2224
YATENDRA PALVARSHNIAND RAMESHCHANDRA SHUKLA Experimental
The lead chloride was prepared by precipitating it with hydrochloric acid from a lead nitrate solution, which previously had been filtered to remove an unknown black residue. It then was dried and melted for 24 hours under hydrogen chloride gas. The gas previously had been treated by running it over charcoal a t 250°, powdered alumina, and anhydrous aluminum trichloride. The apparatus is shown in Fig. 1. The molten metal and lead chloride were contained in a Solidex glass cup. Electrical contact with the metal, which served as a cathode, was maintained by a tungsten wire sealed into the bottom of the cup. The platinum microelectrode was sealed into a glass tube and could be positioned in the melt with the moveable shaft seal. The electrode had an area of approximately 3.5 mm.2. This assembly was contained in a Solidex tube through which purified nitrogen flowed, and wap heated with a wire-wound resistance furnace. For the measurements of the polarographic current, voltages ranging from 0 to 0.6 volt were supplied by a 2 v. storage battery in conjunction with a 100 Q potentiometer. Currents were measured with a Multiflex galvanometer connected through an Ayrton shunt. After the current I Ohad been measured for lead chloride coexisting with pure lead, known amounts of bismuth or gold were added for measurements corresponding to values of aPb < 1. A stirrer shown in Fig. 1 was used in order to obtain a liquid alloy of uniform composition. Only values obtained with gold alloys are considered t o be relevant for reasons discussed below.
Results and Conclusions Figure 2 shows two current us. voltage curves obtained with the apparatus. The top curve represents the current obtained from pure lead in equilibrium with lead chloride. Currents below 0.2 volt could not be measured because the platinum electrode alloys with lead at these low voltages. The bottom curve shows the residual current obtained from lead chloride in contact with pure gold. This current was subtracted from all measured currents. Tungsten was tried as microelectrode material but the limiting currents were not constant with increasing voltage. Measurements with successive additions of gold were made a t 518’. Values of I/Io us. UPb for an applied voltage of 0.40 volt are plotted in Fig. 3 with values based on mole fractions and the results of e.m.f. measurements published by Kleppa.’O No experiments were performed below a lead activity of 0.3, where the lead-gold alloys are solidl’for the tem(10) 0. J. Kleppa, J . Am. Chem. Soc., 71, 3275 (1949).
Vol. 65
peratures used here. The results are in agreement with reaction 2. Thus Pbz++ions are the predominant form in which excess lead dissolves in lead chloride. T o some extent, however, Pbz++ions dissociate into 2 Pb++ ions and two excess electrons, as has been shown recently by Herzog and KlemmI2 with the help of transference measurements. Attempts also were made to use Pb-Bi alloys but it was found that pure bismuth gave currents slightly higher than pure lead. These measurements were disregarded. It is believed that bismuth reacts to a small extent with lead chloride, then a small amount of excess bismuth dissolves in the resulting bismuth trichloride producing a current a t the microelectrode. Karpachev, Stromberg and JordanI3have studied the cell Pb 1 PbCli I I PbClz I C (+Pb) ( S P b )
a t 700” and with varying amounts of P b in the right-hand compartment. They conclude that P b + is the predominant subhalide species. At present there is no explanation of this discrepancy. It can be seen easily from equation 4 that such a polarographic method may be used to measure activities in alloys if the species of subhalide is known. The alloying elements must be more noble than the metal forming the subhalide so that no reaction will take place with the salt. Also subhalides of the second metal must not be formed as was believed to have happened with the Pb-Bi system. Acknowledgments.-The aut8hor would like to express his appreciation to Professor Carl Wagner for suggesting this problem and for helpful discussions concerning the experiments and manuscript. He also is indebted to Dr. George Simkovich for supplying the lead chloride. A grant from Associated Universities, Inc. also is very gratefully acknowledged. (11) M. Hansen and K. Anderko, “Constitution of Binary Alloys,” MoGra-iv-Hill Book Co., New York, N. Y., 1958. (12) W.Hersog and A. Klemm, 2. hiaturforsch., Ma, 523 (1961). (13) S. Karpachev, 8 . Stromberg and E. Jordan, Comp. rend. m a d . sei. U . R . S . S . ,36, 101 (1942).
