The Journal of Physical Chemistty, Vol. 83, No. 10, 1979
Molal Volume arid Compressibility of Aqueous Phosphate
same direction as for pyridine and IDMF. Based on the calculated and experimental results given in Table IX, it appears that cyclohexane behaves anomalously to pyridine rather than C C 4 and benzene. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. The authors also acknowledge Cindy College, a Project SEED student, for her assistance in many phases of this work.
1255
(16) J. N. Spencer, R. S. Harner, and C. D. Penturelli, J. Phys. Chem., 79, 2488 (1975). (17) R. S. Drago, L. B. Parr, and C. S. Chamberlain, J. Am. Chem. Soc., 99, 3203 (1977). (18) A. Allerhand and P. v. R. Schleyer, J . Am. Chem. Soc., 85, 371 (1963). (19) J. N. Spencer, J. E. Gleim, M. L. Hackman, C. H. Blevins, and R. C. Garrett, J . Phys. Chem., 82, 563 (1978). (20) J. N. Spencer, R. S. Harner, and C. D. Penturelli, J. Phys. Chem., 79, 2488 (1975). (21) J. N. Spencer, R. A. Heckman, R. S. Harner, S. L. Shoop, and K. S. Robertson, J . Phys. Chem., 77, 3103 (1973). 122) J. N. Soencer. J. R. Sweiaart. M. E. Brown. R. L. Bensina. T. L. Hassinger, W.'Keliy, D. L.-Housel, and G. W. Reisinger, J: Phys. Chem., 60, 811 (1976). (23) R. Fuchs and P. S. Salvia, Can. J . Chem., 54, 3857 (1976). (24) A. Kolbe and H. Pracejus, Adv. Mol. Relaxation Processes, 5 , 65 11973). (25) M. S. Nazri and R. S. Drago, J. Am. Chem. Soc., 94, 6877 (1972). (26) I?. S.Drago and T. D. Epley, J. Am. Chem. Soc., 91, 2883 (1969). (27) T. J. V. Findlay, J. S. Keniry, A. D. Kdman, and V. A. Pickles, Trans. Faraday SOC.,63,846 (1967). (28) G. L. Bertrand and T. E. Burchfiekl, Anal. Calorimetry,3, 283 (1974). (29) L. 0.Hepler and D. V. Fenby, J. Chem. Thermodyn.,5 , 471 (1973). (30) W. Partenhelmer, T. D. Epley, and R. S. Drago, J . Am. Chem. Soc., 90, 3886 (1968). (31) G. C. Kresheck and I. M. Klotz, Biochemistry, 8, 8 (1969). (32) S. D. Christian, E. E. Tucker, and D. R. Brandt, J. Phys. Chem., 82, 1707 (1978). (33) P. Debye, "Polar Molecules", Dover Publications, New York, 1929. Reprinted by permission of Reinhold Publishing Corp. (34) C. P. Smyth, "Dielectric Behavior and Structure", McGraw-Hill, New York, 1955. (35) J. N. Spencer, J. R. Sweigart, M. E. Brown, R. L. Bensing, T. L. Hassinger, W. Kelly, D. L. Housel, G. W. Reisinger, D. S. Reifsnyder, J. E. Gleim, and J. C. Peiper, J. Phys. Chem., 81, 2237 (1977). (36) J. D. Cox and 0. Pllcher, "Thermochemistry of Organic and Organometallic Compounds", Academic Press, New York, 1970. .
References and Notes (1) E. M. Arnett, T. S. S;. R. Murty, P. v. R. Schleyer, and L. Joris, J . Am. Chem. Soc., 89, 5955 (1967). (2) E. M. Arnett, L. Joris, E. Mitchell, T. S. S. R. Murty, T. M. Gorrie, and P. v. R. Schleyer, J Am. Chem. Soc., 92, 2365 (1970). (3) W. C. Duer and G. L. Bertrand, J. Am. Chem. Soc., 92, 2587 (1970). (4) E. M. Arnett, E. J. Mitchell, and T. S. S. R. Murty, J . Am. Chem. SOC.,96, 9875 (1974). (5) R. A. Pieraitti, J . Phys. Chem., 69, 281 (1965). (6) R. A. Pierotti, J. Ph,vs. Chem., 67, 1840 (1963). (7) C. V. Krishrian and H L. Friedman, J. phys. Chem., 75,3598 (1971). (8) R. B. Hermann, J . Phys. Chem., 76, 2753 (1972). (9) M. J. Harris,, T. Higuchi, and J. H. Rytting, J. Phys. Chem., 77, 2694 (1973) (10) R. Fuchs, T. M. Young, and R. F. Rodewald, J . Am. Chem. Soc., 96, 4706 (1974). (1 1) E. Grunwakland E. K. Ralph, 111, J. Am. Chem. Soc., 89,4405 (1967). (12) E. M. Arnett and J. R. Carter, J. Am. Chem. Soc., 93, 1516 (1971). (13) A. Weissborger, Ed., "Techniques of Chemistry", Vol. 11, "Organic Solvents", 3rd ed., Wiley-Interscience, New York, 1970. (14) C. V. Krshnan and H. L. Friedman, J . Phy5. Chem., 73, 1572 (1969). (15) A. L. McClellan, "Tables of Experimental Dipole Moments", W. H. Freeman aind Co., San Francisco, 1983.
