I
J. E. Ruark, J. I. Ivers. ond J. 1. Roberts1 University of Redlands Redlonds, California 92373
The DCI-HBr Isotope Exchange Equilibrium A statistical a n d experimental exercise for the physical chemistry laboratory
A simple experimental confirmation of a statistically predicted equilibrium constant is not an exercise normally encountered at the undergraduate level. This is particularly true when one stipulates that the theoretical background and experimental techniques be within the grasp of the average physical chemistry student using equipment commonly available in undergraduate laboratories. However, many recent textbooks (cf. ( I , 2, 3)) and curricula include an introduction to the principles of statistical mechanics in their treatment of thermodynamics, and an experiment which can provide concrete verification of these principles may be useful. I t is with this in mind that we have designed the following exercise. We selected the isotopic exchange reaction DCI
+ HBR
=
HCI
+
of DBr because of interference by strong water absorption peaks, unless a high resolution spectrometer is available. Furthermore, the fundamental and overtone spectra do not provide enough information about the vibrational states to calculate accurate dissociation energies from a Birge-Sponer plot. For these reasons, we found it convenient to use molecular parameters given in the literature to calculate the zero point energy differences using eqn. (4) ( 6 )
and setting the vibrational quantum number, u, equal to zero to obtain
DBR
(all molecules in the gaseous state) because it is simple enough so that the statistical calculations are not overwhelming and because a measurement of the equilibrium constant can be made in a simple and direct way.
Using eqn. (51,the differences in the dissociation energies of the DCI/HCI and DBr/HBr isotopic pairs are seen to be given by D,'
- Do =
-(E;
- E,) = -[x(w,'
- w,l - %(wsx; - W,.X~)]
(61
Calculation of K, From Statistical Mechanics Following the derivation given by Dickerson (3), we may write the expression for the equilibrium constant of the exchange reaction as
where qjo is the standard state zero point partition function of component j, u j is the stoichiometric coefficient of component j, and AEo is the internal energy change (difference in ground state energies) associated with the reaction. In order to compute the value of K,, we must evaluate the qjo for each component and the AEOof the reaction. For the system we have chosen, the rigid rotor-harmonic oscillator approximation assuming no electronic excitation is a reasonably good approximation (3) and we may write the zero point partition function as
where m is the mass of molecule, a is the symmetry term, I is the moment of inertia, ujO is the fundamental frequency of vibration, and the other constants have their usual significance. The moments of inertia, fundamental frequencies, and dissociation energies can in principle be obtained by the methods described by Stafford (4) and summarized by Hollenberg (9,and in this sense this experiment can be regarded as an extension of the experiments they describe. However, there are practical difficulties in obtaining good fundamental and overtone spectra
'To whom correspondence should be addressed 758
/ Journal of Chemical Educarion
where we is the harmonic frequency in cm-1 of the "normal" ('HX) molecule, weL is the harmonic frequency of 2HX, and were and wereL are the anharmonicity terms for the 'HX and 2HX molecules, respectively. The validity of this approximation rests on the fact that the potential energy curve for a diatomic molecule is dependent mainly on the nuclear and electronic charges and is relatively independent of the nuclear masses (7). For systems a t high temperatures, or for systems where electronic excitation is present, the student may be referred to reference (8)for a more exact treatment. Conceptually, AEo may be thought of as the sum of the energy changes associated with the reactions displayed in ~ the overall reaction can be calculatTable 1, and A E for ed from eqn. (6)as AE,(erg/mole) = Nhe[lD,,'(DCIl - D$HCI)I IDdfDBr)
- DdHBrIJ]
(7)
using the we and w,x, values in Table 2. In calculating the value of (Doi - Do) for the DCI-HCI pair from eqn. ( 6 ) ,no serious error will be made by choosing either the 35C1 or 37C1 isotope or taking the weighted average of the 35.37C1 isotopes. Although Do values for the four molecules are available in the literature and are listed in Table 1, the A E value ~ calculated from eqn. (7) is undoubtedly more accurate, because the net energy change is much smaller than the uncertainty in the experimental values. Table 1. Experimental Dissociation Energies
-+ ++ -
.
