The Third-law Method

To find the value of by this method, it is necessary to know only 

In this case, a plot drawn in the In / vs 1/T or InA vs 1/T coordinates can yield merely the value of (from the slope of the straight line), because 

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

The equilibrium constant, Kp, can de expressed through the experimentally determined equilibrium pressure of the gaseous product, Pp With the use of Eqs. 3.20 and 3.21, Eq. 4.10 can be rewritten as  

Equations 4.11 and 4.12 allow a simplification by excluding the correction term, aR In [3, which takes into account the differences in molar masses and stoichiometric coefficients between the products of decomposition. This can be done by replacing the partial pressures PA/(a M ) and Pb/( Mo ) in Eqs. 3.14 and 3.16 with a generalized quantity P/(/ / M ) 

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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... 

Equilibrium constants involving each compound were evaluated using the partial pressures by the third law method. Accepting the heats of formation of WF5 and WF obtained from bomb calorimetry, the values for WF (n = 1 to 4) could be extracted by iterative fitting to partial pressure data. The W/02/F2 and W/S/F2 systems were also examined to give heats of formation of tungsten oxo- and thiofluorides. This experimentally simple technique yields thermodynamic data on high-temperature species inaccessible to conventional calorimetry. 

Statistical mechanics affords an accurate method to evaluate ArSP, provided that the necessary structural and spectroscopic parameters (moments of inertia, vibrational frequencies, electronic levels, and degeneracies) are known [1], As this computation implicitly assumes that the entropy of a perfect crystal is zero at the absolute zero, and this is one of the statements of the third law of thermodynamics, the procedure is called the third law method. 

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. 

Another application of intermediate coupling calculations has been to use the calculated results to reevaluate dissociation energies derived using the third-law method and mass- spectral data. Balasubramanian and Pitzer have shown how this can be accomplished in their calculations on Sn2 and Pb2 (90). This method requires the molecular partition function, which can be written... 

A eiei is the number of atoms of element i in the crystalline substance and (j m (298.I5 is the standard molar entropy of element i in its thermodynamic reference state. This equation makes it possible to calculate Af5 ° for a species when Sm ° has been determined by the third law method. Then Af G° for the species in dilute aqueous solution can be calculated using equation 15.3-2. Measurements of pATs, pA gS, and enthalpies of dissociation make it possible to calculate Af G° and Af//° for the other species of a reactant that are significant in the pH range of interest (usually pH 5 to 9). When this can be done, the species properties of solutes in aqueous solution are obtained with respect to the elements in their reference states, just like other species in the NBS Tables (3). 

Use of the third law is not the only way to get large molecules of biochemical reactants into tables of species properties. If apparent equilibrium constants and heats of reaction can be determined for a pathway of reactions from smaller molecules (for which Af G° and Af H° are known with respect to the elements) to form the large molecule, then the properties of the species of the large molecule can be determined relative to the elements in their reference states. This method has its problems in that it is very difficult to determine apparent equilibrium constants greater than about 10 to 10 and the number of reactions in the path may be large and some of the reactants may not be readily available in pure form. Thus it is fortunate that the third law method is available. 

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. 

The third-law method is based on a knowledge of the absolute entropy of the reactants and products. It allows the calculation of a reaction enthalpy from each data point when the change in the Gibbs energy function for the reaction is known. The Gibbs energy function used here is defined as... 

The experimental data presented by Rawling and Toguri in the temperature range of 800 to 900 K have been analysed by the third-law method using the selected heat capacity functions for H2Se(g), H2(g) and Se(cr, I). The derived enthalpy of formation for H2Se(g) from this analysis is Af//°(H2Se, g, 298.15 K) = (29.5 1.5) kJ-mol. ... 

Enthalpy changes can additionally be evaluated according to the third-law method... 

Table 10. Wu [282] identified the molecules Li O and LijO for the first time and determined their enthalpies of atomization (see Table 10) according to the third-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). 

If the Gibbs energy or the equilibrium constant of a reaction is determined experimentally at a temperature >298.15 K (e.g. by emf measurements), then every measurement yields a value of A//r (298.15). However, a condition for this is that 5(7 ) and [//(T)-//(298.15)] are known. This precondition means that the increment of the Gibbs energy, i.e. G(r)-G (298.15) must be known. This reveals that the function G T) is sufficient for analysis by the Third Law method . Nevertheless, the changes in G T) and AG(T) are greater than is the case for Gef(T) and AGeffT), so that these latter are often preferred for a Third Law analysis. 

The data have been processed by the third-law method to give the value for Af//° (TI1F4, cr, 298.15K) given in Table VIII-4, where the uncertainty has been increased to allow for the two factors mentioned above. 

These data were analysed by the third-law method, using the thermal functions for ThF4 discussed above, to give the results shown in Table VIII-5. 

The vapour pressures of ThCLt(cr) and ThCUCl) have been measured by a number of investigators, as suimnarised in Table VIII-17. These data have been analysed by the third-law method, to give the results shown in the Table VIll-17. For the measurements of [1979S1N/PRA], the data measured by the two techniques were consistent, and were thus combined. 

Application of the third-law method for determination of the reaction enthalpy [58]... 

The parameters A and E (or AS and AR) are determined respectively with the use of an Arrhenius plot or the second-law (Knudsen-Langmuir) method. The third-law method, which has received wide recognition in equilibrium thermochemical studies [4], has not been used in kinetics investigations at all. The first publication on this subject appeared only in 2002 [5]. Studies of L vov and his colleagues still remain the only attempts in this area (see review [6]). 

The lack of values of A S, or of tabulated data necessary for their calculation, would seem at first glance to curtail seriously the application of the third-law method. In reality, for most substances, the absolute values of the entropy for standard conditions (Ar gg) and the corresponding temperature corrections S — Aggg) can be found tabulated in many reference handbooks [4, 9-11]. Nevertheless, for some compounds, even quite common ones, for instance for low-volatility salt molecules in the gas phase, these data are lacking. In these cases, it is sometimes possible to evaluate the entropy from a comparison with the values of Ai.5 available for similar molecules of other metals. This approach was used, in particular, to evaluate the value of A S for the molecules of Li2S04, CaS04, and CUSO4 [12]. 

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... 

Thus, the error inherent in the use of the third-law method is indeed an order of magnitude lower than that provided by the two other methods. 

The maximum of this distribution lies at the value Tmax/AT = 10, with an average of Pmax/AP obtained from the analysis of 220 publications being 16. As can be seen, there are some cases when this ratio and therefore a loss of precision (in comparison with the third-law method) reach values of 30-50. The results of measurements obtained under such conditions certainly inspire very little confidence. 

An analysis of publications that have appeared in recent years [5, 12-24] reveals that, where the molar enthalpy (the E parameter) was determined by the third-law method, the random error (relative standard deviation Sr), which... 

Another reason that can account for the systematic differences between the values of A H (II), measured by the second-law and Arrhenius plot methods, and those of A H (III) measured by the third-law method, is the systematic decrease of the contribution of the condensation energy to the reaction enthalpy with increasing temperature and the ensuing slight increase of A H (III) and a substantially larger decrease of A H (II). This effect will be considered in detail in Sect. 8.2. 

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