F. J. KELLY,R. -4.ROBIKSOX A N D R. H. STOKES
1958
T'ol. 65
TABLE 17 MOLALACTIVITYCOEFFICIESTS OF SUCROSE (B) ANI) M.~NSITOT, (C) IN MIXEDSOLUTION AT 25"" -
M C
I\. c
.1In
1
0
1
2
I ,000 1,177 1.$2-1 1.731 1,000 1.094 1.216 1.351 B 1.023 1,214 1.471 1.785 0.3 C 1.003 1.100 1.22i 1,368 0.7 B 1.056 1.267 1.540 1.862 C 1,008 1.110 1.244 1.393 1.0 B 1,082 1.310 (1.595) (1.925) C 1.013 1,120 (1.259) (1,414) V:tlues in parentheses refer to solutions supersaturated to mannitol.
B
0
C
5
calculated by equation 23 and the last column gives the percentage error in the molality of the reference solution necessary to account for the difference between the observed and calculated A values. Activity Coefficients in the Sucrose-MannitolWater System.-Integration of equation 23 gives In
YB
+ 1 mc -4 + R ~ +B Cmsz + Dmn3 + ~ E v w ] (24) 1
= In
YBO
for the activity coefficient of sucrose in the mixed solution, and In yc = In yco trig
[
A
+
+2R 7ng + 3c -
- Vln2
+ D4 vzn3 + ~ T i i c ]
(25)
for the activity coefficient of mannitol in the mixture. Solubility Relations.-The solubility of mannitol in sucrose solutions is determined by the condition (TnC5CO)sat
= (mc-Yc)sat
where the quantities on the left refer to a saturated solution in water and those on the right to saturated solutions containing sucrose. The saturated solution in mater is knomn'l to be 1.186 &I; a t this (11) J. RI
Braham J . Am. Chem. Soc., 41, 1707 (1919).
2,082 1.485 2.138 1.507 2.217 1.541 (2.282) (1.569)
1
2.458 1 .606 2.513 1,636 2,592 1.681 (2,656) (1.717)
5.9
2.806 1.707 2.862 1,i 4 5 (2.943 j (1.799) (3,009) (1.843)
molality, yc0 = 1.017 and the solubility product is 1.207 mole kg.-'. m, now can be calculated for any given sucrose molality! yc being obtained by equation 23 : a few successive approximations are needed. The solubility of sucrose in mannitol solutions likewise can be calculated, the saturated solution in water4 being 6.053 111. Tab!e IV gives some solubilities calculated in this way. S o direct measurements of these solubilities are available but Fig. 2 compares the behavior of this system a t 25" with that of the sucrose-lactoscwater ~ y s t e m a' ~t 0". The molal solubility of lactose in water at 25" is less than that of maiinitol, and this difference is further enhanced by the lower temperature (0') to which the data in Fig. 2 refer. It is clear that the relative magnitudes of the solubility lowering in the presence of sucrose are very similar. Finally, Table T' contains values of the activity coefficients of both mannitol and sucrose a t round concentrations in mixed solutions. It will be noted that each component is (lsalted'' out by the other. We wish to thank the Colonial Sugar Refining Company for the supply of purified sucrose and for the gift of a semi-micro balance. (12) P. N. Peter, J . Phys. Chem., 32, 1856 (1928).
THE THERMODYNAMICS OF THE TERNARY SYSTEM MAKNITOLSODIUAMCHLORIDE-WATER A T 95' FROM SOLUBILITY AND VAPOR PRESSURE MEASUREMENTS BY F. J. KELLY,R. A . ROBINSOX'AND R. H. STOKES Department of Chemastry, University of S e w England, .4nnzdale, S . S . W., Australia Recezzed March I S , 1861
Isopiestic measurements 0'1 the ternary ST stem mannitol-sodium chloride-water a t 25' are used to derive equations for the activity coefficients of each solute. Solubilities of mannitol in sodium chloride solutions and of sodium chloride in mannitol solutions are measured and found to be in fair agreement with those computrd by extrapolation of the activity data to the saturated solutions. Each solute is "salted-in" by the other.