DCI D C1 Doi (DCI) H CI HCI -Do fHCI1 D Br + DBr -Do' (DBr) HBr H Br Do (HBrI DCI HBr HCI DBr
-+ + -
+
.
4.487 -4.431 -3.80 3.75 AEO= 0.006
..
* 0.02
+ 0.02 i0.04
Table 2. Molecular Parameters for HCI-DCI and HBr-DBr HC1
DCI
HBP
DBP
spectrum were corrected to the final temperature. The index, m, is a label which identifies a particular peak in the overtone snectrum of DC1 as described bv Shoemaker & Garland (9). Usine data for 19 neaks from initial and final DCI overtone spectra, K , was calculated to be 0.78 0.06. In order to ensure the ~recisionof measurement of K,, a number of pairs of peaks in the initial and final spectra were cut out and weighed on an analytical balance. Since K , has a value close to 1.0, it may be shown by a simple error treatment that
*
Tabvlatad values ape for the weighted average of the isotopk -masses. Ref. (61, p. 114. Estimatd from d a b in Ref. (11). HCI-35. Ref. (6). p. 534. DC1-35, Ref. (12); the value given for DCI-35 in Ref. (6).p. 534, is apparently the band center frequency rather than ma. b
e
I Ref. ( 2 1 ) . Ref. (13).
The value of A& from eqns. (4-7) is calculated to be 42.5 + 2 cm-' leading to a value of 0.81 f 0.01 for the exponential term in eqn. (1). This uncertainty was estimated by assuming a maximum error of 1 cm-' in the we values listed in Table 2. Using this result and the values for molecular parameters contained in Table 2, K , is calculated from eqns. (2)-(6) to be 0.80 & 0.01, with the error in the exponential term being the dominant source of error. Experimental Determination o l K,
A quantity of pure DCI a t known temperature and pressure is introduced into a 10-cm pathlength cell and the overtone spectrum is recorded. The cell is reattached to the vacuum line, the pressure of DCI is again noted (this second pressure will be slightly lower), and a precisely eaual oartial oressure of nure HBr is introduced into the cell. The system is allowed to stand to permit mixing by diffusion and the establishment of chemical equilibrium. Then a second overtone spectrum of DC1 is' recorded. Fmm the two overtone spectra and the known pressures and temperatures, a value for the equilibrium constant of the exchange reaction can be calculated. Since the exchange reaction produces equal amounts of HC1 and DBr, we may write
where AR and AK, are respectively defined as the standard deviations of R and K,. With this result and the computed standard deviation of R from Tahle 3, we may estimate the uncertainty in K , as + 0.07, in agreement with the calculated standard deviation of K , as shown in Tahle 3. We chose to compare relative peak areas rather than relative absorhances because the peak area was a more linear function of the DC1 pressure than was the absorbance. This is shown in Figure 1 where the absorbance and peak areas of the m = 5 peak of the DC1 overtone spectrum are plotted as a function of the DC1 pressure. This effect is probably caused by an insufficiently small spec-
. .
where Pr represents the initial pressure of reactant and PI represents the final (equilibrium) pressure of reactant. In general P,(HBrI = PWBrl - PI(DBr) = PiHBrl P,W.X P,(DcII (91
+
More conveniently, if PL(DCI) = Pi (HBr), then
Figure 1. Beer's law plots for t h e m = 5 peak of the DCI overtone spectrum. The peak area (measured by the cut-and-weigh method) and absorbance are plotted yersus the DCI pressure. Table 3. Integrated Peak Areas of DCI Overtone Spectrum by Cut-and-Weigh Method
Using the definition of K , and eqns. (8) and (101, we may write for K ,
Uneorreded initial wt ( 8 ) P = 436 torr m
T
= 296.gPK
Initial wt (gl corrected to P = 386 tor. T = 292.2'K
Final wt (g)
Pi(DC11 = Pi(HBr1 Pi = 386 torr T = 292.2'K
R
Kn
It must be kept in mind that eqn. (11) contains the implicit assumption that exactly stoichiometric amounts of DC1 and HBr are mixed, but this condition is easily fulfilled experimentally. If we define the ratio, R, where R = P, (DCl)/Pl (DCI), then eqn (11) may be written as 0
Equation (12) is a convenient form to use, because the ratio, R, is determined experimentally by measuring the relative peak areas (integrated intensities) for corresponding peaks in the initial and final DC1 overtone spectra. In our data, displayed in Table 3, the initial overtone spectrum was taken a t a different temperature than the final overtone. spectrum and the peak areas from the initial
...
