MOLARVOLUMES AND TEMPERATURES OF SUCROSE SOLUTIONS
363
Apparent Molar Volumes and Temperatures of Maximum
Density of Dilute Aqueous Sucrose Solutions by John E. Garrod and Thelma M. Herrington Department of Chemistry, The University, Reading, Bsrlcshire, England
(Received June 9, 1969)
Apparent molar volumes for solutions of sucrose and water are determined over a large range of concentration using both pycnometric and dilatometric techniques. The temperatures of maximum density of dilute sucrose solutions are found. For the interaction between sucrose and water, a rigid particle model combined with an attractive square-well potential is assumed. On this model the attraction between sucrose and water increases with temperature. It is also shown that a solution of sucrose and water cannot be considered to be "semiideal dilute" since a necessary criterion between the activity coefficients is not satisfied. There has been considerable interest recently' in the effect of hydrogen-bonding solutes on the temperature of maximum density of water. Whether a solute raises or lowers the temperature of maximum density depends on the rate of change of the solutesolvent intermolecular forces with temperature. The partial molar volume of the solute at infinite dilution is a measure of these intermolecular forces. Sucrose was chosen as its behavior should be typical of a solute which hydrogen bonds with water. The apparent molar volumes of dilute sucrose solutions were determined so that accurate extrapolation to infinite dilution could be made. Previous density measurements2 on sucrose solutions were carried out on solutions too concentrated to allow an accurate extrapolation. The dependence of the lowering of the temperature of maximum density of water on the molality of the solute is directly related to the rate of change with temperature of the partial molar volume of the solute. Our results were compared with previous measurements of the temperature of maximum density of sucrose solution^;^ good agreement was obtained between our own measurements of the temperature of maximum density of dilute sucrose solutions and the temperature dependence of the partial molar volume of sucrose.
and p is that of the solution. For a 1 m solution, the error in molality, 6m, must be -loq4 mol kg-l and 6(plo - p ) , -10-5 g cm? to give an accuracy of -0.01 em3 in +V, whereas, for a 0.1 m solution, 6m must be -loe5 mol kg-l and 6(pIo - p ) , g cmW3for the same accuracy in 4V. Thus, at concentrations greater than 0.5 m, density determinations with an Ostwald pycnometer give @V with sufficient accuracy, but at lower concentrations another method must be used. Consider mixing a small volume of a concentrated solution containing n2 moles of solute and of apparent molar volume @VTInit. with a much larger volume of pure solvent; then the apparent molar volume of the resulting solution is given by
Theory
search department of Tate and Lyle, Ltd. It was dried in a vacuum oven at 60" before use. The conductivity water used for preparing the solutions was made by passing distilled water through a cation-anion exchange resin: the specific conductance was 5 X ohm-' cm-'. As a further check on any impurities, a large sample (approximately 1000 g) was evaporated slowly to dryness; it was found that any nonionic impurities present were less than one part in lo6.
The apparent molar volume, ' V , is defined by ,b
V
=
(V - nlV1")/nz
( T constant)
(1)
where V is the total volume of the solution, Vlo, is the molar volume of pure solvent and nl and n2 are the number of moles of solvent and solute respectively. In principle, @Vcan be determined from density measurements; the appropriate equation is
'I/' =
roQ0
+P mM2 - E}/m P1"
(2)
where m is the molality of the solution, M2 is the molar mass of solute, p1O is the density of pure solvent,
(3)
where AV is the volume change on mixing. Thus, if @VInit.is known with an accuracy of 0.01 em3 mol-' from pycnometer densities, and AV/nz can be determined with a similar accuracy, 'VFinal is known to *0.02 em3 mol-' in the final dilute solution.
Experimental Section Materials. The sucrose was supplied by the re-
(1) F. Franks and H. T. Smith, Trans. Faraday Soc., 64,2962 (1968) ; F. Franks and B. Watson, ibid., 63, 329 (1967). (2) I. F. Schneider, D. Sohliepake, and A. Klimmek, Zucker, 16, 17 (1963). (3) G. Wada and 8 . Urneda, Bull. Chem. SOC.Jap., 35, 646 (1962). Volume 74, Number d
January $2, 1970
364
JOHN
E. GARROD AND T H ~ L M M,AHERRINGTON
with a 0.15-om capillary. Flask A was used for a final concentration of less than 2 X rn and to 1 X 10-1 m. flask B from 2 X To determine AV/n2 to 0.01 em3 mol-l, 6nz must be -10-5 mol and 6A.8 known to one part in lo7 for an initial 1 m solution and a change in meniscus height of 10 cm. To have an error in 6n2 of &lo-5 mol, the number of moles in one cubic centimeter of the initial solution must be known to and the capsule volume to +0.01 cm3. The total volume of the flask is only required to calculate the final molality of the dilute solution; it was in fact determined to h 0 - 1cm3.
