Absolute entropies at

In 1877, the Austrian physicist Ludwig Boltzmann proposed a molecular definition of entropy that enables us to calculate the absolute entropy at any temperature (Fig. 7.6). His formula provided a way of calculating the entropy when measurements could not be made and deepened our insight into the meaning of entropy at the molecular level. The Boltzmann formula for the entropy is... 

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... 

Unfortunately, there axe at this time no low temperature heat capacity data on polymers of known crystallinity so that absolute entropies at 25° C can be calculated. This gap in our knowledge of polymers represents a scientific vacuum that will rapidly be filled1. 

This is a usefiil equation because the third law sets the zero of entropy for every pure substance and thus permits calculations of absolute entropies at temperature T by Eq. (5) of Chapter 4. 

By examining the following graphs, predict which element—copper or gold—has the higher absolute entropy at a temperature of 200 K. 

As the temperature of a substance increases, the particles vibrate more vigorously, so the entropy increases (Figure 15-14). Further heat input causes either increased temperature (still higher entropy) or phase transitions (melting, sublimation, or boiling) that also result in higher entropy. The entropy of a substance at any condition is its absolute entropy, also called standard molar entropy. Consider the absolute entropies at 298 K listed in Table 15-5. At 298 K, any substance is more disordered than if it were in a perfect crystalline state at absolute zero, so tabulated values for compounds and elements are always positive. Notice especially that g of an element, unlike its A// , is not equal to zero. The reference state for absolute entropy is specified by the Third Law of Ther-... 

TABLE 15-5 Absolute Entropies at 298 Kfor a Eew Common Substances... 

Would you expect an increase or decrease in the entropy of the system as the complex is formed The standard-state absolute entropy at 25°C is 72.68 J/K mol for Ag+, 111.3J/KmolforH, and 253.1 J/Kmol for [Agl3]. Calculate for the reaction and confirm your prediction. 

The heat capacity of TiSe2(cr) single crystals was measured in the temperature range 1.3 to 300 K by Craven, Di Salvo and Hsu [78CRA/SAL]. Unfortunately, the measured values are only presented in a small graph and no attempt was made to evaluate the absolute entropy at 298.15 K. A second order transformation at 200 K complicates the evaluation. No value is selected in this review. 

The heat capacity of a-CdSe was measured in the temperature range 4.2 to 300 K. The heat capacity showed no anomalies except in the range 45 to 50 K where a small deviation from smooth behaviour was observed. The absolute entropy was presented only in a graph of poor resolution and no value was given at 298.15 K. The entropy was therefore evaluated by the review from the heat capacity values given in Table 1 of the paper. The resulting absolute entropy at 298.15 K was S°(CdSe, a, 298.15 K) = (86.76 + 1.00) J-K -mol. The error limit was estimated from the general accuracy for this type of measurements and includes the effect of estimated systematic errors. 

All these correlation formulae were obtained from fuels for which both the lower heating value and the absolute entropy at (T0,P0) are known. Because the latter is not known for the generally used solid and fluid fuels such as bituminous coal, lignite, cokes, oil, tar, and so on, the equations are offered as plausible approximations for these based on analogy of behavior. 

Equation [18] enables one to obtain the absolute entropy at if the absolute entropy at Tj is known note that for a classical system, Tj = 0 cannot be chosen because C is finite at this temperature and the integral is undefined. Calcu-... 

Entropy plots such as Figure 19.13 can be obtained by carefully measuring how the heat capacity of a substance (Section 53) varies with temperature, and we can use the data to obtain the absolute entropies at different temperatures. (The theory and methods used for these measurements and calculations are beyond the scope of this text) Entropies are usually tabulated as molar quantities, in units of joules per mole-kelvin ( J/mol-K). 

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. 

The third law of thermodynamics states that the entropy of a pure crystalline substance is zero at T = 0 K. This defines the reference point for the specific entropy in process simulators and makes it consistent with respect to chemical reactions. Entropies using this reference point are called absolute entropies, which are indicated by the subscript "abs." For convenience, in the reference tables the standard entropies, that is, the absolute entropies at standard conditions, are listed ... 

In Chapter III we reviewed the work of Criss and Cobble (3-9). Applying a technique known as the correspondence principal, a relationship between the "absolute" entropy at temperature 12 and the "absolute" entropy at 25 C was developed. The relationship is ... 

Cyclopropane and propylene are isomers of C3Hg. Based on the molecular structures shown, which of these isomers would you expect to have the higher absolute entropy at 25°C ... 

Rank each group of substances in order of increasing absolute entropy at 2 5°C. Give reasons for your ranking. 

This convention establishes a scale of absolute entropies at temperatures above zero kelvins called third-law entropies, as explained in the next section. 

The absolute entropy of Kr at 298.15 Kis 163.97 J/mol K. What is its absolute entropy at 200.00 K if the volume remains unchanged Assume that the heat capacity is constant. 

Fig. 2.9 The absolute entropy (or Third-Law entropy) of a substance is calculated by extending the measurement of heat capacities down to T= 0 (or as close to that value as possible) and then determining the area of the graph of C/T against T up to the temperature of interest. The area is equal to the absolute entropy at the temperature T.

If we can measure the heat capacity C of some pure species (element or compound) from 0 to 298.15 K, then we can compute its absolute entropy at 298.15 K from the general relationship... 

This is analogons to Eq. (3.34) for the absolute enthalpy except the reference temperature chosen is different and there is a temperature and pressure dependence. Because entropy is a measure of the number of possible microstates available, it is only zero at absolute zero, where there is no molecular motion. Thus, the reference point of the entropy of every substance is 0 K, 1 atm. The absolute entropy at some state 2 is the entropy caused by departure from 0 K and 1 atm. 

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