J . Phys. Chem. 1988, 92, 1312-1314
1312
Wolfsberg's notation): (1) K(eoELE)b,a correction to the BornOppenheimer electronic correction term; (2) (MMI)corr,a correction to the rotational contribution due to Are; (3) (EXC),,,,, a correction term for vibrational energy shifts due to Are for vibrational levels with v L 1; (4) (ZPE),,,, a factor similar to (EXC),,, for the ground-state vibrational level ( u = 0); ( 5 ) (AZPE),,, a correction to the anharmonic term in the zero-pint energy for the Are shift; (6) (QMRC),,,, a correction for shifts in K due to the effects of Are on quantum-mechanical rotation. From values reported in ref 5 , the value of each of the corrections (1-6) has been calculated, either from the shift in energy (A€) values given or from the equilibrium constant corrections given for the reaction HD HX DX H2
+
+
where X can be D or C1. The values of each of the correction factors to the equilibrium constant of reaction 1 are listed in Table 11. Because some of the correction terms are greater than one, and others less, the net effect is rather small. The product of all of these correction terms (Kwn)
is K,,, = 1.00199 f 0.00003, where the error limit has been inferred from the number of significant figures reported by Kleinman and Wolfsberg. The final, calculated equilibrium constant, which takes into account anharmonicity, coupling of vibration and rotation, as well as the failure of the Born-Oppenheimer approximation, is K = 1.965 at 20 OC. The agreement between the calculated and measured values of the equilibrium constant for reaction 1 using Kleinman and Wolfsberg's corrections for Born-Oppenheimer approximation failure (1.965 vs 1.959 0.011) is within the experimental error and is much better than that between the calculated and measured values (2.024 vs 1.959) when the Born-Oppenheimer approximation is invoked. Consequently, we conclude that our experimental results support the theory of isotope-exchange reactions of Kleinman and Wolfsberg in which the Born-Oppenheimer approximation is not adequate for the most precise calculations of equilibrium constants for certain hydrogen-isotope-exchange reactions.
*
Registry No. D2, 7782-39-0; HCI, 7647-01-0.
Thermodynamlcs of Hydrogen-Isotope-Exchange Reactions. 4. Determination of K and AS" for the Reaction D2(g) H20(I) H2(g) D,O(I)
+
+
Rong-Sen Zhangt and Peter A. Rock* Department of Chemistry, University of California, Davis, Davis, California 95616 (Received: April 30, 1987; In Final Form: September 27, 1987)
This paper reports our results for the direct experimental determinationof the equilibrium constant and its temperature dependence for the hydrogen-isotope-exchangereaction Dz(g) + H20(1) + Hz(g) + D20(l). The reaction was studied in an electrochemical double cell without liquid junction of the type Pt(s)lDz(g)lNaOD(daq,m)lHgO(s)IHg(l)lHgO(s)INaOH(aq,m)lHz(g)IPt(s), where (daq,m) denotes a DzO solution. The experimental value of the equilibrium constant at 25.0 "C for the reaction is K = 12.30 0.09; the experimental value of ASo at 25.0 OC for the reaction is -8.1 f 1.7 J-K-'. The experimental results are compared with theoretical calculations within the framework of the Born-Oppenheimer approximation. The comparison suggests a failure of the Born-Oppenheimer approximation of about 4% in the value of the calculated equilibrium constant for the reaction. When Kleinman and Wolfsberg's corrections for the failure of the Born-Oppenheimer approximation are taken into account, the experimental result for the equilibrium constant is in excellent agreement with the calculated value.
Introduction This paper reports part of a continuing investigation of the thermodynamics of hydrogen-isotope-exchange rea~tionsl-~ directed toward (1) providing experimental data to test the theoretical predictions of Kleinman and Wolfsberg, that the BornOppenheimer approximation can introduce an error of several percent in the calculated value of the equilibrium constant for certain hydrogen-isotope-exchange reactions and (2) providing direct experimental data to assess the operational effectiveness of the Girdler-Spevack sulfide process for the industrial-scale separation of hydrogen isotope^.^ We have used the electrochemical double cell2
Pt(s)lD,(g)INaOD(daq,m)lHgO(s)lHg(l)lHgO(s)INaoH(aq,m)lH*(s)lPt(s) (1) for the direct experimental determination of the equilibrium constant and the entropy change for the reaction D2(g) + H20(1) * H2(g) + D20(1) The results obtained suggest a failure of the Born-Oppenheimer approximation of about 4% in the equilibrium constant for this reaction.
