J Therm Anal Calorim (2014) 115:1851–1856 DOI 10.1007/s10973-013-3401-z
Equilibria in systems condensed substance–gas Additions to interpreting and processing tensimetric data Vladimir I. Belevantsev • Alexandr P. Ryzhikh Kseniya V. Zherikova • Natalia B. Morozova
•
Received: 12 February 2013 / Accepted: 3 September 2013 / Published online: 22 October 2013 Ó Akade´miai Kiado´, Budapest, Hungary 2013
Abstract The information about the reasons for linearity of the empirical functions lnK°(T) = f(1/T) in the local temperature range, where K°(T) is polythermal data on equilibrium constants of homogenous and heterogeneous processes (in the latter case, if at least one chemical species presents among reagents), were generalized. During this work the conclusions obtained were used to develop a procedure of tensimetric evaluating thermodynamic parameters of individual substance evaporation process, realized in the area of co-producing two molecular species in saturated vapor composition (Pexp = P1 ? P2). In particular, it was shown that regression analysis of tensimetric data for these systems was informative due to admissibility of the linear model (for the functions lnPi(T) = f(1/T)) to describe the contribution of each of the molecular species in the measured pressure. In addition, it was shown the constructiveness of replacing the variable x = 1/T by x0 = DT/T = (T – T*)/T which was related with reassignment of conceptual meaning for the first of two liner parameters from the standard entropy of the evaporation reaction (DrSi8*) to the standard Gibbs free energy (DrGi8*) (the superscript * refers both of these characteristics and the value T* to the investigated temperature range center calculated in terms of inverse scale). The procedure developed allows ones to realize stable evaluating two pairs of thermodynamic characteristics of DrGi8* and DrH*i (i = 1, 2). The third pair (DrSi8*) is strictly calculated using wellknown fundamental relationship. Procedure informativity
is particularly illustrated by a simple example (evaporation of SeO3solid in the temperature range 360–392 K). Keywords Chemical species as reagent Thermodynamic characteristics Polythermal data Processing tensimetric data Vapor composition over SeO3solid
Introduction and problem formulation Typically, polythermal data on saturated monomolecular vapor pressure (P) of condensed individual substance without phase transitions1 in the temperature range under study are adequately described by a linear dependence in the coordinates lnP = f(1/T). Three limitations are included in the interpretation: the admissibility of the ideal gas approximation, the ability to ignore the influences of total pressure in the system on thermodynamic characteristics of the initial condensed substance and temperature on enthalpy of the evaporation process. However, the last pointed limitation is not necessary. This conclusion strictly follows from the approximation [1–3] widely used for the thermodynamic descriptions of any processes involving chemical species2 [4, 5]: Dr Go ðT Þ=T ffi Dr H =TDr So Dr Cp ðlnðT=T ÞDT=T Þ; ð1Þ *
where DT = T – T , DrCp value is assumed constant over the whole range of inverse temperatures (1/T) realized in a V. I. Belevantsev A. P. Ryzhikh K. V. Zherikova (&) N. B. Morozova Nikolaev Institute of Inorganic Chemistry, Siberian Branch of RAS, Lavrentiev Ave. 3, 630090 Novosibirsk, Russia e-mail:
[email protected]
1
Here, we mean only the transitions with keeping a condensed state (that is, transitions solid0 ? solid00 or solid ? liquid). 2 Naturally, it is applied to both homogeneous and heterogeneous processes involving any (i.e. not only gaseous) chemical species.