I
Molal Volume and Adiabatic Compressibility of Aqueous Phosphate Solutions at 25 O C Antonio Lo Surdo, Kerstln Bernstrom, Carl-Ake Jonsson, and Frank J. Mlllero" Rosenstiel School of Marine and Atmospheric Science, University of Mlami, Miami, Florida 33 149 (Received November 20, 1978) Publictition costs assisted by the Office of Naval Research
The apparent molal volumes (4v)and apparent molal adiabatic compressibilities [c#JK(s)] of aqueous solutions of NaH2PO4,Na2HP04,Na3P04,KH2P04,K2HP04,K3P04,and H3P04in the molality range 0 Im I1.0 have been determined from precise density and sound speed measurements at 25 "C. The values of I$V and $K(s) for Na3P04and K3P04have been corrected for hydrolysis and the values for H3P04have been corrected for ionization. The partial molal volumes and compressibilities have been extrapolated to infinite dilution with the aid of an extended Debye-Huckel equation. Values of = 47.71 cm3 mol-' and Rso = -8.6 X cm3 mol-' bar-' were obtained for H3P04. Ionic values of 9" = 36.92, 16.54, and -13.95 cm3 mol-' and lO4Ko= -55.4, -104.8, and -192.6 cm3mol-' bar-' were obtained, respectively, for HzPO,, HPOZ-, and PO4+ions. The for the phosphate species decreases by 10.7, 20.4, and 30.5 cm3 mol-'; while the 104R0decreases by 46.8, 49.4, and 87.8 cm3 mol-' bar-' as the charge is increased from -1 to -3. This decrease in 9" and Ro is due to the increase in hydration as the charge is increased (hydration numbers of nH = 5.5, 13.0, and 25.5 have been estimated for HzPOd-, HPOZ-, and PO-: ions). The volume and compressibility changes for the first, second, and third ionization of H3P04have been determined from the data and used to calculate the effect of pressure on the ionization. The calculated values of Kp/KO for the first ionization are in good agreement with the direct measurements of Disteche and Disteche and Ellis and Anderson.
vo
v"
Introduction Although numerous measurementgl have been made on the effect of pressure on the ionization of weak electrolytes in aqueous solutions,
HA -,H++ A--
(1)
few measurements have been made a t high concentra0022-3664/79/2083-1255$01 .OO/O
ti on^,^,^ or in various electrolyte media.4-8 The effect of Pressure on the thermodynamic ionization constants (KHA) in solutions RT(a In KHA/aP)T = - A T H A (2) AR"HA= - ( a h T H A / a P ) = RT(a2In K H A / ~ ~ '(3) ) can be determined from partial molal volume and iso0 1979 American Chemical Society
1256
Lo Surdo et al.
The Journal of Physical Chemistry, Vol. 83, No. IO, 1979
thermal compressibility data AVOHA = B"(H+) + VOW) - Po(HA)
(4)
AJP"A = Ro(H+)+ P ( A - ) - R"(HA)
(5)
[where B"(i) and P(i) are the infinite dilution partial molal volume and adiabatic compressibility of i, respectively, T is the absolute temperature, and R is the ideal gas constant]. Values of a In KHA/aPcalculated from eq 2-5 have been found to be in excellent agreement with direct measurements for a number of a c i d ~ . ~ - Since ~,~J~ direct pressure measurements are difficult for many systems of weak electrolytes, it is frequently easier to estimate the effect of pressure on the ionization process by using molal volume data (determined from density measurements) and molal compressibility data (determined from sound speed meas~rernents).~-~J'J~ Some years ago, Smith13 investigated the dissociation of in water a t 25 "C utilizing apparent molal volumes. More recently Disteche and Disteche14and Ellis and Anderson15 determined the effect of pressure on the first dissociation constant of H3P04from conductance and emf measurements in dilute aqueous solutions at 25 "C, and pressures up to 2000 bar. In recent years,2-4Jo-12J6-23 we have attempted to calculate AB" for the ionization of a number of weak electrolytes of practical importance in the study of the chemistry of marine systems. The volume (AB") and compressibility ( A P ) data at infinite dilution have been used to examine the ionization process for waterz3and boric a~id.~-~ The phosphate system was chosen for this study due to its practical importance, e.g., the dissociation of phosphoric acid in seawater as a function of temperature and pressure is needed for studies of apatites and phosphorite minera l ~ ,and ~ theoretical ~ , ~ ~ interest, e.g., the thermodynamics of phosphoric acid and phosphate salts is important because of the analytical behavior they exhibit due to dissociation and h y d r o l y ~ i s . ~To ~ - better ~ ~ understand the physicochemical properties of the various components of the phosphate system, we have measured the apparent molal volumes and compressibilities of H3P04,and salts of sodium and potassium phosphates in water at 25 "C. In this paper we present our results for the various components of the phosphate system, and discuss general procedures used to estimate the various thermodynamic functions for the ionization process of phosphoric acid.
Experimental Section Stock solutions of phosphoric acid, and of the mono- and dibasic sodium and potassium phosphate salts were prepared by weight from reagent grade chemicals. Stock solutions of and Na3P04 were prepared by neutralization of H3P04with standard KOH and NaOH solutions. The concentrations were determined by gravimetric phosphate analysis as magnesium p y r o p h ~ s p h a t e . ~ ~ Solutions of known molalities were prepared by weight dilution using ion-exchanged (Millipore Super Q)water. All weights were vacuum corrected. The densities were measured at 25 "C to f 3 X g cm-3 with a "vibrating tube" flow densimeter described by Picker, Tremblay, and Joli~oeur.~'The values of Ad = d - do, where d and do are the densities of the solution and water are listed in Table I.32 The system was calibrated with N2gas and ion-exchanged water by using the densities of Kell.33 The accuracy ( f 3 ppm) of the system has been determined by measuring the densities of standard seawater solutions.34 For NaH,P04 the densities were also measured with a high precision magnetic float densimeter
described elsewhere.35 These densities are also listed in Table I.32 The sound velocities were measured at 2 MHz to a precision of f0.02 m s-I by using a "sing around" sound velocimeter (Nusonic, Inc.). The procedure has been described elsewhere.36 The system was calibrated with pure water by using the sound velocities of Del Grosso and Mader.37 The accuracy (f0.1 m s-') has been determined by measuring the speed of sound in seawater The relative sound velocities (Au = u - uo,where u and uo are the speed of sound in the solution and in water) of the aqueous solutions at 25 "C are listed in Table II.32 The temperature of the various thermostated baths regulating the densimeter and sound velocimeter was set to f0.002 "C with a platinum resistance thermometer (calibrated by the National Bureau of Standards), and a G-2 Mueller bridge.
