J . Phys. Chem. 1989, 93, 3355-3360
potential energy curve in the very low energy can be approximated in solid hydrogen by the Eckart potential. Second, trapped H and D atoms may have discrete energy levels in solid hydrogen at ultralow temperatures. In the reactions H 2 H and D2 D, the energy of the reactants is the same as that of the products. The rate for these reactions may be enhanced by resonance tunneling.'* Third, H2 and D2 have ortho and para forms that do not interchange except by reaction. As a result, there is a considerable population of excited rotational states in H2 and D2 but not HD. para or para ortho conversion proThe exothermic ortho cesses could also be an important effect that would contribute to the reaction rate.19 Effect of Light Illumination on Decay of H and D Atoms. The decay of H atoms in the H2 solid in the light is much faster than that in the dark (cf. Figure l ) , while the light enhances slightly the decay of D atoms in the D2 solid (cf. Figure 2). The decay of D atoms in the H D solid, however, is not affected by the light
+
+
-
-
-
(18) Brickmann, J.; Zimmerman, H. J. Chem. Phys. 1969, 50, 1608. (19) H atoms repeat H + H2 H2 + H reactions by 105-106times until they encounter other H atoms. It is an unsolved problem whether or not the exothermic ortho para conversion process plays a role in the repetition of the H H2 H2 H reactions.
+
- -+
3355
(cf. Figure 3). The phonon energy of solid hydrogen, obtained by the light illumination, may assist the migration of H and D atoms in the H2 (or D2) solid. The different effects of the light on the decay of hydrogen atoms in the different isotopes cannot be explained here. The following two factors may be related to this phenomenon. First, H2, HD, and D2 have different specific heats at 4.2 K.20 The increase of temperature upon the light illumination may be different in these hydrogen isotopes. Second, H atoms in the H 2 solid, D atoms in the D2 solid, and D atoms in the H D solid migrate different distances until these atoms disappear by recombination or abstraction reactions. The effect of phonon-assisted tunneling may be related to the difference of the migration distance. Since D D2 atoms in the H D solid decay by one reaction of D + H D H without migration, the decay of the D atoms may not be affected by the light illumination.
-
+
Acknowledgment. We thank Prof. B. Brocklehurst of the University of Sheffield, Sheffield, U.K., for his fruitful discussion. Registry No. H, 12385-13-6; H2, 1333-74-0; D, 16873-17-9; D,, 7782-39-0;HD, 13983-20-5. (20) Roberts, R. J.; Daunt, J. G.J . Low Temp. Phys. 1972, 6, 97.
Vapor Pressure of Tetradeuteriomethane J. C. G . Calado,* M. Nunes da Ponte, L. P. N. Rebelo, Centro de Qdmica Estrutural, Complex0 I , Instituto Superior TCcnico, 1096 Lisboa, Portugal
and L. A. K. Staveley The Inorganic Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR, U.K. (Received: July 5, 1988)
A new apparatus suitable for measuring both absolute and differential vapor pressures is described. The absolute vapor pressure of CH,(I) and the differential vapor pressure between CH,(L) and a sample of CD,(l) with 98.9 mol % deuterium content have been measured simultaneously at 12 temperatures between 95 and 122 K. In this range the vapor pressure of CD4 is higher than that of CH4 by 1.6%at 95 K, increasing to 3.2% at 122 K. The corrected values for 100% isotopically pure samples were fitted to the equation T log r = A. + B o / T + COT,where r is the ratio of the vapor pressure of CD, to that of CH4. The fit yielded the following values for the parameters: A. = 6.9641 K; Bo = -526.16 K2;C, = -0.007 939. This equation was used in conjunction with the Clapeyron equation to calculate the differencein molar enthalpies of vaporization between CD, and CH,, A(AfH,). At 100 K the enthalpy of vaporization of CD4 exceeds that of CH4 by 65.6 J-mol-', decreasing to 25.4 J.mo1-I at 120 K. Additionally, the triple-point pressure, pt,of isotopically pure CD, was determined: p t = 10.505 f 0.01 3 kPa. Our results are in accord with the previous results of Armstrong et al. obtained 50 years ago, and an extrapolation of our fitted equation up to almost the critical point gave good agreement with other published data. This lends support to the theoretical foundations of the above equation and indicates the high internal consistency of our results.
