LORENG . HEPLER
1426
The limits of solid solubility in InP and In were studied by examining by slow scanning the highest angle lines in the Cu K powder X-ray spectra. One would expect in this case rather large changes in the lattice constant of either phase if appreciable solid solution existed. However, it appears that the extent of solid solution in both In and InP is below the limit which can be determined by these X-ray methods, and therefore is probably a few tenths of a per cent. or less. Experimental difficulties are acute in compositions between I n P and phosphorus, as the vapor pressure of phosphorus is 44 atm. at its melting point of 589". Silica glass tubing will not withstand
Vol. 61
these pressures except in occasional runs and most other materials that could be used as a container are subject to attack by these liquids. For these reasons no consistent results could be obtained in that part of the system in which the phosphorus concentration exceeded 557,. Heat effects at an eutectic of about 560" were observed, however, with an approximate phosphorus pressure of 50 atmospheres. Acknowledgment.-The authors are indebted to Professors R. Roy and 0. F. Tuttle for many suggestions regarding experimental methods and to RCA Laboratories, Radio Corporation of America, for financial support of this study.
PARTIAL MOLAL VOLUMES OF AQUEOUS IONS BY LORENG. HEPLER Contribution from the Cobb Chemical Laboratory, University of Virginia, Charlottesville, V u . Received May Q, io67
The way in which partial molal volumes of aqueous ions vary with ion charge z and radius r has been investigated. On the basis of a particulgr physical model, it has been deduced that the partial molal volumes of ions should be given by an equation of the form Vion = Ara Bz2/r. The coefficients A and B have been found to be 5.3 and 4.7, respectively, for cations and 4.6 and 19 for anions. The numerical magnitudes of these constants are of some physical significance and have been discussed in detail. The model on which the above equation is based is one that leads to a linear dependence of ionic entropies on z 2 / r . This is considered to constitute additional evidence in favor of the Born or Latimer and Kasper treatments of ion entropies.
-
Because of the complicated nature of the many interactions in aqueous solutions of electrolytes, attempts to deduce theoretically the way partial molal entropies of ions depend on ion charge and radius are necessarily approximate. Various approximations have led to a variety of equations for ion entropies as a function of ion charge and radius and sometimes mass. These equations have contained at least one empirically evaluated parameter. The fact that several equations, based on different treatments of the ion-solvent interaction, more or less satisfactorily represent the experimental entropies suggests that entropies (and/or free energies of hydration) are not likely to be a sufficient criterion for testing the validity of a given theory. Therefore, the present study of the way partial molal volumes of ions vary with ion charge and radius has been undertaken. There has recently been some interest in this problem for its own sake,l but this paper is chiefly concerned with the information we may obtain about the nature of solutions of ions in water and the support given certain treatments of the thermodynamics of ionsolvent interaction and ion solvation. Laidler2 recently has studied the dependence of entropies of aqueous ions on ion charge and radius. He has explained why it is necessary to consider the absolute, rather than relative, partial molal entropies of ions in order to make meaningful correlations of the way ion entropies vary.with charge. It is emphasized here that it is equally necessary, for the same reasons, to consider as best we can the (1) A. M. Couture and K. J. Laidler, Can. J . Chem., 84, 1209 (1956). (2) K.
J. Laidler, ibid., SB, 1107 (1956).
absolutea partial molal volumes of ions rather than volumes relative t o some arbitrary standard or reference volume. It follows that the procedure of Couture and Laidler' which involved empirical correlation of the dependence of relative volumes on charge and radius and subsequent adjustment of relative volumes by making cations and anions fall on the same line is open to several objections. In fact, as discussed later in this paper, cations and anions do not and should not be expectedrto fall on the same line. Laidler2 has shown that the absolute entropies (Sabs. H+(aq) = -5.5) of aqueous ions vary linearly with z 2 / r where z represents the charge on the ion and r the radius. This is the charge and radius dependence that has been predicted several times on the basis of the Born equation 1 for the electrostatic free energy of solution and the thermodynamic relation ( d A F / d T ) , = - AS.2,4 Latimer and Kasper6 also concluded from considerations of the decrease in entropy of the water due to the pressure set up in the polarizable medium by the charge on the ion that partial molal entropies of ions should vary linearly with z 2 / 7 . The following treatment of partial molal volumes of ions at infinite dilution is intended to parallel the (3) The word absolute is used here in the same senne that it is used when referring t o absolute ion entropies;j.e., partial molal volumes of individual ions not arbitrarily based on Vioo for H + equal t o zero. (4) W. M. Latimer, K. 5. Pitzer and C. M. Slansky, J . Chem. Phys., 7, 108 (1939). (5) W. M. Latimer and C. Kasper, J . Am. Chem. Soc., 61, 2293 (1929).
