Standard-state Conversions

We have seen that the standard state of AGs i(H+) must be taken into account to produce reliable results. In this last part of this section we outline the details of standard state conversions. 

If we think of Aj-G as providing a measure of the thermodynamic stability of products relative to reactants, then ArG° < 0 means that products in their standard states are thermodynamically more stable than reactants in their standard states. Conversely, ArG° > 0 means that products in their standard states are thermodynamically less stable than reactants in their standard states. Because the values of ArG° and K are related, K may also be considered a measure of the thermodynamic stability of products relative to reactants. A large value of K implies that products are thermodynamically more stable than reactants and, thus, equilibrium favors products. A small value of K implies that products are thermodynamically less stable than reactants and equilibrium favors reactants. 

Enthalpy of Formation The ideal gas standard enthalpy (heat) of formation (AHJoqs) of chemical compound is the increment of enthalpy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard states, defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.3 kPa). Sources for data are Refs. 15, 23, 24, 104, 115, and 116. The most accurate, but again complicated, estimation method is that of Benson et al. " A compromise between complexity and accuracy is based on the additive atomic group-contribution scheme of Joback his original units of kcal/mol have been converted to kj/mol by the conversion 1 kcal/mol = 4.1868 kJ/moL... 

The equilibrium constant can be determined at any temperature from standard state information on reactants and product. Considering the synthesis of NH3, the equilibrium conversion can be determined for a stoichiometric feed of Hj and Nj, at the total pressure. These conversions are determined by the number of moles of each species against conversion X by taking as a basis, 1 mole of N2. 

In order to have a consistent basis for comparing different reactions and to permit the tabulation of thermochemical data for various reaction systems, it is convenient to define enthalpy and Gibbs free energy changes for standard reaction conditions. These conditions involve the use of stoichiometric amounts of the various reactants (each in its standard state at some temperature T). The reaction proceeds by some unspecified path to end up with complete conversion of reactants to the various products (each in its standard state at the same temperature T). 

The obtained A 7 a() value and the energy equivalent of the calorimeter, e, are then used to calculate the energy change associated with the isothermal bomb process, AE/mp. Conversion of AE/ibp to the standard state, and subtraction from A f/jgp of the thermal corrections due to secondary reactions, finally yield Ac f/°(298.15 K). The energy equivalent of the calorimeter, e, is obtained by electrical calibration or, most commonly, by combustion of benzoic acid in oxygen [110,111,113]. The reduction of fluorine bomb calorimetric data to the standard state was discussed by Hubbard and co-workers [110,111]. 

It is normally a very good approximation to assume that the titration process under study occurs under a pressure of 0.1 MPa. Therefore, the pressure corrections involved in the conversion of AT/icp to the standard state are usually negligible, and in many cases, it is licit to make A77icp = A/T p. When appropriate, other corrections, such as those related to solution standard states, can be applied as described by Vanderzee [129,130]. 

Fuel Cell Efficiency The theoretical energy conversion efficiency of a fuel cell e° is given by the ratio of the free energy (Gibbs function) of the cell reaction at the cell s operating temperature AG, to the enthalpy of reaction at the standard state AH°, both quantities being based on a mole of fuel ... 

Although this equation reduces to an identity whenever solute-solvent interactions are embodied in the definition of the Henry s law standard state (cf section 10.2), it must be noted that K[ is the molar ratio of trace element i in the two phases and not the weight concentration ratio usually adopted in trace element geochemistry. As we will see later in this section, this double conversion (from activity ratio to molar ratio, and from molar ratio to weight concentration ratio) complicates the interpretation of natural evidence in some cases. To avoid ambiguity, we define here as conventional partition coefficients (with the same symbol K ) all mass concentration ratios, to distinguish them from molar ratios and equilibrium constants. 

Provided that the same reference standard state (for example the pure solid) is used in Kth and Z for each reactant, we can combine these to give the free energy of the solid-to-solid conversion according to Equation 4 ... 

After the initial transient, NO and ammonia signals pointed to a steady-state conversion of about 40%. Again the standard SCR stoichiometry was respected. At time of about 300 s, the NF13 feed was shutoff. Consequently its outlet signal slowly decreased, while NO, after a short transient, reached its feed value of 1,000 ppm. Again, a good correlation between experiment and predictive simulation on the integral monolith reactor level was obtained the model was able to correctly predict both the steady-state levels of the reactants and their temporal evolution. 

F (jE°(/=o.2) — J5°(/=o)) (1) This cycle holds for every medium. Equation 1 connects the four basic quantities T Ks0, Ks0, E°(j=o>, and E° 1) without any nonthermo-dynamic assumptions. This enables us to evaluate solubility constants for every appropriate ionic medium using data which have been determined with reference to the usual aqueous standard state. The conversion of Ks into TKs and vice-versa may also be performed using the Debye-Hiickel equation and its extensions as shown in Equation 2. 

Radical heats of formation are defined in the usual way, that is, as enthalpy of formation of the radical in question from the elements in their standard states. The heats of formation and the bond dissociation energies are derivable from each other and are based on the same data. Thus, in Reaction 9.7, the heat of formation of R- is readily found from the bond dissociation energy by means of the enthalpy cycle shown in Scheme 3 if heats of formation of R—X and X are known conversely, D( R—X) may be found once heats of formation of RX, R-,... 

