Calculation of Entropy

A second way of dealing with the relationship between aj and the experimental concentration requires the use of a statistical model. We assume that the system consists of Nj molecules of type 1 and N2 molecules of type 2. In addition, it is assumed that the molecules, while distinguishable, are identical to one another in size and interaction energy. That is, we can replace a molecule of type 1 in the mixture by one of type 2 and both AV and AH are zero for the process. Now we consider the placement of these molecules in the Nj + N2 = N sites of a three-dimensional lattice. The total number of arrangements of the N molecules is given by N , but since interchanging any of the I s or 2 s makes no difference, we divide by the number of ways of doing the latter—Ni and N2 , respectively—to obtain the total number of different ways the system can come about. This is called the thermodynamic probabilty 2 of the system, and we saw in Sec. 3.3 that 2 is the basis for the statistical calculation of entropy. For this specific model... 

The calculation of entropy is required for compression and expansion calculations. Isentropic compression and expansion is often used as a reference for real compression and expansion processes. The calculation of entropy might also be required in order to calculate other derived thermodynamic properties. Like enthalpy, entropy can also be calculated from a departure function ... 

This means that, to obtain the least variance in a multistage calculation, the intermediates should be constructed to have equal entropy difference for all stages. This criterion differs from the often used but unjustified rule of thumb that free energy differences should be equal in all stages [22,42], Simulation tests show that the entropy criterion leads to a great improvement in calculation precision compared to its free energy counterpart [26]. The same optimization criterion holds for calculation of entropy and enthalpy differences [44],... 

The symbol 9 is called the characteristic temperamre and can be calculated from an experimental determination of the heat capacity at a low temperature. This equation has been very useful in the extrapolation of measured heat capacities [16] down to OK, particularly in connection with calculations of entropies from the third law of thermodynamics (see Chapter 11). Strictly speaking, the Debye equation was derived only for an isotropic elementary substance nevertheless, it is applicable to most compounds, particularly in the region close to absolute zero [17]. 

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

Calculation of Entropy and Heat Capacity from Statistical Mechanics... 

Table 3.3 Coefficients of structural components valid for polynomial 3.79. Coefficient K(, appears only at integration for calculation of entropy. The resulting Cp is in J/(mole X K) (from Robinson and Haas, 1983). ...

Calculation of Entropy and Gibbs Energy Change from... 

Calculation of entropy and Gibbs energy change from heat capacities From Eq. (3.7) and for the limiting case of an infinitely small step in temperature... 

CVM is a more powerful formalism which can include the mutual interaction of all the atoms in sets of clusters, so as to properly reflect a greater variety of atomic interactions (de Fontaine 1979, 1994). The smallest cluster that should be used to describe a three-dimensional lattice is the tetrahedron (Kikuchi 1951), but the complete calculation of entropy using CVM requires a consideration of all subsidiary clusters (van Baal 1973). This means pairs and planar triangles have to be included for the tetrahedron approximation, and all these in turn have to be included when larger clusters such as octahedra are considered. Various mathematical techniques... 

In this section the standard molar entropies of a small selection of cations and anions are tabulated and the manner of their derivation discussed. The values themselves are required in the calculation of entropies of hydration of ions, discussed in Section 2.7.2. 

The entropy, Spontaneous vs non-spontaneous, Reversible and irreversible processes, Calculation of entropy changes (Isothermal, isobaric, isochoric, adiabatic), Phase changes at equilibrium, Trouton s rule, Calculation for irreversible processes... 

The translational partition function is a function of both temperature and volume. However, none of the other partition functions have a volume dependence. It is thus convenient to eliminate the volume dependence of 5trans by agreeing to report values that use exclusively some volume that has been agreed upon by convention. The choices of the numerical value of V and its associated units define a standard state (or, more accurately, they contribute to an overall definition that may be considerably more detailed, as described further below). The most typical standard state used in theoretical calculations of entropies of translation is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely, y° = 24.5 L. 

Considerable attention has been given here to heats (enthalpies) of formation, because there are extensive tabulations of these, e.g. [205] and papers on their calculation appear often in the literature, e.g. [201]. However, we should remember that equilibria [147] are dependent not just on enthalpy differences, but also on the often-ignored entropy changes, as reflected in free energy differences, and so the calculation of entropies is also important [206]. 

A further discussion of these approaches is made in Frame 15, section 15.4 when we consider in more detail the calculation of entropy, S. 

Calculation of Entropy Change, AS, for Reversible Expansion Process... 

The addition of the hard wall entropic confinement free energy to the interaction energy, as under (2), only raises somewhat the minimum, but almost does not modify its position. In contrast, in all the other cases, which take into account the other interactions in the calculation of entropy, considerable shifts of the equilibrium distance and of the value of the free energy at the minimum occur. This observation indicates that a common procedure, to add to the conventional interactions, calculated as for planar layers, the free energy due to the hard wall entropic confinement is inaccurate. 

Although derived for a reversible process, this equation relates properties only, and is independent of the process causing the change of state. It is therefore a general equation for the calculation of entropy changes of an ideal gas. 

Both the classical and statistical equations [Eqs. (5.22) and (5.23)] yield absolute values of entropy. Equation (5.23) is known as the Boltzmann equation and, with Eq. (5.20) and quantum statistics, has been used for calculation of entropies in the ideal-gas state for many chemical species. Good agreement between these calculations and those based on calorimetric data provides some of the most impressive evidence for the validity of statistical mechanics and quantum theory. In some instances results based on Eq. (5.23) are considered more reliable because of uncertainties in heat-capacity data or about the crystallinity of the substance near absolute zero. Absolute entropies provide much of the data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 15. 

Example 4.6 Entropy production in a packed duct flow Fluid flow and the wall-to-fluid heat transfer in a packed duct are of interest in fixed bed chemical reactors, packed separation columns, heat exchangers, and some heat storage systems. In this analysis, we take into account the wall effect on the velocity profile in the calculation of entropy production in a packed duct with the top wall heated and the bottom wall cooled (Figure 4.7). We assume... 

Example 8.6 Calculation of entropy production for a reversible reaction Consider the following reaction between a substrate S and a product P ... 

The calculation of entropies for gaseous species generally requires detailed knowledge of geometry, bond distances and vibrational frequencies. We have developed a statistical mechanical model which allows an estimation of the entropy of an unknown molecule using as input only the atomic masses and Interatomic distances. Details of the development of the model have been given in previous publications (4). A brief summary of the important assumptions and equations is given below. 

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