PARTIAL MOLALVOLUMESOF AQUEOUSIONS
Oct., 1957
treatment of partial molal entropies of ions as carried out by Laidle? and Latimer and Kasper.6 We consider ions to be contained in spherical cavities in the water and take the volumes of these cavities t o be proportional to the cubes of the crystal radii of the ions. The volumes of these cavities are a positive contribution to the partial molal volumes of the ions, The volume change due to compression (electrostriction) of water about an ion may be obtained from equation 2 where p represents the compressidV = DdP
(2)
bility of the solvent and P the internal pressure due t o the ion charge. The total volume decrease about an ion is obtained by integrating (2) with respect to the pressure and then integrating over the total volume of water outside the ion cavity as indicated by (3). AV =
d P 4rr2 dr
(3)
Following Latimer and Kasper6 and Zwicky6z7 we take the pressure set up by the ion charge t o be proportional to z2/r4 and also use an average constant compressibility. The result of the integration of (3) is that the volume decrease due to compression of the solvent by the internal pressure set up by the ion charge is proportional to z2/r. From the Born equation 1 and the thermodynamic relation (baF/bP)~ = AV, we deduce that the volume decrease due t o compression of the dielectric medium is also proportional to z 2 / r . On the basis of the above, the total volume decrease due to ion-solvent interaction is taken as proportional to z z / r and the volume of the ion cavity as proportional to r3. We therefore write down equation 4 for the partial molal volumes of ions a t
1427
I are based on this value for the volume of C1and partial molal volumes of aqueous solutes as given by Fajans and J o h n ~ o n , Harned ~ and Owenlo and Couture and Laidler.’ For highly charged ions and ions known to be associated with halide ions in solution, the partial molal volumes have been based on reported volumes of the nitrates whenever possible. The crystal radii in Table I are those given by Goldschmidt as tabulated by Couture and Laidlerl and Pau1ing.l’ It might be mentioned that Pauling’s own radii lead to a very satisfactory correlation except for Li+ and Al+3. TABLE I
Li + Na +
K+ Rb +
cs
+
Ag T1+ Mg++ Ca++ +
Sr++ Ba++ Cd++ Pb++
0.78 0.98 1.33 1.49 1.65 1.13 1.49 0.78 1.06 1.27 1.43 1.03 1.32
-
0.9 1.4 8.8 13.8 21.2 0.9 14.8 -20.7 -17.5 -18.0 -12.1 -15.0 -15.9
-
Zn++ Fe++ Co++ Ni++ Mn++
Al+++ La+++ Th+4 OH F-
c1-
Br-
I-
0.83 .83 .82 -78 .80 .83 1.22 1.10 1.40 1.33 1.81 1.96 2.20
-21.9 -21.2 -18.5 -24.4 -18.1 -44.6 -38.0 -54.2 5.4 2.2 18.0 24.9 36.5
-
Equation 4 may be rearranged to give equations 5 and 6 that are suitable for testing graphically the pzedicted dependence of Tion on x and r. Plots of Vion
r/x2 VS. r4/z2 riionr/za= Ar4/z2 - B Pi,,/rJ = A - Bz2/r4
(5) (6)
