Standard States Again

In most of this chapter we have expressed equilibrium in terms of Eq. 8.3, which involves a standard state fiigacity  

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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... 

In a general case of a mixture, no component takes preference and the standard state is that of the pure component. In solutions, however, one component, termed the solvent, is treated differently from the others, called solutes. Dilute solutions occupy a special position, as the solvent is present in a large excess. The quantities pertaining to the solvent are denoted by the subscript 0 and those of the solute by the subscript 1. For >0 and x0-+ 1, Po = Po and P — kxxx. Equation (1.1.5) is again valid for the chemical potentials of both components. The standard chemical potential of the solvent is defined in the same way as the standard chemical potential of the component of an ideal mixture, the standard state being that of the pure solvent. The standard chemical potential of the dissolved component jU is the chemical potential of that pure component in the physically unattainable state corresponding to linear extrapolation of the behaviour of this component according to Henry s law up to point xx = 1 at the temperature of the mixture T and at pressure p = kx, which is the proportionality constant of Henry s law. 

The null hypothesis test for this problem is stated as follows are two correlation coefficients rx and r2 statistically the same (i.e., rx = r2)l The alternative hypothesis is then rj r2. If the absolute value of the test statistic Z(n) is greater than the absolute value of the z-statistic, then the null hypothesis is rejected and the alternative hypothesis accepted - there is a significant difference between rx and r2. If the absolute value of Z(n) is less than the z-statistic, then the null hypothesis is accepted and the alternative hypothesis is rejected, thus there is not a significant difference between rx and r2. Let us look at a standard example again (equation 60-22). 

Another key feature of redox thermodynamic cycles is that the free energy change in solution is still defined to involve a gas-phase electron, that is, the solvation free energy of the electron is happily not an issue. And, once again, redox potentials in soludon typically assume 1 M standard states for ad species (but not always in this chapter s case study, for instance, all redox potentials were measured and computed for chloride ion concentrations buffered to 0.001 M). So, free energy changes associated with concentration adjustments must also be properly taken into account. 

The standard free energy of formation of a compound is A Gy. It is the standard-state free energy G° of the compound minus the standard-state free energy of the elements from which the compound is formed. Again, from this definition, it must be that case that AG°j(T) = 0 for the elements (in their most stable form) at every T. However, unlike H° for the elements, which are defined to be zero at one temperature Tr, in general, G°(T) 0 for any particular reference condition. 

The quantity gk T) in Equation (7.67) is again a molar quantity, characteristic of the individual gas, and a function of the temperature. It can be related to the molar Gibbs energy of the fcth substance by the use of Equation (7.67). The first two terms on the right-hand side of this equation are zero when the gas is pure and ideal and the pressure is 1 bar. Then gk(T) is the chemical potential or molar Gibbs energy for the pure fcth substance in the ideal gas state at 1 bar pressure. We define this state to be the standard state of the fcth substance and use the symbol 1 bar, yk = 1] for the... 

The definition is completed by assigning a value to m and (f>c in some reference state. To conform with the definitions made in Sections 8.9 and 8.10, the infinitely dilute solution with respect to all molalities or molarities is usually used as the reference state at all temperatures and pressures, and both m and c are made to approach unity as the sum of the molalities or molarities of the solutes approaches zero. The standard state of the solvent is again the pure solvent, and is identical to its reference state in all of its thermodynamic functions. 

We again choose to use Equation (10.30) to express the chemical potential of the component in the liquid phase in terms of the mole fraction and choose the pure liquid to be the standard state. Then we have... 

Figure IV. A. 3. is a heat of combustion plot of the elements with fluorine as the oxidizer. One can carry through the same arguments as those made for the oxygen plot except it is well to remember that many of the metal fluorides are gaseous under the combustion chamber conditions that prevail. Since BF is a gas, the significance of boron compounds as fuels with fluorine oxidizers particularly changes for the better. Comparison of figures IV. A. 1. and IV. A. 3. will show that H2 -02 and H2 - F2 have the same standard state heat release. Yet above it was stated that the specific impulse with hydrogen-fluorine was the greater. This comparison again points out the limitations of the plots, which do not take into account the diss-...

Eqn.(3.25) follows immediately from this equation, if we again choose for the same standard state (the pure solute) in both phases (i.e. s = m). 

It is clear that the equilibrium constants for these activated complexes, which may have a life of only about 10 13 second, can not be determined by direct experimental means. Sometimes important conclusions may be drawn without even attempting to calculate this equilibrium constant—making the necessary estimates from considerations of entropy, or comparing different reactions in such a way that this term cancels out. Again in using activities instead of concentrations one can arbitrarily specify standard states for reactions in different phases in such a way as to simplify the calculations. 

If the electrode reactants and products are not in the standard state, the equilibrium cell voltage will be the difference between the E values (i.e. the corresponding activity terms have to be included). Consider again the cell consisting of the half reactions Ag+/Ag and H+/H2. The equilibrium potential difference between the two electrodes, E - E(2) - E(l), is represented in Equation (20). Considering that czAg - 1 and 0(H+/H2) = 0 V, Equation (20) is identical to Equation (18). 

Solve the problem again, this time using uniform standard states... 

One should not adhere to the mistaken notion that the analysis of subsection (a) can be used in the upper range of xA values and that the analysis of subsection (b) for the same solution can then be used in the lower range. The two approaches are based on different standard states (i.e. P versus P, as discussed earlier) and therefore are not interchangeable or interrelated. Further, one must stay with one scheme or the other to obtain internally consistent results. It is just more convenient to apply methodology (b) if one is interested primarily in the thermodynamic characterization of solutes in dilute solution, and methodology (a) if one wishes to analyze the properties of solvents. This discussion again points to the fact (see also subsection (e) below) that it is only the differences in the chemical potential themselves that are unique in value all other quantities must be determined self-consistently. 

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