17436
J. Phys. Chem. 1996, 100, 17436-17438
LETTERS Measurement of Enthalpy Differences in Cryosolutions: Influence of Thermal Expansion B. J. van der Veken* Department of Chemistry, UniVersitair Centrum Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium ReceiVed: June 13, 1996X
Cryosolvents such as liquid nitrogen, argon, krypton, and xenon show a substantial thermal expansion. Its influence on the value of enthalpy differences for simple chemical equilibria in dilute solutions in such solvents, determined using infrared spectroscopy, is discussed. Corrections to account for the thermal expansion are proposed.
Introduction For the chemical equilibrium described by the following general reaction between species A and B
nA + mB h AnBm
(1)
in vibrational spectroscopy in general,1 and in cryospectroscopy in particular,2,3 the reaction enthalpy ∆H° is determined from the relation
-RT ln K ) ∆G° ) ∆H° - T∆S°
(2)
The equilibrium constant K is a function of the concentrations of the species involved, and for dilute solutions in general molarities are used as concentration units. The concentrations are related to observed spectral intensities via Beer’s Law, and the observed temperature dependence of the intensities is exploited1 to derive a value for ∆H°. This approach is valid when the temperature variation of the spectral intensities is fully accounted for by the shift in the chemical equilibrium as a consequence of a change in the temperature. For cryosolutions such as solutions in liquified nitrogen, argon, krypton, and xenon, this condition is not fulfilled, as a consequence of the pronounced variation in liquid density of these solvents over the available temperature interval. In this letter we will discuss the magnitude of the error on ∆H° introduced by neglecting the solvent expansion, and we will indicate how this effect can be taken into account when relation 2 is used to determine the reaction enthalpy. Experimental Section Infrared spectra were recorded on a Bruker 66v vacuum interferometer equipped with a globar source, a Ge/KBr beam splitter, and a liquified N2 cooled MCT detector. The solutions were investigated in a copper cell of 4 cm path length, coupled to the IR beam via wedged silicon windows. The use of this cell has been described before.3 Of importance to the present study is the pressure range of 0-15 bar under which the cell can be operated. Spectra were obtained from interferograms averaged over 200 scans, Happ-Genzel apodized and Fourier transformed using a zero filling factor of 4. Integrated intensities * E-mail:
[email protected]. X Abstract published in AdVance ACS Abstracts, October 15, 1996.
S0022-3654(96)01747-9 CCC: $12.00
were obtained from least-squares band fitting using GaussLorentz sum functions. The argon used as a solvent has a stated purity of 99.9999%. The ethylene was supplied by Aldrich (nr. 29,532-9), with a stated purity of 99.5%. Discussion Cryosolvents such as liquified argon (LAr), krypton (LKr), xenon (LXe), and nitrogen (LN2) have a very narrow liquid range at 1 bar. For LAr, for example, at this pressure the liquid range covers a mere 3 K. The extended temperature range necessary for thermodynamical studies can only be obtained by allowing higher pressures. Cryogenic solutions are, therefore, usually investigated along their vapor/liquid coexistence line, and the upper pressure limit at which the cryogenic cell can be safely operated determines the highest temperature that can be used for each solvent. For the pure inert gases and for a pressure limit of 15 bar, the temperature ranges are, approximately, 78110 K for LN2, 84-124 K for LAr, 116-169 K for LKr, and 162-231 K for LXe. The liquid phase densities for these solvents vary considerably over these temperature ranges. For instance, the density of LAr varies from 1.416 g cm-3 at 83.81 K to 1.131 g cm-3 at 124.22 K, and the other cryosolvents show similar variations.4,5 The thermal expansion of a solution in such a solvent is not necessarily equal to that of the pure solvent. However, at the concentrations used in cryospectroscopy, typically 10-3 M, the expansion cannot differ greatly from that of the pure solvent. For this study we will, therefore, assume that the thermal expansion of the solutions equals that of the pure solvents. The integrated intensity Ii of a band in the infrared spectrum of solution i, obtained by integrating the decadic absorbance over the complete band, is related to its molarity Ci by Beer’s Law
Ii )
1 ACd 2.303 i i
(3)
in which Ai is the infrared intensity of the band and d is the path length of the cell. Because of this relation, and because of the thermal expansion, for cryosolutions the integrated intensity of a band will decrease with increasing temperature, i.e. with decreasing density. This is illustrated by the behavior of the ν7 (B1u) fundamental at 948 cm-1 of ethylene in a 2 × © 1996 American Chemical Society
Letters
J. Phys. Chem., Vol. 100, No. 44, 1996 17437
Figure 1. Integrated infrared intensity of the 947 cm-1 fundamental of ethylene in a 2 × 10-3 M solution in liquid argon as function of the solvent density. The dots represent the experimental points; the solid line is the result of a linear regression.
