Second-law method

The need for entropy values is bypassed when the van t Hoff equation (d In K/dT) =AH/RT2 is used. This can be integrated, either assuming AH is temperature-independent, or by incorporating a specific heat-temperature variation. This is the so-called second law method which contrasts with the third law method. In the latter method, the standard enthalpy is obtained from each equilibrium constant using free-energy functions of all the species present, for example... 

The previous comparison between first and second law methods may give the false impression that the latter should always be preferred. Second law methods can be unreliable, even if a sophisticated equation is used to fit the experimental data. The problem lies in the evaluation of the equilibrium constant, which must be defined in terms of the activities of reactants and products [ 1 ]. The activity concept... 

The application of the second law method to gas-phase reactions is less problematic than for reactions in solution. As described, a, = pt jp° can be used when the perfect gas model is valid (at low enough pressures). For higher pressures, the real gas model implies a, =f/p°. Either one of these relationships can be... 

Besides the second law method, there is another way of extracting reaction enthalpies from gas-phase equilibrium constants. This alternative involves the determination of a single value of an equilibrium constant at a given temperature and the calculation of the reaction entropy at the same temperature. From equations 2.54 and 2.55, we obtain... 

It is generally agreed that the third law method yields more accurate values than the second law method because it does not require any assumption regarding the temperature variation of the reaction enthalpy and entropy. The usual procedure to obtain third law data is to calculate the reaction enthalpy and entropy for each experimental value of Kp and take the average of all the values derived for a given temperature. 

A general discussion of the second and third law methods, including their advantages and limitations relative to first law techniques, was presented in sections 2.9 and 2.10. Now, after a summary of that introduction, we examine some examples that apply the second law method to the thermochemical study of reactions in solution. Recall that the third law method is only practical for reactions in the gas phase. 

What main conclusions can we draw from the three examples discussed here First, although van t Hoff plots should involve Km rather than Kc data, the use of the latter may afford sensible and possibly accurate thermochemical values (always under the assumption of ideal solutions ), particularly if the density term of equation 14.5 is considered in the calculation of the reaction entropy. Second, due to the lack of gas solubility data, the second law method is much... 

Metals. Kruglikh, et al. (104) measured saturated vapor pressures of erbium, samarium, and ytterbium by the Knudsen effusion method, and standard (average) sublimation entropies of 18.4, 20.7, and 25.6 cal./(gram atom °K.) were derived. Nesmeyanov, et al. (146) studied the vapor pressure of yttrium by an integral variant of the effusion technique. Similar studies at higher temperatures by Herrick (70) on samarium metal have been interpreted in good accord by both first and second law methods. Ideal gas thermodynamic functions have been derived from 100 °K. to 6000°K. at 100° intervals for both actinide and lanthanide elements by Feber and Herrick (45). 

The effects of such corrections may be of significance. It should be noted that the second law cannot be applied to a single observation, but the third-law method, which is described below, can be so used. The second-law method also can be applied when only relative values of the equilibrium constant are available, for example, from mass-spectroscop-ic intensity measurements. 

From this equation the enthalpy and entropy of reaction at the mean temperature, 415 K, are calculated to be (157 3) kJ-mol and (327 6) J K mol, respectively. By use of the second-law method with the estimated A C° = -46 J-K -mol, the following values were obtained for standard conditions ... 

The high sensitivity of the Knudsen cell-mass spectrometer system was, for example, also used to determine the enthalpy of dissociation of Cu2 according to the second-law method with high accuracy and precision Hilpert [209]. The enthalpy of dissociation shown in Table 2 is obtained by taking into account the second- and third-law values agreeing well. 

The dissociation energy of SiO(g) determined by Kvande and Wahlbeck [364] according to the second-law method, = 805.8 + 10.9 kJmol", is in agreement with the values adopted in the tables of Gingerich [21] and JANAF... 

There is no basis for a revision of these values as follows from a critical assessment of the literature data (cf. Ref. 364). Plies [367] showed the existence of the molecules GeWO and GeW207 and determined enthalpies of dissociation by the second-law method (Table 16). The dissociation energies of(PbO) (g) (n = 2, 3, 4. .. 6) were determined for the first time by Drowart et al. [395]. They were redetermined by Semenikhin et al. [369] (Table 14) by the use of the second- and third-law methods. The data obtained by the two groups agree... 

Younes et al. [392] studied different systems (Table 17) to understand the high-temperature behavior of nuclear fuels. The enthalpy changes of different isomolecular gas-phase oxygen-exchange reactions between the molecules listed in Table 17 were determined according to the second-law method. 

The enthalpies of dissociation of the homo-complexes (Table 23) were evaluated according to the second- and third-law methods if they are tabulated at 298 K. Otherwise, only the second-law method was used. The values obtained by the use of the second- and the third-law methods in general agreed excellently (see references quoted in Table 23). 

The application of Second Law methods to design is in its infancy. However, it is clear from the few simple cases where it has been used for these purposes, and for decision-making in system... 

In this equation Av// is the molar change in enthalpy for the conversion of substance from the equilibrium liquid to the equilibrium vapor phase. AVF is the molar change in volume when the substance changes from the liquid to the gas. This equation allows calculation of the enthalpy of vaporization from vapor pressure, and it is the second law method. Measurement of enthalpy of vaporization with a calorimeter is the first law method. The quantities AWH and AVF are functions of temperature along the phase boundary. Equation (1.1) can also be written as,... 

The results were processed by the review (Section Vlll.2.1.6) by the second-law method to give the enthalpy of formation and standard entropy of the oxychloride. The oxychloride is appreciably more stable with respect to Th02(cr) and ThCl4(cr) than is ThOF2(cr) with respect to the similar reaction. 

Just as the second-law method, the third-law method is based on use of the relation... 

As evident from an analysis of thermodynamic data (primarily of the enthalpies of formation and sublimation) listed for several hundreds of substances in a reference book [4], determination of these constants by the third-law method yields values more precise, on the average, by an order of magnitude than those obtained using the second-law method. This can be traced to A, H depending differently on random and systematic errors in determination of the true reactant temperature and measurement of the variables P, J, or k, a point which becomes obvious when comparing Eqs. 4.10-4.12 with Eq. 4.18 below... 

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