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particular study, enthalpy and entropy characteristics are attributed to its center (1/T*). The analysis of [1–3] has shown that the reason for the linearity of the empirical functions RlnK8(T) = u(1/T), where K8(T) are polythermal data on equilibrium constants of any of the processes under study,3 is not usually the absence of objectively occurring temperature effects (at jDrCpj [ 0) on the DrH(T) and DrS8(T) values, but high compensation levels of these effects. This fact has already been included by us to develop a number of additions to the processing procedures of polythermal data on simple systems [1–3]. It is shown that the admissibility of the linear approximations of the experimental data is connected with clouding the uncompensated part of jDrCpj [ 0 consequences by errors. At the same time the DrH* and DrS8* estimates remain correct. This result can be used much broader than it was recommended in the above papers. In particular, they can underlie the procedures of the tensimetric evaluation of thermodynamic characteristics referred to T* for the individual substance evaporation processes complicated by the formation of two (or more) molecular species in the gas. The aim of this article is to consider this possibility and to illustrate one of the variants of its application by a simple example.
Theory The meaningful consequence of Eq. (1) sufficient to interpret tensimetric data for equilibria of the evaporation process (leading to the monomeric species (Agas) formation): Acondensed ¼ Agas
ð2Þ
is the following:4 R ln ðPA ðT Þ=atmÞ ffi Dr So ðT ÞDr H ðT Þ=T þ Dr CP ðlnðT=T ÞDT=T Þ:
3
ð3Þ
We would remind readers that, here, only and only processes having at least one or more chemical species among reagents are referred to. It should be noted that well-known equation: DrG8(T) = –RTlnK8(T), where the superscript ‘‘zero’’ has no relation to the standardization of not only temperature (it may be clear from context), but also total pressure (whose influences on the considered equilibria can be neglected), serves the basis for the description of such processes. Nontrivial complex of meaningful and formal aspects of standardization this superscript displays has been discussed in detail in previous studies [1–4]. 4 Naturally, choice of the dimensions of all thermodynamic characteristics included in equations (1, 3) should be mutually agreed. Besides, the relationship among the standardization of the partial pressure values of gaseous molecular species and consequently thermodynamic characteristics of the processes involving them is of primary importance [2].
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It has already been shown in [2, 3] that using leastsquares processing experimental data on these systems the quadratic function:5 ln ðPA =TorrÞ ffi a þ b ðDT=TÞ þ c ðDT=TÞ2 ;
ð4Þ
based on expanding the term ln(T/T*) of Eq. (3) in a Taylor series is preferred. At the same time the linear one: ln ðPA =TorrÞ ffi a þ b ðDT=TÞ
ð5Þ
is usually also adequate. The preference to the quadratic approximations (4) of the lnPexp % ln(PA/Torr) values in above simple case (2) was described in detail earlier [2, 3]. However, vapor pressures measured in systems with formation of two or more molecular species can be approximately equal to the partial pressure values of the individual chemical species only locally. In general, we have the following: Pexp ffi
l X
Pi ;
ð6Þ
i¼1
where i is the index marking both the stoichiometry of the molecular species under study and its partial pressure, l is the number of different molecular species presented in commensurable amounts in the vapor in the considered temperature interval. Based on the above, it is quite obvious that both the approximations (4, 5) as a rule will be acceptable to parameterize connection of any of the Pi values with temperature. However, it is reasonable to confine ourselves to the function (5) analyzing the complex (l C 2) systems. The reason for this limitation is often a sharp increase in the instability of the components of the desired parameter evaluation vector with the growth of its dimension. Thus, if l = 2, the dimension of the desired parameter evaluation vector will be equal to four even in case of a linear model. If the model (4) is preferred for each of two processes under consideration, the regression analysis will need to find six adjustable parameters, and, as a rule [2, 3], two of them will be statistical zero. Naturally, also the stability of the other four parameters will sharply decrease.6 It is useful to note that change of the variables x = 1/T to x0 = DT/T at widely used linear least squares
5
Here, the record [Torr] means that there is only a numerical (dimensionless) value for the denominate quantity of PA measured in Torr under ln. The last one (incidentally, the dimensions and everything associated with the denomination of any of the characteristics under review) should be properly taken into account at subsequent correlating dimensionless adjustable parameters a, b and c with the meaningful equivalents. 6 Perhaps, this is why such increase in the instability of the desired parameter estimates are usually called ‘‘curse of the dimensionality’’ [6].