Figure 2. Logarithm of the liquid density of liquid xenon (LXe), liquid krypton (LKr), liquid argon (LAr), and liquid nitrogen (LN2) as a function of 1/T. The dots represent the experimental points; the solid lines are the result of the linear regression.
10-3 M solution in LAr. The integrated intensity of this band was measured at 17 temperatures between 86.4 and 124.8 K. The densities of LAr at these temperatures were interpolated from a third-order least-squares polynomial through the experimental density/temperature couples as listed in ref 4. In Figure 1 the intensities are plotted against the solvent densities. They show the expected variation with solvent density, and the linearity of the plot appears to justify the use of the solvent densities for the solution. Thermodynamically the equilibrium constant K for a chemical equilibrium in a solution must be written in terms of activities of the species. The latter usually are unknown, so we will assume that the solution behaves ideally. Then Raoult’s Law can be used to express K in terms of the mole fractions xi. For the above reaction this leads to
TABLE 1: Values of Rb for Some Cryogenic Solvents
K)
xAnBm xAn xBm
Ma ) M/N
(5)
The mass of the solution can also be expressed using the density F and the volume V of the solution
M ) FV
(6)
Combining these two allows the volume of the solution to be expressed as
V)
NMa F
ni ni F F ) xi ) V N Ma Ma
0.555 0.457 0.623 1.359
K)
n+m-1 IAnBm
IAn IBm
Fn+m-1
IAnBm ∆H° ) ln n m + (n + m - 1) ln F + C RT I I
-
(10)
(11)
A B
with
[
( )
AAn ABm d C ) ln AAnBm Ma
]
n+m-1
-
∆S° R
(12)
It is usually assumed1 that only the first term on the righthand side of eq 11 changes with the temperature, so that from the slope of the Van’t Hoff plot of ln(IAnBm/IAn ImB ) versus 1/T, ∆H° can be calculated. The presence of the term (n + m - 1) ln F, however, shows that for solutions with substantial thermal expansion this procedure does not lead to the correct value for ∆H°. It will be shown below that for the temperature ranges envisaged, the logarithm of the density of the cryosolvents is approximately inversely related to the temperature
ln F ) a +
b T
(13)
in which a and b are temperature-independent constants. With this, eq 11 can be written as
ln
(9)
( )
AAn ABm d AAnBm Ma
The equilibrium constant K is related to thermodynamical quantities via eq 2, which leads to
(8)
This expression is used to rewrite K
CAnBm Fn+m-1 K ) n m n+m-1 CACB Ma
LN2 LAr LKr LXe
(7)
Then the definition of the molarity Ci of species i, of which ni mol are present in the solution, can be written as
Ci )
Rb/kJ mol-1
Next, Beer’s Law, eq 3, is used to substitute the molarities for integrated intensities
(4)
In order to link observed spectral intensities to K, the xi must be related to the molarities of the species, which we accomplish as follows. For a solution of mass M composed of N mol of substance, the average molar mass Ma is given by
solvent
IAnBm IAn IBm
(
)-
)
∆H° + (n + m - 1)Rb 1 - (C + a) R T
(14)
This relation shows that the slope of the Van’t Hoff plot, multiplied by R, equals -(∆H° + (n + m - 1)Rb), and not ∆H° as in the simple approximation. The value of b is the slope of the plot of ln F versus 1/T. In Figure 2 such a plot is
17438 J. Phys. Chem., Vol. 100, No. 44, 1996
Letters
