Other Thermodynamic Properties

Properties other than free-energy changes are usually considerably more difficult to evaluate to an equivalent level of accuracy. One approach is simply to attempt a brute force calculation for different systems analogous to that outlined for V in Eqs. (12.9) and (12.10). However, this approach has little value in any but the simplest of systems owing to the large uncertainties in the absolute values of the thermodynamic quantities. 

Another approach is to carry out free-energy simulations at several different temperatures, and then construct the equivalent of a van t Hoff plot to separate, say, the enthalpic and entropic contributions to the free energy. This approach is obviously extraordinarily demanding of resources, since every temperature point requires a new free-energy simulation, and unless there are many points, the error in the temperature dependence of the free energy determined by linear regression of the latter on the former may be rather large. 

Other Thermodynamic Properties.—Determination of the other critical exponents from thermodynamic measurements is less common and usually less precise. 

The exponent 8 is obtainable from the slope of the critical isotherm, but unlike the p, p plot for a one-component fluid the A, x (or A, j ) diagram is not constructed from directly measured quantities px and (and hence A) must be deduced from vapour-pressure measurements. The few values reported so far (Table 5) lie between 4 and 5, and are consistent with either the Ising value (5.1 0.1) or the best pure-fluid value (4.4 0.4). 

The exponent is deduced from the quantity (0J/0x)-r, p evaluated at = jc for a series of temperatures again A must be extracted from measurements of vapour pressure. Alternatively measurements of or can be used a log-log plot of (b yml x )T,v,x- xo or i Hjal x h,9,x-xo against t yields a slope (y+ - 1). The majority of y+ values are obtained from light scattering measurements, as we shall see in the next subsection the few thermodynamic studies are summarized in Table 6. No experimental information about y for binary mixtures is yet available. 

The exponent a is the most elusive of all. In principle it can be determined [see equation (12)] by measurements at constant x = x of the enthalpy H or the volume K as a function of temperature at constant pressure (yielding Cp.a.m or os,.,) or the volume as a function of pressure at constant temperature (yielding Kr. ). While volumetric measurements leading to a,. may ultimately yield the best values of a, early experiments (before the concept of critical exponents was fully developed) yielded little useful information, and most 

It is clear from the reports of these experimenters, many of whom tested for both positive a (log-log plots) and zero a (semi-log plots), that they could not distinguish between the two evidently all that one can say is that 0 a 0.1. 

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Vapor-pressure data and other thermodynamic properties. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

All other thermodynamic properties for an ideal solution foUow from this equation. In particular, differentiation with respect to temperature and pressure, followed by appHcation of equations for partial properties analogous to equations 62 and 63, leads to equations 191 and 192 ... 

Thermodynamic and physical properties of water vapor, Hquid water, and ice I are given ia Tables 3—5. The extremely high heat of vaporization, relatively low heat of fusion, and the unusual values of the other thermodynamic properties, including melting poiat, boiling poiat, and heat capacity, can be explained by the presence of hydrogen bonding (2,7). 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), apphed to an ideal solution with replaced by Gj, can be written... 

The endothermic radical lO has also been studied in the gas phase the interatomic distance is 186.7 pm and the bond dissociation energy 175 20kJmol . It thus appears that, although the higher oxides of iodine are much more stable than any oxide of Cl or Br, nevertheless, lO is much less stable than CIO (p. 849) or BrO (p. 851). Its enthalpy of formation and other thermodynamic properties are A//f(298K) 175.1 kJmol", AGf(298 K) 149.8 kJmol-, 5°(298 K) 245.5 J K- mor . 

For pure substances, n is usually held constant. We will usually be working with molar quantities so that n = 1. The number of moles n will become a variable when we work with solutions. Then, the number of moles will be used to express the effect of concentration (usually mole fraction, molality, or molarity) on the other thermodynamic properties. 