Equilibria in systems condensed substance–gas
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approximations of the experimentally recorded functions y % ln(PA/Torr) = f(1/T) in the simple cases (l = 1) is actually equivalent to centering the variables x to their arithmetical means ( x) conjugated with scaling. In this case, the correlation coefficients of adjustable parameters after standard centering any linear regression y/x are known to be equal to zero [3, 6], which leads to a substantial increase in the stability of their point estimates. The addition of centering by conjugated scaling, that is, the inclusion of 1/T* value in the angular coefficient b does not cancel the above consequence. It is also easy to see that the replacement of 1/T by DT/T leads to the equivalences of the values a to DrG8*instead of DrS8* (accurate within multiplier). Therefore, the regression estimates of the first ones (referred to the center T*) are the most stable in the simple case considered here. Significance of such consequences from centering conjugated with scaling sharply increases when changing simple cases (l = 1) by more complex ones (already at l = 2). Exactly these consequences considered in detail above by the example of l = 1 makes it possible to set and solve the problem of estimating four thermodynamic characteristics of DrGi8*, DrH*i (i = 1, 2) in cases where l = 2. After correct rounding the determination results of these two pairs of thermodynamic characteristics, the third one (DrSi8*) useful in the analysis of such cases could be strictly calculated using well-known fundamental relationship.7 In light of the above statement the idea of our proposed approach will be considered by an example, where system dimension is limited by the equality l = 2 in the analyzed temperature interval. In this case, a basis set of the processes will come to just two heterogeneous reactions: n1 Asolid ¼ An1gas ;
ð7Þ
n2 Asolid ¼ An2gas ;
ð8Þ
where the n1 and n2 values are the integers which represent the chemical species stoichiometry in the vapor composition in terms of the number of fragments having the stoichiometry of the initial individual substance. To parameterize connection among partial pressures of either of two molecular species (An1 and An2) and temperature, we will obtain the functions consistently addressed to the Pi values: ln P1 ffi a1 þ b1 ðDT=TÞ;
7
ð9Þ
We may remind the fact that the rejection of the model (5) in the simple case l = 1 and the usage of the model (4) results in an additional informativity. It has been adequately substantiated above that already in the case l = 2, the use of the model (4) becomes unpromising.
ln P2 ffi a2 þ b2 ðDT=TÞ;
ð10Þ
based on Eq. (5). Total vapor pressure of this system is rational to express by the formulas: Pexp ffi P1 þ P2 ;
ð11Þ
lnPexp ffi ln P1 þ ln ð1 þ P2 =P1 Þ:
ð12Þ
The last one shows that the linear approximation of function ln(Pexp/Torr) = f (DT/T) is adequate in the temperature range where P2 P1. In such cases it is possible to carry out sustained and rapidly convergent to desired solution iteration for estimating the parameters of (9), (10). In principle, this problem can be solved, firstly, with less acute constraints of the level of local inequality P2 \ P1, and secondly, by many methods. We recommend the simplest of known variants used in chemical kinetics and radiochemistry (for expansion of two or more exponent sums on components). Using our model, it is reasonable to distinguish two subgroups with centers at T*1 and T*2 within the general array of experimental points. Then, using them two pairs of the parameters a1, b1 and a2, b2 of the first approximation cycle should be evaluated iteratively, and the result should be applied as a start in the second one. During the last cycle, the whole array of points is included in the least squares approximation (at least in the approximation of the P^1 values if just they are attributed to the form prevailing in the whole temperature range under study), and centering conjugated with scaling is performed at a general value of T*. The evaluations of desired thermodynamic characteristics for the processes (7, 8) can be calculated from two pairs of the parameters ai and bi (i = 1, 2) resulted from the second cycle of approximations using following equations: Dr Go i ¼ ðT R=1000Þ ðai 6:633Þ;
ð13Þ
Dr Hi
ð14Þ
¼ ðT R=1000Þ bi ;
Dr So i ¼ Rðai 6:633 þ bi Þ; Dr Go i
¼
Dr Hi T
o
Dr Si =1000;
ð15Þ ð16Þ
where R = 8.3144 J mol-1 K-1, and the divisor 1000 has dimension of J kJ-1 and provides it in kJ mol-1 for DrGi8* and DrH*i , while R and DrSi°* has dimension of J mol-1 K-1.8 The manner of reaching the decisions on the n1 and n2 values (Eqs. 7, 8), that is, the stoichiometric coefficients and numbers expressing the stoichiometry of chemical
8
This set of equations clearly demonstrates the reason why the evaluations of thermodynamic characteristics DrGi8* and DrH*i determined by the present procedure are not correlated with each other (see also Footnote 6 to Eq. (5)) while the estimate of DrSi°* is correlated with each of them.