1. Introduction In the last few years we have studied a number of binary mixtures of liquefied gases, mainly with the purpose of assisting in the development of modern statistical theories of solutions. The good agreement obtained, for simple systems, between experiment (phase equilibria, equation of state) and a variety of perturbation theories and computer simulations shows that the majority of the theoretical problems have been solved and that we now have a reasonable understanding of the thermodynamic behavior of mixtures of small, quasi-spherical molecules. Some problems, however, remain. Even for small molecules, shape seems to play an important part and cannot be ignored. While, for instance, methane can be approximated by a sphere, carbon tetrafluoride has to be modeled by a tetrahedral or at least a four-center structure. A truly quantitative account of the thermodynamic properties of simple systems requires a detailed knowledge of the 0022-365418912093-3355$01.50/0
intervening intermolecular potentials. A good test of the adequacy of a particular form of potential would be its ability to account for the observed differences between two isotopic forms of the same simple molecule. At the same time a mixture of two isotopic species of the same compound would be one of the simplest in molecular terms and would thus constitute a very strict test of the validity of a particular theory. For this reason we have decided to embark on a program to study the thermodynamic excess properties of a variety of simple mixtures of isotopic species. The effect of isotopic substitution on the vapor pressure of a compound is an old problem that Lindemann was the first to tackle from the theoretical point of view more than 60 years ago.' There have been, in the meantime, important theoretical contributions, among them from Bigeleisen and his group,2 from Stern, Van (1) Lindemann, F. A. Philos. Mag. 1919, 38, 173.
0 1989 American Chemical Society
3356 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 Hook, and Wolfsberg? and from Grigor and The various developments have been described in the excellent review article of JancsB and Van Hook.5 Recently, interest has turned to mixtures of isotopic molecules and their excess thermodynamic properties, and therefore theoretical models6,’ and experiments8 have been developed. This paper describes measurements of the vapor pressure @) of methane, and of the difference in the vapor pressure ( A p ) of methane and tetradeuteriomethane, from 95 to 122 K. Since the departure from ideality of a mixture of two isotopic species inevitably will be relatively very small, the vapor pressure measurements from which the degree of nonideality is assessed must be of the highest possible precision. The primary purpose of the experiments described here was to test the accuracy and precision capabilities of our apparatus, and CH4 and CD, were chosen for this as there already exist two sets of data on these substances, namely, those due to Armstrong, Brickwedde, and Scott9 and to Grigor and Steele.Io The latter authors also made molar volume measurements. Using the simple cell model, Bigeleisen et al.ll successfully applied their theory to tetradeuteriomethane. Pollin and Ishida12J3have also obtained good results with the mediumcluster model. A further reason for the choice of CH, and CD4 for these test measurements was that we are proceeding with a study of the CH4 + CD4 system, on which we hope to report in due course. The vapor pressure measurements were then used to calculate the differences in the molar enthalpies of vaporization A( AfH,,,) between the two isotopic species. Detailed comparisons with the above mentioned results of other authors were performed. 2. Experimental Section The vapor pressure measurements were performed with a new apparatus containing three cells. In a study of a liquid mixture of two isotopic species A and B, one cell can contain A, a second B, and the third the mixture A B, so that absolute vapor pressure measurements can be made on pure A or B and simultaneously differential measurements can be carried out on any pair of the three liquids. The essential part of the apparatus, the cryostat, is shown in Figure 1 and will be briefly described. It was derived from an apparatus used by Clusius et aL14 for the measurement of isotope effects on the nitric oxide species. The three cylindrical cavities, which had a volume of about 1.7 cm3 (one of which is shown in Figure IA), were located in a cylindrical copper block (B) to avoid local thermal gradients. This copper block has a fourth cavity to house a platinum resistance thermometer (C). An adiabatic shield (D) completely surrounds the block and is maintained at a temperature slightly higher (=0.2 K) than that of the block itself. It is essential that there is no “cold spot” on the inlet tubes (E) to the cylindrical cavities where condensation could occur and thereby completely falsify the measurements, and accordingly independent heating coils and thermocouples (F) were provided on the block, shield, and inlet tubes, as well as on a copper vessel (G) surrounding the inlet tubes above the shield. G is a thickwalled hollow vessel, provided with an inlet tube, and not a solid block, so that the desired temperature difference between the inlet tubes passing through G and the block (B) can be achieved in two
+
(2) Bigeleisen, J. J . Chem. Phys. 1961, 34, 1485. (3) Stern, M. J.; Van Hook, W. A,; Wolfsberg, M. J . Chem. Phys. 1963, 39. 3 179. Grigor, A. F.; Steele, W. A. J . Chem. Phys. 1968, 48, 1038. Jancsb, G.; Van Hook, W. A. Chem. Rev. 1974, 74, 689. Jancd, G.; Van Hook, W. A. Physica A: (Amsterdam) 1978,91,619. Singh, R. R.; Van Hook, W. A. J. Chem. Phys. 1987, 86, 2969. Lewis, D. G.; Staveley, L. A. K.; Lobo, L. Q.J . Phys. Chem. 1986.90, 5456.