Introduction The activity coefficients of mannitol and sodium chloride in mixed solutions are required in conneetion with our current studies of diffusion in tllreecomponent systems. ~h~ data can he obtained from isopiestic measurements of the vapor pres(1) Notional Bureau of Standards, Washington 25, D. C.
sures of mixed solutions, and such measurements are reported in Table 11. This method has the advantage of being in Principle applicable at any ComPosition of the mixture, but in practice experimental limitations confine its usefulness to cases where the total concentration of the two solutes is fairly large. The study of the solubility of one solute in the presence of varying concentrations of the other
yields the activity coefficient of the first more directly, hiit only over a limited range of molalities of the first' m l i i t P , P ~ L ,thosc of its saturated solutions. Experimental Mannitol ~ r n spurified by three recrystallizations from conductance water, and dried in 2rucuo a t room temperature. Sodium chloride xas of Analytical Reagent, qiiality, dried a t 400". Conductance water was used for all solutions. I . Solubility of Sodium Chloride in Aqueous Mannitol Solutions.-Xlannitol solutions of the compositions s h o m in Table I were made up by weight, and portions were sealed up in 6" X I " Pyrex tubes along with excess sodium chlo. ride. The tubes w x e rotated end over end in a 25' thermostat for several days. Without removing t,he lower part of the tube from the thermostat, the ends were opened and a SOLUBILITIES IS
TABLE I SI-STEXMANSITOL-SODIUM CHLO-
THE
RIDE-WATER AT 25"" (a) ---mFJ-
Ohsd.
90111s. satd. wit!i manxito1 ( R )
ih) S o h . sntd. with sodium chloride (C)
r'mC----
--7
Calcd.
mc
mu
Obsd.
Calod.
1.185 (1.185) 0 0 6.147 (6.147) 1.206 1.'205 1.549 0,4601 6.214 6.216 1.'242 3.202 ,8166 6.270 6.272 1.240 1.332 1.1342 5.146 1,1947 6.322 6.334 1.381 L.395 5.772 1.3956 6.360 6.366 1.411 1.427 6.089 0 Calculated values obtained by extrapolation from the isopiestic vapor pressure results on mixed solutions. m = moles solute/'kg. water. pipet, fitted with a sinteredoglass filter tip, previously warmed t o slightly above 25 , was used t o withdraw a sample of the liquid phase. This was weighed and then analyzed gravimetrically for chloride as silver chloride. The results, expressed on the molality scale, are given in Table I. The value otitained for the solubilit,y in water alone, 6.147 mole/kg., is in good agreement with published solubility data2 and also agrees exactly to the fourth significant figure with unpublished measurements made by us by the isopiestic method described by Scatchard, Hamer and Wood.3 11. Solubility of Mannitol in Sodium Chloride Solutions. -Because of' the difficulty of precise chemical analysis for mannitol in the presence of large amounts of sodium chloride, the following technique, not involving analysis, was used. A sodium chloride solution of known molality was made up by weight. Six portions of this solution were weighed into sample tubes, and known weights of mannitol were added in regularly increasing amounts, so that the first tube would certainly form a solution unsaturated with mannitol, while the sixth tube would contain a readily visible excess of solid mannitol. 'The sealed tubes were rotated end over end in a glass-fronted water-thermostat a t 25" until equilibrium was reached, t,he rotation being periodically stopped for visual inspection of the tubes. Thus the composition of the solution saturated to mannitol could be estimated as lying between the compositions of two successive tubes in the series, in one of which no excess solid was visible, while the next contained a small but visible amount. On the basis of this estimate a further set of six tubes \vas made up, in which the interval betu-een the successive amounts of niannitol added was much smallcr; this enabled the solubility to be brackrted within O.2q0,. This method proved simple and reliable, a determination of the solubility of mannitol in water alone at 25' yielding 1.186 molal in good agreement m-ith the value of 1.186 M calculated from the data of Braham.l The method is of course applicable only t o substances of relatively high solubility, since one can scarcely detect less than a milligram of excess solid visually. The results are given in Table I. 111. Isopiestic Measurements on the System MannitolSodium Chloride-Water .-Thc Pxperimmtal technique and ( 2 ) A . Seidell, "Solubilities of Inorganic a n d Metaloreanic Compounds," 3rd ed., D. Van Nostrand. Y e w York, N. P.. 1940. (3) G. Scatchard. W. J. Aamer a n d E. E. Wood, J . A m . Chem. S o c . , 60, 3061 ilY381. (4) .J. 31. Biaham, i h i d . , 41, 1707 11919).