infinite dilution (Pion)as a function of x and r. The first term on the right side of equation 4 represents the positive contribution to the partial molal volume due to the space occupied by the ion. Stokes and Robinsons recently have considered this part of the partial molal volume in an interesting way, Their treatment is discussed later in this paper. The second term on the right side of equation 4 represents the decrease in volume due to compression of the solvent about the ions (electrostriction). It is important that both of these terms be included in equation 4 because it is the net sum of the effects these terms represent that is the observed partial molal volume. The physical significance of the proportionality constants A and B is of interest and is also discussed later in this paper, particularly with respect to differences between anions and cations. Fajans and Johnsonghave investigated the partial molal volumes of ions in aqueous solution and found evidence that the partial molal volume at infinite dilution of chloride ions is about 18.0 cc./mole. The partial molal volumes of ions given in Table
and Vion/ra vs. z2/r4 are shown in Figs. 1 and 2, respectively. The plot shown in Fig. 1 has all the data for multiply charged ions compressed into a small area near the Vion r/z2 intercept but shows clearly the difference between singly charged cations and anions. Figure 2 (only cations are plotted here) shows the best over-all confirmation of equation 4 but the data for the singly charged ioss are so compressed into a small area near the Vion/r3 intercept that it is difficult to see the important differences between cations and anions on this plot. It is unfortunate that suitable data are not available for multiply charged monatomic anions. From the slopes and intercepts of the lines in the two plots the constants A and B have been evaluated as 5.3 and 4.7, respectively, for cations and 4.6 and 19 for anions. It is of interest to consider carefully the magnitude of the constant A for both cations and anions. The simplest approach, namely, that the volume of an ion cavity equals 4nr3/3 where r represents the crystal radius, leads t o A = 2 . W cc./mole with r expressed in iingstroms. The values for A actually found are about twice as large because of the following factors: (1) true ion radii in solution are larger
( 6 ) F. Zwicky, Physik. Z., 26, 664 (1926). (7) F. Zwicky, Proc. Nut. Acud., 12, 86 (1926). (8) R. H. Stokes and R . A. Robinson, Trans. Faradnv Soc., 63, 301 (1987). (9) K. Fajans and 0. Johnson, J . Am. Chsm. Soc., 64, GG8 (1942).
(IO) H. S. Harned and B. B. Owen, “Physical Chemistry of Electrolytic Solutions,” 2nd Ed.. Reinhold Publ. Corp., New York, N. Y., 1950. ( 1 1 ) L. Pauling, “The Nature of the Chemical Bond,” Cornell University Press, Ithaca, N. Y., 1940.
Vion =
Ar8 - Bz2/r
(4)
LORENG. HEPLER
1428
I
l
l
4
l
l
8
l
l
12
l
l
16
l
l
l
l
20
r4/22.
Fig. 1.
1
z2/r4.
Fig. 2.
than crystal radii because of differences in distortion of the ions by different environments, and (2) there is an additional free volume associated with ions in solution.1*2J2 I n this connection it is interesting t o consider the conclusions of Stokes and Robinson.* From considerations of the way spheres pack together they concluded that the contribution t o the ion partial molal volume corresponding t o the first term on the right side of equation 4 should be 4.35r3. It is gratifying to find this value in agreement with A = 4.6 as found above for anions. Stokes and Robinson8 further assumed that the larger anions cause no significant electrostriction cf surrounding solvent. This assumption leads to Vion for I- equal t o 43.8 cc./mole as com(12) H. 5. Frank and M. W. Evans, J . Chem. Phya., 13,507 (1945).