Results and Calculations The apparent molal volumes, &, and the adiabatic apparent molal compressibilities, 4K(S), of the various phosphate salts and phosphoric acid solutions were determined, respectively, from the density, d , and adiabatic compressibility, Ps, of the solution with the equations M (do - d)103 4v = 2 (6) mddo +
4K(S)
=
1000(Psdo - @sod) PsM mddo d
+-
(7)
where do is the density of water, m is the molality, M is the formula weight of the solute, and = 44.7735 X lo4 bar-136 is the adiabatic compressibility of water. The adiabatic compressibilities of the aqueous solutions at 25 "C were calculated from the sound speeds ( u ) by using ps = l / u 2 d (8) The densities and sound velocities were determined from the relative densities and sound speeds given in Tables I and I1 by using do = 0.997045 g 33 and uo = 1496.69 m s-'.~' The values of 6" and +K(S) are listed, respectively, in Tables I and II.32The uncertainties in 4v and 4K(S) in the concentration range 0.05 Im I1.0 are, respectively, less than f0.1 cm3 mol-' and f0.5 X cm3 mol-' bar-' for NaH2P04,Na2HP04,KH2P04,and K2HP04solutions. For Na3P04and K3P04solutions the uncertainties are f0.8 and f1.8 X cm3 mol-' bar-' in in 4v and k3.3 X c ~ I ~ ( respectively, ~ ) , for mT = 0.05 and 0.5. These larger errors are due to hydrolysis which is discussed later. The apparent molal (4) data for the mono- and dibasic salts (Tables I and 11) were fit to the extended DebyeHuckel limiting law (DHLL) equation38 4 = 4" SDHI'/z[(l 1'/2)-'- ~(1)/3]+ BI + CP +
os"
(os)
+
+
DP
(9) where SDHis the Debye-Huckel limiting slope39 [SV = 2 . 8 0 2 ~ d ~ SK '/~= , 4.22 X 10-4wdo1/2, and w = '/2&Z?]: ~ ( l=) (3/13/2)[(1 W2)- (1 F/2)-1- 2 In (1 + 11/')3
+
+
(10) I is the molal ionic strength ( I = l/zCv,miZ?), and B, C, and D are adjustable parameters. Plots of [4 - EDHLL] vs. I for the Na and K phosphate salts are given in Figures 1and 2. Large positive deviations from the limiting law are observed for all the salts. The smooth curves represent the least-squares lines. The value_sof the infinite dilution partial molal volumes (+v" = V " ) and adiabatic compressibilities = &"I, and the adjustable parameters
The Journal of Physical Chemistty, Vol. 83, No. 70, 1979
Molal Volume and Compressibility of Aqueous Phosphate
TABLE 111: Values of @', B, C,and D for the Apparent Molal Volumes, and Adiabatic Compressibilities of Phosphate Salts and Phosphoric Acid in Water at 25 "C Fit to Eq 9'
NaH,PO, Na,HPO, Na,PO, KH,PO, K,HPO, KPO, H,PO,
30.21 3.13 -34.12 40.44 23.57 -3.34 47.71
4.510 4.637 10.706 4.856 4.369 9.220 0.491'
-1.046 -0.764 -2.773 -1.164 -0.850 -2.363
0.303 -0.104 0.256
I
6
1257
L l6
0.03 0.03 0.43 0.04 0.04 0.34 0.02 -35-1 0
05
' IO
15
20
n
I
25
30
'
- 35
40
35
I = 2m,( 3 - 4
NaH,PO, Na,HPO, Na,PO, KH,PO, K,HPO, K,PO, H3PO4
89.27 172.81 2911.27 82.19 167.96 275.61 8.61
27.285 37.081 69.842 30.666 30.375 63.663 0.595'
-0.717 -16.722 - 18.808 -9.295 -9.234 -17.423
0.31 41.261 0.21 3.959 0.98 0.44 3.376 0.32 l.829 0.87 0.32
a The Least-squares program used to obtain these coefficients had a weighting factor related to the errors in @V Standard error in @v,cm3 and @K(s) at a given molality. mol-'. C Experimental slope for un-ionized H,PO,. Standard error in @K(s), cm3 mol-' bar-'.
TABLE IV: Comparison of the Infinite Dilution Partial Molal Volume for H,PO, and Sodium and Potassium Phosphate Salts Obtained in This Study with Literature Values @ V o ,cm3 mol-' compid
this work
'
0
05
15
IO
20
25
I30
30
I 450
7
,450
lit. values
30.21 30.39,' 27.gb NaH,PO , 3.13 3.01 5.3b Na,HPO, 39.22' KH,PO, 40.44 23.22d K,HPO, 23.57 47.71 47.67e H3P04 ' Refit data from ref 13. From ref 41. Refit data from re€ 13 afl,er reported concentrations were corrected by 1.4%(see text). From ref 40. e From ref 13.
B, C , and D are listed in Table I11 along with the standard errors (u) of the fits. The infinite dilution ]partial molal volumes of the various phosphate salts reported in the l i t e r a t ~ r e ' ~ are , ~ ' compared with our resultEi in Table IV. Our results are in excellent agreement with the literature values. A similar comparison of the 4°K(S) values cannot be made at thiis time since compressibility data are lacking in the literature. It should be noted that Smith13 reported a value of $vo = 0.42 mL mol-' for Na2HP04which was based on concentrations made up from a stock solution whose concentration was determined by gravimetric phosphate analysis as Mg2P2O7. Evaporation of the Na2HP04stock solution and subsequent ignition to Na4P207gave analytical results which were 1.4% higher.13 If we apply a 1.4% correction to Smith's data, we obtain 4"' = 3.01 cm3 mol-1 for Na2HP04which is in excellent agreement with our value of 3.13 cm:! mol-' (Table IV). In dilute solutions, the observed values of 4v and c $ ~ ( ~ ) for Na3P04 and K3P04decrease sharply due to the hydrolysis of P043-ion given by Po43- H 2 0 HP042-+ OH(11)
+
30
-
and had to be corrected for hydrolysis prior to the
'
430
-k7
420
H 5,
410
(0
rnlN
30.5 300
I
0
02
04
06
08
IO
I300 12
I
Figure 1. Concentration dependence of the apparent molal volume (corrected for the Debye-Huckel limiting law) of NaH,PO,, Na,HPO,, Na3P0,, KHzP04,K2HP0,,, and K,PO, in H,O at 25 O C . For NaH2P0, the density was measured with a vibrating densimeter (0)and a magnetic float densimeter (0). The 4 of Na,PO, and K3PO4 were corrected for hydrolysls.