ON THE FROST-MUSULIN REDUCED POTENTlAL ENERGY FUNCTION B Y YATENDRA P A L VARSHNI’
AND
RAMESHCHA4NDRASHUKLA
Department of Physics, Allahabad University, Allahabad, India Received July 18, 1961
The values of cye and uezehave been calculated for 23 molecules by the Frost-Mwulin reduced potential energy function for diatomic molecules, and discussed with reference to the question of the existence of a “universal” potential energy curve.
Introduction Like the reduced equation of state, Frost and Musulin2 have investigated the possibility of the existence of a reduced potential energy function for
diatomic molecules. To define a reduced potential energy (u’)as a function of reduced internuclear one possibility is distance @’I,
(1) Division of Pure Physics, National Research Council, Ottawa 2, Canada. (2) A. A. Frost and B. bIusulin, J. A m . Chem. 76, 2045 (1954).
where De is the dissociation energy and Re is the equilibrium internuclear distance. In terms of
m.,
U’ = U / D , and R’
=
R/R,
(1)
FROST-MUSULIN REDUCED POTENTIAL ENERGY FUNCTION
Dee., 1961
2225
TABLE I (obsd.) em. -1
ae
Diatoms
H:, + H? CH OH HCI HC1+ KIi ZnH HBr CdH HI HgH Liz
1.4 2.993 0.534 .714 ,3019 ,3183 .0673 ,2500 ,226 ,218 ,183 .312 ,00704 ,01579 ,01984 ,00436 ,00079 ,00142 .0017 .000219 .000275 .000536 .000117
OS 0 2
ae
+
ClF Na?
Pz Clz
Kz Brz IC1 I2
(cded.) cm.-l
0.578 1.4628 0.4174 .5508 ,2521 ,2114 ,0644 ,3586 .2102 .3098 .1677 .4368 .008243 ,01486 ,01612 .004763 .001199 .001493 ,001625 .0003491 .0003664 ,0005755 .0001335
Average
% error
+
In case of HZ arid Hzf,R' of (2) is identical with R' of (1) since Ri, = 0. Assuming li' a s a universal function of R' (eq. 2), independent of which molecule is being considered
_:Lyi
=
62.00 117.995 64.3 82.81 52.05 53.5 14.65 55.14 45.21 46.30 39.73 83.01 2.592 12.073 16.53 4.00 0.726 2.804 4.00 0.354 1.07 1.465 0.6127
+ +
and R' = 1
= -1
K , a dimension-less constant
Since d2U =
(obsd.),
cm.-1
-58.7 -51.1 -21.8 -22.9 -16.5 -33.6 - 4.3 +43.4 - 7.0 +42.1 - 8.4 +40.0 $17.1 - 5.9 -18.7 9.2 $51.8 5.1 - 4.4 +56.1 +33.2 7.4 +14.1 24.9
these variables the potential energy minima would be a t U' = -1 and R' = 1. To express the potential energy function as a function of more general reduced internuclear distance they argued as follows: The curvature at the minima which is related to the force constant (k,) will vary considerably from molecule to molecule, since the inner shell repulsion for atoms other than hydrogen should influence the internuclear distance a t the minimum. To take this into account they used the reduced distance variable U' = U / D , and R' = (R - Ril)/(Re - R J (2) where Rij is a constant for a given molecule and is a measure of the inner shell radii of atoms i and j. Here again the minima is given by I;'
weze
oeze
(calcd.), cm. -1
47.04 102.398 55.78 73.544 48.154 38.40 12.592 67.266 45.077 67.388 41.584 102.934 2.911 12.090 13.30 5.904 0.850 2.973 3.155 0.406 1.320 1.693 0.7392
- 0 i j ) xZ X
WRZO.IIA
% error
(Re
-24.1 -13.2 -13.3 -11.2 - 7.5 -28.2 -14.0 +22.0 - 0.3 +45.5 4.7 +24.0 +12.3 0.1 -19.6 $47.6 +17.1 6.0 -21.1 +14.7 $23.4 +15.6 +20.6 17.7
+
+
+
10'6
35.447 30.980 30.983 30.248 29.043 37.459 31.279 22.039 26.965 18.478 25.687 21.692 23.958 26.865 33.418 18.214 22.975 25.355 34.083 23.433 23.324 23.265 22.283 26.847
ficients of the higher terms such as L/6, M/24 in the expansion U'
= -1
+ (K/2) (R' - l ) z + (L/6) (R' - ) l a + ( ~ 4 / 2 4 )(R' - )14 + . . . .