(9) Armstrong, G. T.; Brickwedde, F. G.; Scott, R. B. J . Res. Natl. Bur. Stand. 1955, 55, 39.
(IO) Grigor, A. F.; Steele, W. A. J . Chem. Phys. 1968, 48, 1032. ( I 1 ) Bigeleisen, J.; Cragg, C. B.; Jeevanandam, M. J . Chem. Phys. 1967, 47, 4335. (12) Pollin, J. S.; Ishida, T. J . Chem. Phys. 1977, 66, 4433. (13) Pollin, J. S.;Ishida, T. J . Chem. Phys. 1977, 66, 4442. (14) Clusius, K.; Schleich, K. Helu. Chim. Acra 1958, 41, 1342
Calado et al. N
d
M P
4
Q -H
-i
G -
-E
-A -B
Figure 1. Cryostat for both absolute and differential vapor pressure measurements: A, one of three cylindrical cavities for condensed samples; B, copper block; C, platinum resistance thermometer; D, adiabatic shield; E, one of three inlet tubes to cavities; F, one of seven thermocouples, the other six being indicated by the short vertical lines; G, copper vessel; H, copper cylinders supporting thin-walled stainless steel tubes of reduced thermal conductivity; I, outer brass can; J, liquid nitrogen level; K, inlet tube to copper vessel; L, thin-walled stainless steel tubes with holes to minimize thermal conduction; M, brass disk with four holes through which inlet tubes emerge; N, plastic disk with 29 holes through which electrical wires emerge; 0, connection to vacuum line or helium globe; P, iron support; Q,thin-walled German silver tube; R, thin-walled Monel tube.
ways. One of these is, with G empty, to control its temperature by adjusting the electrical energy input and monitoring the temperature difference between G and B with the thermocouples. The alternative is to condense a gas into G (which can be one of the substances under examination, e.g., methane in the work described here) and arrange for this liquefied gas to boil off slowly at a constant pressure such that a predetermined temperature difference between G and A is maintained. In the work described here, the first method was used. The electrical wires were wound twice around three copper cylinders (H) (two of them are shown) immediately after entering the cryostat for heat sinking. The shield is surrounded by a brass can (I) that can be evacuated up to about 0.1 Pa. Cooling of the block is achieved by immersing the can in a cryostatic fluid (liquid nitrogen was used in this work) up to level J, with helium as the exchange gas. Additionally, heating coils and two thermocouples are provided around the inlet tubes above this copper vessel. It was therefore possible to measure and control the temperature at any section of the cryostat assembly. Any temporary temperature changes that developed within the
Vapor Pressure of CD4 block were found to be lower than 6 pK-s-I, small enough to get good thermal equilibrium between the liquid samples and the platinum resistance thermometer. The inlet tubes (E) to the block cavities are made of German silver with a 1-mm i.d. below the copper vessel and of stainless steel with a 1.6-mm 0.d. above the vessel and fitted together with a low-temperature melting solder. The outer brass can (I) has a diameter of 85 mm and a height of 330 mm. In operation, the liquid samples occupied about 65% of the cylindrical cavities, a value that was easily reproduced between different runs and remained almost constant over the range of temperatures studied, due to a compensation between the small dead spaces and the expansion of the liquid samples. Absolute vapor pressure measurements were obtained with a calibrated Texas Instruments fused quartz Bourdon tube with a resolution of 4 Pa, while differential vapor pressure were monitored using a capacitance sensor with a 0.13-Pa resolution. Temperatures were read on the IPTS-68 with a four-wire platinum resistance thermometer connected to a Keithley 197 multimeter with an estimated accuracy of fO.O1 K. Tetradeuteriomethane was prepared in our laboratory from carbon tetrachloride, powdered zinc, and heavy water, by the procedure described in Leblanc et aI.l5 The experimental setup used in this preparation is similar to that reported by Deiters.I6 The CD4 was further purified by a low-temperature fractional distillation to a purity of at least 99.99 mol % with respect to nonisotopic impurities. The isotopic impurities of CD4 were determined by mass spectrometry, showing the existence of 4.3 mol % trideuteriomethane (CD3H) that corresponds to a deuterium content higher than 98.9 mol %. Additionally, the purity of CD4 was checked by examining the constancy of the triple-point pressure during melting. Two different samples of CHI were used. One (minimum purity 99.9 mol %) was supplied by Air Liquide. The other was prepared in the apparatus built for the preparation of CD4, substituting normal water for heavy water. This crude methane obtained was further distilled, giving a triple-point pressure of 11.694 f 0.008 kPa. This methane was redistilled with no detectable change in its triple-point pressure. The commercial CHI was also fractionated, giving a triple-point pressure of 11.692 f 0.008 kPa, to be compared with 11.691 f 0.008 kPa recommended by Staveley et al.,I7 1 1.696 f 0.020 kPa of Kleinrahm and Wagner,ls or 11.696 f 0.002 kPa of Pavese et al.,’9,20and a triple-point temperature of 90.69 f 0.01 K, also in excellent agreement with 90.69 f 0.01 K recommended by Staveley et a1.I’ and 90.685 f 0.001 K from Kleinrahm and Wagner.18 These findings seem to indicate that one single distillation is enough for the purification of CD4 with respect to nonisotopic impurities. With this procedure we found a value of 10.514 f 0.01 3 kPa for the triple-point vapor pressure of our CD4 sample, as compared with 10.519 f 0.013 kPa of Clusius et aLZ1or 10.569 kPa from Cradduck et aLz2 Armstrong et aL9 recorded a value of 10.546 f 0.013 kPa corrected to apply to CD, of 100% isotopic purity. Assuming, with Armstrong et al., that the triple-point temperatures and pressures are a linear function of the number of deuterium atoms per molecule, our value for the triple-point pressure of CD4 corrected to 100% isotopic purity would be 10.505 f 0.01 3 kPa.