Vol. 61
pared to the smaller value (36.5) on which the present treatment, allowing for the electrostriction of water about all ions, is based. It may be seen from the second term on the right side of equation 4 that the volume decrease due to ion-solvent interaction is inversely proportional t o ion radius but is by no means negligible even for iodide ions. It already has been pointed out several times that radii of ion cavities are larger than corresponding crystal radii and also that the differences between cavity radii and crystal radii are greater for cations than a n i ~ n s . ~ ~ - This l s is in excellent accord with the present observation that A is larger for cations than anions and is one reason why ?ne should not expect plots of various functions of Vi,*, x and T to give only one line for both cations and anions. The fact that B is larger for anions than cations indicates that the ion-solvent interaction is greater for anions than for cations of the same size and charge. This idea is supported by a comparison of the energies and entropies of hydration of anions and cations. The energy and entropy of hydration of F- and K+ (ions of about the same size) have been given, respectively, as - 123 kcal./mole, -24.7 e.u., -76 kcal./mole and - 11.5e.u. Powell and Latimer14 have suggested that ionsolvent interaction might best be considered as the interaction of a point charge with a number of oriented dipoles a t some fixed distance. This model is an appealing one, partly because it recognizes the molecular nature of the solvent rather than treating it as a continuous dielectric medium. Several unsuccessful attempts have been made t o correlate partial molal volumes of ions with z and T on the basis of the Powell and Latimer14 equation for the free energy of interaction and the thermo= AT'. An explanadynamic relation (baF/bP)~ tion of the failure of this charge-oriented dipole interaction model t o account satisfactorily for the partial molal volumes of ions may be given in terms of saturation effects that follows the reasoning of Frank and EvansI2 as applied to considerations of the entropies of ions. According t o this reasoning, the field near the "surface" of a large and singly charged ion, such as I-, is so very strong that multiplying it by about 50, which is what happens in going to a small highly charged ion such as A1+*,does not appreciably increase the effect of the ion-solvent interaction for those water molecules immediately surrounding the ion. We conclude that the ion-solvent interaction for the water molecules very close t o the ion, for which the charge-oriented dipole model is most applicable, is nearly independent of ion size and charge. It might also be expected that some combination of the Born and Latimer and Kasper treatments would be reasonably realistic for the interaction of the ion and the more distant solvent. Acknowledgment.-It is a pleasure to acknowledge support of t,his work by the National Science Foundation. (13)
W. M. Latimer, ibid., 33, 90 (1955).
(14) R. E. Powell and W. M. L a t h e r , ibid., 19, 1139 (1951). (15) R. E. Powell, THISJOURNAL, 68, 528 (1954).
I
c
1
HEATOF FORMATION OF AMMONIUM DICHROMATE
Oct., 1957
1429
THE HEAT OF FORMATION OF AMMONIUM DICHROMATE BY CONSTANTINE A. NEUGEBAUER AND JOHN L. MARGRAVE Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received June $4, 1967
The heat of formation of (NHa)&r207was determined calorimetrically as -429.1 i 0.3 kcal./mole through the reaction (NH&Cr2O7(s) = CrZOa(s) Nz(g) + 4Hz0(1). From solution data on chromates and dichromates which can be related to this heat, one deduces a heat of formation for CrO,(s) of -142.1 i 1 kcal./mole.
+
Literature values for the heat of formation of ammonium dichromate are in poor agreement. The National Bureau of Standards Circular 500 lists -425.5 kcal./mole as the standard heat of formation based on the work of Berthelot' and Moles and Gonzales. Kapustinskii and Shidlovskiia have reported the heat of formation as -430 =I= 6 kcal./mole. Since the intramolecular combustion is a simple reaction with well-defined products, a very precise determination should be possible.
