Absolute entropy, determination

Draw the curve of Cp vs. T and Cp/T vs. T from the following heat capacity data for solid chlorine and determine the absolute entropy of solid chlorine at 70.0 K... 

Thermodynamics is concerned with the relationship between heat energy and work and is based on two general laws, the 1st and 2nd laws of thermodynamics, which both deal with the interconversion of the different forms of energy. The 3rd law states that at the absolute zero of temperature the entropy of a perfect crystal is zero, and thus provides a method of determining absolute entropies. 

The entropies of individual ions in solution are determined by setting the entropy <>1 II in water equal to 0 and then defining the entropies of all other ions relative to this value hence a negative entropy is one that is lower than the entropy of H in water. All absolute entropies are positive, and no sign need he given all entropies of ions are relative to that ot H+ and are listed here with a sign (either + or —). 

C14-0064. Without doing calculations or looking up absolute entropy values, determine the sign of A S ° for the following processes ... 

Boltzmann, following Clausius, considered entropy to be defined only to an arbitrary constant, and related the difference in entropy between two states of a system to their relative probability. An enormous advance was made by Planck who proposed to determine the absolute entropy as a quantity, which, for every realizable system, must always be positive (third law of thermodynamics). He related this absolute entropy, not to the probability of a system, but to the total number of its possibilities. This view of Planck has been the basis of all recent efforts to find the statistical basis of thermodynamics, and while these have led to many differences of opinion, and of interpretation, we believe it is now possible to derive the second law of thermodynamics in an exact form and to obtain... 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

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... 

How can the absolute entropy be determined Equation 1 defines entropy in terms of the transfer of heat so it can be used to calculate only changes in the entropy of a substance. In 1877, the Austrian... 

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. 

Equation (16-2) allows the calculations of changes in the entropy of a substance, specifically by measuring the heat capacities at different temperatures and the enthalpies of phase changes. If the absolute value of the entropy were known at any one temperature, the measurements of changes in entropy in going from that temperature to another temperature would allow the determination of the absolute value of the entropy at the other temperature. The third law of thermodynamics provides the basis for establishing absolute entropies. The law states that the entropy of any perfect crystal is zero (0) at the temperature of absolute zero (OK or -273.15°C). This is understandable in terms of the molecular interpretation of entropy. In a perfect crystal, every atom is fixed in position, and, at absolute zero, every form of internal energy (such as atomic vibrations) has its lowest possible value. 

In view of the above survey, neither correlation nor estimation method has been developed to determine the enthalpy of formation and the absolute entropy of coal-derived liquids. 

Equation (5.20) is the basis for calculation of absolute entropies. In the case of an ideal gas, for example, it gives the probability ft for the equilibrium distribution of molecules among the various quantum states determined by the translational, rotational, and vibrational energy levels of the molecules. When energy levels are assigned in accord with quantum mechanics, this procedure leads to a value for the energy as well as for the entropy. From these two quantities all other thermodynamic properties can be evaluated from definitions (of H. G,... 

While absolute entropy values can now be determined absolute values of Internal Energy and Enthalpy cannot be conceived. For ease of calculation, related especially to metallurgical reactions (constant pressure processes), a suitable reference point of enthalpy is conventionally chosen and that is - for pure elements, the enthalpy is zero when in Standard State . Standard... 

To use this value of S to obtain the individual ionic entropies of other ions in solution, it is necessary to toow values for the entropy of hydration of a number of electrolytes containing H. Thereafter, the value of the entropy of the counterion can be obtained. It can then be used in conjunction with entropies of hydration of electrolytes containing the counterion to determine the absolute entropies of partner ions in the electrolyte containing the constant anion. Of course, in all cases, the value of the entropy of the ion in the gaseous state must be subtracted from that of the ion in solution to give the entropy ofhydration [i.e., = (S,)so, - (S,)g ]. 

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. ... 

Equation (16.20) for the molar entropy of an ideal gas allows calculation of absolute entropies for tile ideal-gas state. The data required for evaluation of the last two terms on tlie right are tlie bond distances and bond angles in the molecules, and the vibration frequencies associated witli tlie various bonds, as determined from spectroscopic data. The procedure lias been very successful in the evaluation of ideal-gas entropies for molecules whose atomic stractures are known. 

The standard entropy change, AS, of a reaction can be determined from the absolute entropies of reactants and products. The relationship is analogous to Hess s Law. 

Make the same determination as in Example 15-16, using heats of formation and absolute entropies rather than free energies of formation. 

The Gibbs energy of formation of PbSe(cr) was measured in the temperature ranges 490 to 600 K and 650 to 858 K using electrochemical cells. The determined enthalpies and entropies of reaction at the mean temperatures of the temperature intervals are given in Table A-95. The quantities were recalculated to 298.15 K by the review and are tabulated in Table A-96. In addition, the corresponding values of the absolute entropy of PbSe(cr) at 298.15 K were calculated. All calculations employed the selected heat capacity of PbSe(cr), the selected thermodynamic properties of selenium, and the data for Pb(cr, I) in [89COX/WAG]. 

The values of hj(T0,Po) in equation [2c] may be evaluated directly with standard tabular values of (1) enthalpy of formation and (2) absolute entropy alternatively, (3) Gibbs free energy of formation could be employed in lieu of the absolute entropy. What is crucial is this Whichever pair is used to determine hj(To Po) and sj(To,pg), the same pair should be employed to calculate yjo (and, for that matter, the same pair should be used throughout the analysis of a process and/or system). 

However, unlike enthalpy and free energy, the absolute entropy of a substance can be determined by invoking the Third Law of Thermodynamics. This may be expressed The entropy of all perfect crystals is zero at the absolute zero of temperature. Most pure substances form essentially perfect crystals at low temperatures and in such cases we can assume S0 = 0. Therefore... 

Thus the absolute entropies of elements and compounds can be. established. These can be used to determine the entropy changes accompanying chemical reactions. 

As we will see, it is possible to determine the absolute entropy of a substance, something we cannot do for energy or enthalpy. Standard entropy is the absolute entropy of a substance at 1 atm and 25°C. It is this value that is generally used in calculations. (Recall that the standard state refers only to 1 atm. The reason for specifying 25°C is that many processes are carried out at room temperature.) Table 18.1 lists standard entropies of a few elements and compounds Appendix 3 provides a more extensive listing. The units of entropy are J/K or J/K mol for 1 mole of the substance. We use joules rather than kilojoules because entropy values are typically quite small. Entropies of elements and compounds are all positive (that is, S° > 0). By contrast, the... 

The important point about the third law of thermodynamics is that it allows us to determine 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 when it is heated to, say, 298 K. The change in entropy, AS, is given by... 

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