where
For L and M they obtained the relations
The average values of L and M, for 23 molecules, were found to be (- 15.06) and 43.48, respectively. The mean deviations of L and M from their averages were 13.2 and 42%, respectively. However, a more direct test of this concept is to test for the reproducibility of the molecular constants a, (vibration-rotation interaction constant) and w,x, (anharmonicity constant). We have applied this test in the present note. The method of obtaining a, and w,x, from a potential function is explained by V a r ~ h n i . ~By this we get the following expressions for ar and w,z, in terms of L and M
we get by virtue of (2) ke(Re - Rij)'/De
a
K
(3)
or
- (KDR/ke)'/2
where W = 2.1078 x and pA is the reduced They obtained K = 4 from the data for Ht+ mass in atomic weight units. The observed and calculated values for a, and and Hz. Thus Rij can be calculated for each molewexe, for the 23 molecules considered by Frost and cule. Frost and Musulinz have examined the validity Musulin, are compared in Table I. of this idea by examining the constancy of the coef(3) Y.P. Varshni, Reo. Mod. Phys., 29, 664 (1957); 31, 839 (1959). Rij = Re
(4)
K.
2226
W.R. JOHNSON, M. KAHNAND J. A. LEARY
The necessary data have been taken from their pa,per and H e r ~ b e r g . ~ Equations 7 and 8 can be recast as
Vol. 65
which suggests another possibility of testing the constancy of the L.H.S. of the expressions 9 and 10, respectively. This is equivalent to testing the constancy of L and ( L 2 / K - M ) , respectively. As L has been examined by Frost, and Musulin, we have investigated expression 10 only. The results are given in the last column of Table I. It may be noted that our procedure of examining (LZ/K - M ) does not involve ( r e , though L involves %. The average value of the L.H.S. in (10) comes out to be 26.847. Discussion The problem of a “Universal” potential energy function also has been discussed by Varshni13mho found that though rigorously speaking a “Universal” potential energy function does not exist, it is still possible to find approximate “Universal” relations for ole and wexein terms of the Sutherland parameter A( = k e r e 2 / 2 D e ) . The average percentage errors for cye and wexe are seen to be 24.9 and 17.7. These may be cornpared with corresponding average percentage errors
22.1 and 11.1 found by T’arshni3for 23 molecules, 18 of which are common with the present list. The average percentage error for wexe is only 17.7, while for L and M the corresponding values were 13.2 and 42, respectively. wexe depends on both the third and the fourth derivatives. The low error in the case of wex, is due to the fact that for most of the molecules, errors in L and M were in the same direction. A happy cancellation of the errors has led t o the good results for oexe. For most of the molecules, expression 10 is seen to be near the average value. Four molecules, vi2., H2+, HC1+, 02+and Clz give specially high values. Three of these are ionized molecules. The case of C12 is interesting. Varshni3 (see Fig. 6 and Table XII) and Varshni and Shukla5also found that the reported experimental value of wexe for Cln deviates widely from the calculated values by other methods. The experimental value of wexe for Clz has been obtained from levels observed only up to 5’’ = 3. The uncertainty in the determination of wexe from the observed data has been discussed by Varshni and Shukla.b It appears that the reported value wexe = 4 is not sat,isfactory. A value of wexe N 3 would be in accord with the considerations of references 3 and 5 and the present paper. Acknowledgments.-The authors are thmkful to Prof. D. S. Kothari (Delhi) for his kind interest in the work, and to the Council of Scientific and Industrial Research for the financial assistance.
(4) G. IIprzberg, “Spectra of Diatomic Molecules,” D. Van Nostrand Co., Inc., Kew York, N. Y., 1951.
( 5 ) Y. P. Varshni and R. C. Shukla, Trans. Faradag Soc., 67, 537 (1981).
(cueoe/6Be24-1) ( E ,
- Rij)
-L/3K
(9)
and
PHASE EQUILIBRIA ITC’ FUSED SALT SYSTEIMS: BINARY SYSTEMS OF PLUTONIUiU(II1) CHLORIDE WITH THE CHLORIDES OF MAGNESIUM, CALCIUhf, STRONTIUILI: AKD BARIUM’ B Y I
