HERBERTS.HARNED
112
chloric acid. However, in 1.1 M sulfuric acid in which the sulfate concentration was estimated to be 0.45 M , the rate was measurable and only 1.3 times as fast as the rate under comparable conditions of 1.5M perchloric acid a t I.( = 2. These results suggest that the effect of sulfate is similar to that found in the neptunium(1V)neptunium(T’1) reaction which has been extensively studied. l7
Vol. 64
Acknowledgments.-The authors gratefully acknowledge many helpful discussions with Professor G. Scatchard primarily concerning the hydrolysis of cerium(1V). They also acknowledge discussions with Dr. C E. Holley, Jr., and especially with Dr. J. F. Lemons, under whose general direction this work was done. (17) J. C. Sullivan, D. Cohon and J. C. Hindman, J. Am. Chem. SOC., 79,4029 (1957).
THE THERMODYKAIIC PROPERTIES OF THE SYSTEM : HYDROCHLORIC ACID, POTASSIUM CHLORIDE AND WATER FROM 0 TO 40’ BY HERBERT S. HARXED Contribution X o . 1630from the Department of Chemistry of Yale University, New Haven, Connecticut Recezved J u l y 87, 1959
Parameters of equations which permit the calculation of the activity coefficients of both electrolytic components and the activity of water in the system, hydrochloric acid, potassium chloride and water from 0 to 40°J have been obtained. The magnitude and sign of the excess heat of mixing as a function of temperature has been estimated.
In an earlier communication, a thorough investigation of the thermodynamic properties of the system hydrochloric acid, sodium chloride and water from 0 to 50’ has been recorded.’ The present contribution contains a similar calculation for aqueous solutions containing hydrochloric acid and potassium chloride. The calculations are based on the equations log 7 1 = log log 12 = log
- C Y ~ Z ( D ) ~Z h2m?2 ( 0-) orzmml - SzIml1
YNO) ~ ~
(I) (2)
+
to be applied a t constant total molalities m = ml m2. In these equations yl, ml and y 2 , m2 represent the activity coefficients and molalities of the acid and the salt in their mixtures, respectively, y!(o) is the activity coefficient of the acid of molality m in mater and ~ 2 ( ~ the ) , activity coefficient of the salt in water at molality m. The quantities 0(12(c), a 2 1 ( 0 ) , p 1 2 and pzl are the empirical coefficients of the linear and quadratic terms. Calculations of the Parameters of Equations 1 and 2.-The coefficient a 1 2 ( 0 ) was obtained by reconsideration of the electromotive force data of Harned and Hamer2 and Harned and Gancy3 for the activity coefficient of the acid, q l l in the mixtures. The activity coefficients, yl(o),of the acid in water were obtained from the data of Harned and Ehlers4 as tabulated by Harned and Owen.5 The values of log ylc0)and 0(12(0)from 0 to 40’ at 5’ intervals are given in Table I for the system at total molalities 1, 2 and 3. Since all the experimental evidence indicates that log y1 varies linearly with the acid or salt concentration a t constant total molality, it is assumed that p 1 2 equal zero. The parameter azl = a21(o) &ml was computed by the method of ;IlacKay.6 Values of log YZ(C)
+
(1) H. S. Harned, J . P h y s Chem.. 63, 1?99 (1959). (2) H. S. Harned and W. J Hamer, J . Am. Chem. Soc.. 86, 2194 ( 1933). ( 3 ) II. P. Ilarned and A. €3. Gancy, t b ~ d .62, , 627 (1958). (4) H. S. Harned arid R . W.Ehlers, ibad., 56, 2179 (1933). ( 5 ) H. S. Harned and B. B. Owen, “The Physical Chemistry of
Electrolytic Solutions.” 3rd Ed., Reinhold Publ. Corp., New York, N. Y.,1958, p. 716.