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species in vapor composition will be considered at the end of the next section.
Table 1 Initial data and used in the iterative processing coordinates obtained by local centering at T*1, T*2, and T* conjugated with scaling T/K
Pexp/Torr
1/T
(DT/T)1
(DT/T)2
DT/T
360
0.942
0.00278
-0.0194
-0.0694
-0.0389
363
1.081
0.00275
-0.0110
-0.0606
-0.0303
367
1.247
0.00272
0
-0.0491
-0.0191
370
1.45
0.00270
0.00811
-0.0405
-0.0108
374
1.71
0.00267
0.0187
-0.0294
0
377
2.03
0.00265
0.0265
-0.0212
0.00796
381 385
2.46 3.02
0.00262 0.00260
0.0368 0.0468
-0.0105 0
0.0184 0.0286
388
3.80
0.00258
0.0541
0.00773
0.0361
392
4.90
0.00255
0.0638
0.0179
0.0459
Fig. 1 Experimental data (points) and least squares line at the beginning of iterative processing using (9, 10). Here, five points with center at T*1 = 367 K are included in the averaging based on the model (9)
Processing tensimetric data of [7, 8] and results We have applied the approach grounded above (1–16) for processing experimental data on equilibrium vapor pressure over SeO3solid (mp = 394 K) obtained by the static method with a quartz zero-manometer [7] and included in the review by the authors of the monograph [8, p. 222]. Initial data and the variants of centering associated with scaling applied within the procedure described below in detail are shown in Table 1.
T/K 395
390
385
380
375
370
365
360
1.6 1.4
ln(P/ Torr)
1.2 1.0 0.8 0.6 0.4 0.2 lnP1calc.(0) = 15.77 – 5.70*103 (1/T )
0.0 –0.2
0.00255
0.00260
0.00265
T
–1
0.00270
0.00275
0.00280
–1
/K
Table 2 The equilibrium vapor composition over SeO3solid [7] according to the parameters (a1 = 0.4080, b1 = 13.05; a2 = -1.470, b2 = 48.52) of the model (9, 10) and the picture of residues with the total sum of their squares R(DlnP)2 = 0.00186 DlnP
DT/T
Pexp/Torr
P1calc
P2calc
Pcalc
lnPexp
lnPcalc
-0.03889
0.942
0.905
0.035
0.940
-0.0598
-0.0617
0.002
-0.03030
1.081
1.013
0.053
1.065
0.0779
0.0634
0.014
-0.01907
1.247
1.172
0.091
1.264
0.2207
0.2339
-0.013
-0.01081 0
1.45 1.71
1.31 1.50
0.14 0.23
1.44 1.73
0.372 0.536
0.366 0.550
0.006 -0.014
0.00796
2.03
1.67
0.34
2.00
0.708
0.696
0.012
0.01837
2.46
1.91
0.56
2.47
0.900
0.905
-0.005
0.02857
3.02
2.18
0.92
3.10
1.105
1.132
-0.027
0.03608
3.80
2.41
1.32
3.73
1.335
1.317
0.018
0.04592
4.90
2.74
2.13
4.87
1.589
1.583
0.006
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Equilibria in systems condensed substance–gas Fig. 2 Experimental data (lnPexp), the results of their final approximation according to (9) (line lin(I)lnP1calc with parameters a1 = 0.4080, b1 = 13.05 fitted using lnP^1 points) and (10) (line lin(I)lnP2calc with parameters a2 = -1.470, b2 = 48.52 fitted using lnP^2 points), accumulation calculated curve (lnPcalc), and the results of final approximation of initial data with introduced adjustments at the start, see end of the section (line lin(II)lnP1calc with parameters a1 = 0.3804, b1 = 16.71 and line lin(II)lnP2calc with parameters a2 = -1.665, b2 = 48.00)
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T/K 356
360
364
368
372
376
380
384
388
392