3. Results and Discussion The absolute vapor pressure of CH4 and the differential vapor (15) Leblanc, M. E.; Morse, A. T.; Leitch, L. C. Can. J . Chem. 1956, 34, 354. (16) Deiters, U. Ph.D. Dissertation, Ruhr-Universitat Bochum, 1976. (17) Staveley, L. A. K.;Lobo,L. Q.; Calado, J. C. G. Cryogenics 1981, 21, 131. (18) Kleinrahm, R.; Wagner, W. J. Chem. Thermodn. 1986, 18, 739. (19) Pavese, F. Metrologia 1981, 17, 3 5 . (20) Pavese, F.; Cagna, G.; Ferri, D. Proceedings of the Sixth International Cryogenic Conference; IPC Science and Technology: London, 1976; p 281. (21) Clusius, K.; Weigand, K. Z. Phys. Chem., Abi. B 1940, 46, 1. (22) Cradduck, N. .I.Staveley, ; L. A. K., unpublished result, 1972.
The Journal of Physical Chemistry, Vol. 93, No. 8,1989 3357 TABLE I: Experimental Absolute Vapor Pressure @) of Liquid CHI and Differential Vapor Pressure (Ap)), Where Ap = p(0.957(CD4) 0.043(CDfi)) - p(CH,) (EpRepresents the Estimated Uncertainty Associated with the Ap Values)
+
TIK 95.06 97.10 100.61 100.62 102.85 105.25 108.10 110.1 1 113.37 116.19 119.23 121.53
PlkPa 20.006 25.175 36.658 36.684 45.892 57.739 74.686 89.021 116.32 144.99 181.48 213.42
ApplkPa
Ep/kPa
0.3205 0.4594 0.7789 0.7831 1.0536 1.4183 1.9660 2.4355 3.3551 4.3482 5.6313 6.7706
0.0008 0.001 1 0.0013 0.0013 0.0016 0.0025 0.0036 0.0039 0.0041 0.0039 0.0039 0.0040
I
\ Ik
0 58
1
1
I
I
0 5
00
95
10 0
I IO 5
io3 n/r
Figure 2. Plot of T log r against TI, where r = p(CD4, impure)/p(CH4): represents eq 1.
0 , this work; 0,ref 9. The full line (-)
pressure between 0.957(CD4) + 0.043(CD3H) and CH4 were measured at 12 temperatures between 95 and 122 K. The results are recorded in Table I. Small corrections for a “zero shift” in the capacitance sensor were made by determining the apparent vapor pressure difference between two CH4 samples, and the estimates of the possible uncertainty, Ep, in a Ap measurement arising for this source are given in Table I. As regards the absolute vapor pressure of CH,, we found that, according to the calibration curve of our platinum resistance thermometer, our results for the methane vapor pressures were below (0.154.25%) those of Prydz and GoodwinZ3but above (0.1 54.25%) the very recent results of Kleinrahm and Wagner.Is It was difficult to decide which results to use as a reference for methane in view of the fact that both are of high accuracy and internal consistency. Nevertheless, at the triple point of methane, the results of Prydz and Goodwin for the triple-point pressure seem to be too high (by OS%), while the results of Kleinrahm and Wagner are consistent with several experimental determinat i o n ~ , ’ including ~ - ~ ~ this work. So we decided to shift our scale of temperature by 0.01 K in order to obtain an agreement with the “Wagner scale” better than 0.01 K. All temperatures recorded in this work contain this correction. Vapor Pressure Ratio. These results are compared in Figure 2 with all available data in the literature for this system in the range of temperatures from 9 1 to 122 K. It should be mentioned that the published results were obtained from CD, samples prepared in ways different from each other. The full line represents a least-squares fitting to our experimental results of the form Tlog r = A
+B/T+
CT
(1)
where r = p(0.957(CD4) + 0.043(CD3H))/p(CH4)and the parameters have the following values: A = 6.8316 f 0.2606 K, B = -518.14 f 13.99 K2. C = -0.007578 f 0.001 208. Thestandard (23) Prydz, R.; Goodwin, R. D. J. Chem. Thermodn. 1972, 4 , 127.
