August, 1961
CORRECTIOX OF THE POTASSIUM VAPOEPRESSU~E EQUATIOF
1297
CORRECTION OF THE POTASSIUM VAPOR PRESSURE EQUATION BY USE OF THE SECOKD VIRIAL COEFFICIENT’ BYR. J. THORN AND G. H. WINSLOW Chemistry Division, Argonne National Laboratory, Argonne, Illinois Received November 4, lQS0 The available thermodynamic properties of saturated potassium vapor have been used to predict the dissociation energy
of the dimer on the brtsis of a perfect gas treatment including the diatomic molecules, and on the basis of predictions from an imperfect gas treatment in which i t waa assumed that the inclusion of the second virial coefficient in the equation of state was adequate. Recognition of the necessary (although not sufficient) criterion that the measured entropy of vaporization must agree with that derived from absolute entropies establishes that the effusion experiments of Edmondson and Egerton can be acaepted as a reliable starting point for the analysis. A statistically certain experimental distinction between the two treahments cannot be made because of a large discrepancy in the high temperature vapor pressure measurements, though sufficient precision can be obtained to make the distinction. The evidence su ports the imperfect gaa treatment and the spectroscopic value of the dissociation energy. The most consistent set of tfermodynamic properties T - 1.0659 X T 2 10.14506. log pz(idea1) = 4.9800 X is: log p,(ideal) = -4802.27/T - 1.97108 log T -6.98 x 10s/T -- 4.44216 log T 1.04976 X loes T - 2.1317 X loe7 T 2 - 321.28/T9 18.0093. log p(tota1) = log pi(ideal) (1946.9/T) exp( -4351.0/T). AH001 = 21747.0 f 10.3 cal./mole. DO = 11.85 =t0.10 kcal./mole. In
+
+
+
++
the first three equations the pressures are in atmospheres.
Introduction It should be possible to make sufficiently accurate vapor pressure measurements, over a sufficiently wide temperatu:re range, that they provide a means of checking, by itwe of thermal data, those values for dissociation energies which are obtained by extrapolation of spectroscopically measured vibration bands. In particular the easiest application should be to the dissociation energy of metal dimers formed as the temperature of the saturated vapor and, hence, its (density are increased. Unfortunately, there are very few measurements which are sufficiently precise and extensive to justify such an analysis. An examination of the periodic table quickly suggests that of all the elements, the alkali metals are the optimum ones for the purpose. In fact there have been several attempts to study and correlate the vapor pressures and dissociation energies which have been published for In no case, however, has the analysis been founded on any sort of explicitly stated, plausible basis which would allow the judicious selection of the best of the available measurements from a number of discordant values. We support the necessity of looking for a basis for the selection of data by the following arguments. The mere fact that there is a small random error associated with the measuring equipment used in a particular experiment does not mean necessarily that the results are accurate. That is, the small random error does not testify to the absence of a fixed systematic error or to the absence of a systematic error that varies in a precise way with the recognized variables of an experiment. Thus, it is necessary, when there is discordance between experimental results which is greater than their precision, to try to select those which do not contain a systematic error (granting that there are such in the group). It is obvious that while the degree of (1) Based on work performed under the auspices of t h e U. S. 4tomic Energy Commission a n d presented before t h e American Chemical Somety, Division of Physical Chemistry, Septemher. 19.59. (2) W. H. Evans. R. Jacobson, T. R. Munaon and D. n. Q’Rgman, J . Rerenrch h’all. Bur. Standards, 66, 83 (1955). (3) M. Sittig, “Sodium; Its Manufacture, Properties a n d Uses,” Reinhold Publ. Corp., New York, N. Y., 1956. See Chapter by G. W. Thomuon and E. Garehs.