TABLE 115 mrcf
mB
nic
_ _ _I1
mBmc
Diff. yo
0.2465 0,5700 0.0164 +0.03 ,4871 .0107 - .03 .4400 ,0110 ,2324 - .01 ,8640 ,0205 1 2616 ,2297 1.1483 .08 ,0172 ,4713 .08 1.0259 ,0154 ,7159 0.8993 .08 ,9595 ,0138 0.7702 .08 ,2756 1.4870 1 6178 .0098 - .08 1.3621 .0111 - .04 ,5337 .0091 .04 ,7391 1.2592 .ll 1.1740 ,9070 .0093 2 2253 - .01 2.0923 ,3091 .0189 2.0541 3946 .0165 - .04 2,0342 ,0162 - .05 ,4391 1.9316 .6672 ,0154 - .05 1,9013 ,0172 .02 .7362 ,0134 - .03 1.7129 1.1344 - .04 2.9900 0,2856 3 0957 ,0248 2.8905 ,5505 ,0264 .02 2,7872 0 ,8125 ,0239 $0.02 2.6857 1.0622 ,0224 ,0409 3.9097 0.2296 3 9788 .01 3.8348 ,4713 ,0393 .02 3.7631 ,6947 ,0378 .04 ,0370 3.6835 ,9377 .09 .04 5.0613 ,2253 ,0567 5 1119 5,0142 .02 ,4370 ,0581 4.9609 ,0586 ,6707 .05 4.9023 ,9096 .0566 .04 ,2717 .0671 0 5.2843 5 3382 $0.02 5.2249 .0641 .5507 5.1642 ,0613 0 .8191 5,0904 .0604 1,1401 $0.08 5.9159 ,0822 5 9544 0,2584 .02 5.8743 ,0769 .5026 - .02 5.8359 ,0757 - .02 ,7285 1,0482 5.7784 ,0744 .01 5 B = mannitol; C = SaCl; the reference solute was sodium chloride. A: defined by equation 1. ''Diff.%": calculated by equations 1 and 5 with a = -0.0145, b = 0.00184, c = -0.0022, e = 0.0032. 0 io00
+ + + +
-
+
+
+ + + + -
+ +
+
+
the theory of the method for the case of two non-electrolyte solutes have been given in a previous paper from this Laboratory.6 When one solute (C), and the reference solution are 1: 1 electrolytes, the only modification required is the introduction of a factor 2 A = 2mrer vrei - ~ ~ Z B V B' 2?nc.pc0 (1) A b In bln ?C -- = = 2 (2) mmc mc Equation 2, as before, is valid subject to the condition that the cross-differentials may be adequately described by an equation of the form
(z)
Whether this holds must be determined experimentally in rach case; with the present system, it does. Table I1 gives mrefrthe molality of the reference solution, sodium chloride, in equilibrium m-ith mixed solutions of molalities ?ng in mannitol and mc in sodium chloride, and values of A defined by equation 1. In computing A,pc0 ( 5 ) R. A . Robinson a n d R. H. Stokes, J . P h y s . Chem., 66, 1954 (1961).