1.5 1.0 0.5 0.0
ln(P/ Torr)
–0.5 –1.0 –1.5
lnPexp lnP1
–2.0
lnP2
–2.5
lnPcalc
–3.0
lin(I)lnP1calc lin(I)lnP2calc
–3.5
lin(II)lnP1calc lin(II)lnP2calc
–4.0 –4.5
–0.04
–0.02
0.00
0.02
0.04
ΔT/T
Table 3 Thermodynamic characteristics of heterogeneous reactions (7, 8) (A = SeO3) and tetramer dissociation in the gas phase Reaction
DrG°*/kJ mol-1
SeO3solid = (SeO3)gas
DrH*/ kJ mol-1
DrS°*/ J mol-1 K-1
19.36 ± 0.02
40.6 ± 0.7
56.8 ± 2.0
4SeO3solid = (SeO3)4gas
25.2 ± 0.2
150.9 ± 7.4
336 ± 20
(SeO3)4gas = 4(SeO3)gas
52.2 ± 0.3 (lnK°*diss = -16.8)
SeO3solid = (SeO3)gas
11.5 ± 10.2
-109 ± 28
19.44 ± 0.02
52.0 ± 0.7
87 ± 2
4SeO3solid = (SeO3)4gas
25.8 ± 0.3
149.3 ± 8.8
330 ± 24
(SeO3)4gas = 4(SeO3)gas
52.0 ± 0.4 (lnK°*diss = -16.7)
58.7 ± 11.6
I
II
18 ± 32
The calculation was performed with the following parameters: (I) a1 = 0.4080(0.0065), b1 = 13.05(0.24); a2 = -1.470(0.073), b2 = 48.52(2.39); (II) a1 = 0.3804(0.0059), b1 = 16.71(0.22); a2 = -1.665(0.086), b2 = 48.00(2.83), where there is the standard deviation in brackets. Superscript * recalls that these characteristics are addressed to T* = 374 K
At the beginning of processing we divided the data set presented in the coordinates lnPexp = f(1/T) (see graphic illustration in Fig. 1) in two groups of five points at lower and higher temperatures with the centers at T*1 = 367 K and T*2 = 385 K. After estimating the parameters a1(0), b1(0) of zero approximation [by the least squares approximation of the first five points according to Eq. (9) at P^1 (0) % Pexp and T*1 = 367 K], the P1calc(0) values were calculated at higher temperatures (the results are presented in Fig. 1 in terms of anamorphosis obtained by the substitution of initial variable 1/T for DT/T locally used). Then, the P^2 (0) values for five points with center at T*2 were calculated according to equation P^2 (0) = Pexp – P1calc(0); the parameters a2(0), b2(0) were obtained by the least squares approximation of P^2 (0) according to Eq. (10) and used to calculate the values of P2calc(0) and P^1 (1) = Pexp – P2calc(0) in the temperature
interval with a center at T*1. The last ones were the source for the first step of the first cycle of the desired parameter refinements. After five such refinements, we used the parameters a2(5), b2(5) as starting ones for the second cycle of approximations. During this cycle we included all ten points in the least squares approximation of the P^1 values, but the P^2 values were approximated as previously using only the last five ones.9 However, at the same time one general center at T* = 374 K was chosen for the meaningful associating of the parameters (a1, b1, a2, b2). The processing results based on the assumption that there is no significant systematic components in error (version I) are 9
Thus, we reduced the possible, in principle, displacements which arose (if errors contained not only the random components) from including the ranges of the sharp increase of difference ‘‘infinitesimal level’’ in processing (even in weighting least squares variant) [6, 9]. In our case it concerns exactly the P^2 estimates at T \ 377 K.
