850
Harry Donald Brooke Jenkins
G.V. Buxton, F. S. Dainton, T. E. Lantz, and F. P. Sargent, Trans. Faraday Soc., 66, 2962 (1970). D. C. Walker and R. May, Int. J. Rad&. Phys. Chem., 6, 345 (1974). S. A. Rice and M. d. Pilllng, Chem. Rev., submitted for publication.
(16) R. d. Woods, B. Leslgne, L. Gllles, C. Ferradini, and J. Purcheault, J . Phys. Chem., 79, 2700 (1975). (17) G. Bomd.lil, P. Delahay, and I. Levin, J. Chem. phys., 56,5194 (1972). (18) T. Huang and W. H. Hamill, J . Phys. Chem., 78, 2081 (1974).
Comparative Study of the Repulsion Energy in Potassium Hexachloroplatinate and Evaluation of the Total Lattice Energy of the Salt Harry Donald Brooke Jenklns Department of Molecular Sciences, University of Warwick, Coventry CV4 7AL, Warwickshire, United Kingdom (Received March 23, 1976: Revised Manuscript Received December 3, 1976)
This paper considers the literature calculations of the lattice energy of potassium hexachloroplatinate. It discusses the calculation of the repulsion energy in the light of the author's newly proposed equation and suggests that existing calculations based on the Born-Lande equation and supposedly taking into account the distributed charge on the PtCb2-ion give values for the repulsion energy which are approximatelydouble the true magnitude. This contention is further supported by a calculation based on an exaggerated repulsion situation in the complex ion. The total lattice energy of K2PtC16is found to be 1468 kJ mol-' corresponding to the experimentally and theoretically determined charge distribution in the complex ion. 1. Introduction
The literature contains calculations of the lattice potential energy of potassium hexachloroplatinate, UpoT(KZPtCb),by three independent workers. Hartley' in 1972 used both the Born-Mayer2 and the Kapustinskii3 equations and obtained values in fairly close agreement (differing by 4%)and averaging to 1569 kJ mol-'. Lister, Nyburg, and Poyntz? 2 years later, obtained substantially lower values from empirical treatment and by use of the Born-Lande equation5 although their value derived from the Kapustinskii equation agreed well with Hartley's value (within 2%). Quite recently De Jonge6 has used the extended Born-Lande equation5 for the case where the PtC16'- ion has a distributed charge corresponding to a charge (= qcl) equal to -0.44 on each of the chlorine atoms of the complex ion. De Jonge's calculation considers a similar charge distribution to that considered by Lister, Nyburg, and Poyntz and yet their values for Upo~(KzPtCls) differ by 50 kJ mol-'. All the results referred to above are collected in Table I. The values cited for UPoT(KSPt&) span a range of some 325 kJ mol-' and average to 1452 kJ mol-' with a standard deviation of 119 kJ mol-'. The results of De Jonge and Lister et al. which correspond to the experimental7 (qcl = -0.44) and theoretical' (gel = -0.445) charge distribution assigned to the complex ion in the salt span a range of 54 kJ mol-' and average to 1308 kJ mol-'. The present state of the art of lattice energy calculations is such that one should be able to fix the lattice energy more precisely than this, and it is the claim of the present critique to do this. In order to investigate the essential differences which exists between the literature calculations we break down, in Table I, the lattice energy into its component terms, where the equations used and the information given permits. If we consider the repulsion term, UR,this is given in the case of the Born-Mayer equation by
where p is the repulsion parameter (taken to be 0.345 8, in the studies described here). In the case of the KaThe Journal of Physical Chemistry, Voi. 81, No. 9 , 1977
pustinskii equation,
U R takes
the form
1201.6~Iz+llz-1 p
UR
=
r0
(6)
(2)
where u is the number of ions in the stochiometricformula, lZ+lis the cationic charge, 12-1is the anionic charge, and ro is the cation-anion separation. In the case of the Born-Lande equation
(3) and for the extended Born-Lande equation, UR,can be written
(4) In the literature calculations the dipoledipole dispersion term, Ud,.j, only is calculated or else dispersion energies are ignored. In Table I, the electrostatic component of the lattice energy varies considerably (= U E L E C ) but it is possible to accurately calculate the magnitude, as is done here as a function of the distributed charge on the PtCb'ion. The two calculations which consider u d d agree fairly well regarding its magnitude?V6 The most striking feature in Table I concerns the apparent doubt about the magnitude of the repulsion energy in the salt. Empirical based studies based on the Kapustinskii equation suggest that URis about 140 kJ mol-', as does the estimate made by Hartley' from the BornMayer equation. On the other hand, the Born-Lande equation estimates that UR is some 48% higher at 207 kJ mol-' when the eq 3 is used. The extended Born-Lande equation suggests UR is 240 kJ mol-' some 64% higher than the empirical value. De Jonge's calculation (by virtue of the use of a lower value for n) requires UR to be 256 kJ mol-'. The point that emerges is that there are serious differences of opinion as to the value of UR in K'PtCI, and it is the purpose of this present paper to try and resolve the question. The problem is fundamental since the hexachloroplatinate structure is widely exhibited by
85 1
Repulsion Energy in K2PtC16
TABLE I: Lattice Energies of Potassium Hexachloroplatinate (kJ mol-' ) Ref 4
Ref 1 BornMayer eqa
Kapustinskii eqb
EmpiricalC
Extended BornLande eqd
transition metal salts of the type we are currently studying and the calculation of the lattice energies to a reasonable degree of accuracy is the key to the estimation of thermochemical data of octahedral ions of the PtCb2- type for which we have a systematic study in progress.