3358 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 TABLE 11: Values of Differential Vapor Pressure ( A p ) , Where Ap = p(CD4) - p(CHJ, and of the Absolute Vapor Pressure @ ) of Isotopicallv Pure LiQuidCD,
T/K
ApplkPa
pplkPa
T/K
ApplkPa
pplkPa
95.06 97.10 100.61 100.62 102.85 105.25
0.3272 0.4685 0.7929 0.7974 1.0719 1.4419
20.334 25.644 37.451 37.481 46.964 59.181
108.10 110.11 113.37 116.19 119.23 121.53
1.9974 2.4735 3.4055 4.4119 5.7115 6.8654
76.683 91.494 119.73 149.40 187.20 220.28
deviation of the fit was s = 2.148 X Figure 2 shows the good agreement, the more so, the higher the temperature, between our measurements and those of Armstrong et aL9 [with a CD4 sample of 0.924(CD4) + 0.068(CD3H) + O.O08(CDH3)]. The experimental Ap values recorded in Table I were further corrected to 100% isotopic purity from the presence of 4.3 mol % of CD3H, assuming that CD, and CD3H form an ideal solution. The necessary vapor pressure values for the difference between CD3H and CH, were taken from ref 9. These new Ap values between isotopically pure CD4 and CHI are recorded in Table 11, together with calculated values for the absolute vapor pressure of CD4, p(CD4) = p(CH4) Ap. These calculated vapor pressure values of pure CD4 were also fitted to the equation
Calado et al. I 0.4
I
I
I
I
I
I
I
I
I
I
0
\
90
110
130
150
190
170
T/K
I r 1
I
I
I
I
I
1
I
I
I
1
1
+
T log r = A.
+ B o / T + COT
+ B/P +B / P +C
In r = A / T
(3) (4)
where C can play a relatively important role if anharmonicity should be considered. Armstrong et aL9 decided to fit their values (also corrected for 100% isotopic purity) to an equation of the form of eq 3 instead of eq 4, i.e. T log r = 5.159 - 421.1/T
(5)
but an evaluation of the residuals shows that they would have obtained a slightly better representation of their results by using eq 4. In other words their plot of T In r vs 1 / T shows a slight curvature. If our own results for the vapor pressure of CD, are fitted to a type-5 equation involving only two terms, the following values are obtained for the parameters: A = 5.197 15 f 0.01965 K and B = -430.530 f 2.094 K2 for the 0.957(CD4) 0.043(CD3H) sample and A = 5.251 79 f 0.02062 K and B = -434.377 f 2.193 K2 for the isotopically pure CD,. The standard deviations of the parameters themselves are smaller than those obtained with three-term equation ( l ) , but the standard deviation of the overall and s = 5.472 X fitting is, of course, larger: s = 5.224 X respectively. Absolute Vapor Pressure of CD,. We can now compare experimental p values of pure CD, available in the literature with those calculated from eq 2. The various sets of results are plotted in Figure 3a,b. To perform these calculations, we used the same reference with respect to CH,; Le., the CH, vapor pressure was calculated from the Wagner equation'*
+
In ( p / p c ) = ( T , / T ) ( N , t
+ N2t1,5+ N3t2+N4t4,5) (6)
(24) Ishida, T.; Bigeleisen, J. J . Chem. Phys. 1968, 49, 5498.
