Dec., 1961
BISULFATE ACID CONSTAXT AS COMPUTED FROM SOLUBILITY DATA
-
which are approximately proportional to (A A'), made use of this property. The appreciable manifestation of the absorption spectrum in the reflectivity ratio, illustrated in Fig. 4, is impressive testimony of the efficiency of interaction of a given dye molecule with radiation incident upon its geometric area. The reflectivity ratio for the component of radiation parallel to the plane of incidence is, as expected, much more sensitive to the film than the ratio for the perpendicular component. The strikingly faithful reproduction
2247
of the detail of the absorption spectrum by the reflectivity ratio indicates that a direct photometric inference of electronic absorption spectra of molecular films on metals should prove entirely practicable. Acknowledgment.-We wish to thank Mr. B. L. Carroll for assistance in the spectrometer measurements and numerical computations. We also are greatly indebted to the National Institutes of Health for making available the polarimetric apparatus used in this research.
THE BISULFATE ACID CONSTANT FROM 25 TO 8%' AS COMPUTED FROM SOLUBILITY DATA1 BY M. H. LIETZKE, R. W. STOUGHTON AND Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tenn.
T. F. YOUNG Department of Chemistry, University of Chicago, Chicago, Illinois Received August 4, 1981
The bisulfate acid constant has been computed from 25 to 225" from data on the solubility of Ag2SOdin H2S04solutions. In the small temperature range (25-55') where a comparison can be made the results agree very well with those of other investigations. In addition the thermodynamic constants for the reaction HSOd- = H + SOd-are presented.
+
I n a previous series of papers a study of the solubility of AgzS04 in a variety of electrolyte media has been described.2 I n this work it was shown that the concentration dependence of the logarithms of the equilibrium quotients and solubility products could be expressed by single-parameter expressions of the type S d / ( l A d I ) , where S is the appropriate Debye-Huckel limiting slope, I is the ionic strength of the solution, and A is an adjustable parameter. These expressions were shown to hold for ionic strengths as high as 4.0 and from 25 to 275". In the present paper the data on the solubility of Ag2S04 in H2S04solutions are used to compute values of the bisulfate dissociation constant K Z
+
HSO*- = H +
+ Soh'
= 45%
m+s
= 2m
-y+
x
or s+y-x-m=O
(3)
Equations 1, 2 and 3 represent three equations in the three unknowns (s, 2 and y) which may be solved for any particular values of S and &2. In accordance with previous calculations2 it was assumed that
+4s~
e
In QZ = In Kz
- -
(4)
and
Kz
from 25 to 225'. Method of Calculation.-In carrying out the calculations i t was assumed that only the species Ag+, E+, so4-and HS04- existed in a solution of is the molal &,SO4 dissolved in H2SO4. If solubility of AgzS04in HzS04of molality m, and and Y are taken as the so4-and H + concentrations, then the molality solubility product of the Ag2S04 is given by
s
The equation for the conservation of total sulfate is
(1)
By conservation of acid hydrogen, the HS04- concentration is seen to be equal to 2m less the H + concentration y, and the bisulfate dissociation quotient becomes (1) This paper is based upon work performed for the United States Atomic Energy Commiesion a t the Oak Ridge National Laboratory operated by Union Carbide Corporation. (2) M. H. Lieteke and R. W. Stoughton. J . Phys. Chem., 63, 1183, 1186, 1188, 1180, 1984 (1959); 64, 816 (1960).
In
where Kz is the bisulfate acid constant, ST is the Debye-Hiickel limiting slope a t temperature T for a singly charged ion, sois the solubility of AgzS04 in water a t temperature T , P and A are adjustable parameters and I is the ionic strength of the sohtion, given by I=m+s+2x
(6)
Thus, the over-all problem involves the evaluation of In Kz,A and P by a non-linear least squares Procedure, subject to the restrictions represented by equations 1 , 2 , 3 and 6. The criterion adopted in solving the above set of equations was that x(&,bsd. - ~ , ~ lbe~ a &mini~ i
where the summation is taken Over the different solubilities (at different values of m) a t any tem-
2248
1 % .
H. LIETZKE,R. W. STOUGHTON .4ND T . F. YOUNG
perature. Accordingly a series expansion of s was made in terms of the partial derivatives with respect to the three adjustable parameters Kz, P and A as Sobd.
=
Scalcd.
