The Second-law Method

By measuring Kp at different temperatures and plotting nKp vs 1/T, one readily finds both parameters, more specifically, from the slope 

---

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... 

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). 

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... 

In contrast to the second-law method, for the third-law method the influence of self-cooling manifests itself in overestimation of the E parameter. This is evident from Eqs. 4.10-4.12. Therefore, the ratio of these two values measured by different methods is a very sensitive indicator of self-cooling. 

In conformity with the above theoretical estimations (Sects. 6.1-6.3), the temperature difference between the temperature-controlled heater and a sample in a high vacuum amounts to a few tens of degrees (e.g., 20-30 K for the first three reactants in Table6.1). This systematic error in combination with the condensation effect manifests itself in a considerable (up to 15-50%) underestimation of E parameters when the second-law method is used. 

Historically the underestimation of the role and magnitude of the self-cooling effect in kinetic studies of thermal decomposition has turned out to be one of the most important reasons which hindered interpretation of the T-S effect and, to some extent, the compensation effect, and promoted some misconceptions [19, 26-28], based on the confidence in infallibility of the second-law method in determination of thermochemical parameters (the enthalpy and entropy for decomposition reactions). 

Results and Discussion The initial data for calculating the enthalpy by the third-law method were the equivalent pressures measured for both elements under the free-surface vaporization conditions (after Langmuir) in [8]. The quantities were also determined in [8] by the second-law method. 

The results of effusion measurements are in excellent agreement with the results of a thermochemical calculation for the dissociative decomposition of Sn02 to SnO - - I/2O2. The mean molar enthalpy measured by the third-law method (397.0 1.8kJ moP ) is very close to the averaged value of A H /iy for the thermochemical calculation (393.8 0.8kJ moP ). The value measured by the second-law method, 390.3kJ moP is slightly (by <1%) lower than the thermochemical value. 

Fig. 16.1 Results of Knudsen and Langmuir [22] evaporation studies of Sn02 and the molar enthalpies of the corresponding sublimation reactions calculated by the second-law method...

It is noteworthy that all the measurements in question were performed within a short period (1965-1968) in the same laboratory (headed by Searcy). These results show that it is not appropriate to use the second-law method when studying the free-surface vaporization of compounds in a high vacuum. 

Table 16.17 presents also the A H /i/ values determined in the papers cited by the second-law method. These values are less reliable than those determined by the third-law method and are lower in seven cases out of ten. The difference in the A H /v values for BN reaches 60%. This systematic underestimation may be attributing to the self-cooling effect, which is particularly pronounced... 

Some comments on the calculations should be noted. The molar enthalpies of decomposition of sodium, potassium, barium, and lead azides that were calculated by the third-law method are given in Table 16.20. The initial data for these calculations were the absolute decomposition rates J of these azides, measured in [49-51]. The results obtained for NaNs and Pb(N3)2 at different temperatures and in different studies show good agreement. In Table 16.21, these results are compared with the averaged molar enthalpies measured for these and some other azides by the second-law method (the initial values of... 

The experimental values of the molar enthalpy, obtained by the second-and third-law methods, differ by no more than 10kJ moP, which is within the scatter of the A H /v values measured by the second-law method in different studies. 

Among the numerous published papers dealing with the decomposition of carbonates, those by Searcy et al. [110-112] seem to be the most reliable. In these studies, experiments on the decomposition of CaCOg, CaMg(COg)2, and BaCOg were performed in a high vacuum under free-surface vaporization conditions (after Langmuir). However, the enthalpies of decomposition of these carbonates, found by the second-law method (418, 390, and... 

---