The Thermodynamic Functions

we would like to ehange the reference state from the isolated nuelei and eleetions to the elements in their standard states, C(graphite) and H2(g) at 298 K. This leads to the energy of formation at 0 K AfEo, whieh is identieal to the enthalpy of formation AfHo at 0 K. The energy and enthalpy are identieal only at 0 K. Next we would like to know the enthalpy ehange on heating propene from 0 to 298 K so as to obtain the enthalpy of formation from the isolated nuelei and eleetions elements This we will eonvert to from the elements in their standard 

The entire proeedure ean be eanied out in steps. We find the ground-state energy of formation of propene at 0 K from C and H atoms in the gaseous state 

Kij tire 10-5 Eormation of Gaseous Atoms from Elements in the Standard State. 

We now know the energy of the propene thermodynamic state (propene(g)) relative to the state 3 C(g) and 6 11(g) and the energy of the therrnodynarnie standard state of the elements relative to the same state 3 C(g) and 6 11(g)). which is opposite in sign to the summed energies of formation of 3 C(g) and (i IKg). The energy differenee between these thennodynamie states is 

The remaining question is how we got from G3MP2 (OK) = —117.672791 to G3MP2 Enthalpy = —117.667683. This is not a textbook of classical thermodynamics (see Klotz and Rosenberg, 2000) or statistical themiodynamics (see McQuarrie, 1997 or Maczek, 1998), so we shall use a few equations from these fields opportunistically, without explanation. The definition of heat capacity of an ideal gas 

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In Chapter 2 we discuss briefly the thermodynamic functions whereby the abstract fugacities are related to the measurable, real quantities temperature, pressure, and composition. This formulation is then given more completely in Chapters 3 and 4, which present detailed material on vapor-phase and liquid-phase fugacities, respectively. 

The procedure would then require calculation of (2m+2) partial derivatives per iteration, requiring 2m+2 evaluations of the thermodynamic functions per iteration. Since the computation effort is essentially proportional to the number of evaluations, this form of iteration is excessively expensive, even if it converges rapidly. Fortunately, simpler forms exist that are almost always much more efficient in application. 

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

Characteristic bands occur in the 1300-1000 cm region for 3,4- and 3,5-disubstituted isoxazoles (7i PMh(4)265, p. 330), while bands below 1000 cm contain modes for most substitution patterns (71PMh(4)265, p. 332). Total assignments for isoxazole and isoxazole-d have been made (63SA1145, 7lPMH(4)265,p. 325) and some of the thermodynamic functions calculated (68SA(A)361, 71PMH(4)265,p.330). 

The protonation equilibria for nine hydroxamic acids in solutions have been studied pH-potentiometrically via a modified Irving and Rossotti technique. The dissociation constants (p/fa values) of hydroxamic acids and the thermodynamic functions (AG°, AH°, AS°, and 5) for the successive and overall protonation processes of hydroxamic acids have been derived at different temperatures in water and in three different mixtures of water and dioxane (the mole fractions of dioxane were 0.083, 0.174, and 0.33). Titrations were also carried out in water ionic strengths of (0.15, 0.20, and 0.25) mol dm NaNOg, and the resulting dissociation constants are reported. A detailed thermodynamic analysis of the effects of organic solvent (dioxane), temperature, and ionic strength on the protonation processes of hydroxamic acids is presented and discussed to determine the factors which control these processes. 

It follows that although the thermodynamic functions can be measured for a given distribution system, they can not be predicted before the fact. Nevertheless, the thermodynamic properties of the distribution system can help explain the characteristics of the distribution and to predict, quite accurately, the effect of temperature on the separation. 

Consistency between the thermodynamic functions that are obtained from Eqs. (9), (10), and (13) is often used as the criterion for the accuracy of a theory. In this regard it is worth pointing out that, in the HNC approximation, the thermodynamic functions that are obtained from Eqs. (9) and (10) are identical. Thus, a partial degree of consistency is achieved. 