Vol. 65
derivatives bs/b In K 2 and bs/dA were computed for each solubility by incrementation of the current estimate of In K 2 and A by 1% and the assumption that
as A ~ K Z + 3s - A P + G 3.9 AA + m2 3P
(7)
in which and the partial derivatives are evaluated for approximate values of In K z , P and A. The increments A In K2, AP and AA then give approximate corrections to these parameters. Since, however, the solubility of Ag2S04has been measured as a function of temperature in only three different concentrations of H2S04it was decided to omit the evaluation of ds/dP directly. Rather, values of In K 2 and A were obtained over a selected range of values of P. Then that value of P was chosen for the final calculations which gave a most nearly temperature independent value of A (consistent with previous calculations2 in which it was shown that temperature independent values of P and A could be used to describe the system). The procedure used to carry out the computation of In K2 was as follows. (a) Quadratic analytical expressions representing the solubilities of &,So4 in each concentration of H2S04 (0.1, 0.5 and 1.0 m) us. temperature were obtained by the method of least squares. These equations were solved a t 25’ intervals from 25 to 225’ to give the solubilities. (b) At each temperature a preliminary value of I was computed (with equation 6 ) for each solubility from an estimated (first approximation) value of the sulfate concentration x and the experimental values of m and s. (In subsequent iterations the calculated values of s and z are used in computing I.) (c) The preliminary value of I was substituted into equations 4 and 5 along with initial guesses ( i e . , first approximations) of the values of In K Z and A , and the selected value of P . (d) With the values of Q 2 and S so obtained equations l , 2 and 3 were solved simultaneously by the Newton-Raphson method, which lends itself readily to computations on a high speed computer. The method involves linearizing the lion-linear equations 1 and 2 by expansion in a Taylor’s series through the first derivative to give equations 8 and 9, respectively s 4- ( S O / ~ Z O ) Z (SO S/8 SOZO) = 0 (8) YOZ
+ (zo +
+
Q2)Y
- (ZOYO
+ 2mQ2) = 0
(9)
where the subscript 0 refers to initial estimates on the values of s, 2 and y. The solutions of equations l , 2 and 3 then are obtained by solving (itetatively) the linear equations 8, 9 and 3 using standard matrix techniques until converged values of s, x and y result. While equations 8 and 9 are only approximate (because only first derivatives are used in the Taylor’s expansion), no error results, i.e., the final converged values obtained are the correct solutions for the non-linear equations 1 and 2 and the linear equation 3. (e) After the converged values of s, x and y had been obtained approximate values for the partial (3) R. Margenau and G . M. Murphy, “The Mathematics of Physics and Chemistry,” D. Van Nostrand Co., Inc., New York, N. Y . ,1943, P. 475.
(f) The values of the partial derivatives for each solubility a t a particular temperature then were used to form a matrix
where the subscripts 1, 2, denote the solubilities in the three concentrations of H2S04. The matrix X was transposed to give X T and the matrix product XTXwas formed
(g)
A vector V was computed as
As mentioned previously the vector solution, A In K 2 and AA, of the matrix equation A
In Kz
X T X (AA
)=v
constituted corrections for modifying the initial estimates of In KZand A. This procedure (steps b through g) was repeated with the new values of In K 2 and A and with the most recently calculated values of s, z, y and I until two successive values of both In K2 and A differed by less than 0.1%. Then the entire calculation (steps b through g) was repeated with a series of values of P. (It should be mentioned that steps e through g represent an application of the standard least squares technique.) (h) The computations were repeated for 25’ intervals from 25 to 225’ (the highest temperature a t which solubility data were available). Then the values of In K2 were chosen corresponding to that value of P for which the value of A was most nearly temperature independent, i.e., P = 0.72. (Actually the values of In K z and A were not particularly sensitive to the values of P. A variation in the value of P from 0.6 to 0.8 caused a maximum deviation in the value of In K Zof only 2%.) (i) The values of In K2 obtained as a function of temperature were fitted by the method of least squares to give equation 10. In Kt = - ___ 1283‘108
T
+ 12.31995 - 0.04223215T
where T is the absolute temperature.
(10)
THERMODYNAMIC PROPERTIES OF OXIDESOF KITROGEN
Dec., 1961
Results and Discussion The values of In K 2 obtained in this work were compared with those of previous investigators. The value of K2 a t 25’ as calculated from equation 10 is 0.01032, which compares very well with reported values.* The only previous work as a function of temperature was that of Young, Klotz and Singleterry6 in which K 2 was determined over the range of 5 to 55’ and an equation 11 was obtained for K 2 us. temperature to 155’ by utilization of the conductivity data of Noyes.6 l n K 2 = - T 1785‘390
+ 15.99658 - 0.0489236T
each temperature, as calculated from equation 10, and as obtained from equation 11. In Table I1 are summarized the thermodynamic constants for the reaction HSOd-
TABLE I1 THERMODYNAMIC CONSTANTS FOR TRE REACTION HSO,‘ H + SO&-
TABLE I VALUESOF Loa K z AS A FUNCTION OF TEMPERATURE -log Kz (eq. 10)
-log K, (es. 11)
I . 891 2.374 2.699 3.010 3,334 3.688 4.087 4.489 4.941
1.987 2.301 2.636 2.987 3.352 3.728 4.113 4.506 4.905
1.988 2.318 2.677 3.059 3.460 3.876
-1OK
25 50 75 100 125 150 175 200 225
+ SO*-
+
(11)
(from solubilities)
H+
from 25 to 225’ as computed by using equation 10. All calculations in this paper were carried out on an IBM-7090 computer.