precision might be a factor to be considered in choosing the basis to be used for the selection it is also obvious that there can be no fundamental relation between the precision and the basis chosen. Indeed, it could well turn out that the least precise results are the most accurate. It should be emphasized next that if one has applied some rule for selection it is totally illogical to make any sort of combination of the rejected results and the accepted one by use of a weighting procedure based on the precision of the various results. On the other hand, after correction of the rejected results (supposing that this becomes possible via discovery of the source of the systematic error) such that the several sets now appear to be identical within the limits of precision, they can then be combined by appropriate (weighting) techniques. With this in mind, we propose the following “rules” for the present case: 1. Of all the possible numbers that could be used in such an analysis, it will be assumed that the electronic levels of the atoms, the vibrational, rotational, and electronic levels of the molecules (all relative to the corresponding ground levels) and the heat capacities of the condensed phases are the most accurate and/or reliable. 2. Satisfaction of the following criterion, or an equivalent, is necessary for the acceptance of vapor pressure measurements: The measured entropy of vaporization must agree with that calculated from the measurements mentioned in the first rule. 2a. An equivalent which is pertinent to the present problem, is that vapor pressure data must extend over a sufficiently large range that the ratio of monomer to dimer changes significantly but that values of the dissociation energy calculated a t various points in the range are identical within the experimental error. One of the few substances to which the above rules can be applied successfully is potassium but, heretofore, its near uniqueness in this regard has not been recognized. For instance, Evans, et al., attempted t o average all the data together without realizing that of all the measurements a t low t,emperatures (about 440°K.)4only those by Edniond(4) At these temperatures the concentration of t h e dimer IS sufficiently small t h a t it can be calculated accurately enough (with any reasonable dissociation energy) t o be subtracted oub.
son and Egertonj yield an entropy of vaporization to the monatomic form (20.50 i 0.27 e.u.) which is in excellent agreement wit’h that (20.63 rt 0.02 e.u.) calculated from the absolute entropies of the gaseous and condensed phases2 Admittedly this agreement may be fortuitous, but it certainly appears to be better to assume that it is not, than to assume that the others are equally reliable but uiifortuit’ously disagree with the third law of t,hermodynamics and/or all the other results such as t.he specific heats of the condensed phases. Consesequently we attempted to analyze the exist’ing inf ~ r r n a t i o i i ~on~ potassium. ~-~~ It immediately became apparent that the various results were not consistent, to the degree we believe possible1’ when treated within the framework of perfect gas theory, with the equilibrium concentration of diatomic molecules iricluded. We therefore began to look into the application of imperfect gas theory to t’hisproblem as being a possible answer to part of the difficulty. Theoretical Background.-The principal problem to be faced is that^ imperfect gas theory is usually applied to gases in So states and, indeed, to these when above their critical pointsI2; certainly there have been exceptions, particularly to the latter restriction. The present application is t’oa gaseous atom with a single unpaired electron. The process of dimerizat’ion,t’hen, involves an attract’ivepotential which gives rise to non-degenerate bound states. Interaction a t positive energy will involve t’his pot,entialplus a repulsive potent’ial which gives rise to (unbound) t,riply degenerate stat’es, as with hydrogen atoms.I3 Hence, we reached the same conclusion as subsequently published by Sinanoglu 3 nd Pitzcr,l* ie., that the second virial coefficient, should be
+ (3i4)B~
H = (1/4)B.%
(1)
Here BB is calculated with the at,tractive potential function, and BR is calculated with the repulsive pot,ential function, according to ( 5 ) 11.. Ednronrison and .i.l k e r t o n , P r o c . Roy. SOC.(London), 8113, 320 (1927). ( 6 ) T. B. Douglas, i. E’. Bail, 1).C. Ginnings and W. D. Davis, J. Am. Chem. Sac.. 74, 2472 (1952).