110
130
110
170
190 T/K
where r = p(CD4)/p(CH4)and A. = 6.9641 f 0.2728 K, Bo = -526.16 f 14.65 K2, Co = -0.007939 f 0.001 264. The standard deviation of the fit was s = 2.249 X Although several other equations were tried to fit the results (and they all gave similar standard deviations), there is a theoretical reason for choosing the present one (2). In fact, according to vapor pressure isotopic effects t h e ~ r y , ~we , ~might , expect the relation between In r and T to be of the forms In r = A / T
90
Figure 3. Comparison of the experimental values of the vapor pressure (p) of isotopically pure CD4with those calculated from the combination of eq 2 and 6: (a) experimental p of CD4obtained with the experimental p values of CH,; (b) experimentalp of CD4 obtained with the Wagner equation (6) for CH4. Key: 0 , this work; 0,ref 9; 0, ref 10.
with t = (1 - T/Tc) and Tc = 190.551 K,pc = 4.5992 MPa, N 1 = -6.035927, N 2 = 1.412132, N3 = -0.4992592, and N4 = -1.438 047. Returning to the comparison between our results and those of Armstrong et al.? our Ap values are slightly lower (mainly at low temperatures) than theirs. This difference might be associated with an error in the estimated isotopic purity of the two samples, but, on the other hand, the derivative of the CD4 vapor pressure with respect to temperature is higher in our work, which leads to a higher value of the enthalpy ofvaporization, and this quantity increases with increasing deuterium content.25 Perhaps the use of oil manometers in the work of Armstrong et al. can explain the higher Ap values, since dissolution of methane into the oil lowers the oil density. Nevertheless, the discrepancy between the two sets of results is small. There are two possible ways of calculating the absolute vapor pressure of isotopically pure CD,: one consists of adding the observed Ap value (after correcting to 100% isotopic purity) to the experimentally determined vapor pressure of CH,; in the other, we add the experimentally determined Ap value (also corrected to 100%of isotopic purity) to a common vapor pressure curve for CH4 [e.g., the Wagner equation (6)]. This means that, in the case represented in Figure 3a, discrepancies between values arise from errors in the CHI vapor pressures and also differences in Ap values, but in Figure 3b, only the Ap values are tested. It is interesting to observe that, under this second comparison, our results differ from those of Armstrong et al. by between 0.04 and 0.14% in the overlap of the temperature ranges of the two sets of results, but the agreement is slightly better (0.05%) when the experimentally determined vapor pressures of CHI are used (Figure 3a). Nevertheless, the most remarkable feature of both plots is that eq 2 represents quite reasonably the data of Grigor and Steele'O up to 180 K, that is, 60 K above the range of our experimental data. This seems to be a consequence of two things: one is the high internal consistency of our results, and the other, the (25) Colwell, J. H.; Gill, E.K.;Morrison, J. A. J . Chem. Phys. 1964, 40, 2041.
The Journal of Physical Chemistry, Vol. 93, No. 8,1989 3359
Vapor Pressure of CD4
TABLE IIk Values of A(AfHm)= AfH,(CD4) T lK 100 110 120
- AfH,(CH4)'
A(AfH,)/J.mol-' I1 111
I 117 f 40
19.5 37.7 20.9
60.2 42.6
130 140
IV 65.6 44.4 25.4 7.7' -7.6'
"r 30
Key: 1, calorimetric determination from ref 25; 11, given in ref 4; 111, calculated from eq 5;9 IV, this work according to eq 2. Extrapolating eq 2 above 122 K.
soundness of the theoretical basis of eq 2. Enthalpies of Vaporization. We turn now to the evaluation of the difference between the enthalpies of vaporization of CD, and CH,, A(AfHm) = AfH,(CD4) - AfH,(CH4). This difference can be obtained from the Clapeyron equation d p / d T = AfH,/T(V,B-
V,,,')
(7)
in two alternative forms:
UAfH,) = -T-'A[(RT
+ B,p
- V,,,'p) d In p / d ( l / T ) ]
(9)
All the information needed to apply eq 8 or 9 is available. Values of d In p/d( 1/ T ) and p were derived from eq 6 for CH,; the combination of eq 6 and 2 yields the corresponding values for CD,. Orthobaric liquid molar volumes, V,,,', were taken from Kleinrahm and Wagner'* for CH,, and for CD,, from Fuks et a1.26below 1 12 K and from Grigor and Steelelo above 112 K. The second virial coefficients, B,, of CH4 were selected from the compilation of Dymond and Smith2' above 110 K; for lower temperatures an extrapolation was carried out with the equation of Goodwin.% With respect to CD4, the following equation29was used: B"(CD4) - BV(CH4) = 2.08 (cm3.mol-')
+ 400 (cm3.mol-'.K)/T
(10)
This kind of calculation is very sensitive to gaseous imperfection. If we neglect VL with respect to V,g, Le., if we take (Vmg- V,,') = V,g and use for Vmgthe ideal gas value, eq 8 takes the form d In r / d ( l / T ) = -A(AfH,)/R
(11)
Values of A(AfH,) determined in this way would be in error by 4% at 100 K, increasing to 36%at 120 K (being always higher than the estimate obtained if the approximation just mentioned is not made). The presence of 4.3%CD3H has an almost negligible effect in A(AfH,): 0.8% at 100 K decreasing to 0.06%at 120 K. The results of A(AfH,) are reported in Table 111 in column IV. The vapor pressure data yield values of A(AfH,) that are approximately half the value of the only direct calorimetric determinati~n.~~ The most important outcome of the evaluation of the enthalpies of vaporization is that an inversion of an isotopic effect is predicted. In other words, while at low temperatures the enthalpy of vaporization of CD4 exceeds that of CH, (the normal effect), as the temperature increases this difference decreases until a temperature is reached where both enthalpies have the same value. Above this
1
-10
I
I
I
I
7.0
8.0
0.0
10.0
Id K/T
Figure 4. Plot of A(AfH,) = AfH,,,(CD,) - AfH,(CH,) against 7% 0 , values from column IV of Table 111. The dashed straight line (--) represents the combination of eq 2 and 1 1 .