Ref 6 Simple BornLande eqe
Kapustinskii eq f
ud, and Uqd are the dipoledipole and quadrupole-dipole dispersion terms. To evaluate URusing eq 5 we calculate the Madelung (UM)energy corresponding to the process
K2PtC16(c) 2K+(g) + Pt*pt(g) +
2. Calculations
We consider first the question of the magnitude of UR, the repulsion energy in K2PtC&,and then go on to consider the value of Upo~(K2PtC16)for the salt. Jenkinsg has recently derived an equation for the repulsion energy of salts containing complex ions such as K,PtC&, which should provide insight into the magnitude of UR. The equation takes the form
Alter- Extended native BornKapuLande stinskiig eqh
+ 6Cl*Cl(g)
where QPt
+ 6qa = - 2
(8)
and find
UM = 1657.3 + 7522.3qcl
+ 15336.7qcl2
(9)
where the associated Madelung constant (based on a0 = 9.755 takes the form
(5) and is based on the Jenkins and Waddingtonlo equation
M
=
11.636579 + 52.818308~c1 107.687562qCl2
+
(10)
The self-energy, USE, corresponding to the process Pt*Pt(g) + 6Cl*Cl(g) + PtC1, '-(g)
takes the form U S E = -7120.9qcl-
+ uqJ1L
15437*5qc12
(11)
and the total electrostatic energy, UELEC and the corresponding Madelung constant MELEC (based on ao),which refer to the process
I
K,PtCl,(c)
"02
which gives the total lattice potential energy of a salt possessing a complex ion having an extended charge distribution. This equation takes account of nearest neighbor repulsions using the repulsion expression of the form A exp(-alp) and was derived by direct minimization with respect to an external cell parameter, a, while keeping the internal distance within the complex ion, d (= Pt-C1 distance) constant. K is the conversion factor to convert e2 k' to units of kJ mol-' (= 1389.30). a. is the cell parameter at the energy minimum. a, and P,, are the multipole expansion coefficients when the electrostatic Madelung constant is written in the form
(7)
+ 2K+(g)
+ PtCl,'-(g)
are given by
UELEC = UM + USE VEL,, = 1657.3 + 401.4qC1- 10O.7qcl2 MELEC= 11.636579 + 2.818304qc1 -
o.707138qcl2
(12) (13) (14)
The above equations can be checked in several ways. First when qc1 = 0, M = 11.63657 which is the value for the fluorite, CaF2,1atti~e.l~ The charge dependence of M can be established from the work of Simon and ZelleP or from some recently published work carried out by Herzig, Neckel, and Jenkins." The dipoledipole contribution to the dispersion energy can be obtained by adaptation of the London'' formula for this salt. u d d is given by the equation The Journal of Physicel Chemistty, Vol. 81, No. 9 , 1977
852
Harry Donald Brooke Jenkins
and listed values for the coefficients for the KzPtC16 structure. Differentiating eq 17 we have P t atonis
i= 1 C1 atoms i= 1
-.I
and for K2PtC16,d = 0 . 2 4 and ~ ~ thus the only nonzero terms which occur in eq 18 correspond to m = 1 in the summations, whence
(15) where t ~ +ECI-, , and t~ are the characteristic energies of the ions, taken as 90% of the second ionization potential in the case of Kt (= 45.85 erg molecule-'), estimated in the case of Pt4+from the data on Rb SnC127 (= 95.21 erg molecule-') and taken from Mayer 1% in the case of C1- (= 15.7 erg molecule-'). The polarizabilities, LYK+and 1yc1-,of the ions K+ and Cl- were taken from Tessman, Kahn, and Shockleylg (aK+= 0.84 A3, ac1- = 3.29 A3). apt was estimated from the empirical relationship (aR4+/asn4+) = (rR4+/rSn4+)3 (16) taking rsn4+= 0.71 A,rpt4+ = 0.65 and asn4+= 0.5 A3l6 and finding apt4+to be 0.38 A3. The value of Udd so obtained was 119 kJ mol-'. uqd =
1 2 1
-d++SB*
=-d++ [ 2 2
1 + -d--S,-+ d+-Ss+2
K+ ions
x
(17)
(RK+K+)-~]
Using the coefficients recently e~tablished,'~ B1 = 39.9451 and C1 = 22.92206, we find