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presented quantitatively in Table 2. They indicate that the data set processed has high quality in relation to level and sign alteration of the random error component. Figure 2 graphically illustrates not only the version I, but also the version II, where the hypothesis concerning the presence of a systematic error component requiring adjustments has been applied.10 It is evident that to solve the problem of n1 and n2 coefficients is impossible within (9, 10) models only, that is, without applying additional grounds.11 We have chosen the variant where n1 = 1, n2 = 4, although the authors of used [7] interpreted their experimental results conversely.12 There are two factors defining our choice. Firstly, it is easy to see that raising T from 360 to 392 K increased the value of P1calc only by 3 times, whereas the value of P2calc by 61 times (Table 2). This result definitely indicates the growth of the association degree with increasing temperature (due to raising P2 conjugated with P1 by law Kass = P2/P41). And, a slightly smaller increase in the ratio P2(392 K)/ P2(360 K) than it should be expected from the fourth power law (34 = 81) is reasonable to attribute to a certain reduction of the association constant (which is really connected with the fact that the gas dissociation usually occurs with heat absorption). Secondly, the monomer was detected by mass spectrometry [10] at low temperatures, and significant amounts of tetramer were not registered, whereas authors of [11] suggested the presence and even the prevalence of tetramer at mp and above based on electron diffraction data. The final evaluations of desired thermodynamic parameters related to T* = 374 K are shown in Table 3. Without canceling the undisputed (as we suppose) conclusion about the monomer predominance at low temperatures and the tetramer formation in clearly significant quantities only at temperatures above 374 K, it still should be noted that negative entropy value of tetramer dissociation (see Table 3, I) is hardly probable. It can be supposed that the data on the Pexp values in 360–377 K are slightly overstated. In particular, a set of adjustments -0.15 Torr (360 K) with monotonic lowering their modulus up to zero at 385 K (version II) has led to the significant displacement of desired DrS°* estimate of tetramer dissociation (see 10
See end of the section about the reason for introducing adjustments and their levels. 11 This can be knowledge about the regularities in the characteristics of heterogeneous processes under consideration and dissociationassociation reactions in gases, about the average molecular mass of the vapor under studying and/or data on its state using mass spectrometry, electron diffraction, etc. 12 Having overestimated the role of the gas dissociation constants growth with temperature increase they could have underestimated the consequences of species concentrations growth in the gas due to the equilibrium displacement in each of them and a solid phase.
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Table 3, II) to positive region (while retaining the standard deviation of the approximation results at the same level).
Conclusions Information about the reasons for linearity of the empirical functions lnK°(T) = f(1/T) in the local temperature range, where K°(T) is polythermal data on equilibrium constants both homogeneous and heterogeneous processes, is generalized. Naturally, in the latter case the matter is only about the process, where at least one chemical species presents among reagents. In particular, these are the evaporation of any condensed substances stoichiometrically defined. Based on generalizing, the procedure has been developed to evaluate thermodynamic parameters of evaporation processes, realized in the area of co-producing two molecular species. Procedure informativity is particularly illustrated by a simple example.
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