In Table I are summarized values of log Kz us. temperature as computed from the solubilities at
t
2249
(4) R. A. Robinson and R . H. Stokes, “Electrolyte Solutions,” Academic Press. Inc., New York, N. Y., 1955, p. 374. (5) I. M. Klotz and C. R. Singleterry, Theses, University of Chicago, 1940; R. A. Robineon and R. H. Stokes, “Electrolyte Solutions,” Academic Press. Inc., New York, N. Y., 1955 p. 376; T. F. Young, L. F. hlaranville and H. M. Smitb, “The Structure of Electrolytia Solutions,” edited by W. J. Hamer, John Wiley and Sons, Inc., New York. N. Y., 1949, Chap. 4; T. F. Young, unpublished a-ork. (6) A. A. Noyes, “The Electrical Conductivity of Aqueous Sohtions,” The Carnegie Institution of Washington, Washington, D. C., 1907.
t
AFQ8cal.
25 50 75 100 125 150 175 200 225
2.712 3.403 4.200 5.102 6.108 7.219 8.436 9 * 757 11.183
AHO, cal.
ASQ, 8.u.
-18.280
-25.6 -29.8 -34.0 -38.2 -42.4 -46.5 -50.7 -54.9 -59.1
- 4.911 - 6.214 - 7.623 - 9.136 -10.750 - 12.480 -14.310 - 16.240
=
It is interesting to note that the entropy of dissociation of HSOI- is negative and attains a higher negative value the higher the temperature. A similar effect was found2 for the dissociation of Uoz804 into U02++and SO4--, and for the dissolution of Ag2S04. Thus it appears that the formation of SO4-- in water increased the amount of “order” or “structure” shown by the solvent at any temperature and that this effect is much greater the higher the temperature. Acknowledgments.-The authors wish to thank Dr. H. A. Levy and Mrs. M. P. Lietzke for helpful advice on the mathematical procedures and the computer programming.
THERMODYNAMIC PROPERTIES OF SOME OXIDES OF NITROGEN1 BY I. C. HISATSUNE Department of Chemistry, Pennsylvania State University, University Park, Pa. Received Auguat 4, 1961
Available spectroscopic and structural data have been used to calculate the thermodynamic functions for NzOa, NzOd and NzOs, and dissociation equilibria of these oxides. For the N z O dissociation, ~ the necessary functions for NOs radical were estimated from vibrational frequencies calculated with Urey-Bradley force constants. These data together with those obtained from other sources lead to the following estimated properties for ideal gases a t one atmosphere and 25”. CpO(cal./ deg. mole)
SQ(cal./ deg. mole)
AHfQ (kcal./mole)
AFrQ (kcal./mole)
11.22 15.68 18.47 20.22
60.36 73.92 72.73 85.00
16.95 20 00 2.54 3.35
27.36 33.49 23.66 28.18
I
Introduction are of considerable interest in air pollution, geoThe oxides of nitrogen, which form a “happy physics3 and recently in astrophysics4 as well. hunting ground’12afor chemical kineticists and have There are approximately twenty of these oxides been subjected to extensive kinetic investigations,2b (3) L. E. Miller, “The Chemistry and Vertical Distribution of the (1) Supported by the PHS Grant RG-8192 and the Air Force Geophysics Research Directorate. (2) (a) F. Daniela, Chem. Enp. Nema, 33, 2370 (1955); (b) see for recent reviews S. W. Benson, “The Foundation of Chemical Kinetics,” McGraw-Hill Book Co., Inc.. New York, N. Y., 1960.
Oxides of Nitrogen in the Atmoaphere,” U. 5. Air Force Geophysical Research Paper No. 38, AFCRCTR-56-207, 1954. (4) C. C. Kiess, C. H. Corlisa and H. K. Kieae, Science, 131, 1319 (1960); F. J. Heyden, C. C. Kiess and H. K. Kieaa, ibid., 130, 1195 (1959).