(7) J. W . Johnson, H. R. Bronstein a n d M. A . Bredig, a n unpuhlished manuscript privately coinmunicated b y 11. A. Bredig. The data, cited herein for these authors were taken entirely from this manuscript. After the preparation of our present paper, but before its snbmission for publication, Dr. Bredig informed us t h a t additional measurements had turned oiit t o h e significantly different from those in t h e ilianilscript. Tliis rinsatisiactory experimental situation a t the higher temperature (sep ref. 8 and 11) is largely due t o the difficulty of determining t h e true eijuilihriiim temperature for a given pressure. I t Beenis t o make it evpn more certain t h a t t h e equations deduced in o u r p’nlx31 a n d ciillwted. with the other desirable information, in the Bunitnary are thc: best available description of the thermodynamic properties uf potassium vapor. (8)M. XI. Mskansi, A l . hladaen, W. A . Selke and C . F. Bonilla, J . PhUs. Ciiem., 6 0 , 128 (1956). (9) F. FV. Loomis, Phys. Rev., 38, 2153 (1931). (10) F. W ,Looniis a n d R. E. Nushaum, ibid., 39, 89 (1932). (11) The most obvious inconsistency is t h e lack of agreement betaeen the high temperature vapor pressure m e a s u r e r n e n t ~ . ~ - ~ (12) Prohahly the niost extensive discussion t o he found in any one placr is .I. 0. HirsclifPldtlr. C . F. Crirtiss a n d R . R . Bird, “Molecnlar T h w r y of (;asps and I,i 0 @R =
De(1 f bP
+ b/P) exp(--bf)
(4)
as the repulsive potential function. The virial coefficients were calculated by Simpson rule integration on an IBM 650; the simple alteration of eq. 3 to give eq. 4 was easy to program and. we believe, gave a function sufficieiitly good for the purpose. T’irial coefficients were calculated at 900, 1200 and 1500°K. for various values of De. This was in agreement with the attitude that we are using thermal measurements t o determine dissocistion energies as an independent check of extrapolations of spectroscopic measurements. On the other hand, the second derivative of the potential function, a t the minimum. can be determined .i.ery precisely from the spectroscopic observations. Thus, Ivhen De was altered the b of eq.3 and 4 was also altered so that this second derivative was held constant. Derivation of the Vapor Pressure Equations.In order t o determine a free energy functioii for liquid potaGiini to usc for interpolation between tahulated ~-:tlu~s.* vhere needed. we were guided by the work of Douglas, rt aL6 The function used wa? made to conform to the tabulated values. This was done by making slight changes (in the constant and in the term which is linear in the temperature) in the function we had worked out from Dough\’ results. 117 units of cnl. ’mole,” this function i.; (FO - H#)l/7‘
=
-220 6 5 / T - 20.4582 log T 10-3T - 4.877 X 10-’T2 ( 5 )
+ 41.3972 + 2.2787 X
Two alternative equational descriptions of thc tot21 vapor are t o be given. If the mpor is treated as a two component perfect gas, the free energy function (per mole) for the monatoiiiic component is (FO
- liToo)a/?’
=
- R ’ { ( 3 / 2 ) log Jf f ( 3 ’ 2 ) log T
+ log [ ( 2 ~ k ) ~ ’ ~ / h ~+~ 1~0:o ‘R’ ~f] 10%21
(6)
(15) Y. P. Varshni. Reus. Modern Phys., 29, 61i4 (1957). (16) R. Rydberg, Z. Physik, 73,376 (1031). (17) Folloaing Evans et d.,*the constants given Iw n. D. W a ~ r n n n . .I. IC. liilpatrick. W. J . Taylor, K. S. Pitrer and 17. I ) . Rossini. J . Zipsearch Natl. Bur. Standards, 34, 143 (1945) havr been used, except. for the calorie. Here a s with Evans et G I . . 1 cal = 4.1810 abs. j. This means t h a t R = 1.98719 cal. and R‘ (RIn I O ’ = 4..57567.
