Determination of Optimal Vapor Pressure Data by the Second and Third Law Methods

Though equilibrium vapor pressures are utilized to determine thermodynamic properties of not only gaseous species but also condensed phases, the obtained data often disagree by a factor of 100 and more. A new data analysis method is proposed using the so-called second and third law procedures to improve accuracy of vapor pressure measurements. It was found from examination of vapor pressures of cesium metaborate and silver that the analysis of the difference between the second and third law values can result in determination of an optimal data set. Since the new thermodynamic method does not require special techniques and or experiences in dealing with measured data, it is reliable and versatile to improve the accuracy of vapor pressure evaluation.


INTRODUCTION
Equilibrium vapor pressure data are utilized to determine thermodynamic properties of not only gaseous species but also condensed phases. But, vapor pressure measurement generally has a large uncertainty as mentioned in the preface of a handbook on thermodynamic data, 1) "vapor pressures are not infrequently reported which disagree by a factor of 100 or more," or as said in another paper, 2) "experience in vapor pressure measurements, particularly at high temperatures, has shown that large systematic errors are common, even among experienced investigators." Moreover, another thermodynamic data hand book 3) says "when calorimetric data were available, they were given considerably more weight than values determined from vapor pressure data." Based on this situation, the so-called second and third law procedures are considered as a method to improve accuracy of vapor pressure measurement. e second and third law methods 4,5) are commonly used to determine reaction enthalpy from equilibrium constant measured in experiment, for example, to determine vaporization enthalpy in terms of vapor pressure measurement. In addition, examination of the di erence of the determined enthalpies between those calculated by the second and third law methods has been utilized to check whether the Gibbs energy functions and equilibrium data are mutually consistent 4) or to check whether cause of systematic errors are under control. 5) As for Gibbs energy functions, even if in the absence of experimental data, satisfactory estimates can usually be made by a variety of procedures and uncertainties calculated from experimental molecular parameters are said to be small, i.e., ~±0.2 to 0.4 J K −1 mol −1 . 5) ese uncertainties could correspond to a two to ve percent error in the equilibrium constant which is assumed to have the same uncertainty level as the Gibbs energy functions. erefore, such a thermodynamic method based on the second and third laws can be used to improve accuracy of vapor pressure data especially if Gibbs energy functions are determined based on experimental data. is paper concerns about application of the second and third law methods for determination of optimal vapor pressure data. e new evaluation method is practically applied for equilibrium vapor pressure data of cesium metaborate, CsBO 2 , 6) which is one of the most important chemical species having signi cant radiological impact in a severe accident like the one that occurred at Fukushima Daiichi Nuclear Power Station. Vapor pressures of silver given by nine laboratories 7) were also evaluated by the present method.

THERMODYNAMIC METHOD FOR DETERMI-NATION OF OPTIMAL VAPOR PRESSURES
e new thermodynamic method to determine optimal vapor pressures proposed in this study is based on the second and third law methods. In the third law method, 8,9) enthalpy changes of investigated reactions, ∆ r H°, can be determined from each experimental value of equilibrium constant, K p°, using the following relation: where θ is reference temperature of 0 or 298.15 K, R gas constant, T absolute temperature, and gef Gibbs energy function. ∆ signi es the di erence between thermodynamic quantities of products and reactants. e subscript ( r ) denotes reaction. e superscript (°) refers to standard-state pressure of 1 bar. e Gibbs energy function is de ned as where S is entropy. Gibbs energy function of gaseous species can be calculated from molecular parameters by the use of statistical mechanics. While, Gibbs energy function of condensed phases can be obtained from heat capacity Cp° from 0 K to the temperature of interest. Equation (1) is derived through the relation: On the other hand, in the second law method, 10) van't Ho equation: is used. Equation (5) assuming that both of ∆ r H°(T) and ∆ r S°(T) are constant with temperature. us, ∆ r H°(T ave ) at average temperature, T ave , is obtained from measured K p°( T) by a least-squares analysis of an Arrhenius plot: where both of A and ∆ r H°(T ave ) are tting parameters. It is noted that T ave , is determined from the average value of 1/T for the measurements. en, ∆ r H°(T ave ) is reduced to the reference temperature θ K through the use of heat content functions, ∆ r H°(T ave )−∆ r H°2 nd (θ): But this procedure does not take into account for the temperature dependence of ∆ r H°(T) within the measured temperature range. So more precise ∆ r H°2 nd (θ) can be obtained from the so-called Σ plot 11,12) : where T′ and T″ are variables of the integrations and A′ is a tting parameter. is equation is obtained by integrating the van't Ho equation, Eq. (4), so that a Σ plot of measured K p°( T) will have linear dependence on 1/T. Now, this integration can further be changed to the following equation: en, Eq. (9) can be transformed to the following equation: and, when temperature-independent constants of A′ and ∆ r S°(0) are combined into a single constant A″ for simplicity, similar equation to Eq. (1) can be obtained: Actually, this equation can also be derived by integrating the van't Ho equation, Eq. (4), from 0 K to the temperature of interest and we can nd the following equation: by use of the relation: us, in the case of the second-law method, two or more measured K p°( T) are necessary to determine the value of ∆H°2 nd (θ) since the second law of thermodynamics cannot provide the relation: ∆ r S°(0)=0. 13) On the other hand, the third law value can be obtained even from a single measured K p°( T) by use of Eq. (1) since the third law of thermodynamics can provide ∆ r S°(0)=0. In addition, when the temperature is multiplied on both sides of Eqs. (12) or (13), the lehand side is the same as the right-hand side of Eq. (1). en, if both the measured K p°( T) and ∆ r gef°(T) are accurate, A″ can become close to zero and ∆H°2 nd (θ) can also agree well with ∆H°3 rd (θ). erefore, examination of di erence of the values between ∆H°2 nd (θ) and ∆H°3 rd (θ) can result in determination of optimal data set of vapor pressures. In some cases, application of such a thermodynamic method might lead to choice of wrong data set of vapor pressures due to fortuitous compensations of errors between Gibbs energy function and vapor pressure data. However, since vapor pressure data are said to have a large uncertainty and accuracy of Gibbs energy functions is inferred to be commonly higher than that of vapor pressure data, such compensations are unlikely to happen. In other words, even if such compensations happen, vapor pressure data are considered to be the same level of errors as Gibbs energy functions.