"crossover" temperature AfH,(CH,) > AfH,(CD4), as shown in Figure 4. Within the framework of vapor pressure isotope effects theory, the prediction of a crossover occurrence in A(AfHm)is possible. In fact, since in eq 3 or 4 A is usually positive and B negative, a plot of In r as a function of 1/T can exhibit a maximum; the combination of eq 3 or 4 with the approximate eq 11 implies A(AfH,) = 0 at T = -2B/A. On this basis, use of the values A. and Bo of eq 2 gives a crossover at T = 151 K. The important factor in determining this temperature is not the derivative of pressure with temperature [because dp/dT(CD,) > dp/dT(CH,)] but the higher gaseous density of tetradeuteriomethane. The values recorded in column IV of Table 111 and plotted in Figure 4 give a lower and more reliable value of 135 K. Comparison with Theory. Under the Born-Oppenheimer approximation, CH, and CD4 are moving in the same potential energy surfaces. This means that any difference in their physical properties must have a quantum origin. This quantum effect will reflect mainly differences between isotope molecules in the acoustic frequencies associated with hindered translation and rotation in the condensed phase and differences in the zero-point energy shifts of the internal vibrational modes of a gas molecule upon condensation. Within the framework of Bigeleisen's theory2 based on a simple cell model (SCM), the reduced partition function U, ratio between the condensed (c) and vapor (g) phases for the two isotopes is given by the following approximate expression
where vi are the frequencies [internal (int) or external (ext)] with the prime labeling the lighter isotope. On the other hand, this ratio is approximately given by
31, 606.
where Bp is the second virial coefficient of the gas when the equation of state is written as a series of pressures and Vis the molar volume of the condensed phase. The frequencies to be used can be calculated by adjusting the experimental ones to a harmonic force field and to the experimental vapor pressure isotopic effect. This can be done with the Wilson's, FG matrix method30 through the Stern-Van HookWolfsberg treatment.3 Bigeleisen applied a similar treatment to the methane isotopes" and arrived at an expression in In VC/fg)that can be transformed
(27) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases; Clarendon: Oxford, 1969. (28) Goodwin, R. D. NBS Tech. Note (US.) 1974, 653. (29) Fang, A. Y.; Van Hook, W. A. J . Chem. Phys. 1974, 60, 3513.
(30) Wilson, E. B.; Decius, .I. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955.
(26) Fuks, S.; Legros, J. C.; Bellemans, A. Physica (Amsterdam) 1965,
J . Phys. Chem. 1989, 93, 3360-3363
3360 I
7..
I
I
I
I
. I
d n / 1
Figure 5. Comparison of the experimental vapor pressure ratio ( r ) of isotopically pure methanes (CDd/CH,) with that calculated from theory: (-) experimental from this work (eq 2); (--) simple-cell model;" (-.-) medium-cluster nod el.'^^'^
into log ( p / p ? with use of eq 13. The necessary thermodynamic quantities needed to perform this calculation were obtained from the same literature sources used for the enthalpies of vaporization. The results are shown in Figure 5 (dashed line).
Similarly, Pollin and IshidaI2*I3modified slightly the SCM, considering that the cell can now hold not only one simple molecule but a cluster of m molecules freely translating and rotating, with a central molecule and m - 1 shell molecules. With this medium-cluster model (MCM), Pollin and Ishida deduced similar equations, suggesting that the entire liquid phase can be represented by the central molecule of the cell. They applied them to the methane isotopes using a cluster of size m = 9 in a gear geometry. This approach leads to the dot-dash line shown in Figure 5 . Considering the approximations involved, and the fact that anharmonicity was ignored, it is somewhat surprising that both models predict the experimental results quite well. They give similar overall agreement with experiment, with SCM giving results that are too high, whereas those obtained with the MCM are too low; MCM performing slightly better perhaps because of the isotropic nature of the SCM.