and that for the diatomic component is
monatomic vapor and the heat of vaporization to, or, alternatively, the heat of dissociation of the diatomic form. It is the investigation of the thermal + determination of the latter that is the object of the (7) present discussion. As for the former, it was menIn eq. 6 the log 2 appears because of the doublet tioned earlier that we used the observations by nature of the ground state of the atom; corre- Edmondson and E g e r t ~ i i . ~This was done by first spondingly the Q1 in eq. 7 is the internal partition making small corrections to their individual obfunction of the dimer. We take Qi from RIayer servations for diatomic molecules. The corrections and Mayer'*; a factor of 2 is included because the were calculated with eq. 11 and the heat given by molecule is homonuclear, and the terms in log QI, Evans, et aL2 h least squares line was put through log [1 - exp(-u)], [exp(u) -11, and [exp(u) the corrected and weighted observations with the - 1]+ have been expanded. Thus result log &, = -10,. 2 0 -t ~ 0.434294[8y2/u log pl(atm) = 4.48 f 0.06 - (4513.4 rt 0.4)/T (13) + s/u 4-2x/u - s/2 - 22 u / 3 The weights for the individual values of log pl were + u/2 + ~ 6 / 1 2+ 5 ~ ~ -/ u2/24 6 - u ' z / ~ ] (8) determined according to the form of the actual obThe terms in eq. 8 were evaluated by use of the ac- servation equation. From the least squares recepted numbers1gto give sults, it mas determined that the value of log pl log &, = -1.3335 $- 3 log T 28.929/2' which mas determined most accurately was
+ +
(Fa - HoO)m/lT = -R'{(3/2) log (ZM) (5/2) log 7' log [(2Tk)V9/hW4/2] log log & I 1
+
+
+
- 321.28/Ta
+ 5.376 X 10 62'
(9)
From eq. 5 through 9, then log pi = - [226.65 -I- AHDO( l ) ] / R ' T- 1.97108 log 2'
+ 4.9800 x
+ 10.14506
10-4T - 1.0659 X 10-TT2
(10)
and log p , =
-
- j320.93 -k AHo'
(2)]/n'T -4.44216 log 2' 1.04976 X 10-'1' 2.1317 X 10-'T2 - 321.28/T2 18.0093 (11)
+
+
The observed pressure is to be compared with log (Pl Pz). In the above treatment the only interactions recognized are those corresponding to the bound states of two particles. With the aid of the second virial coefficient one accounts for all the states corresponding to free particles and, to some degree of accuracy, for all states of interaction, bound and unbound, between pairs of particles. That is, it is not only recognized that some pairs will be bound as bona fide molecules, but that the behavior of the others will he altered because of their mutual potential energy. If the second virial coefficient is used, the free energy function for the vapor is the same as eq. 6 except that -R' log (1 B / V ) 2R'B/(V In 10) is to be added on the right.20 If the right side of eq. 10 is called log then the vapor pressure equation, when the vapor is treated as an imperfect gas, is log p = log T;II"' log (1 4 B / J 7 ) - 0.8685SU/V- ( 1 2 ) The obviously unknown quantities in eq. 10 through 12 are the heat, of vaporization t o t'he
+
+
log p,(atm)
=
-5.7750 rt 0.0026
at T
= 440.2jOK.
When this information is used in eq. 10 it is found that AH@(l)= 21747.0 =t10.3 ea1 /mole
The error given here includes that in the free energy functions2pz1as well as that in the vapor pressure. Predicted Values of Dissociation Energy.-The description of the calculations of the second virial coefficient was given previously. The results are given here in Table I where, for convenieiice, diswciation energies are given in e.\'. and in cal. 'mole. The various determinations of Do via perfrct and imperfect gas treafnients are shown in Table I1 and the error21 situation is shown in Table 111. The infotmation in the latter table is that on which the meaning of the information in the former is to be judged. A graphical picture of the content of these tables is shown in Fig. 1 in order to make it easier t o grasp their meaning quickly. I
I
I
+
+
(18) J. E. LIay-er a i d AI. G. l f a y e r , "Statistical Mechanics," John N-iley a n d Sons, :'Jew York, N. Y., 1940. See page 164 where corrections, also discussed by Evans, et al.,? for non-rigid rotation and anharmonicity are given. Note here t h a t if the problem were merely one of determination of t h e dissociation energy a n d of the numerical differences between t h e perfect a n d imperfect gas treatments, it could be doni? uia the tabulations in ref. 2 , except for t h e determination c f A H 8 (1) t o be described later. On the other hand, we a i s h t o summarize );his work by proposing what appears t o us t o be the best v a ~ i o rv r e s w r e equations: we approach t h a t by giring all these introductory equations. 119) See ref. 2 , Table I and the accompanying text for definitions, values. reference and comment. (20) See ref. 18. p. 2!12. Xote t h a t B = --p1/2. For the hIayers' u under the In in their eq. 13.49, (RT/NoP) (1 B / V ) has been substituted. ComDare, for instance ref. 12, p. 230.
