Standard State Transformations

The conversion factor Vs relating Kx to K, and the correction term -/JrinVs relating AG° to AG° are given in Table 1.8. 

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Polynomial expressions are conveniently used to represent a condensed phase which is stable in the whole temperature range of interest and which does not undergo any structural, electronic or magnetic transformations. The Gibbs energy of a compound is in the CALPHAD approach represented relative to the elements in their defined standard state at 298.15 K as a power series in terms of temperature in the form of [16] ... 

Adsorption of TU on active and passive iron electrodes was studied by Bockris et al. using radiotracer and Fourier Transform Infrared Spectroscopy techniques. The high values of AG° (the standard states 0° = 0.582 and c° = 1 mol T ) equal to -17.20 kJ/mol and -18.98 kJ/mol have been obtained for the active and passive iron electrodes, respectively. They are markedly higher than those obtained for other metals (Table 1). 

In addition to the correction for the new standard state, it is necessary to correct for the transformation from the real gas at standard pressure to the ideal gas at standard pressure, which is defined as the standard state. ... 

As the Gibbs function is a thermodynamic property, values of AG do not depend on the intermediate chemical reactions that have been used to transform a set of reactants, under specified conditions, to a series of products. Thus, one can add known values of a Gibbs function to obtain values for reactions for which direct data are not available. The most convenient values to use are the functions for the formation of a compound in its standard state from the elements in their standard states, as given in Tables 7.2... 

The relationships between the two different states and between the enthalpy of formation from the elements at standard state (H°) and the lattice energy (U) are easily understood by referring to the Born-Haber-Fayans thermochemical cycle. In this cycle, the formation of a crystalline compound from isolated atoms in the gaseous state is visualized as a stepwise process connecting the various transformations. Let us follow the condensation process of a crystal MX formed from a metal M and a gaseous molecule X2 ... 

We then close the cycle by transforming crystal MX into its component elements in the standard state. To do this, we must furnish energy corresponding to the enthalpy of formation from the elements ... 

The standard potential of equation 8.176 is = 1.228 V. At standard state, the activity of gaseous oxygen is 1 by definition, and standard potential thus refers to H2O in equilibrium with an atmosphere of pure O2 at T = 25 °C and P = bar. Applying the Nernst and Faraday relations to equation 8.176 and transforming natural logarithms into base 10 logarithms, we obtain... 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

Interaction parameter for the p phase standard state Standard free energy of phase transformation of A from a to P G°g(a, p) Standard free energy of phase transformation of B from a to P... 

The superscript ° indicates that this equation is for a gas at the standard state of 1 bar. Standard states will be discussed in greater detail later in this chapter. For a solid, the calculation just includes the first integral (to 7), and for a liquid, the first two terms plus the second integral (to 7). If there are any solid-phase transformations at temperatures T, with heats of transformation A H, additional terms of the form A H/T must be included. 

Figure 3.6 shows schematically the molar entropy of a pure substance as a function of temperature. If a structural transformation occurs in the solid state, an additional increase in the molar entropy comes from the heat of the transformations. As shown in the figure, the molar entropy of a pure substance increases with increasing temperature. In chemical handbooks we see the tabulated numerical values of the molar entropy calculated for a number of pure substances in the standard state at temperature 298 K and pressure 101.3 kPa. A few of them will be listed as the standard molar entropy, s , in Table 5.1. Note that the molar entropy thus calculated based on the third law of thermodynamics is occasionally called absolute entropy. 

Similar effects arise from varying particle size. The transformation of a macroscopic crystalline solid phase with negligible specific surface area to a finely divided powder whose particles have a significant specific surface area produces an additional term, proportional to the specific surface area, in the expression for ArG° describing the reaction in Eq. 3.1.15 This term increases the value of the Standard-State chemical potential of the solid phase but does not alter Kdis, such that the equilibrium IAP increases also. The effect on Eq. 3.3 is to increase Kso according to the following equation 15... 

Here, AH(A-B) is the partial molar net adsorption enthalpy associated with the transformation of 1 mol of the pure metal A in its standard state into the state of zero coverage on the surface of the electrode material B, ASVjbr is the difference in the vibrational entropies in the above states, n is the number of electrons involved in the electrode process, F the Faraday constant, and Am the surface of 1 mol of A as a mono layer on the electrode metal B [70]. For the calculation of the thermodynamic functions in (12), a number of models were used in [70] and calculations were performed for Ni-, Cu-, Pd-, Ag-, Pt-, and Au-electrodes and the micro components Hg, Tl, Pb, Bi, and Po, confirming the decisive influence of the choice of the electrode material on the deposition potential. For Pd and Pt, particularly large, positive values of E5o% were calculated, larger than the standard electrode potentials tabulated for these elements. This makes these electrode materials the prime choice for practical applications. An application of the same model to the superheavy elements still needs to be done, but one can anticipate that the preference for Pd and Pt will persist. The latter are metals in which, due to the formation of the metallic bond, almost or completely filled d orbitals are broken up, such that these metals tend in an extreme way towards the formation of intermetallic compounds with sp-metals. The perspective is to make use of the Pd or Pt in form of a tape on which the tracer activities are electrodeposited and the deposition zone is subsequently stepped between pairs of Si detectors for a-spectroscopy and SF measurements. 

In this example the standard heat of formation of H20 is available for its hypothetical standard state as a gas at 25°C. One might expect the value of the heat of formation of water to be listed for its actual state as a liquid at 1 bar or l(atm) and 25°C. As a matter of fact, values for both states are given because they are both frequently used. This is true for many compounds that normally exist as liquids at 25°C and the standard-state pressure. Cases do arise, however, in which a value is given only for the standard state as a liquid or as an ideal gas when what is needed is the other value. Suppose that this, were the case for the preceding example and that only the standard heat of formation of liquid H20 is known. We must now include an equation for the physical change that transforms water from its standard state as a liquid into its standard state as a gas. The enthalpy change for this physical process is the difference between the heats of formation of water in its two standard states ... 

Initially, a moles of species A and b moles of species B are contained in cylinders shown at the top of the box in Fig. 15.2. Each is stored in its cylinder t pure gas at temperature T and at a fugacity of 1 bar, i.e., in its standard state. following series of steps transforms these reactants into l moles of L and mm of Af, the pure product species in their standard states at temperature T and a fug of 1 bar. They are collected in the lower cylinders shown in Fig. 15.2. 

H°) standard enthalpy. Compare with enthalpy change. A change in enthalpy associated with a reaction or transformation involving substances in their standard states. 

The current or stationary values of chemical potentials of catalytic inter mediates are of principal importance for analyzing the role of the inter mediates in catalytic processes. For example, in the stationary mode of catalytic reactions, the relevant chemical transformations, the reactant-active center complexes—should be described as transitions between the stationary chemical potentials rather than the traditionally considered minimums of potential energy that relate to the standard state of the parti cipants of the stepwise transformation (see, for example, Figure 4.1). 

For convenience of calculations, biochemists therefore define a different standard state, in which the concentration of H+ is 10-7 M (pH 7) and the water is 55.5 M. Physical constants based on the biochemical standard state are called standard transformed constants and are written as AG° or K to distinguish them from the untransformed constants used by chemists and physicists. By con-... 

Application of the standard Legendre transformations then yields the remaining functions of state ... 

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