Combining the results together into eq 5 we obtain
(L )
UR = - [2478.5 + 1766.7qc1- 884.2qc12] (23) For the charge distribution in PtC12- corresponding to qc1 = -0.44 the equation for URtakes the form
UR = 1530.0(p/ao)
(24) Thus if we assume p = 0.345 A and that a. correspondsg to the anion-cation distance, K-Pt (= 4.224 A), we find p/ao = 0.08 and for the above charge distribution UR = 125 kJ mol-' from eq 24, while from eq 13 UELEC = 1461 kJ mol-'. These values combined with the total dispersion energy (= 132 kJ mol-') give the total lattice potential energy of &PtC&
i= 1
+ d+-[2 K +iions (Rc~-K~+)-~] =l The quadrupole dipole term may be estimated using eq 17 of Mayer;"estimating the coefficients d++, d--, and d,to be similar to those for KC1. Consequently d,, = 24 X erg cm6,d-- = 250 X lo7' erg cm6,and d,- = 73 X erg cm6. The inverse eighth power summations were carried out to exhaustion and u q d was found to be 13 kJ mol-' whereupon the total dispersion energy amounts to some 132 kJ mol-', this value being in close agreement with the estimates of Lister, Nyburg, and Poyntz* and De Jonge.' In the above equations RClCl*, (= 2d) refers to refers to the internal C1-C1 distance in the ion and RclcL,B the "through space" distance between adjacent C1 atoms. We now proceed to evaluate the final term in eq 5, the derivative of the electrostatic Madelung constant. In our recent work15we have expanded the electrostatic Madelung constant (based on ao) in the form
The Journal of phvsicsl Chemistry, Vol. 81, No. g7 1977
Upo~(K~Ptc16) = 1468 k J mol-' (25) We can also express the total lattice potential energy as a function of the charge qcl on the terminal chlorine atoms of the ion, by combining eq 13 and 24 with the dispersion energy, using p/ao = 0.08 we find Uw~(K2PtC16)= 1591.0 + 2 6 0 . l q a - 30.Oqc1' (26)
3. Discussion We consider the results obtained in this paper along with the other literature values under four headings. (a) The Electrostatic Lattice Energy, UELEC.In the present work the charge dependence of the electrostatic component of the lattice energy is given by eq 13 and the value of UELEC at the particular value of q 1 of interest (-0.447p8)is 1461 kJ mol-'. De Jonge's values agrees with this and the checks we have made on the general e 13 confirms it to be correct. Lister, Nyburg, and Poyntz9 on the other hand obtain a value of qcl = -0.44, some 76 kJ mol-' lower in energy, and a further worrying feature about the other value they quote (for qcl = -0.445) is that it is higher than the value quoted for qcl= -0.44 by 55 kJ mol-' whereas eq 13 demands that the value of U r n for a higher negative charge on the chlorine atoms should produce a lowering of U E L E C by about 3 kJ mol-' for qcl = -0.445! The Born-Mayer equation applied to the true value of UELEC predicts UR to be (~/CZO)UELEC and equal to 119 kJ mol-'. The extended Born-Mayer including dispersion terms (Ladd and Lee2' equation) in the form
s
Repulsion Energy in K,PtCls
853
(27) from which the value of
UR is
distance in the ion. On the basis of these parameters we find the total repulsion between the six chloride ions and the two K+ ions amounts to 236 kJ mol-’, the major repulsion arising from K+-Cl- repulsions. Consequently it is reasonable to conclude that the repulsion energy in the true salt is a good deal less than this value. Since the UR values of both the above-mentioned workers actually lie above this value they have, we conclude, somewhat overestimated U R by their choices of n. ( c ) The Dispersion Energy, UD.All three estimates of the dispersion energy in K2PtC16from the three laboratories concerned agree.
