under appropriate pressures. Here we have evidence that the third law is applicable to liquid helium and is not restricted to crystalline phases. 

Solutions and glasses do not follow the third law. If a solid solution continues to persist on cooling a sample to the lowest possible temperature, then at this temperature the molecules of the several components are distributed in some fashion in the same crystal lattice. Under such conditions all of the molecules of the substance could not attain the same quantum state on further cooling to 0 K in the sense of the required extrapolation. Only if the solid solution was separated into the pure components would the value of zero be obtained for the entropy function at 0 K. If the molecules of the components were randomly distributed in the crystal lattice, as in an ideal solution, then the entropy of the substance at absolute zero would be equal to the ideal entropy of mixing, so... 

We thus can obtain a consistent set of absolute values of the entropy function for pure substances from thermal measurements alone on the practical basis of assigning the value of zero to the entropy function at 0 K with the exclusion of nuclear and isotopic effects, within the understanding of the third law as discussed in Section 15.4. The calculation of the entropy function of pure substances in the ideal gas state by the methods of statistical mechanics must be consistent with the practical basis. In addition to obtaining absolute values by the methods that have been discussed, values can also be obtained from equilibrium measurements from which ASe can be determined for some change of state. If all but one of the absolute values in the equivalent sum VjSP are known, then the value of that one can be calculated. 

We have seen that absolute values of the entropy function based on the third law can be obtained from measurements of the heat capacity and heats of transitions. A more general equation than Equation (15.9) may be written as... 

Figure 3.6 shows schematically the molar entropy of a pure substance as a function of temperature. If a structural transformation occurs in the solid state, an additional increase in the molar entropy comes from the heat of the transformations. As shown in the figure, the molar entropy of a pure substance increases with increasing temperature. In chemical handbooks we see the tabulated numerical values of the molar entropy calculated for a number of pure substances in the standard state at temperature 298 K and pressure 101.3 kPa. A few of them will be listed as the standard molar entropy, s , in Table 5.1. Note that the molar entropy thus calculated based on the third law of thermodynamics is occasionally called absolute entropy. 

If the heat capacity is not constant, it must be used as a function of temperature for Equation (1.70), for which the integration must then be carried out. When the temperature nears the absolute zero temperature, Cp=aT3 where a = 2.27x 10 4cal mol1 K-4. If there are some phase changes before reaching the temperature T, the entropy of phase transitions must be incorporated into the calculation for the absolute entropy ... 

The third-law method is based on a knowledge of the absolute entropy of the reactants and products. It allows the calculation of a reaction enthalpy from each data point when the change in the Gibbs energy function for the reaction is known. The Gibbs energy function used here is defined as... 

Tables for this defined reference state, including the heat capacity, the heat content relative to 298.15° K., the absolute entropy, and the free energy function at even 100° intervals from 298.15° to 3000° K. have b n assembled for the first 92 elements. These tables are arranged alphabetically beginning on page 36. The choice of 298.15° K. as the reference temperature is made because the low temperature heat capacities of many elements and compounds are not known. Most of the thermodynamic data now reported in the literature refer to 25° C., which, when combined with the recent international agreement on 273.15° K. for the ice point (319) gives a reference temperature of 298.15° K. The figure 298° K. quoted in the tables and text should be understood to be the reference temperature, 298.15° K. For those who prefer to use 0° K. as the reference temperature, we have included, for cases in which it is known, the heat content at 298.15° K. relative to 0° K.

Rubber Company Handbook (Weast, 1987) is one of the more commonly available sources. More complete sources, including some with data for a range of temperatures, are listed in the references at the end of the chapter. Note that many tabulations still represent these energy functions in calories and that it may be necessary to make the conversion to Joules (1 cal = 4.1840J). Because of the definition of the energy of formation, elements in their standard state (carbon as graphite, chlorine as CI2 gas at one bar, bromine as Br2 liquid, etc.) have free energies and enthalpies of formation equal to zero. If needed, the absolute entropies of substances (from which AS may be evaluated) are also available in standard sources. 

The following general statements may be made about entropy. It resembles the total energy, U, in that It is a function of the state of a thermodynamic system. Only changes of entropy are of practical significance since absolute entropies are unknown. This arbitrariness may, however, be removed by choosing some standard condition as a point of reference. In general... 

The theorem not only allowed the calculation of chemical equilibria, it was also soon recognized as an independent third law of general thermodynamics with many important consequences. One such consequence was that it is impossible to reach the absolute zero. Another consequence was that one could define a reference point for entropy functions, such that the entropies of all elements and all perfect crystalline compounds were taken as zero at the absolute zero. 

The quantity 5 is the entropy function and is a function of state. The quantity T, the thermodynamic temperature, is identical to the absolute temperature. 