least-squares fit of the data. The observed apparent molal [dv(obsd)]volume of Na3P04and K3P04can be considered to be comprised of three components 4v(obsd) = aC#JV(MOH) ah(M2HPO4) + (1 - 44v(M3POJ (12) where M+ denotes Na+ and K+ ions, and 4V(MX)is the apparent molal volume of the electrolyte MX at the ionic strength I = 2mT(3 - a) of the solution. Defining 4v(i) = [4V(MOH)+ C#JV(M,HP04)] and rearranging eq 12, we obtain the apparent molal volume of M3P04 C#JV(M~PO~) = [4v(obsd) - aC#Jv(i)l/(l- a) (13) The values of C#Jv(i)a t I = 2mT(3 - a) can be calculated
+
1258
The Journal of Physical Chemistry, Vol. 83,
-303
10
20
Lo Surdo et al.
No. 10, 1979
30
40
I-303
I: 2 m T ( 3 - a : )
[where r+(MX) is the mean activity coefficient of the electrolyte MX at I = 2mT(3 - a ) ]using the generalized equations developed by Pitzer and M a y ~ r g to a ~calculate ~ r+(MX). The values of a were determined by an iterative procedure by using KH = K w / K 3= 2.13 X (obtained at 25 "C from Kw = 1.008 X 10-1446147 and K3 = 4.732 X Values of a = 0.042 and 0.275 are obtained, respectively, for mT = 0.5 and 0.05 for both Na3P04and K3P04solutions.29 The corrected molal volumes of Na3P04 and K3P04 (minus the EDHLL contribution) is plotted vs. I = 2mT(3 - a ) in Figure 1. The coefficients for eq 9 are given in Table 111. The larger errors in $vo of Na3P04and K3P04 are due to errors in estimating a. The KH values of H3P04 calculated from various values of K 3 in the literature@vary from 0.0211 to 0.0239. These errors in KH yield errors in $v of f0.55 and f0.04 cm3 mol-l, respectively, for mT = 0.05 and 0.5. By differentiating eq 13 with respect to pressure (at constant entropy), one obtains the observed apparent molal [$K(s)(obsd) = -(a$v(obsd)/aP)s] adiabatic compressibility $K(s)(obsd)
Q$K(s)(~) + (1- ~ ) $ K ( S ) ( & P O-~ )
A $ d a a / a P ) (16) where is the apparent molal volume change for the hydrolysis reaction. Rearranging eq 16 yields $K(S)(M3P04)= -174
I
05
IO
15
20
25
-174 30
I
-91
I o
I 02
04
06
oa
10
12
Figure 2. Concentration dependence of the apparent molal adiabatic compresslblllty (Corrected for the Debye-Huckel llmiting law) of NaH2P04, Na&IPO4, Na3P04,KH2P04,K2HP04,and in H 2 0 at 25 "C. The 4K(S,of Na3P04 and were corrected for hydrolysis and -A4 ,(a a l a P ) term.
from the 4v of NaOH and KOH taken from the work of Miller0 et al.,23942and Akerlof et al.,43p44and the $V values of Na2HP04and KzHP04listed in Table 111. The fraction ( a )of HP042-and OH- in the Na3P04and K3P04 solutions at the total molality mT can be determined from the thermodynamic hydrolysis (KH)constant29
++)
$K(s)(obsd) - & J K ( s ) ( ~ ) A$" (1 - a ) (1- a ) aP (17)
where &S)(i) = -[a$v(i)/aPl~ = [$K(s)WOH) + $K(s)(M,HPO4)I and $K(s)(M~PO~) = - [ ~ $ v ( M ~ P ~ ~ )are, /~PIs respectively, the apparent molal adiabatic compressibility of i and M3P04(e.g., Na3P04and K3P04). The values of $K(S)(i)can be determined from the $K(S) of NaOH and KOH published elsewhere,49and Na2HP04and K2HP04 listed in Table 111. The apparent molal volume change, A$v = [&,(i) - $V(M3P04)],can be estimated from the $V of the i components, by using the parameters given in Table 111. The effect of pressure on a can be estimated by the differentiation of eq 14 with respect to pressure:
The term (a In KH/aP) = - A T H is given by A v o H= P ( M O H ) P ( M 2 H P 0 4 )- P ( M 3 P 0 4 ) (19)
+
The effect of pressure on x is given by a In x/aP = [ A v H - A P H ] / R T
(20)
where AVH = V(M0H) + V(M2HP04)- P(M3P04) (21) The P(i) a t the ionic strength of the solution I = 2mT(3 a ) of electrolyte i can be estimated from the @v data
-
P = @v + I(a$v/al)
(22)
The corrected values of @K(S) for Na3P04and K3P04minus the DHLL term are plotted vs. I in Figure 2. The pa(1- 4" rameters for the fit of $K(S) to eq 9 for Na3P04and K3P04 solutions are given in Table 111. The contributions of where yi is the activity coefficient of species i. The activity $K(s)(obsd)due to $K(S)(i)and A$v(aa/aP) tend to comcoefficient factor, r, can be estimated from pensate each other; thus, the extrapolated $"K(S) obtained by fitting either the corrected or observed values of $K(s) P = Y*~(MOH)Y+~(M~HPO~)/Y+~(M~P~~) (15)
KH =
ff2mT YOHYHPOl (1- f f ) YPO4
-
---ff2mT
(14)
The Journal of Physical Chemistry, Vol. 83, No.