TABLE I
DATAFOR COJIPUTINGTHE ACTIVITYCOEFFICIENTSOF HYDROCHLORIC ACID, yl, IS
t
A N D POTASSIUM CHLORIDE,yzJ SOLUTIONS OF 1, 2 A N D 3 m TOTAL MOLALITIES 021 = -0.006
log
Yl(0)
m
o J
10 15 20 25 30 35 40
i.9253 1.9224 1.9188 i- ‘3153 1 9118 1 9079 -1,9041 1 8999 1.8957
log YZ(0) = ml + m a
1.7694 -i.7745 1.7767 i.7188 -1.7810 1.7824 -1 ,7810 -1.7810 1.7803
m = ml
0 5 10 15 20 25 30 35 40
0.0326 .0286 ,0224 .016ti ,0103 - ,0039
-1.9967 1.9894 i.9824
5 10 15 20 25
UlItO)
0.0670 .0650 .0625 ,0605 ,0585 ,0558 ,0540 .05?0 ,0500
-0 -
-
0740 0714 ,0681 ,0655 ,0629 0602 0586 0568 0560
+ m2 = 2
1 7380 1.7435 -1.74‘37 1 7*54:3 -1.7582 1 . 7604 1 . 7619 1 7627 1.7619
+ +
0
alz(o1
+1
ml rn? 3 0.1620 1.7419 .1544 1.7396 ,1464 -1.7451 ,1377 1.7497 .1287 -1.7526 .1192 1.7566
0.0680 ,0665 ,0645 ,0625 .Of305 . 0.590 ,0570 ,0550 ,0530
-0 -
0640 0615 0585 0555 0535 0514 0485 0470 - 0435
= -0.005
0.0690 ,0680 -0660 ,0650 ,0640 ,0630
-0.0585
-
.(‘Si5
.b545 - ,0510 - ,0485 - ,0440
required for this calculation were obtained from the data of Harned and Cook’ and are recorded in the third column of Table I. (6) H. A. C. MacKay. Trans. Faraday Soc., 61. 903 (1955). See also ref. 5, p. 628, 629. (7) H. 6. Harned and AI. A. Cook, J. Am. Chem. SOC.,69, 1920 (1937). See also ref. 5 , p. 727.
Jan., 1960
THERMODYNAMIC PROPERTIES OF THE SYSTEM
HCL-KCL-HzO
113
The method of evaluation of a21(0)and ,921 is shown in Fig. 1 in which -ayz1is plotted against m,. The solid graphs in the figure are straight lines with a slope of -0.006 which is taken to be the value of p 2 1 a t all temperatures. This result is quite consistent over the temperature range from 10 to 40'. On the other hand, the results at 0 and 5' are less consist,ent as shown by the dashed lines. This discrepancy may be due to experimental errors which are more likely to occur at these lower temperatures. However, the results a t ml equals 1 and 1.5 agree closely with the assumed straight lines. Values of aZl(o),obtained from these graphs, are given in Table I. 0.04 It was shown in the earlier communication1 0 0.5 1.0 1.5 2.0 that thermodynamics imposes two restrictions on rnl . the use of the quadratic equations 1 and 2. The Fig. l.--azl versus ml. first requirement is that at a given temperature 0.015 I 1 (PIS - &I), is not a function of the total molality m. Since in present instance plz is zero, P21 must 5 E also be independent of m. Thermodynamics also N + 0.01 M requires that, a t constant temperature, (a12(0) A l m crzl(o)) ampzl must be constant.8 In Table 11, values of this latter quantity a t 1, 2 and 3 total molalities are recorded. It is clear from the data in this table that the required constancy occurs at 1 and 2 tot4almolalities over the entire temperat,ure range. At 25', a similar constancy occurs a t 1, 2 and 3 total molalities. At temperatures from 0 10 20 30 40 20 to Oo, the results at 3 total molality indicate an T. increasing departure from constancy. Our cal) )+ 2m2 versus T at 1 and 2 total culations indicate that as the total concentration Fig. 2.--(a1z(0,+ c u ~ ~ ( oconcentrations. increases above 2m, increases and has a value latter quant'ity at 1 and 2 total molalities are of -0.005 a t 3m. shown in Fig. 2. The nature of these curves indiTABLE I1 cates that the excess heat of mixing will be positive ((YIZ~O) ~ z I ( o ) ) + 2mBzl AT 1, 2 AND 3 TOTAL MOLALITIESbelow 25' and negative a t the higher temperatures. 821 = -0.006 At 1 m total molality, Young, Wu and Krawitzs 1 m = l m - 2 m = 3 obtain -3 cal. for the excess heat of mixing per 0 -0.0190 -0.0200 -0.0265 mole of solute a t 25'.