Note that in the terminology of the theory of statistics, a function, such as (x), that is defined over fi, is called a random variable the brackets <> denote a statistical average defined with the Boltzmann PD. We now define a total absolute entropy S that depends on both the coordinates and the momenta (see Eqs. [IJ and [2], ... 

Further detailed argument of a purely mathematical nature shows that in this case there is one universal factor, T, for the separate systems and for the joint system, and that dS — d 8 - -S, that is, that there is an absolute temperature and an entropy function for aU substances. From this point the derivation of the principles of thermodynamics follows the same course as before. 

The use of partition functions of the type just derived, or in appropriate cases more complex ones, in the calculation of absolute entropies, leads to incorrect results. The fault lies, as has been stated, with the mode of definition of the probability. The question of a modified definition and a reconsideration of what constitutes the number of assignments to states must now be considered. 

The discussion of absolute entropies has shown that there is no statistical significance in the distinction represented by (l) j(2) and i/(j,(l) 3(2). As we have seen, there is sense in saying that we have two particles, one in state a and the other in state b, but no significance in the further specification as to which individual is in which state. Thus the two wave functions under consideration, although mathematically correct, do not correspond to the needs of the situation. 

In these equations, the referenee states of H tq) are, by convention, equal to zero as are the functions AHf e[g ) and AG ( ( ,). The absolute entropies for the gaseous ions are calculated from statistical mechanics (Bratsch and Lagowski 1985a) and agree fairly well with the experimental values reported by Bertha and Choppin (1969), who interpreted the S-shaped dependence of standard state entropies on ionic radius in terms of a change in the overall hydration of the cation across the lanthanide series. Hinchey and Cobble (1970) proposed that this S-shaped relationship was an artifact of the method of data treatment and calculated a set of entropies from lanthanide... 

For perfect, crystalline solids, the entropy at 0 K is zero, Sc° = 0. Thus, absolute entropies can be calculated directly. Information to the available data is given in Table 1. For crystalline, linear macromolecnles, the derived thermodynamic functions are reported as follows... 

From the absolute entropy andi T - the free enthalpy (Gibbs function) Gt -... 

Laws of Thermodynamics The laws of thermodynamics have been successfully applied to the study of chemical and physical processes. The first law of thermodynamics is based on the law of conservation of energy. The second law of thermodynamics deals with natural or spontaneous processes. The function that predicts the spontaneity of a reaction is entropy. The second law states that for a spontaneous process, the change in the entropy of the universe must be positive. The third law enables us to determine absolute entropy values. 

Figure 6.1 shows some possible distributions of outcomes. Each distribution function satisfies the constraint that pin) + pie) + pis) + piiv) = 1. You can compute the entropy per spin of the pencil of any of these distributions by using Equation (6.2), Sjk = - X =i Pi In Pi. The absolute entropy is never negative, that is S > 0. 

There are several important ramifications of equation 3.26. First, it introduces the concept that an absolute entropy can be determined. Entropy thus stands alone among state functions as the only one whose absolute values can be determined. Therefore, in large thermodynamic tables of At/ and AH values, parallel entries for entropy are for S, not AS. It also implies that the entropies found in tables are not zero for elements under standard conditions, because we are now tabulating absolute entropies, not entropies for formation reactions. We can determine changes in entropies, AS s, for processes up to now we have dealt exclusively with changes in entropy. But Boltzmann s equation 3.26 means that we can determine absolute values for entropy. 

The definition of entropy ultimately brings us to an idea that we call the second law of thermodynamics Any spontaneous change occurs with a concurrent increase in the entropy of the universe. The mathematical definition of entropy, in terms of the change in heat for a reversible process, allows us to derive many mathematical expressions we can use to calculate the entropy change for a physical or chemical process. The concept of order brings us to what we call the third law of thermodynamics that the absolute entropy of a perfect crystal at absolute zero is exactly zero. We can therefore speak of absolute entropies of materials at temperatures other than 0 K. Entropy becomes—and will remain—the only thermodynamics state function for a system that we can know absolutely. (Contrast this with state variables like p, V, T, and n, whose values we can also know absolutely.)... 

At such low temperatures, most matter is solid, and the best type of solid sample to study is a crystal. Studies of crystals showed some intriguing thermodynamic behavior. For instance, in the measurement of entropy it was found that absolute entropy approached zero as the temperature approached absolute zero. This is experimental verification of the third law of thermodynamics. But a measurement of the heat capacity of the solid showed something interesting The heat capacity of the solid approached zero as the temperature approached absolute zero, also. But for virtually all crystalline solids, the heat-capacity-versus-temperature plot took on a similar shape at low temperatures, typified by Figure 18.3 The curves have the distinct shape of a cubic function, that is, y = x. In this case, the variable is absolute temperature, so experimentally it was found that the constant-volume heat capacity Cy was directly related to T ... 

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