traces of unreacted material were found. The solution produced in the reaction was titrated for ammonia with 0.01 N acid to the phenolphthalein end-point. The product gases were analyzed mass spectrometrically. The nitrogen produced in the reaction was assumed to behave like oxygen for the purpose of the non-ideality corrections. Thus, ( ~ ~ E / Wwas ) T taken as -1.574 ca1.jatm.l mole. This is an entirely valid approximattion since the total correction from this source in calories is only of the order of 0.5 cal. All corrections for the heat capacity of the contents before and after the reaction were made according to the method outlined by Hubbard, Scott and Waddington.7
Experimental
Results and Calculations
The apparatus used exclusively in this investigation is essentially that described in various previous publi~ations.~--B The results of four determinations of the heat of A rotating bomb calorimeter was used, but no use was made reaction 1 are given in Table I. of the rotating mechanism, since, In these experiments TABLE I large quantities of solution were not produced. R243 R244 R 241 R242 The reaction whose heat was measured was identical to that of previous investigators, and can be represented as Moles of dichromate, n 0.067749 0.066935 0.068049 0.0072740
+
+
Ndgl 4HzOCI) (1) (NH&CrzO-i(s) = CrzOds) The reaction did not proceed entirely as written and was accompanied by the side reaction 1/2 Nz(g) 3/2 HzOW = NHdaq) 3/4 02(g) AEO = +82.9 kcal./mole Tt should be noted that the mechanism of the production of ammonia and oxygen is probably entirely different, but for the purpose of thermodynamic calculations this reaction is stoichiometrically correct. The procedure in making the thermal measurements and the analysis for the extent of the reaction will be described. Sixteen to eighteen grams of the ammonium dichromate was weighed accurately into a clay crucible. The dichromate waa Merck reagent quality and was free frommaterials which would interfere in the self oxidation-reduction. The rrucible was put in place in the loop electrode of the conventional double valve Parr bomb and a short coil of nichrome knition wire was just placed under the surface of the dichromate. One ml. of water was then put into the bomb, which was closed and flushed and filled with high purity helium to a pressure of 10 atm. to prevent a leak in the sealing mechanism of the bomb. The bomb was now placed in the calorimeter as described in previous publications. The reaction was initiated by pnqsing a current of 3 amp. at 25 v. through the wire for about 1 sec. After the reaction the solid residue was analyzed for residual dichromate by precipitation with barium; only
+
+
(1) M . Berthelot, Ann, chim. phys., 1, 92 (1884). (2) E. Moles and F. Gonzales, Anales $ 8 . y quim. (Madrid),21, 206 (1923). (3) A. F. Kapustinskii and A. A. Shidlovskii, Izseat. Sektora PEatiny i Drug. Blaoorod. Metal., Ins.?. Obshchei i Naoro. Khim., Akad. Nauk., S.S.S.R., S O , 31 (1955); see also C.A , , SO, 9849c (1956). (4) C. A. Neugebauer and J. L. Margrave, THIS JOURNAL, 60, 1318 (1956). .+4 (5)'C. A. Neugebauer and J. L. Margrave, J . Am. Chsm. SOC.,19, 1338 (1957). (6) C. A. Neugebauer, Ph.D. Thesis, Univ. of Wis., 1967.
Moles NHs AT9 " C . AEb, oal. C(cont.) AT, oal. AE cor. for "3, cal. AEipnroal. AEnon id,, cal. nAE8 cul. AE2 kcal./mole
traces 1.0577 7771.49 6.82 0 4-14.4 +0.51 7753.8 114.45
0.000410 0,000204 traces 1.0604 1.1298 1.0405 7791.64 8311.40 7655.35 8.95 7.14 6.59 0 34.1 16.9 +18.0 C13.45 +17.9 +0.51 +0.51 f0.51 7780.4 8338.7 7680.4 114.33 114.64 114.99
Here A E b stands for the uncorrected heat evolved is the standard heat of in the calorimeter, and AEOo reaction at constant volume at 25". Run 242 was discarded since an unusually large amount of oxygen was evolved here, a d not enough ammonia was found t o account for it. The average of the remaining runs is AEoo = -114.47 f 0.09 kcal./ mole, where the error quoted is the standard deviation from the mean. Applying the AnRT cor0.09 kcal./mole. rection, AHco = -113.87 Then, with the heat of formatiop of Crz03equal to -269.7 kcal./mole, one calculates the standard heat of formation of ammonium dichromate at 25' to be -429.1 -I: 0.3 kcal./mole. This is to be contrasted with a value of -425.0 kcal./mole recently deduced by Muldrow and Hepler* from heat of solution 'data on dichromates and the heat of formation of Cr03(s). Since the only heat'with any large uncertainty used in their calculations was that for Cr03(s), it appears that the heat of formation of CrOa is"probab1y -142.1 + 1 kcal./mole, i.e., about 4 kcal./mole more stable than previously believed.
*
(7) W. N . Hubbard, D. W. Scott and G. Waddington, THE JOUR152 (1954). .~ (8) C. N. Muldrow, Jr., and L. G . Hepler, J . Am. Chsm. SOC..79, 4045 (1957). NAL, 68.