Molal Volume arid Compressibility of Aqueous Phosphate
50.0 I
are not too different. The larger uncertainties in the fits of Na3P04 and 1K3PO4are due to errors in the hydrolysis correction. The hydrolysis constants for H P 0 2 - (1.63 X and indicate that the corrections to $v H2P04-(1.14 X and 4K(s)for hydrolysiis are less than the experimental error of the observled values. The observed app,arent molal [$v(obsd)] volume of a weak electrolyte (H3I3O4)which undergoes the ionization
-
H3P04
H+
+ H2P04
'_
(23)
a2mT Y H Y H ~ P O ~ a2mT =-= -(1 - a ) YH3P04 (l-a)T
IO
15
20
42.0
E
fs
38.0
(24)
34.0
'
(25)
(26)
where yi is thle activity coefficient of species i at the ionic strength of the solution, I = amT. The activity coefficient factor, R , can be estimated from T
1
u
The value of $V(H3P04)can be calculated from eq 25 provided $v(i) and can be estimated by some independent method. Since the thermodynamic ionization ( K H A ) constants are normally known,28the fraction ( a )of the free ions (H+,H2P04-)in a H3P04 solution of total molality (mT)can be determined from KHA
I
m
where CY is the fraction of free ions, and 4V(H+,H2P04-)= c#q(i) and $V(H3P04)are the apparent molal volumes of the free ions (H+,H2P04-)and un-ionized acid (H3P04), respectively. Rearranging eq 24 we have $ v ( H ~ P O J [$v(obsd) - a & v ( i ) l / ( l - a )
I
I
-
in dilute solutions can be divided into two components $v(obsd) = a$v(H+,112P04-)+ (1 - a)$V(H3P04)
IO, 1979 1259
= y+'((H+,H2PO4-)/y(H3PO4)= yh2(i)/yU (27)
where yh(i) == y+(Hf,H2P04-)is the mean molal ionic activity coefficient ofthe free ions, and yu = y(H3P04)is the activity coefficient of the un-ionized acid (H3P04). Values of yh(i) and yu can be calculated from the equations of Pitzer and Silvester,28and a estimated by an iterative method. The apparent molal volume, $,(H3P04), of the unionized phosphoric acid were calculated from eq 25 assuming the apparent molal volume of the free ions, &(i) = $V(H+,H2P04-),can be determined from
30.0 0
I
05
I
I
rn Figure 3. Concentration dependence of for H,PO, in HO , at 25 OC.
4 v(obsd), 4 v(i), and 4 d u )
phosphoric acid in dilute solutions decreases dramatically due to ionization. The volume contribution of two components in solutions, Le., 4v(H3P04) = &(u) and $V(H+,H2P04-)= $V(i), is shown by the upper and lower curves of Figure 3. The $V(H3P04)have been fit to the equation
~ v U W "=~$voWM"4) ) +B d l -
a)mT
The values of $"O (H3P04)and Bv are given in Table 111. Our +v0 = 47.71 cm3 mol-' for H3PO4 is in excellent agreement with 4v0= 47.67 cm3mol-' obtained by Smith.I3 Differentiation of eq 24 with respect to pressure yields the equation for the observed apparent molal, 4K(s)(obsd) = -[a$v(obsd)/aP]s, adiabatic compressibility for a weak electrolyte (H3P04) $K(S)(obsd) = C"K(s)(H+,H2P04-)+ (1 - 44K(S)(H3P04)- A $ v ( H ~ P O ~ ) ( ~ ~(30) /~P) Rearranging eq 30 we get
$dH+,H2PO4-)= $,ANaH2P04) + &dHCl) 4dNaCl) = $v(KHJ'OJ + $v(HCl) - $V(KCl)
(31)
(28)
The values of' dVa t I = amT for the various electrolytes were determined by using the coefficients given in Table I11 and the $v data of HC1, NaC1, and KC1 given elsewhere.42 The values of a were determined from eq 26 by using K H A = K1 = 7.1425 x 10-3,28and two different factors of T , Le., (1) R = yh2(H+,H2PO4-)/y,(H3PO4), where the values of y d i ) = y+(W,H2P04-)and yu = yu(H3P04)were determined from the equations of Pitzer and Silvester,28 and (2) R = y*2(H+,C!1-),where yu = y,(H3P04) = 1, and y d i ) = y+(H+,Cl-)were calculated from the equations of Pitzer and M a y ~ r g a . 'Values ~~ of a did not differ significantly, for example, a t mT = 0.05, CY = 0.352 and 0.347; while at mT = 1.85, a = 0.116 and 0.077, respectively, for methods (1)and (2). The concentration dependence of the &(obsd), 4v(i), and &(u) for aqueous solutions of H3P04 is shown in Figure 3. The observed apparent molal volume of aqueous
(29)
where ~K(s)(H+,H~PO,-) = 4K(S)(i)= -[adv(H+,H2PO4)/ = - [ ~ ~ v ( H ~ P O ~ )y/e~, P I S apls, and ~ K ( s ) ( H ~ P=O$K(s)(u) ~) respectively, the apparent molal adiabatic compressibilities of the free ions (H+,H2P04-)and un-ionized acid (H3PO4). Differentiation of eq 26 with respect to pressure yields the (aa/aP)term
aP where (a In KHA/aP)= -APoo/RT andla In ,lap) = ( A V of AVO and AV can be calculated
- A V o ) / R T . The values
from AVO = Vo(H+,H2P04-)- V0(H3P04)
(33)
AP = P(H+,H,PO4-) - P(H3PO4)
(34)
The values of P(i) at I = amT can be calculated from eq
1260
The Journal of Physical Chemistry, Vol. 83, No.
IO, 1979
Lo Surdo et al.
TABLE V: Infinite Dilution Part@ Molal Volume ( A and Adiabatic Compressibility (aKso) Changes for the Ionization of H,PO, in H,O at 25 "C
12.0
T)
AT, 10~ai?,~, cm3 cm3 mol-' mol-' bar-' H,PO,+H' + H,POL -16.26 i 0.01 -38.3 f 0.8 H,POL + H + t H P 0 L 2 -25.85 + 0.02 -40.9 t 2.1 HPO;, + H C t POL3 -35.96 f 0.05 -79.3 + 3.9
reaction
60
L
0
n
-
'0 E
TABLE VI: A Comparison of the Infinite Dilution Partial Molal Volume Change for Ionization of Phosphoric Acid with Literature Values at 25 "C
0
nl
5
-$
-AT, cm3 mo1-l
-6.0
compd
8 c
o_
H3PO4 H,POL -12 0
z
a
15.
this study 16.26 25.85
References 13 and 51. Reference 52.
lit. values
16.2,a 16.6bvC 28.1,a 24.1,d 24.0b Reference 14.