and using u d d = 119 kJ mol-’ and u q d = 13 kJ mol-’ assigns UR = 182 kJ mol-l. Since the equation proposed in this paper (eq 5) as being suitable for the calculation 4. Conclusion of UR, includes the extra term (P/u~)K(~MELEC/~U)~,,=,~ We conclude by restating our contention that literature (arising from the complex ion having an extended charge values for the repulsion energy in K2PtC16in the region distribution), which evaluates to 61 kJ mol-’, we arrive at of 240 kJ mol-’ seem to be far too high. The true lattice the prediction that UR should be about 120 kJ mol-’. energy of K2PtC16(corresponding to Q C ~= -0.44) has a (b) The Repulsion Energy, UR. This paper is really value 1468 kJ mol-’ arising from an electrostatic ener making the statement that the repulsion calculations of of some 1461 kJ mol-l, a dispersion energy of 132 kJ mol , Lister, Nyburg, and P0yntz4 and De Jonge‘ are incorrect. and a repulsion energy of 125 kJ mol-’. An improved While their values suggest URfor K2PtC16having qcl = method for the calculation of the lattice energies of these -0.44 should be 238 and 258 kJ mol-’ our present work salts is being developed23and is it hoped to confirm the suggest roughly half this value. As a final justification of present studies. the unreasonable magnitude of their values, we calculate the repulsion energy caused by placing six chloride ions References and Notes each bearing a charge of -1 at the chlorine atom sites in (1) F. R. Hartiey, Nature(London), Phys. Sci., 236, 75 (1972). the crystal and calculate the resulting U,. This situation, (2) M. Born and J. E. Mayer, 2. Phys., 75, 1 (1932). representing as it does a totally unrealistic situation as far (3) A. F. Kapustinskii, 0.Rev. Chem. SOC., 10, 283 (1956). (4) M. W. Uster, S.C. Nyburg, and R. B. Poyntz, J. Chem. Soc., Faraday as K2PtC16is concerned, is treated using the Huggins and Trans. 1 , 70, 685 (1974). Mayer equation,22which takes the form (5) M. Born and A. Lande, Verh. Dtsch. Phys. Ges., 20, 210 (1918).
v
where ijp and i‘cl-are the Huggins “basic” radii for the ions concerned, b is a constant (= lo1’ erg molecule-’), c++ = 1.25, c-- = 0.75, and c+- = 1.0, the appropriate Huggins coefficients for the presence of six C1- ions at the C1 sites in PtCl?-. ‘r+= 1.185 8, and Pel- = 1.475 A2’ and the summations indicated were carried out to exhaustion. R c ~ =c4.6824 ~ ~ 8,~ and ~ is the longest C1-C1 distance in = 3.3109 8, and is the shortest C1-C1 PtC& and RclcLi,,,
(6) R. M. De Jonge, J . Inorg. Nucl. Chem., 36, 1821 (1976). (7) M. Kubo and T. Nakamura, Adv. Inorg. Chem. Radiochem., 8, 256 (1966). (8) F. A. Cotton and C. B. Harris, Inorg. Chem., 6, 376 (1967). (9) H. D. B. Jenkins, J. Chem. Soc., faraday Trans. 2,72, 1569 (1976). (10) H. D. B. Jenkins and T. C. Waddington, Chem. Phys. Lett., 31, 369 (1975). (1 1) R. W. G. Wyckoff, “Crystal Structure”, 2nd ed,Interscience, New York, N.Y., 1965. (12) R. H. Busch, E. E. Gaikmi, and C. R. Hark, Ann. Assoc. a i m Argent., 39, 55 (1951). (13) T. C. Waddington, Adv. Inorg. Chem. Radbchem., 1, 158 (1959). (14) G. Simon and C. R. Zeller, J. Phys. Chem. Solids, 35, 187 (1974). (15) P. Herzig, H. D. B. Jenkins, and A. Neckei, J. Chem. Phys., 80, 1608 (1976). (16) F. London, Z. Phys. Chem. (Leipig),B11, 222 (1930). (17) H. D. B. Jenkins and B. T. Smith, J . Chem. SOC.,Faraday Trans. I , 72, 353 (1975). (18) J. E. Mayer, J . Chem. Phys., 1, 270 (1933). (19) J. R. Tessman, A. H. Kahn, and W. H. Schockley, Phys. Rev., 92, 890 (1953). (20) R. T. Sanderson, “Inorganic Chemistry”, Reinhold, New York, N.Y., 1967. (21) M. F. C. Ladd and W. H. Lee, Trans. Faraday, Soc., 54, 34 (1958). (22) M. L. Huggins and J. E. Mayer, J . Chem. Phys., 1, 643 (1933). (23) H. D. B. Jenkins and K. F. Pratt, Proc. R . SOC.(London), in press.
The Journal of physlcal Chemistry, Vol. 81, No. 9 , 1977