F. J. KELLY,R. A. ROBINSON AND R. H. STOKES
1960
Vol. 65
TABLEI11 MOLALACTIVITY COEFFICIENTS OF MANNITOL (€3) AND SODIUM CHLORIDE (c)IN MIXEDSOLUTION AT 25’ MB
1
0
0
B
0.3
B
0.7
B
1 .o
B
1.000 1 .ooo 1.003 0.9980 1.008 0.9953 1.013 0.9936
c c c
c
0.986 .6569 .989 .6555 .996 .6538 1.002 0.6526
2
0 969 .6676 .974 ,6656 .981 .6633 .988 ,6616
for sodium chloride was taken from standard tables,G and for mannitol was calculateds by the equation pg0 = 1 0.0034m~ 0.0042m~e (4) Equations for the Activity Coefficients in the Mixed Solutions.-A three-dimensional graph was prcpared on c ) represented by metal rods which the quantity A / ( ~ B v ~was of appropriate length, set into a wooden baseboard representing the mB-mC plane. Inspection of the surface so formed suggested the equation A - a h YB 2 ?I In __ -)mB z ! (-bn.B?-C) = mBmc mc a bmc cmc2 em8 (5) The constants a, b, c and e then were determined by the method of least squares to have the values shown in Table 11. The final column of this table shows the accuracy with which the experimental results are reproduced by equation 5; the quantity “diff .yo” is defined as follows mref(cale) = ( P B O ~ B 4-2 d m c 4- Aeaie)/(2vref) where A is given by equation 5 and “Diff Yo’’ = mref(”hs) _- m rf(oaio.) _ x 100~ mref(oi,s) The “difference” column of Table I1 therefore gives the percentage accuracy with which the molality of the reference solution in equilibrium with a given mixture can be predicted with the aid of equations 1 and 5. Alternatively, it may be regarded as giving the percentage difference between the calculated and observed vapor pressure lmerings of the mixed solution. The mean difference for the 37 solutions is +0.04%, which is well within the estimated accuracy of the measurements, since each value of A/( mBmc) really involves three measurements, one for each solute by itself and one for the mixture. Integration of equation 5 and conversion to common logarithms yields the following expressions for the activity coefficients of each solute 1 log Y B = log YB” 0 . 4 3 4 3 ~ u~ $mc
+
+
+
+
+
~
+
[ +1 +
3Cmc2 -i-m B ]
log yc = log
YCO
(6)
+ 0.21715mB
R. H. Stokes, “Electrolyte Solutions,” 2nd ed.. Butterworths London. 1969. ( 6 ) R. A. Robinson and
M. 3
4
0.946 .7137 .952 .7107 .960 .7068 .968 .7040
0.914 .7832 .920 .7782 .929 .7720 .938 .7676 [a
6
0.868 .8740 .875 .8662 .885 ,8561 .894 .8487
6
0.809 .9862 .816 .9742 .826 .9585 .835 .9471
1 + bmc + m c 2 + ,em,]
(7)
where log YB” is obtainable from equation 4 by the GibbsDuhem equation and log yc0 from the Etandard tables for eodium chloride. Inserting the numerical values we have log YB = 0 . 0 0 2 9 5 ~ ~0~. 0 0 2 7 4 ~ ~-~mc[0.00630 ’ 0.00040??~~0.000318mC2- 0.00139mBI ( 8 ) log yc = log yc” - ~ ~ ~ [ 0 . 0 0 3-1 0.00040mc 5 0.000478mc2 - 0.00035mB] ( 9 )
+ +
+
Calculation of Solubilities.-The solubility of mannitol in water being 1.185 molal, a t which concentration the activity coefficient is 1.017, we can calculate its solubility in sodium chloride solutions (Table I) by the relation
1%
0 0811
- log YB(8s.t)
(10) where log y ~ ( ~ is given ~ t ) by equation 8 with mc equal to the sodium chloride concentration of interest. A first approximation for YB is obtained by using a n estimated value of mR in equation 8; this then IS substituted in equation 10 to yield a better approvimstion to m ~ ( ~ ~ ~ ) ; this cycle is repeated, convergence being very rapid. Similar calculations from equation 9 give the solubility of sodium chloride in mannitol solutions. Since the solutes are mutually salted-in, the values of BO and yco in equations 6-9 must be evtrapolated beyond the range of the experimental data for aqueous solutions of the pure substances. In the case of sodium chloride, the standard tables show that log ycO is nearly linear in mc in the 6 M aqueous solution, and the extrapolation ww made by assuming the formula
1
nzB(mt) =
+ log *,c0 = 0.9940 + 0.0545(mc - 6)
Agreement between the measured and Calculated solubilities is excellent. For mannitol, we have had to assume that the equation log ?go = 0.00295m~ 0 . 0 0 2 7 4 m ~ ~
+
(derived from equation 4 by the Gibbs-Duhem equation) remains valid above the saturation molality. This extrapolation is clearly less satisfactory than that for sodium chloride, and may be a reason for the discrepancies of the order of 1% between the measured and calculated mannitol solubilities a t the higher concentrations of sodium chloride. Table 111 gives activity coefficients for both components in the mixtures at round concentrations, calculated from rquntions 8 nnd 9.