In summary, a reference state or standard state must be defined for each component in the system. The definition may be quite arbitrary and may be defined for convenience for any thermodynamic system, but the two states cannot be defined independently. When the reference state is defined, the standard state is determined conversely, when the standard state is defined, the reference state is determined. There are certain conventions that have been developed through experience but, for any particular problem, it is not necessary to hold to these conventions. These conventions are discussed in the following sections. The general practice is to define the reference state. This state is then a physically realizable state and is the one to which experimental measurements are referred. The standard state may or may not be physically realizable, and in some cases it is convenient to speak of the standard state for the chemical potential, for the enthalpy, for the entropy,... 

From Table 7-1, the formation of diamond from graphite (the standard state of carbon) is accompanied by a positive AH of 1.88kJ/mol at 25°C. From Problem 16.1(f), AS for the same process is negative. Since 25°C is not the transition temperature, the process is not a reversible one. In fact, it is not even a spontaneous irreversible process, and (16-2) does not apply with the inequality sign. On the contrary, the opposite process, the conversion of diamond to graphite at 1 atm, is thermodynamically spontaneous. The AS for this process would obey (16-2) with the inequality sign. This means that diamonds are NOT forever The term spontaneous does not cover the speed... 

We define the standard state of a real gas so that Eq. (51) is general (i.e., so that it also applies to ideal gases). For ideal gases, the standard state is at 1.0 bar pressure. For real gases, we also use a 1.0-bar ideal gas as the standard state. We find the standard state by the two-step process shown in Fig. 6. First we extrapolate the real gas to very low pressure, where / —> P and the gas becomes ideal (Step I). We then convert the ideal gas to 1.0 bar (step II). The convenience of an ideal gas standard state is that it allows temperature conversions to be made with ideal gas heat capacities (which are pressure independent). Conversion to the real gas state is then made at the temperature of interest. 

The equilibrium constant K is independent of pressure with standard states. The effect of the pressure is shown in Equation 6-6. Ky is usually insensitive and may either increase or decrease slightly with pressure. When (r + s) > (a + b), the stoichiometric coefficients, an increase in pressure P results in a decrease in conversion of the reactants to the products (i.e., A + B <-> R + S). Alternatively, when (r + s) < (a + b), an increase in pressure P results in an increase in the equilibrium conversion. In ammonia synthesis (N2 + 3H2 <-> 2NH3), the reaction results in a decrease in the number of moles. Therefore, an increase in pressure causes an increase in equilibrium conversion due to this factor. 

Leakage rates are generally expressed in units of p V throughput (see Equation 2.2 and comments). Mass flow rate (Equation 2.1) and molar flow rates (Equation 2.4) can also be used. Conversion factors for leak rates (qpV leak) and mass flow rates are often given for gases in their standard states (pn, Tn). In practice, the difference between room temperature and Tn is not significant when compared to the uncertainties that can occur in the measurement of leakage. 

Example 15.4 Devise a gas-phase process for the reversible conversion of reactant species A and B in their standard states into product species L and M in their standard states in accord with the reaction... 

The overall change in the Gibbs energy for the entire process, i.e., the sum of the changes for the five steps, is also the standard Gibbs energy change of reaction, because the overall result of the process is the conversion of reactants to products, all in their standard states. Therefore... 

Sensor. The control of the exhaust composition was essential to maintain the air-to-fuel ratio close to stoichiometric for simultaneous conversion of all three pollutants. This control came about with the invention of the 02 sensor.21,22 The sensor head of this device was installed in the exhaust immediately at the inlet to the catalyst and was able to measure the 02 content instantly and precisely. It generates a voltage consistent with the Nemst equation in which the partial pressure of 02 (P02)exhaust in the exhaust develops a voltage (E) relative to a reference. The exhaust electrode was Pt deposited on a solid oxygen ion conductor of yttrium-stabilized zirconia (Zr02). The reference electrode, also Pt, was deposited on the opposite side of the electrolyte but was physically mounted outside the exhaust and sensed the partial pressure (P02)ref in the atmosphere. E0 is the standard state or thermodynamic voltage. R is the universal gas constant, T the absolute temperature, n the number of electrons transferred in the process, and F the Faraday constant. 

The conversion of values corresponding to different p is described in [65]. The newer value of p = 105 Pa is sometimes called the standard state pressure. 

U and V respectively. Systeme International (SI) units, described in Appendix B, are used extensively but not slavishly. Chemically convenient quantities such as the gram (g), cubic centimeter (cm ), and hter (L = dm =10 cm ) are still used where useful—densities in g cm , concentrations in mol L , molar masses in g. Conversions of such quantities into their SI equivalents is trivially easy. The situation with pressure is not so simple, since the SI pascal is a very awkward unit. Throughout the text, both bar and atmosphere are used. Generally bar = 10 Pa) is used when a precisely measured pressure is involved, and atmosphere = 760 Torr = 1.01325 X 10 Pa) is used to describe casually the ambient air pressure, which is usually closer to 1 atm than to 1 bar. Standard states for all chemical substances are officially defined at a pressure of 1 bar normal boiling points for liquids are still understood to refer to 1-atm values. The conversion factors given inside the front cover will help in coping with non-SI pressures. 

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