Reference
I
__
-18 0
0
03
09
06
12
15
mT
Flgure 4. Concentration dependence of q5 K(s,(ObSd), Aq5 ,(a a l a P ) for H3P04 in H20 at 25 "C.
4 K(s)(i),and
Discussion The infinite dilution partial molal volume (V") and adiabatic compressibility (&O) for the phosphate salts and phosphoric acid can be used to calculate the AVO and AK," for the ionization of H3P0,. The values of AYio (where Y = V or K) have been calculated from AYio = P(HC1) + P(MHZPO4) - p(H3P04) P(MC1) (37)
AYzo = P(HC1) + P(M2HPOJ AY3"
= F(HC1)
-
F"(MHZPO4) p(MC1) (38)
+ Y"(M3P04) - P(MZHPO4) YO(MC1) (39)
-90'
0
'
02
I
I
I
06
08
IO
I
I
04
12
14
(1-a)m T
Flgure 5. Plot of q5Kn vs. (1 corrected for ionization.
- a)m, for
aqueous H3P04at 25 OC
22. The @K(S)(i)of the free ions (Ht,H2P04-)can be estimated from
@~(s)(H+,HzPo4-) = @K(s)(HC~)@K,s)(MHzPO~) @K(s)(MCU(35) where M is Na+ or K+. The values of @ K ( ~ for ) the various electrolytes were determined from the parameters in Table I11 and the published data49for HC1, NaC1, and KC1. The concentration dependence of the various components for H,PO,, q5Kcs)(obsd),+K(S)(i),and correction term [-A@v(aa/aP)],is shown in Figure 4. As is quite apparent from this figure, the contribution to q5K(s)(obsd)due to the free ions and the correction term, A&,(aa/aP), tend to compensate each other. It is thus fortuitous that @K(s)(obsd), uncorrected for ionization, extrapolate to approximately the correct infinite dilution value. The corrected @K(s) for H3P04were fit to @K(S) =
@OK@)
+ BK(1
a)mT
(36)
A plot of c $ ~ (for ~ )&PO4 vs. (1- a)mT is shown in Figure 5. The line represents the least-squares best fit. The infinite dilution value of q5°K(s)(H3P04) and the slope BK are given in Table 111.
where M is Na' or K+. Using V O and Rsodata for HC1, NaC1, and KCl given e l s e ~ h e r e l ' , and ~ ~ , the ~ ~ values of To and Et" given in Table 111, we have calculated AVio and AKio for the ionization of H3P04. The average values obtained from the Na+ and K' salts are given in Table V. A comparison of AVlo and AVzo obtained in this study with literature data13-15y51r52 is shown in Table VI. Our results for AVl0 and AVzo are in good agreement with the literature data. To the best of our knowledge, no literature data are available for AV30 or AKiO to compare with our results. The ratios of AVio and AKio for the three ionizations are, respectively, 4.3 x IO3,6.3 x lo3, and 4.5 x lo3 bar. These values can be compared to the correlation values of 4.7 x lo3 bar for monobasic acids53and 3.7 X lo3 bar for ion pair formation.10 The larger value of AVio/AKio for the second ionization indicates that the water molecules around H P 0 2 - are more tightly packed than around H2P04-and PO>- (the value of AVi"/AKio is proportional to the volume change of transferring water from the bulk phase to a region close_to thejon).1° The ionic values of Vo and Kso for phosphate ions can be determined fromAVo and AKO using ionic values for H+.50954 Values of V"(HzP04-) = 36.92, Vo(HPo42-) = 16.54, and Qo(P02-)= -13.95 cm3 mol-' and Ko(HzPO,-) = -55.4 x Ro(HP042-)= -104.8 X and P(PO?-) = -192.6 X cm3 mol-l bar-l were found. By assuming the intrinsic molal volume of the ions is equal Lo the value of H3P04,the electrostriction molal volumes V" (elect) of -10.9, -31.3, and -61.9 cm3 mol-' are, respectively, found for HzP04-, HPOt-, and P043-. A similar calculation for K ogives Ro(elect)of -46.9 x W4,-96.4 x lo4, and -184.3 x cm3 mol-l bar-l for H,PO;, HPO:-, and Pod3-. The decrease in volume does not follow a Z2 relationship as
The Journal of Physical Chemistry, Vol. 83, No. 10, 1979
Molal Volume and Compressibility of Aqueous Phosphate I
I
perature and in various ionic media.
I
Acknowledgment. A. Lo Surdo and F. J. Millero acknowledge the support of the Office of Naval Research (N00014-75-C-0173)and the oceanographic section of the National Science Foundation (OCE73-00351-A01) for this study. K. Bernstrom and C.-A. Jonsson from the Royal Institute of Technology, Stockholm, Sweden acknowledge the sponsorship of the CHUST 77 committee.
35 30 UP i$
25 20 15
10
1201
0
200
400
600
800
1000
P, bars
Figure 6. Plot of ( K y I K p )vs. Pfor the first (I), second (2), and third (3) ionization process of t-I,P04 in H,O at 25 "C:.
TABLE VII: C o m p a r i s o n of t h e Measured a n d Calculated E f f e c t of Pressure on t h e I o n i z a t i o n of H,PO, in Water Kp/Ko P. bar calcd ref 15 r e f 14 1.00 1.00 0 1.00 1.36 1.37 500 1.37 1.89 1.80 1.83 1000 '1500 2.29 2.34 2.98 2000 2.82 ~~
expected from the continuum model,M)but varies in a ratio of 1:3:6 for volumes and 1:2:4 for compressibility. The values of V"((e1ect) and R"(elect) can be used to estimate the number of water molecules hydrated to phosphate ions.1° The compressibility data gives values of 6, 12, and 23 water molecules hydrated _to H2P04-, HPO:-, and I'043-ions [ n =~ -R(elec)/PHzOVHzO,where PH~O and VHZoare the compressibility and volumes of water].1° The molal volume data give values of 5, 14, and 28 for the water molecules hydrated to H2P0,, HPOt-, and PO:- ions [nH = -Vo(elect)/-2.2].'0 The effect of pressure on the ionization of H3P04, H,PO;, and HPOt- ccrnbe estimated from AV: and AK,O by using 111 (KP,IK?)