Experimental Section Details of our general experimental procedures have been published.'-3 Thus, we confine our discussion to those aspects that are unique to the experiments described in this paper. The double-cell system was set up such that H2(g) and D2(g) were supplied continuously by electrolysis of 0.7 M Na2C03in H20(1) and D20(1), respectively, in specially constructed self-contained gas generatow6 Oxygen was removed by passing the gases over hot Ni-Cr wire coils. The H2 and D2 generators are connected in series to a constant-current power supply. Therefore, H2and D2 have the same flow rate, which is regulated to the desired value by adjusting the current. Appropriate experiments showed that the cell voltage is constant over the electrolysis current range of 0.2-1.0 A in the generators. The H 2 0 used in the experiments was prepared by distillation of deionized water to which dilute NaOH(aq) and KMnO, were (1) Silvester, L. F.; Kim, J. J.; Rock, P. A. J . Chem. Phys. 1972, 56, 1863. (2) Rock, P. A. ACS Symp. Ser. 1975, No. 11, 13 1. (3) Contreras-Ortega, C. H.; Nash, C. P.; Rock, P. A. J. Solution Chem. 1976, 5, 133.
(4) Kleinman, L. I.; Wolfsberg, M. J. Chem. Phys. 1974, 60, 4740, 4749; 1973.59, 2043. Wolfsberg, M.; Kleinman, L. ACS Symp. Ser. 1975, No. 1 I, 64.
*Author to whom correspondence should be addressed. 'Work carried out a t UCD while on leave from the Department of Chemistry, Peking University, Beijing, China.
0022-3654/88/2092-1312$01.50/0
( 5 ) Rae, H. K. ACS Symp. Ser. 1978, No. 68, 1. (6) Zhang, R . 6 ; Chang, 0.-K.; Rock, P. A,, submitted to J . Appl. Electrochem.
0 1988 American Chemical Society
Thermodynamics of Hydrogen-Isotope-Exchange Reactions
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1313
TABLE I: Results of the Measurement on the Cell P t ( ~ ) l D z ( ~ ) I N a o D ( d a ~ ~ m ) l H g O ( ~ ) l H g ( ~ ) lIPt(s) Hg0~~~l~~~~~~~~~~l~z~~~
measuring period/h
m/mol.kg-'
14 18 18 24 20 34 26 111 72 11 7 9 24 3 22 6 8 9 5
0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.100 0.200 0.200 0.020 0.100 0.100 0.100 0.200 0.200 0.200
ca/mV
t / T 15 20 25 30 35 25 35 25 35 25 25 25 25 15 25 35 15 25 35
32.48 32.30 32.19 32.09 31.98 32.41 31.99 32.36 32.02 32.18 32.37 32.34 32.22 32.57 32.19 31.80 32.86 32.38 31.82
f 0.03 f 0.07 f 0.05 f 0.03 f 0.09 f 0.02 f 0.12 f 0.04 f 0.09 f 0.02 f 0.15 f 0.06 f 0.06 f 0.01 f 0.01 f 0.01 f 0.06 f 0.04 f 0.14
eo
b/mV
32.45 f 0.03 32.26 f 0.07 32.13 f 0.05 32.02 f 0.03 31.89 f 0.09 32.36 f 0.02 31.90 f 0.12 32.31 f 0.04 31.93 f 0.09 32.12 f 0.02 32.31 f 0.15 32.28 f 0.06 32.16 f 0.06 32.54 f 0.01 32.12 f 0.01 31.71 f 0.01 32.83 f 0.06 32.32 f 0.04 31.73 f 0.14
-(AcO/AT)/mV.K-'
0.028 f 0.001
1
0.046 f 0.014
1
0.038 k 0.013
I
0.042 f 0.001
I
0.055 i 0.010
"The quoted uncertainties are the average deviations of the readings taken at 1-h intervals over the measuring period. bCalculated from eq 3. added. The specific conductivity of the water was 2.5 X 10" ohm-km-' at 25 O C . The deuterium content of the H 2 0 was that of natural abundance. The D 2 0 used was prepared by deuterium enrichment to 99.96 atom % D of commercially available D20(1) (99.8 atom % D) in the D2 generator: followed by distillation in the presence of NaOD(aq) and KMnO,(aq). The distillate collected had a specific conductivity of 3 X 10" ohm-lcm-' at 25 OC. The NaOH and NaOD used were prepared by the reaction of clean dry sodium metal (obtained from glass ampules sealed under vacuum) with HzO and DzO, respectively. The HgO(s) used in the cells was prepared by thermal decomposition of reagent-grade Hg(N03)2. The resulting HgO was washed with purified water, filtered, and oven-dried at 160 OC. The hydrogen electrodes were prepared as described previously.' Sealed glass H cells equipped with two-stage gas presaturators were used. The lower sections of the interiors of the mercury electrode compartments were pretreated with Dow-Corning 200 silicone fluid. The cells and presaturators were immersed in a fO.l OC thermostat. Voltage measurements were made with a certified L & N K-3 potentiometer.
