Absolute entropy, experimental

It should be noted that what one measures in experiments is the difference in the entropy, and not the absolute entropy. Assuming that the entropy is zero at absolute zero in accordance with the Nemst-Planck postulate, one can determine the absolute entropy experimentally. However, it is well known that SCL is a metastable state, and there is no reason for its entropy to vanish at absolute zero [16]. Indeed, it has been demonstrated some time ago that the residual entropy at absolute zero obtained by extrapolation is a nonzero fraction of the entropy of melting [43 ], which is not known a priori. Therefore, it is impossible to argue from experimental data that the entropy indeed falls to zero, since such a demonstration will certainly require calculating absolute entropy though efforts continue to date [61, 62]. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

Two points concerning the evaluation of the first integral in Equation (15.9) require further discussion. In most experimental determinations of absolute entropies, the lowest temperature attained ranges from 1 to 15 K ... 

Over the years, many experiments have been carried out which confirm the third law. The experiments have generally been of two types. In one type the change of entropy for a change of phase of a pure substance or for a standard change of state for a chemical reaction has been determined from equilibrium measurements and compared with the value determined from the absolute entropies of the substances based on the third law. In the other type the absolute entropy of a substance in the state of an ideal gas at a given temperature and pressure has been calculated on the basis of statistical mechanics and compared with those based on the third law. Except for well-known, specific cases the agreement has been within the experimental error. The specific cases have been explained on the basis of statistical mechanics or further experiments. Such studies have led to a further understanding of the third law as it is applied to chemical systems. 

The condition discussed in the previous paragraph demands certain care in the experimental determination of absolute entropies, particularly in the cooling of the sample to the lowest experimental temperature. In order to approach the condition that all molecules are in the same quantum state at 0 K, we must cool the sample under the condition that thermodynamic equilibrium is maintained within the sample at all times. Otherwise some state may be obtained at the lowest experimental temperature that is metastable with respect to another state and in which all the molecules may not be in the same quantum state 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... 

The hydration entropy can also be deduced experimentally (Latimer 18) as the difference between the standard entropy of the hydrated ions (deduced from measurements of the specific heat on the basis of Nernst s Heat Theorem or the Third Law of Thermodynamics) and the theoretically calculated absolute entropy of the gaseous ion, both reckoned per unit volume at constant concentration. This entropy can also be calculated (Eley and Evans18). 

Estimate absolute entropy of crystalline n-hexanol. Since no experimental data are available below 18.3 K, estimate the entropy change below this temperature using the Debye-Einstein equation. Use the crystal entropy value of 1.695 cal/(g mol)(°K) at 18.3 K to evaluate the coefficient a. Hence a = 1.695/18.33 = 0.2766 x 10 3. The A term in Eq. 1.1 therefore is... 

Values of AG° for matty formation reactions are tabulated in standard references. The reported values of AG are not measured experimentally, but are calculated by Eq. (13.16). The detennination of A5 may be based on the tliird law of thennodynamics, discussed in Sec. 5.10. Combination of valnes from Eq. (5.40) for the absolute entropies of the species taking part in the reaction gives the valne of AS. Entropies (and heat capacities) are also commonly determined from statistical calcnlations based on spectroscopic data. ... 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

Three points are worth making about this third law. First, if it weren t true, it would not be a big deal. We would just have to tabulate entropies at absolute zero as we already do for enthalpies and forego the expression "absolute entropy." Second, some compounds have "residual entropies" at absolute zero as it is, and we can cope with and understand them easily. Third, there are no perfect crystals, there never will be, and there don t need to be in order for the entropy at absolute zero to measure zero. What is the maximum experimental precision of an entropy measurement Perhaps 10 5 eu How big can the degeneracy of a crystal be before its entropy becomes 10 eu ... 

We conclude this brief discussion of deviations from the third law by stating that, although the cases of nonconformity are frequent, we can usually understand their origin with the aid of molecular concepts and quantum statistics. The latter discipline permits calculation of thermodynamic quantities, thereby providing a useful check on experimental data indeed, it often supplies answers of greater accuracy. In this way, it is possible to use the third law to build up tables of absolute entropies of chemical substances. 