+
0.5
1
Johnson et at7 Spectroscoplc
f
o P e r f e c t Gas M o d e l
x I m p e r f e c t Gas Model
1
R. J. THORN AND G. H. WINSLOW
1300 TABLE I
SECOND VIRIAL COEFFICIENTS FOR POrASSIUM V A P O R Do - B o (cm.a/mole) O.V. cal./mole QOO'K. 1000OK. 1100'K. 120O0K.
0.477 11000 8 7598.0 4298.7 2626.1 1703 1 ,494 11392.8 9519.6 5287.4 3192.8 2056.7 .514* 11854.1 12334.2 6702.0 3988.8 2545.9 .529 12200.0 14887.7 7957.8 4683.2 2966.8 .546 12592.1 18592.3 9742.4 5654.4 3547.8 ,563 12984.1 23203.5 11914.6 6815.8 4232.7 .580 13376.2 28943.5 14558.4 8204.6 5040.1 a The values originally calculated on the IBM 650 were st 900, 1200 and 1500%. The values listed here a t 1000 P/T and 1100'K. were interpolated ma In IBl = CY y In T . b This is the spectroscopic value. See ref. 9 and 10.
+
+
TABLE I1 PREDICTED DISSOCIATION ENERGY (E.V.) -Perfect gas-Imperfect gasJohnson, Makansi, Johnson, Makansi, T,OK. et al.7 et al.8 el al.7 et al.8
900 1000 1100 1200
AD^'^^^
- 0.5141
0.510 ,517 .530 ,543
0.631 ,609 .595 .589
0.500 ,504 ,512 ,523
0.604
0.033
0.042 0.092
0.023 0.004
0 050 0.061
0.011
,578 .563 .554
TABLE I11 ERROR PICTURE (E.v.) Exptl. vapor pressure error
T, "IC.
Total error Johnson, Ivfakansi, et al.7 et al.8
son, et al.7
900 1000 1100 1200
0.031 0.012 .022 .013 ,029 .om ,056 ,040
0.008 0.005 .003 .004 .001 ,002 .002 .001
John-
Makansi, et al.8
IADol between models Johnson, Makansi, et al.7 et a1.8
0.010 0.027 ,013 ,031 ,018 ,032 ,020 ,035
weighted treatment of the pressure; it appears that each group of workers measured pressures directly. The results are: for Johnson, et a1.' log p = 4.176 & 0.007 - (4332.3 f 7.6) (1/T)
and for Makansi, et aL8 log p = 4.927
* 0.011 - (4243.4 f 13.8)(1/T)
From these equations and other necessary information which is developed during the calculations we also determined p for each set of observers a t each of the temperatures shown in Tables I1 and 111and the errors in those pressures.21 The perfect gas results in Table I1 were obtained by finding p l , subtracting it from the experimentally observed p and calculating A H o o ( 2 ) ,and hence Do [= 2aHoO(l)- AHOO(2)], from eq. 11 or the corresponding tabulations in ref. 2. The imperfect gas results were obtained graphically, as follows. By a method of successive approximations on an IBM 610 simultaneous solutions of eq. 12 and p v = RT(1
+ B/V)
(14)
for p and V were obtained at each of the several values of B. [That is, it is via B that DO(or De) enters the imperfect gas model.] The resulting pressures were plotted us. the trial values of DO and the solution obtained by graphical interpolation a t the experimentally observed pressures. The errors given in Table I11 were all calculated on the basis of the perfect gas model. These are
Vol. 65
errors which arise from various sorts of experimentation and would be expected to propagate about the same in each model. A systematic (theoretical) error which would arise from using a perfect gas model when an imperfect gas model is required is one of the items t o be examined here via the tabulated differences. Such a difference would be expected to be larger for materials with larger values of dissociation energy; the larger Dowould mean increased interaction (imperfection) even in unbound states. In addition, as here, the difference will also increase with temperature because a high T will correspond to a higher density (smaller molar volume) of the (saturated) vapor. On the other hand, the accuracy of either model as used here will be expected to go down for greater interaction, or a t higher temperature, because of the restriction to only diatomic molecules or, alternatively, only the second virial coefficient. Mathematically, in the second case for instance, one can see that a complete factor such as the (V - b)-l of van der Waals' equation ie needed to turn the calculated pressure back up at small values of the volume (condensed phase); the use of V--I(l - B / V ) leads to a maximum (corresponding to the one in van der Waals' equation) a t V = 2/BI,but then to zero pressure a t V = IB1. It should be noted, however, that in the present example, the molar volume, 1.58 X lo4 cm.