By a statistical model of a solution we mean a model which does not attempt to describe explicitly the nature of the interaction between solvent and solute species, but simply assumes some general characteristic for the interaction, and presents expressions for the thermodynamic functions of the solution in terms of an assumed interaction parameter. The quasi-chemical theory is of this type, and we have noted that a serious deficiency is its failure to consider the vibrational effects in the solution. It is of interest, therefore, to consider briefly the average-potential model which does include the effect of vibrations. 

Values for the thermodynamic functions as a function of temperature for condensed phases are usually obtained from Third Law measurements. Values for ideal gases are usually calculated from the molecular parameters using the statistical mechanics procedures to be described in Chapter 10. In either... 

Example 9.3 Calculate the pressure of atomic chlorine in Cl2(g) at a total pressure of 1.00 bar and a temperature of 2000 K. Do the calculation (a) using the thermodynamic functions in Table 4.3, and (b) using equation (9.58), which requires Cp m expressed as a function of T, obtained from Table 2.1, and compare the results. 

This value can be compared with K= 0.543 obtained from the thermodynamic functions given in Table 4.3. The agreement is quite reasonable. Using this value of K gives pC = 0.515 bar instead of 0.514 bar as obtained using Table 4.1. 

The thermodynamic functions of primary interest in chemistry are Cp.m, Sm, and Gm-Ho.m- The translational, rotational, and vibrational contributions are summarized in Table 10.4.u We will not attempt to derive all the equations in this table but will do enough to show how it is done. 

P10.5 The thermodynamic functions for solid, liquid, and gaseous carbonyl chloride (COCL) obtained from Third Law and statistical calculations... 

Another distinction that we make among the thermodynamic functions is to describe p, V, T, U, and 5 as the fundamental properties of thermodynamics. The other quantities, H, A, and G are derived properties, in that they are defined in terms of the fundamental properties, with... 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

The thermodynamic function used as the criterion of spontaneity for a chemical reaction is the Gibbs free energy of reaction, AG (which is commonly referred to as the reaction free energy ). This quantity is defined as the difference in molar Gibbs free energies, Gm, of the products and the reactants ... 

A note on good practice Notice that we have used the "molar convention for the thermodynamic functions, as this convention is required by Eq. 8. 

In systems with different components, the values of the thermodynamic functions depend on the nature and number of these components. One distinguishes components forming independent phases of constant composition (the pure components) from the components that are part of mixed phases of variable composition (e.g., solutions). 

The electrode processes that are reversible provide values for the equilibrium emfs of cells, which are related to the thermodynamic functions. The condition of reversibility is practically obtained by balancing cell emf against an external emf until only an unappreciable current passes through the cell, in order that the cell reactions proceed very slowly. It may, however, be pointed out that for many of the applications of electrometallurgy, it is clearly necessary to consider more rapid reaction rates. In that situation there is necessarily a departure from the equilibrium condition. Either the cell reactions occur spontaneously to produce electric energy, or an external source of electric energy is used to implement chemical reactions (electrolyses). 

The thermodynamic functions (AH, AS, AG(298 K)) of hydrogen peroxide reactions with transition metal ions in aqueous solutions are presented in Table 10.1. We see that AG(298K) has negative values for reactions of hydroxyl radical generation with Cu1+, Cr2+, and Fe2+ ions and for reactions of hydroperoxyl radical generation with Ce4+, Co3+, and Mn3+. 

A homogeneous open system consists of a single phase and allows mass transfer across its boundaries. The thermodynamic functions depend not only on temperature and pressure but also on the variables necessary to describe the size of the system and its composition. The Gibbs energy of the system is therefore a function of T, p and the number of moles of the chemical components i, tif. 

The contribution of the transition to the thermodynamic functions can be evaluated once the coefficients B and d have been determined. Experimental determination of the transition temperature and one additional thermodynamic quantity at one specific temperature is sufficient to describe the transition thermodynamically using this model. 

The defect interaction energies appearing in Eq. (33) are, for the purposes of the present article, assumed to be known either from theory or experiment. Certain other quantities appear in the final expressions for the thermodynamic functions and must therefore be known. The quantity defined by the relation... 

Up to densities of about 0.001 fm-3 density effects can be neglected. This way we describe an ideal mixture in chemical equilibrium. The composition as well as the thermodynamical functions can be calculated immediately... 

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