produced for the pure solvents LAr, LKr, LXe, and LN2, in the temperature ranges in which the equilibrium pressure is below 15 bar. In this figure also the result of the linear regression for each solvent is shown. The values of Rb derived from the linear regressions are collected in Table 1. As implied above, we will assume that these plots also describe the behavior of dilute solutions in the solvents. The simplest equilibria that can be investigated are conformational equilibria of the type
conformer 1 h conformer 2
(15)
For these, n ) 1 and m ) 0, so that the density correction drops out of eq 11. Thus, when measuring the conformational enthalpy difference from a temperature study, no correction for the thermal expansion of the solution is required. From a straightforward generalization it is easily shown that density corrections are not required for equilibria in which the number of solute molecules is not changed by the equilibrium reaction. For an equilibrium of the type encountered with 1:1 van der Waals complexes
A + B h AB the correction equals Rb and can be seen from Table 1 to vary from 0.457 kJ mol-1 in LAr to 1.359 kJ mol-1 in LXe. Values of ∆H° determined for such complexes3,6 typically range upward from 3 kJ mol-1, with uncertainties for the more accurate determinations on the order of 0.2-0.3 kJ mol-1. Hence, in LAr, LKr, and LN2 the value of Rb is small compared to ∆H° but exceeds the average uncertainty, which shows that the correction is statistically significant.
Close scrutiny of the data in Figure 2 shows that the correct description of the relation between ln F and 1/T requires not only a first-order term but small higher order terms as well. In view of the typical uncertainty in ∆H°, these small higher order terms will not markedly upset the linearity of the Van’t Hoff plot, and ∆H° can be obtained in the way suggested by eq 14, i.e. by constructing the Van’t Hoff plot using ln(IAn ImB /IAnBm), and correcting the slope by the term (n + m - 1)Rb. For complexes with n and/or m higher than 1, or in LXe, the density influences are more important. In pronounced cases this may cause the simple Van’t Hoff plot to deviate from linearity. In such cases it is better to follow the procedure suggested by eq 11: a linear plot will be obtained by using the sum of the first two terms of the right-hand side of this equation as the ordinate in the Van’t Hoff plot. Evidently, the slope of this plot must not be corrected to produce the correct ∆H°. Acknowledgment. The NFWO (Belgium) is thanked for grants toward the spectroscopic equipment used in this study. G. Everaert is thanked for the help with recording the spectra. References and Notes (1) Leipertz, A.; Spiekermann, M. In Infrared and Raman Spectroscopy, Methods and Applications; Schrader, B. Ed.; VCH: Weinheim, 1995, p 685. (2) Tokhadze, K. G. In Molecular Cryospectroscopy; Clark, R. J. H., Hester, R. E., Eds.; John Wiley and Sons: Chichester, 1995, p 133. (3) Van der Veken, B. J.; De Munck, F. R. J. Chem. Phys. 1992, 97, 3060. (4) Braker, W.; Mossman, A. L. Matheson gas data book; Matheson Gas Products, Inc.: Secancus, NJ, 1980. (5) Fastovskii, V. G.; Rovinskii, A. E.; Petrovskii, Yu. V. Inert Gases; Israel Program for Scientific Translations: Jerusalem, 1967. (6) Tokhadze, K. G.; Tkhorzhevskaya, N. A. J. Mol. Struct. 1992, 270, 351.
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