*, for the saturated vapor a t 1200°K. and Do = 0.580 e.v., the last entry in the tables, is still greater than 2jBI ( = 1.01 x 104 cm.8). Of principal interest here, then, is the effect of the experimental error situation on conclusions about the distinction between the present two models and about the value of the dissociation energy. The statements to be made can be verified by referring to Tables I1 and 111and to Fig. 1. The most obvious point is, as has been mentioned, the large difference between the results derived from the two sets of high temperature vapor pressure measurements. The difference is quite large compared to the standard deviation. We believe the principal difficulty to be temperature measurement (as do Makansi, et aL8) rather than pressure measurement. Wez2are in the process of making a third set of measurements of the vapor pressure in this region, and are obtaining results closer to those of Johnson, Bronstein and Bredig.' The next point to note is that our criterion 2a is not strictly satisfied in any case. Indeed, it is not, satisfied within the error by the results from Makansi, et al.,8~29 although it is by the results from Johnson, et aZs7 Johnson's results are improved by the use of the imperfect gas model, whereas Makansi's are worsened, in the sense of criterion 2a. Although, in both cases, the difference between models is greater than the error which arises only from the random error in the pressure measurement a t high temperatures, the over-all error is such that one (22) R. J. Thorn, R. R. Walter8 a n d G. H. Winslow, t o be published. (23) The propagation of error8 t o the error in Do involves, a t m s n y points, the presence of PI in the denominators of fractions or. more importantly, the presence of the ratio piJpr. The higher the observed pressure t h e smaller will be t h e propagated error for a given pi.
CORRECTION OF THE POTASSIUM VAPORPRESSURE EQUATION
August, 1961
cannot conclude unequivocally that the imperfect gas model is required to treat the present data. An illustration of the contributions to this over-all error is shown in Table IVa21 The principal difficulty is that it is necessary to extrapolate the liquid free energy function outside the region in which it was measured in order to reach the region in which the distinction between the perfect and imperfect gas models becomes important. This problem cannot be dismissed unless one is satisfied merely with empirical tables of vapor pressures. As temperatures of interest become higher, the imperfect gas treatment and values of the liquid free energy function measured a t those temperatures will become more important. The final items to notice are the comparisons with the spectroscopically observed dissociation energy. The results which come closer to satisfying our criterion 2u (regardless of model) also tend to confirm that value. Although it is an event which is well bounded by i,he size of the errors, the application of the imperfect gas model to the results by Johnson, et ul.,’ does improve their agreement, on the average, with the spectroscopic value. Our conclusion is that the vapor pressure of potassium is best described, a t the present time, by use of the imperfect gas model with the spectroscopically observed dissociation energy used for evaluation of the second virid coefficient. Iv Do (E.V.) FOR THE GASMODEL,JOHNSOS’S DATA,1000’K. T-4BLE
C O N T R I B U T I O N S TO IZRROR I N
PERFECT
Error
Source
Pressure meapurement 0 0027 Monomer free energy function 0016 0087 Dimer free energy function Liquid free energy function 0179 AHoo( I)“ .0092 Accumulated error 022 0 Because of the Large difference in temperature between 1000’K. and the temperature (440.25’K.) a t which AHoO(1) was determined, the contribution of the liquid free energy function error to the error in AHoo(l) was considered to be independent of that entered in the table.
summary If B / V is treated as a small number in eq. 12, the last two terms cam be combined to give BPIR‘T. In the range up to 1200°K. it is sufficient to represent B by a term of the form a exp(P/T). Since p is also of this form, quite closely, the product BP can be combined into one term of that form. The constants can be evaluated via the values of log p determined by successive approximations a t 900” I