Results The results of the measurements on double cells of the type (1) are given in Table I. Reversibility tests (voltage vs current plots) of the cells gave no evidence of hysteresis, which is strong evidence for the reversible operation of the cells. The cells exhibited excellent long-term stability. The electrode reactions occurring in the double cell are postulated to be as follows: D2(g) + 2OD-(daq,m)
* 2D20(1) + 2e-
+ D20(1) + 2e- Hg(1) + 2OD-(daq,m) Hg(1) + 20H-(aq,m) + HgO(s) + H20(l) + 2e2H20(1) + 2e- * H2(g) + 20H-(aq,m)
HgO(s)
Summation of the electrode reactions yields the net cell reaction D2(g) + H20(1) * H 2 ( d + D20(1) Application of the Nernst equation to eq 2 yields RT 2F
to=t+-In
1 1
PHPD~O
PD~aH~O
(2)
t is the measured cell voltage, and t o is the standard cell voltage at temperature T. At equal molalities of NaOH and NaOD, the ~ O( )M ~ ~ R T / ~ ~ ~ ~ F )-( M H , o value of (RT/2F) In ( ~ D ~ O / ~ H = MD,0)(where q5 is the osmotic coefficient and Mi is the molar mass of i) is less than 0.01 mV for all of the cells in Table I. The values of PH,and PD2 were obtained by subtracting the equilibrium vapor pressures of H20(g) and DzO(g) over the respective cell solutions from the barometric pressure measured in the laboratory. Virial coefficients for Hz(g) and D2(g)' were used to calculate the activity coefficients of H2(g) and D2(g). At 25 O C and 1 atm we compute yD,= 1.00055 and Y~~ = 1.000 53. These deviations from ideality contribute less than 0.0003 mV to t o and are thus negligible. The average of the t o values at 298.15 K in Table I is 32.32 f 0.09 mV. This average value of t o yields an equilibrium constant for reaction 2 of K2 = exp(t0,,/12.846 mV) = 12.30 f 0.09 at 298.15 K. The average of the AtO/ATvalues in Table I is -0.042 f 0.007 mV.K-'; however, the average uncertainty in the measured Aeo/AT values is f0.009mV.K-', and therefore we have (Ae"/AT),, = -0.042 f 0.009 mV.K-'. This value of (AEO/AT')~~ yields an entropy change for reaction 2 of ASo = 2F(Ato/AT) = -8.1 f 1.7 J-K-'.
Discussion Reaction 2 can be obtained as the summation of the following two reactions D2k) + H 2 W D2W)
H2O(g)
+ D20(1)
(4) (5)
net: D2(g) + H20(1) * H2(g) + D20(1) (2) Bron, Chang, and Wolfsberg (BCW) published8 the gas-phase partition function ratios QD,o/QH,o and Qb/QH2 calculated within the framework of the Born-Oppenheimer approximation. They expressed the logarithms of these ratios as a power series in 1/T, where Tis the absolute temperature. The power series reproduced the partition function ratios with a maximum error or 0.1% over the temperature range 200-600 K. The equilibrium constant of the gas-phase hydrogen-isotopeexchange reaction (4) can be expressed in terms of the partition .function ratios as K4
(3)
where PHIand PD2are the partial pressures of H2 and D2, respectively, aDZ0is the activity of D 2 0 in the NaOD(daq,m) solution, aHzO is the activity of H 2 0 in the NaOH(aq,m) solution,
+ H20(1)
* Hz(g) + D2O(g)
= (QD,o/QH~o)/(QD~/QH~)
(6)
(7) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Gases; Clarendon: Oxford, 1969. (8) Bron, J.; Chang, C.-F.; Wolfsberg, M. 2.Narurforsch.,A 1973,28(2), 129.