[-AVoiP
+ 0.5AK",P2]/RT
(40)
where P is the applied pressure. The results are shown in Figure 6. Comparisons of the calculated values of Kp/K" for the ionization of H3P04using eq 40 with direct measurements14J5are given in Table VII, The calculated results from P = 0 to 1.000 bar are in good agreement with the direct measurements of Disteche and Disteche14 and Ellis and Anderson.15 The larger differences above 1000 bar are probably due to the fact that we have equated the isothermal compressibility AI? with A&". The differences in AK" ,and AKso are given at 25 "C by54955 104(AK0- AKs") = 36.76AE" - 1.134 X 10-3AC,0
(41)
where aEo = a A V o / a T is the expansibility change and AC," is the heat capacity change for the ionization. A value of AC," = -212 J mol-l 51 has been determined for H3P04 and does not contribute significantly to the difference. The AE" has not been determined for H3P04. A value of AE" = 0.1 cms mol-l for H3P04,which is the value for boric acid,l' gives AK" - Uso = 3.9 X cm3 mol-' bar1. A value of AKa = AKso + 3.9 X gives Kp/KO = 2.31 a t P = 1500 bar and K P / P = 2.85 at P = 2000 bar, which are in better agreement with the measurements of Ellis and Anderson.15 In our future work we plan to study the volume properties of phosphate solutions as a function of tem-
Supplementary Material Available: Two tables (4 pages) consisting of experimental data for mono-, di-, and tribasic sodium and potassium phosphate salts, and phosphoric acid in water a t 25 "C. The data are relative density and apparent molal volume (Table I), and relative sound speed and apparent molal adiabatic compressibility (Table 11). Ordering information is available on any current masthead page.
References and Notes S.D. Hamann, "Physical Chemical Effects of Pressure", ButterworUls, London, 1957. G. K. Ward and F. J. Millero, J. Solution Chem., 3, 417 (1974). A. Lo Surdo, G, K. Ward, and F. J. Millero, manuscript in preparation. G. K. Ward and F. J. Millero, Geochim. Cosmochim. Acta, 39, 1595 (1975). C. H. Culberson, D. R. Kester, and R. M. Pytkowicz, Science, 157, 59 (1967). C. H. Culberson and R. M. Pytkowicz, Limnoi. Oceanogr., 13, 403 (1968). A. Disteche and S. Disteche, J. Electrochem. Soc., 114, 330 (1967) A. Disteche in "The Sea", Vol. V, E. D. Goldberg, Ed., Wiley, New York, 1974, pp 81-122. S. D. Hamann, Dw. Applied Chemistry Technical Paper (3), CSIRO, Australia (1972). F. J. Mlllero, G. K. Ward, F. K. Lepple, and E. V. Hoff, J. Phys. Chem., 78, 1636 (1974). F. J. Millero, Chem. Rev., 71, 147 (1971). F. J. Millero and R. A. Berner, Geochim. Cosmochim. Acta, 38, 92 (1972). J. S.Smith, Ph.D. Dissertation, Yale University, 1943. A. Disteche and S. Disteche. J. Eiectrochem. Soc.. 112. 350 (1965). A. J. Ellis and D. W. Anderson, J . Chem. SOC.London, 1765 (1961j. F. J. Mlllero, J . Phys. Chem., 74, 356 (1970). F. J. Millero, Limnoi. Oceanogr., 14, 376 (1969). F. J. Millero, Geochim. Cosmochim. Acta, 34, 1261 (1970). W. L. Masterton, H. Welles, J. H. Knox, and F. J. Millero, J . Solution Chem., 3, 91 (1974). F. J. Millero and W. L. Masterton, J . fhys. Chem.,78, 1287 (1974). F. J. Millero, F. Gombar, and J. Oster, J . Solution Chem., 6, 269 (1977). C.-T. Chen and F. J. Millero, J . Solution Chem., 8, 589 (1977). F.J. Millero, E. V. Hoff, and L. Kahn, J. Solution Chem., 1, 309 (1972). R. S. Dietz, K. 0. Emery, and F. P. Shepard, Buii. Geoi. SOC.Am., 53, 815 (1942). C. E. Roberson, "Solubility Implications of Apatite in Seawater", M.S. Thesis, Univerisity of California, San Diego, 1965. R. H. Wood and R. F. Platford, J . Solution Chem., 4, 977 (1975). R. F. Platford, J . Chem. Eng. Data, 21, 468 (1976). K. S. Pitzer and L. F. Silvester, J. Solution Chem., 5, 269 (1976). F. J. Millero, W. C. Duer, E. Shepard, and P. V. Chetirkin, J . Solution Chem., 7, 877 (1978). I. M. Kolthoff, E. B. Sandell, E. J. Meehan, and S. Bruckenstein, "Quantitative Chemical Analysis", Macmillan, New York, 1969, pp 638-639. P. Picker, E. Tremblay, and C. Jolicoeur, J. Solution Chem., 3, 377 (1974). Available as supplementary material. See paragraph at the end of text. ' G. S. Kell, J. Chem. Eng. Data, 20, 97 (1975). F. J. Millero, D. Lawson, and A. Gonzalez, J . Geophys. Res.. 81, 1177 (1976). F. J. Millero, Rev. Sci. Instrum., 38, 1441 (1967). F. J. Millero and T. Kubinski, J. Acoust. SOC.Am., 57, 312 (1975). V. A. Del Grosso and C. W. Mader, J . Acoust. SOC.Am., 52, 961 (1972). B. B. Owen and S. R. Brinkley, Jr., Ann. N . Y . Acad. Sci., 51, 753 ( 1949). F. J. Millero in "Activii Coefficients in Aqueous Electrolyte Solutions", R. M. Pytkowicz, Ed., CRC Press, W. Palm Beach, Fla., Chapter 13, in press. H. E. Wirth and S.Shapiro, unpublished data; H. E. Wirth, personal communication. A. M. Conture and K. J. Laidler, Can. J. Chem., 35, 207 (1957).