In Chapter 10, we will make quantitative calculations of U- U0 and the other thermodynamic properties for a gas, based on the molecular parameters of the molecules such as mass, bond angles, bond lengths, fundamental vibrational frequencies, and electronic energy levels and degeneracies. 

Fugacity, like other thermodynamics properties, is a defined quantity that does not need to have physical significance, but it is nice that it does relate to physical quantities. Under some conditions, it becomes (within experimental error) the equilibrium gas pressure (vapor pressure) above a condensed phase. It is this property that makes fugacity especially useful. We will now define fugacity, see how to calculate it, and see how it is related to vapor pressure. We will then define a related quantity known as the activity and describe the properties of fugacity and activity, especially in solution. 

We are interested in describing and calculating AmixZ, the change in the thermodynamic variable Z, when liquids (or solids) are mixed to form a solution. We will begin by deriving the relationship for calculating Amjx<7. Changes in the other thermodynamic properties can then be obtained. 

From the Debye-Hiickel expressions for lny , one can derive equations to calculate other thermodynamic properties. For example L2, the relative partial molar enthalpy,q and V2, the partial molar volume are related to j by the equations... 

A chart which correlates experimental P - V - T data for all gases is included as Figure 2.1 and this is known as the generalised compressibility-factor chart.(1) Use is made of reduced coordinates where the reduced temperature Tr, the reduced pressure Pr, and the reduced volume Vr are defined as the ratio of the actual temperature, pressure, and volume of the gas to the corresponding values of these properties at the critical state. It is found that, at a given value of Tr and Pr, nearly all gases have the same molar volume, compressibility factor, and other thermodynamic properties. This empirical relationship applies to within about 2 per cent for most gases the most important exception to the rule is ammonia. 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

Care must however be taken with the method. The final value is a sum of many, often small, contributions. Errors in these values can quickly lead to qualitatively incorrect results. The gas phase energy is furthermore the difference of two large numbers, and the ab initio calculations must therefore be of sufficient accuracy. The importance of zero-point energy and other thermodynamic properties must also be checked. 

Changes in phase have important consequences for other thermodynamic properties and thus geophysical implications. For example, the bulk modulus at any pressurep in the static limit is given by the value of V(d2(//dV2) (at that pressure) for CaO this increases markedly across the phase boundary. 

Given A, V and T (or G, p and T), all other thermodynamic properties can be obtained by appropriate thermodynamic manipulations. For example, given the variation of volume with temperature (at given pressure) it is straightforward to calculate the isobaric expansivity a given by... 

Before considering different theoretical approaches to determining the free energies and other thermodynamic properties of ionic solvation, it is important to be aware of a problem on the experimental level. There are several methods available for obtaining these quantities for electrolyte solutions, both aqueous and nonaqueous some of these have been described by Conway and Bockris162 and by Padova.163 For example, enthalpies of solvation can be found via thermodynamic cycles, free energies from solubilities or galvanic cell potentials. However the results... 

Besides the difference in the expressions for activity coefficients and other thermodynamic properties from those published and used by the hydrocarbon processing industries, it is more important to realize the need to describe the ionic and... 

Other thermodynamic properties of aqueous solutions are being evaluated. A recent publication reports values calculated for the association constants of aqueous ionic species at 298 K for alkaline earth salts (Staples, 1978). 

Other thermodynamic properties are similarly evaluated from T fo)-... 

Like stability constants and other thermodynamic properties of metal ions in solution, hydrolysis constants are affected by ionic strength and temperature, and these should be specified when quoting precise pfta values. For the ballpark figures cited here, 25 °C and high dilution are assumed. 

In addition to the equation of state, it will be necessary to describe other thermodynamic properties of the fluid. These include specific heat, enthalpy, entropy, and free energy. For ideal gases the thermodynamic properties usually depend on temperature and mixture composition, with very little pressure dependence. Most descriptions of fluid behavior also depend on transport properties, including viscosity, thermal conductivity, and diffusion coefficients. These properties generally depend on temperature, pressure, and mixture composition. 

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