-1
O.OW2
-2 -3 -4 -5 -6 -7 -8
0.0783
Average
0.0927
0.1039 0.0956 0.0785 0.0599
0.0417
Standard Deviation
Volume51. Number 11. November 1974 / 759
Figure 2. V a c u u m line a p p a r a t u s used t o m e a s u r e t h e DCI-HBr i s o t o p e exchange equilibrium. The r e a c t i o n f l a s k v o l u m e is 2 5 0 m i and t h e DCI reservoir has a v o l u m e of 5 0 0 m l . The cell h a s a 7 0 - c m pathlength.
tral bandwidth leading t o absorbance values which a r e too small, t h e error increasing a s t h e absorbance increases. Experimental Procedure DCI overtone spectra were obtained on a Cary Model 14 spectrophotometer in the 2450-2360 nm region using a 10-cm pathlength cell with 4-em diameter KBr windows. (Pyrex windows could also he used in this region.) The overtone spectrum was used for all quantitative measurements because a smaller spectral bandwidth could he obtained in the overtone region than in the fundamental region, leading to a more linear relation between absorbance and DCI pressure. Although we used the cut-and-weigh method because of its simplieity, the peak areas could he integrated by planimeter or by electronic means if the absorbance signal is readily available. On spectrophotometers such as the Cary Model 14, an electrical signal corresponding to the absorbance can be obtained only with some modifications to the instrument or by use of a retransmitting slidewire on the recorder. Using KBr or NaCl windows, the fundamental spectra of HCI and DBr can also be observed and provide confirmation that isotope exchange has taken place. A Beckman IR-12 spectrophotometer was used to obtain the fundamental spectra, using KBr windows. Figure 2 shows the vacuum line used to prepare the DCI and introduce the DC1 and HBr into the cell. All standard taper joints were sealed with Apiezon W black wax except for the side arm of the reaction flask which was sealed with stopcock grease. The DCI was prepared by a modification of a method described by Shoemaker and Garland (9). The reservoir and cell were shut off under vacuum, and the rest of the system was isolated from the manifold and hackfilled with dry air or nitrogen. A magnetic stirring bar, 25 ml of henzoyl chloride, and 2 ml DzO were added through the side arm, and the system was evacuated. A Dry-Ice/ isopropanol bath was placed around the reaction flask during evacuation to partially freeze the reactants and minimize humping. When all of the air had been removed from the system, the stopcock to the vacuum pump was closed, the reservoir was opened to the manifold, and the reaction vessel was heated using an infrared lamp or heating mantle with stirring provided by a magnetic stirrer. The reaction is rather slow at first, and sometimes takes as long as an hour. The condenser minimizes the escape of reactants from the reaction flask. The reaction was allowed to proceed until the final DC1 pressure was about 750 tarr, at which point the reaetion was halted by immersing the reaetion flask in a Dry-Ice/isopropanol bath. Unreaeted benzayl chloride and DzO were frozen out while the rest of the system was gently heated with a hot air blower. The reaction flask and condenser 760 / Journalof Chemical Education
were then shut off and the rest of the system was allowed to cool toraom temperature. The cell was opened to the r e s e ~ o i rand filled with DCI gas to a pressure of about 400 tom. The temperature and cell pressures were recorded, the reservoir and cell stopcocks were closed, the cell was removed, and the overtone spectrum of the pure DC1 was recorded from 2450-2360 nm. An evacuated compensating cell may he used in the reference beam; otherwise, a few "negative" water peaks may be observed in the spectrum if the partial pressure of water in the atmosphere is more than a few tom. If desired, the calibration