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... 

An investigation was made of the thermodynamic properties of gallium phosphide using the emf method in the temperature range 320 430 K. The experimental results were used to calculate the free energy, dhe endialpy and the entropy of formation, as well as the absolute entropy and the energy of atomization of gallium phosphide at 298 K. 

Table 2 lists our values of the standard heats of formation, together with published values deduced by various workers from the experimental values of the dissociation pressure and from calculations of the absolute entropy. 

The third law sets a zero for the entropy so that, experimentally, measurement of heat capacity of a thermodynamically stable crystal can give the absolute entropy at any temperature T. 

For simple molecules it is possible to calculate absolute entropies theoretically, and for most systems the theoretical value very closely agrees with third-law entropies calculated experimentally. However, in some situations the third-law entropy (determined using Equation 8.27 and accounting for any phase transitions) does not agree with the theoretical value. The origin of this discrepancy is generally due to defects or impurities that are frozen in the system at low temperatures. The third law does not apply for such a system because the system is not in its thermodynamically most stable state. The difference between the experimental entropy and the theoretical entropy for such a system is referred to as the residual entropy, which is the value of the entropy at 0 K for systems for which the third law is not applicable. 

The third law of thermodynamics states that the entropy of a perfect oystaUine solid at 0 K is zero. From this reference point the absolute entropies of pure substances at temperatures above absolute zero can be calculated from experimental data. 

In Section 5.5 we discussed how calorimetry can be used to measure AH for chemical reactions. No comparable, easy method exists for measuring AS for a reaction. By using experimental measurements of the variation of heat capacity with temperature, however, we can determine the absolute entropy, S, for many substances at any temperature. (The theory and the methods used for these measurements and calculations are beyond the scope of this text.) Absolute entropies are based on the reference point of zero entropy for perfect crystalline solids at 0 K (the third law). Entropies are usually tabulated as molar quantities, in units of joules per mole-Kelvin (J/mol-K). 

The ability to obtain the complete set of vibrational modes for large polyatomic systems is of considerable importance. Experimentally this information is very difficult to determine and, once available, it becomes possible to compute thermodynamics quantities such as absolute entropies. Where necessary, improvements on the harmonic approximation have been computed by introducing cubic and quartic terms in studies of a variety of organic molecules. ... 

The experimental value of S and its calculated value using statistical thermodynamics in the above example are virtually identical At this point in our development of statistical thermodynamics, absolute entropy is our best evidence that the ideas behind statistical thermodynamics are valid and useful in understanding the thermodynamic behavior of systems (at least systems of gases). Table 17.1 compares experimental values with calculated values of S for several monatomic gases. You can see that the agreement is very, very good. 

Explain why the calculated value for the absolute entropy of Kr at 120 K might not be very close to the experimental value, even though the boiling point of Kr is 119.8 K. 

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 ... 

Estimate absolute entropy of crystalline n-hexanol Since no experimental data are available... 

Student Annotation You will find that tables, including Appendix 2, contain negative absolute entropies for some aqueous ions. Unlike a substance, an individual ion cannot be studied experimentally. Therefore, standard entropies of ions are actually relative values, where a standard entropy of zero is arbitrarily assigned to the hydrated hydrogen ion. Depending on an ion s extent of hydration, its standard entropy may be positive or negative, relative to that of hydrogen ion. 

The significance of the third law of thermodynamics is that it enables us to determine experimentally the absolute entropies of substances. Starting with the knowledge that the entropy of a pure crystalline substance is zero at 0 K, we can measure the increase in entropy of the substance as it is heated. The change in entropy of a substance, AS, is the difference between the final and initial entropy values ... 

A number of temperature ranges may be used. The first and second derivatives of this quantity with respect to temperature are related to the absolute entropy and heat capacity of the compound at the same temperature. Experimental values for heat capacities can thus be directly used in the optimisation and will be... 

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