1262 The Journal of Physical Chemistry, Vol. 83, No. 10, 1979
Harbison et
(42) F. J. Miiiero, A. Laferriere, and P. V. Chetirkin, J. Phys. Chem., 81, 1737 (1977). (43) G.Akerlof and G. Kegeies, J. Am. Chem. SOC.,61,1027 (1939). (44) 0. Akerlof and P. Bender, J. Am. Chem. SOC.,63, 1085 (1941). (45) K. S. Pitzer and G. Mayorga, J. Phys. Chem., 77, 2300 (1973). (46) H. S. Harned and 8. B. Owen, "The Physical Chemistry of Electrolyte Solutions", Reinhold, New York, 1958. (47) R. A. Robinsonand R. H. Stokes, "Electrolyte Solutions", Butterworths, London, 1959. (48) L. G. Sillen and A. E. Martell, "Stability Constants of Metal-Ion Complexes", The Chemical Society, London, 1964.
al.
(49) F. J. Miliero, G. K. Ward, and P. V. Chetirkin, J. Acoust. SOC.Am., 61, 1492 (1977). ( 5 0 ) F. J. Miilero in "Water and Aqueous Solutions", R. A. Horne, Ed.,
Wlley-Interscience, New York, 1972. (51) H. S.Harned and B. B. Owen, ref 46,p 405. (52) R. LinderstromLang and C. F. Jacobsen, C . R. Trav. Lab. Cai-fsberg, Ser. Chim., 24, l(1941). (53) D. A. Lown, H. R. Thirsk, and Lord Wynnedones, Trans. Faraday SOC., 64,2073 (1968). (54)J. G. Mathieson and B. E. Conway, J. Solotion Chem.,3, 455 (1974). (55) J. E. Desnoyers and P. R. Philip, Can. J. Chem., 50, 1094 (1972).
Solute Infinite-Dilution Partition Coefficients with Mixtures of Squalane and Dinonyl Phthalate Solvents at 30.0 OC M. W. P. Harblson,? R. J. Laub," D. E. Martlre,?J. H. Purneli,t and P. S. Willlamst Departments of Chemistry, Georgetown Universky, Washington, D.C. 20057, The Ohio State University, Columbus, Ohio 432 IO, and University College of Swansea, Swansea, Wales SA2 BPP, United Kingdom (Received February 16, 1979) Publication costs assisted by the National Science Foundatlon
Using gas-liquid chromatography (GLC), activity and partition coefficients at 30.0 "C were obtained for aliphatic, alicyclic, and aromatic solutes with 12 mixtures of squalane + bis(3,5,5-trimethylhexyl)phthalate (hereafter called dinonyl phthalate) over the mole fraction range 0-1. The random error in the partition coefficients is estimated to be less than 1.0%. The average difference between the GLC results and those determined by a static technique is found to be f0.5%, thus establishing (for the first time) the validity and attainable accuracy of the GLC method for thermodynamic studies of ternary solutions. Theoretical expressions derived from an extension of a conventional (Tompa) solution model are next employed to describe the variation of the solute partition coefficient ( P R ) with the volume fraction ($1 of dinonyl phthalate, and are found to fit the data to within experimental error. However, although the fitted solution parameters are physically reasonable and internally self-consistent, independent tests of the solution model prove it to be inconclusive in the present instance. The data are then examined in light of the diachoric solutions model, wherein P R is said to vary linearly with $ over the entire solvent composition range. Deviations from linearity of up to 9% are observed for the systems at hand and so, it cannot here be claimed that the diachoric solutions hypothesis applies unless it is postulated that other concurrent solution phenomena (such as solvent dimerization) obtain. The success of the diachoric solutions equation in terms of analytical applications (i.e., GLC separations), which is based upon prediction of relatiue partition coefficients, can, however, be rationalized in terms of the current data and is shown to be accurate to &2.5% for the systems herein examined. It is concluded that, in view of the importance of mixed solvent systems in thermodynamics, in chromatography, and in spectroscopy, in light of the apparent conformity to diachoric behavior of hundreds of mixtures of a remarkable range of solute and mixed solvent types, and because of the still-unresolved nature of the situation, additional appropriate experiments are called for.
In an analysis of the great majority of quantitatively useful GLC-based data allowing evaluation of the infinite-dilution partition coefficient (KORm)of a solute component (A) distributed between a binary liquid mixture (B + C) and the gas phase, Laub, Purnell, and cow o r k e r ~ found ~ - ~ that, with few exceptions and for a wide variety of system types, the results were described by the linear relation KOR(M,=
$B@R(~,
+ $c@R(~)
(1)
to within experimental error (which in a few cases reached &lo%),where $i represents the volume fraction of solvent component i and P R G ) (i = B or C) pertains to A in either pure solvent. Solutes of almost all types, and solvent mixtures ranging from those where only van der Waals 'Department of Chemistry, Georgetown University. *Author to whom correspondence should be addressed a t the Department of Chemistry, The Ohio State University. t Department of Chemistry, University College of Swansea. 0022-365417912083-1262$0 1 .OOlO
interaction is expected to those where there is spectroscopic evidence of complexation were found to conform to eq 1. Laub and PurnelP proposed that systems which are described by eq 1 be termed diachoric. Noteworthy a t this point is that expressions equivalent to eq 1have, in the past, been employed empirically with success in analytical GLC, beginning with Primavesi6 who showed in 1959 that the solute specific retention volume (vOgcM,)varied linearly with the weight fraction (wi)of solvent component i, viz. =
u.rBvog(~)i"Cvom
(2)
for mixtures of triisobutylene with silver nitrate-ethylene glycol. Another version of eq 1in terms of capacity factors h' [= ( t -~t A ) / t A ] presented by Hildebrand and Reilley6 in 1964 (3) k ' ( ~=) W B ~ ' ( B ) + W C ~ ' ( C ) was found to describe solute elution behavior with mixtures of the solvents, silicone oil with polyethylene glycol (mol 0 1979 American Chemical Society