-
+
+
+
5 10 15 20 25 30 35 40
-
-
,0184 ,0176 ,0170 - .0164 - ,0164 - .0166 - ,0168 - ,0180
-
.0190 .0180 ,0170 .0170 .0164 ,0155 ,0160 ,0175
.0255 ,0245 ,0220 ,0205 ,0170
d[(0112(0)
AH:! = 2.303RT2mlnz2
----m
......
dT
(3)
an equation derived in the earlier communication.' Values of (CU~!(O) amo)) and ( ~ I z ( o ) ~ZI(G) 2pZ1m2)are given in Table I11 and graphs of the
+
+
+
(8) In the earlier contribution (Ref. 1) equations (16), (17) and (18) are in error. The term 2m d ( h &)/dm should be m d(&, f &I)/ dm. -4sa rerult equation (18)should read (oin[o~ CCZI~Q)) -I- m(Bn -I-
+
821)
+
(815 + B n i ) d m = constant.
+
This reduces t o equation (18)
+
&I) is not a function of m. In the present case, since81: is if (PIS & I \ , &I is zero and thermodynamics requires the constancy of (PI; not a function of m and ( a l , ( a i m1[0,) 2m &I is independent of m. Table I1 in the previous contribution is correct.
+
+
-
=- 1
+
(aIZ(0) anl(0))
t
......
+ a m ) + 2Bzlm21
AT
0 5 10 15 20 25 30 35 40
+
+
azl(o)) 2m~P2~ WHEN ?nl = 1 AND 2 m TOTAL MOLALITIES
(al2(,,) a z l ( ~AND ) ) (aim
The Excess Heat of Mixing.-The estimation of the excess heat of mixing, AH!, will be restricted to the case where both electrolytic components are a t the same concentration. Thus when ml = m2
TABLE I11
+
......
-0.0070 - ,0064 - .0056 - ,0050 - ,0044 .0044 .0046 - ,0048 - .0060
-
(aizm azllo))
2Bz1
7 - r n
++
= 2---
+
(aIZ(0)
OLII(0)
(alz(o)
-0.0130 - .0124 - ,0116 - ,0110 - ,0104 - ,0104 - ,0106 - ,0108 - ,0120
m2
+
f
azl(o1)
2BZl
+0.0040
-0.0080
-
.0050
,0070 ,0060 ,0050 - ,0050 - ,0044 - ,0035 - ,0040 - ,0055
.OOGO ,0070 ,0070 ,0076 ,0085 ,0080
.0065
Osmotic Coefficients.-The osmotic coefficients of mixtures (b and subsequently the activity a,, of the third component, water, may be computed by the equations
+-
+2(0)
=
2.303m [(
2m ( P I S -
alU(0)
+
mla
~
2
m,, $-
azl(0))
4
2 ~ ) - 5 m,(~lz -
mi %l(O)
is]
+
m3
(4)
(8) T. F. Young, Y. C. Wu and A. A. Krawitz. Trans. P'aradau Sor., 24, 37 (1958).
D. STIGTER
114
Vol. 64
and 9
- 91(0)
TABLE IV =
2
2.303m [(
2m
aZl(0)
+
OrlZ(0))
m+
mZ2 m2 7 - 2a12 ml
mP2 4
- ~ 1 2 )2 - 3 m ( P Z ~ - LM
( ~ 2 1
g]
(5)
and In a, =
OSMOTICCOEFFICIENTS OF HYDROCHLORIC ACID, +l(o), POTASSIUM CHLORIDE, +2(o) m
1 2
3
w -2 55.51
0"
10'
20'
25'
30'
AND
40'
1.055 1.049 1.043 1.040 1.036 1.030 1.216 1.206 1.197 1.189 1.183 1.170 1.394 1.374 1.357 1.348 ... ...