Acknowledgment. This work has been supported by a grant from the Volkswagen Foundation, which we gratefully acknowledge. Registry No. CHI, 74-82-8; CD4, 558-20-3.
Reduction Potential of the 'COP- Radical Anion in Aqueous Solutions Parminder S. Surdhar, Stephen P. Mezyk, and David A. Armstrong* Department of Chemistry, University of Calgary, Calgary, Alberta, Canada T2N 1 N4 (Received: January 26, 1988; In Final Form: August 29, 1988)
The reduction potential for the 'COT radical anion has been determined by equilibration of formate with sulfhydryl radicals of @-mercaptoethanol,penicillamine, and lipamide in aqueous solutions at pH 3-6. The reaction 'CO, + e- + H+ = HCOT yields the value Eo9 = 1.49 V with an uncertainty of i0.06 V. On the basis of this value and the known free energies of C02(aq) and HC0,-(aq), E O 1 9 for C 0 2 + e- = T O 2 - was found to be -1.85 V.
Introduction Hydrogen-transfer reactions have frequently been utilized in the gas phase for obtaining bond dissociation energy data.'S2 Unfortunately the halogen systems, such as HI, which are so useful in the gas phase, cannot be employed in solution because of the very small pK values of these hydrides. However, sulfhydryl systems do not suffer from this restriction, the pK value of a typical aliphatic sulfhydryl being about 8-10.394 Recently, we determined the redox potentials of several sulfhydryls in aqueous s o l ~ t i o n . ~ In the current investigation, we have utilized the equilibrium depicted in reaction 1 to obtain values of the redox potential for PSH 'C02- = PS' + HC02(1)
+
the 'COY radical anion. Three different sulfhydryl molecules were used: P-mercaptoethanol, penicillamine, and lipoamide. The latter species has the advantage that the PS' radical form has a high absorbance coefficient, similar to that of the protonated lipoic acid radical studied earlier by Hoffman and Hayon6 and Farragi, Redpath, and Tal.' This is also true of the penicillamine radical,*v9 (1) Benson, S. W. Thermochemical Kinetics 2nd 1976.
ed.; Wiley: New York,
(2) Alfassi, 2.B.; Golden, D. M. J . Phys. Chem. 1972, 76, 3314. (3) Szajewski, R. P.; Whitesides, G. M. J . Am. Chem. Soc. 1980, 102,
2011.
(4) Whitesides, G. M.;Houk, J.; Patterson, M. A. K. J . Org. Cbem. 1983, 48, 112. ( 5 ) Surdhar, P. S.; Armstrong, D. A. J . Phys. Cbem. 1987, 91, 6532. (6) Hoffman, M. Z.; Hayon, E. J . Am. Chem. Soc. 1972, 94, 7950.
0022-3654/89/2093-3360$01.50/0
and for these two radicals, their concentration was followed by observing the absorbance of the pulse-irradiated solutions. For the P-mercaptoethanol system, the equilibrium was observed from the photostationary ratio of flavin and dihydroflavin in a y-radiolysis experiment, which will be described under the Experimental Section and Results. Since the equilibrium of reaction 1 lies to the right, the formate ion was present in excess (0.01-0.5 M) and the sulfhydryl molecule M). Radicals at a considerably lower concentration ( 104-2 X were produced by the radiolysis of the aqueous solutions, with N20 added to convert the solvated electrons to hydroxyl radicals.I0 The following reactions then take place: 'OH (or 'H)
+ HCOT = H 2 0 (or H2) + T O 2 -
'OH (or 'H)
+ PSH = H,O
(or H2)
+ PS'
(2) (3)
Experimental Section Penicillamine (PenSH) and @-mercaptoethanol(P-RSH) were purchased from Sigma Chemical Co. Dihydrolipoamide [L(SH),] was prepared from lipoamide (LS,) by the method of Reed et al." (7) Faraggi, M.; Redpath, J. L.; Tal, Y. Radial. Res. 1975, 64,
452.
(8) Purdie, J. W.; Gillis, H. A.; Klassen, N. V. Can. J . Cbem. 1973, 51, 3132. (9) Hoffman, M . Z . ; Hayon, E. J . Pbys. Cbem. 1973, 77, 990. (IO) Draganit, I. G.; Draganit, Z . D. The Radiation Chemistry of Water; Academic Press: New York, 197 1.
0 1989 American Chemical Society