1314 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
Substitution of the BCW partition function ratio expressions into eq 6 yields -0.31074
N
where T is the absolute temperature. Equation 8 was obtained from equilibrium vapor pressure measurements and is valid over the temperature range 268-353 K. Because K2 = K4K5,we obtain, using eq 7 and 8, the following expression for K2 +
1.657274 X lo5 -
P
1.865268 X lo7 + 1.191753 X lo9 T3 P and for AS20 we obtain
1.657274 X lo5 + 3.730536 X lo7 - 3.575259 X lo9 T2 T3 74
The calculated value of K2 = 12.79 is about 4% greater than the measured value of K2 = 12.30. Bardo and Wolfsbergio found that the isotope-dependent adiabatic first-order correction to the Born-Oppenheimer approximation, evaluated at the equilibrium Born-Oppenheimer configuration of the respective molecules, K(BOELE), for the equilibrium constant of the isotope-exchange reaction (11)
is KIl(BOELE) = exp(-5.5 KIT). The Born-Oppenheimer correction to the energy change for reaction 1 1 , ACll, is given by4.i0.ii
(9) Jancso, G.; Van Hook, W. A. Chem. Rev. 1974, 74, 689. (10) Bardo, R.D.; Wolfsberg, M. J . Chem. Phys. 1975, 62, 4555
K4(BOELE) = KIl2(BOELE) (16) Using eq 16, we have as our estimate for the first-order correction to the Born-Oppenheimer approximation for reaction 4 K4(BOELE) = 0.963. The theoretical value for the equilibrium constant for reaction 2 is thus K2 = 12.79 X 0.963 = 12.32, which is in excellent agreement with the measured value. Because the correction for Born-Oppenheimer approximation failure is temperature independent: the value of ASZ0calculated within the framework of the Born-Oppenheimer approximation (AS20 = -8.46 J-K-I) is directly comparable to the experimental value of ASZ0 = -8.1 f 1.7 JaK-'; the agreement is within the experimental error. Rolston, den Hartog, and Butleri3 determined experimentally the equilibrium constant as a function of temperature for reaction 1 1 . Their value of KI1is 3.53 f 0.04 at 25 OC (see also ref 11). This value of Kll can be combined with our experimental results to obtain a value of the equilibrium constant for the reaction +
HD(g)
+ D,O(g)
(17)
Because Ki7 = K4fKll and K4 = K2fK5, we have
K2 = 12.79 and AS2' = -8.46J.K-'
+ C ( H D 0 ) - C(HD) - C(H2O)
(13)
whereas for reaction 4 we obtain
D2k) + HDO(g)
Using eq 9 and 10, we calculate at 298.15 K
HD(g) + H,O(g) == H 2 k ) + H D O W
i= 1
and AC4 = 2ACll. Thus,the value of the correction term to the Born-Oppenheimer approximation for reaction 4, K4(BOELE), is given by
209.412 57320.6 0.202524 - T
401.073 +T
C(x) = C W ) / m l
where bI does not depend on isotopic mass. Application of eq 1 3 to reaction 1 1 yields
Equation 7 is used to give the BCW theoretical values for K4 calculated within the framework of the Born-Oppenheimer approximation over the temperature range 200-600 K. The equilibrium constant for reaction 5 is equal to the equilibrium vapor pressure ratio PHlof PDz0,which can be calculated via the equation reported by Jancso and Van Hook9
AC1, = C(H2)
Bardo and Wolfsberg have showni0that the C term for a species depends on the inverse first power of the nuclear masses
610.485 + 1.084068 X lo5 +T P
1.865268 X.107 + 1.191753 X lo9 T3 P
-0.108216
Zhang and Rock
K i 7 = K2/K5K11 (18) and at 25 OC Ki7 = 12.30/(1.156 X 3.53) = 3.01 f 0.06. The results reported here, together with those we obtained in a separate study,12 provide strong quantitative support for the theoretical predictions of Kleinman and Wolfsberg4 resulting the failure of the Born-Oppenheimer approximation for certain hydrogen-isotope-exchange reactions.
Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for the partial support of this research. Partial support by the Committee on Research of UCD is also acknowledged. We also thank Prof. M. Wolfsberg for helpful discussions on the theory of isotopic-exchange reactions.
(12) ( 1 1 ) Bardo, R. D.; Wolfsberg, M. J. J . Phys. Chem. 1976, 80, 1068. (12) Postma, J. M.; Silvester, L. F.; Rock, P. A. J . Phys. Chem., in press. (13) Rolston, J. J.; den Hartog, J.; Butler, J. P. J . Phys. Chem. 1976, 80, 1064.