~
Values of the osmotic coefficients of the acid and potassium chloride, +z(c), required for this calculation are given at suitable temperatures in Table IV. Summary (1) By means of equations 1 and 2 and the data in Table I, the activity coefficients of hydrochloric acid and potassium chloride may be calculated precisely in their mixtures in water a t 1, 2 and 3 total molalities and from 0 to 40'. (2) By equations 4, 5 and 6, the osmotic coefficients and the activity of water in these mixtures may be computed.
1 2
3
9W) 0.879 0.888 0.894 0.896 0.897 0.898 .880 ,897 ,908 ,912 ,914 .919 .910 .923 ,933 .937 .939 ,943
(3) Our estimate of the excess heat of mixing per mole of solute is in reasonable agreement with the calorimetric data. (4) The special value of the present contribution resides in indicating the behavior of the system over a wide range of concentration and of temperature. This contribution was supported in part by The Atomic Energy Commission under Contract AT(30-1)1375.
INTERACTIOXS I N AQUEOUS SOLUTIONS. I. CALCULATION OF THE TURBIDITY OF SUCROSE SOLUTIONS AND CALIBRATION OF LIGHT SCATTERING PHOTOMETERS BY D. STIGTER Contribution from the Western Regional Research Laboratory,1800 Buchanan Street, Albany 10, California Received July $9,1969
The thermodynamic part of the light scattering equation for two-component systems is connected with the osmotic coefficient and the solution density in a closed expression. Also the reciprocal turbidity is expanded in powers of the volume concentration of the solute. The coefficientsin this expansion are given up to that of the fourth power of the concentration. Calculations for aqueous sucrose solutions at 25' are based on literature data on the osmotic coefficient and the solution density. The results are compared with turbidity data by Maron and Lou.4 Data are tabulated for concentrations u p to 0.7 g. of sucrose per cc. of solution.
Introduction Recently Hill2 has given a treatment of twocomponent solutions in which various thermodynamic functions are expanded in powers of the weight concentration of the solute. The results have been applied to the light scattering of twocomponent system^.^ In the following, this light scattering expansion is extended by deriving the coefficients of the next three higher powers of the volume concentration of the solute. The resulting expression is used to evaluate the turbidity of aqueous sucrose solutions a t 25". I n order to investigate the convergence of the expansion, the turbidity is derived also from a closed expression. In both cases the relevant thermodynamic information is obtained from literature data on the osmotic coefficient and on the density of sucrose solutions. The object is to provide data for the (absolute) turbidity of sucrose solutions over a large concentration range which may be utilized for the calibration of light scattering photometers. (1) A lahoratory of the Western Utiliaation Research and Development Division, Agricultural Research Service, U. s. Department of Agriculture. (2) T. L. Hill, J . A n . Chem. Soc., 79, 4885 (1957). (3) T. L. Hill, J. Chem. Phys., 30, 93 (1959).
Maron and Lou4 have calculated the turbidity of sucrose solutions with the help of osmotic pressure data. Their results have been used by Princen and Mysels6 for calibration purposes. However, Hilla has shown that the thermodynamic information obtained from osmotic pressure is not identical with that involved in light scattering. This invalidates the exactness of Maron and Lou's procedure. Theoretical The general light scattering equation6-8 specialized for two-component systems may be written as
r is the turbidity due to concentration fluctuations for isotropic particles small compared with the (4) S. H. Maron and R. L. H. Lou. J . Phys. Chem., 69, 231 (1955). (5) L. H. Princen and K. J. Mysels, Office of Naval Research, Contract Nonr-274(00). Project N R 356-254, 11th Technical Report, Sept. 1958. L. H. Princen. Thesis. Utrecht, 1959. 03) H. C. Brinkman and J. J. Hermans, J . Chem. Phys., 17, 574 (1949). (7)J. G. Kirkwood and R. J. Goldberg. ibid., 18, 54 (1950). (8) W. H. Stockmayer, ibid., 18, 58 (1950).