APPLICATION TO VAPOR PRESSURE MEA-SUREMENT OF CESIUM METABORATE AND SILVER
Vapor pressure of CsBO 2 e detailed procedures of the sample and the Knudsen e usion mass spectrometric, KEMS, measurement are described in another paper. 6) Brie y speaking, CsBO 2 sample used for the KEMS measurement was prepared from Cs 2 CO 3 and H 3 BO 3 mixed powders by heating at 873 K in air. In the KEMS measurement, the sample was heated and cooled stepwise in 30 K steps and the heating process was repeated two times. According to the paper 14) which examined an ionization behavior of CsBO 2 by a KEMS method, detected Cs + ion results from the dissociative ionization of CsBO 2 molecule. So only the ion current of Cs + ionized under 50 eV electron impact, I Cs , was monitored until the end of the KEMS measurement, as shown in Fig. 1. e measured ion current was converted into absolute vapor pressure by using the following equation 6) : where ∆W CsBO2 is the weight of total amount of CsBO 2 e using through the ori ce, R the gas constant, a cross section of e usion ori ce, L the Clausing factor, 15) T absolute temperature, i an isotopic species, M the mass number of CsBO 2 , γ isotopic abundance ratio of CsBO 2 . ∆W CsBO2 is assumed to be the same as the weight di erence of the samples before and a er the KEMS measurement. Ion currents used in conversion into absolute vapor pressures were chosen for the last one minute before the next heating or cooling step. Figure 2 shows the determined vapor pressures plotted in Arrhenius form. As shown in this graph, there is a certain di erence of vapor pressures between the heating and cooling processes. e ion currents at each temperature ultimately seem to be independent of time from Fig. 1. So it is hard from the time dependence of the ion current to determine which data set of vapor pressures is more accurate. erefore, sublimation enthalpies of CsBO 2 by the second and third law methods are examined to nd optimal data set of the vapor pressures. e Gibbs energy function of gaseous CsBO 2 required in this method is derived from molecular parameters given by Ezhov and Komarov 16) assuming a rigid rotor harmonic oscillator model. However, it is found that there is a negligible di erence between the Gibbs energy functions calculated from other molecular parameters reported in the past. 6) While, the Gibbs energy function of solid CsBO 2 is computed from its standard entropy of formation and heat capacity recommended by Cordfunke and Konings,17) which are regarded as good quality. erefore, accuracy level of both of Gibbs energy functions of solid and gaseous CsBO 2 are expected to be better than that of vapor pressures of CsBO 2 . Table 1 shows examination results of the second-law and third-law sublimation enthalpies. As shown in the table, the di erence between the second-law and third-law values for the 2nd cooling stage of Nakajima et al. is the smallest. Indeed, Fig. 2 indicates the vapor pressures increase in the order of the 1st heating, 2nd heating and 2nd cooling stages. Further, when we take a closer look at Fig. 1, the ion currents at the heating stages, especially at the 2nd heating stage, still seem to increase with time and then the equilibrium might not be reached yet. Consequently, since the present thermodynamic method requires neither special technique nor experience when dealing with measured data, it can be useful to determine an optimal data set of vapor pressures. Table 1 also includes the second-law and third-law sublimation enthalpies evaluated from vapor pressures of CsBO 2 reported in the past. 18,19) Further, the values derived from the le -hand side of Eq. (12) and from the right-hand side of Eq. (1) are plotted in Figs. 3 and 4, respectively. Figure 3 also includes regression lines extrapolated to near the origin. As shown in these gures, the data of our work is the most reasonable because the intercept of the regression line is the nearest of the origin and temperature dependence of the third law values is the weakest. us, the second and third law methods can also allow us to nd a more preferable data set among literature data.

Vapor pressure of Ag
Vapor pressure measurements of silver were conducted  at nine laboratories to establish vapor pressure standard reference materials. 7) ese data sets of vapor pressures were used to con rm validity of the second and third law meth-ods. Gibbs energy functions of silver given by Cox et al. 20) and Barin 21) were used to identify their e ect on the di erence between the second and third law values. e examination results are summarized in Table 2. As shown in this table, the di erences between the second and third law values using Gibbs energy functions given by Cox et al ese gures show that, as the di erence between the second and third law values becomes smaller, the intercept of the regression line approaches the origin and the temperature dependence of the third law values becomes weaker. Further, these plots can help us to easily nd wrong data points which are largely deviated from the regression line or the average value. erefore such a thermodynamic method including not only examination of the di erence between the second and third    law values but also the plots such as shown in Figs. 6 and 7 can contribute to improvement of accuracy of vapor pressure measurement.

CONCLUSION
In this study, a new data analysis method is proposed to improve accuracy of vapor pressure measurements. e key of the new method is to use Eq. (12), which is derived from the van't Ho equation based on the second law of thermodynamics. e proposed method using the second and third law methods is found to be helpful to determine an optimal data set of vapor pressures when it is hard to know only measured data such as time dependence of ion currents in KEMS measurement. Further, examination results for vapor pressures of silver indicate that a smaller di erence between the second and third law values can lead to nd more accurate vapor pressure data. erefore, such a thermodynamic method can contribute to improve accuracy of vapor pressure measurement.