spectra necessary to establish the relationship between peak area and the DCI pressure can he obtained by returning the cell to the line, evacuating the connecting lines, and carefully pumping off some of the DCI in the cell to obtain a lower pressure. In this manner we obtained spectra at several DCI pressures. To measure the equilibrium constant, a DC1 pressure of about 350 torr was admitted to the cell, then the stopcock t o the manifold was closed, and the needle valve on the lecture hattle of HBr was carefully opened to allow a precisely equal pressure (+ 1ton) of HBr to enter the eell. Since mixing by diffusion in the narrow connecting tubing is not rapid, we allowed about 20 min for the reaction mixture to become homogeneous. We then recorded the temperature and pressure of the mixture in the cell, and recorded the DCI overtone spectrum. The peaks of a number of pairs of absorption lines were cut out and weighed, although it would probably be sufficiently accurate and less time consuming to use the measured absorbance. In the data of Table 3, all peaks with m > 0 include both 3W and W 1 , while all those with rn < -2 include just peaks for the W I isotope. The 7931Br isotopic peaks were not resolved under our experimental conditions. Because the spectrum for pure DCI was taken at a pressure and temperature different from the initial DCI pressure in the mixture of DCI and HBr, it was necessary for us to correct the integrated peak areas to the initial conditions of temperature and pressure used in the DCl/HBr reaetion. We should mention that an initial attempt to dry the DCI with a desiccant tube in the line caused erroneously low DCI pressure values in the final equilibrium mixture because residual water in the desiccant exchanged with the DCI. For this reason, we adopted the method of freezing unreacted DzO out with the Dry-Ice/ isopmpanol bath.2 Conclusions W e obtained a n experimental value for K , t h a t agrees with t h e theoretical value, within t h e specified uncertainties. T h i s agreement appears to justify t h e harmonic oscillator approximation which was assumed in t h e calculation ofK,. T h e equilibrium constant for t h e exchange of DC1 with HI could probably b e measured in a n analogous fashion. A calculation for t h i s system using d a t a available in t h e literature (10) predicts a value of K, = 0.65 for t h e exchange reaction at 298°K. Literature Cited 111 Bsrmw. G. M.. "Physical Chemistry." 2nd Ed., MeGraw-Hill Bmk Co.. New York. I.. W ..l, nn-. 7?-7&C . . .. . ..
121 Mmre. W. J.. "Physical Chemistry." 4th Ed., Pientice-Hall h e . . Englewood Cliffs. N.J., 1972. pp 296-299. 131 Dickerson, R. E., "Molecular Thermodynamics." W. A. Benjamin. h e . . New York. 1969. o. 289. 141 stsffo~d.Fred E., H d t . C. W..and Paulnon, G. L., J . CHEM. EDUC.. 40. 245 (19631. 15) Hollenherg,,I.L.. J. CHEM.EDUC..47.2119701. 161 Herrbere. G., "S~ectraof Diatomic Molecuios." 2nd Ed.. D. Van Nostrand Ca.. Princeton, N.J.;I~SO.p.92. (71 Gaydon. A. G., "Disnociatlon Energies..' Chapman & Hall Lfd.. London. 1968. p. 249. 181 Lewis. G. N . . Randall. M.. Pitrcr. K . S.. and Brewer L.. "Thermodynamics." 2nd Ed..McGrav-Hill Bmk Co..lnc.. New York,1961, p.419. 191 Shwmaker. D. P.. and Garland. C. W., "Experiments in Physicel Chemistry." 2nd Ed., McGmw-HillBwkCo., New York, 1961, p.332. 1101 Hurloek. S. C.. Alexander, R. M., Rso. K . N.. andureska. N . . J. Mol. Specfrose.. 31,373119711. (111 Thompson. H. W.. Williemn. R. L.. and Callomon, H. J.. Sprrtroehim. Acta, 5. 313 11952). 1121 Rsnk. D. H.. Eastman, D. P.. Rao. B. S.. and Wiggins, T. A . J. Opt. Soc ~ m s r . 52.1i1862). (131 Keller. F.L..and Nlelsen. A . H . J . Chem Phys.. 22. 294 11954).
*A reviewer has suggested that a desiccant might be used if it were exposed several times to DCI, pumping off in between.