Technique Figure 1. The dilatometer (not to scale).
The conductivity water was outgassed on a water pump and then further degassed by keeping at 10" above thermostat temperature for several hours for the measurements at 25". Apparatus. The thermostat used was an insulated copper tank containing 220 1. of demineralized water; a plate glass window was mounted in one side of the tank so that manipulation of the contents of the dilatometer could be readily observed. The tank was thoroughly stirred and its temperature controlled by a mercury-decane regulator. For measurements below room temperature, chilled water was passed through a coil immersed in the tank. The temperature of the tank was determined with an K.P.L. certified platinum resistance thermometer and a transformer ratio bridge to ~ 0 . 0 0 1 " . Ostwald-Sprengel type pycnometers made of Pyrex glass, of volume 25 cm3 and with 0.1-cm diameter capillary tubing were used to determine the densities of the concentrated sucrose solutions. Duplicate density determinations agreed to 1 X 10-6 g ~ m - ~ . The dilatometer illustrated in Figure 1 is similar in construction to that described by Hepler, Stokes, and Stokes.4 It is essentially a large flask of known volume with a precision bore capillary tube attached which can be mounted vertically in the thermostat. Changes in volume of the contents of the dilatometer are observed by measuring the height of liquid in the capillary tube with a cathetometer. A glass tube joined to the stopper of the flask forms part of a capsule to hold the concentrated solution. The glass lid of the capsule, shaped like a plano-convex lens, is a ground fit; it is lightly greased with stopcock grease, or pure petroleum jelly for the lowest temperatures, and held in place with light springs. The same capsule of v01ume 11 cm3 was used throughout but two sizes of flask were used; flask A was of volume 1100 cm3 with a 0.075-cm diameter capillary and flask B was 250 em3
*
The Journal of Physical Chemistry
All solutions were used within 3 days of making up, as it was found that in certain samples biological action could be observed within 2 weeks (see also Robinson and Stokes5). The capsule of the dilatometer was filled with a concentrated solution of known density at thermostat temperature and inserted into the flask filled with degassed conductivity water also at thermostat temperature (for water, ( b p / ? ~ T = )~ 4 atm OK-'). The dilatometer was then carefully checked for entrapped air bubbles and left to equilibrate in the thermostat; a magnetic stirrer was used to speed up the attainment of equilibrium. After about 30 min, the level of the meniscus was observed every 5 min until it was constant to *0.001 cm for two consecutive readings. This took about 1 or 2 hr, depending on the temperature. The capsule lid was then slid off by pulling on the attached Teflon encased magnet with a powerful permanent magnet adjacent to the outside of the flask. The solution was stirred t o mix thoroughly. Observations of the meniscus height were taken as before until two consecutive readings agreed to hO.001 cm; this was achieved within about 1 hr of mixing. Lines etched on the capillary tube were used to check that no movement of the dilatometer assembly OCcurred. A volume change of one part in lo8 was detectable with the large dilatometer, and one part in lo' with the small dilatometer. However, a volume change of one part in 107 is equivalent to a change in the average temperature of the thermostat of 5 X a change in room temperature of 10"(0.5") or a change in atmospheric pressure of 133 N m-2. Except on days of very rapidly changing barometric pressure, these conditions were easily satisfied for the time of 1 hr taken in complete mixing. The principal sources of error were (a) the change in pressure due to changes in the hydrostatic head of (4) L. G. Hepler, J. M. Stokes, and R. H. Stokes, Trans. Faraday SOC., (1965). (6) R. A. Robinson and R. H. Stokes, J . Phys. C h ~ m .6, 5 , 1954 (1961).
61,ZO
MOLARVOLUMESAND TEMPERATURES OF SUCROSE SOLUTIONS liquid in the capillary and (b) the change in temperature of the liquid displaced from the flask into the capillary. I n (a) a change in height of 1 ern of liquid in the capillary is theoretically equivalent to a volume change of 1 part in 10'. The maximum change in height observed was 10 em. The effect of pressure on the meniscus height was determined experimentally and was in fact found to be about 3 parts in lo8 for a 1 cm change in meniscus height. Calibration curves were plotted for both flasks and appropriate corrections applied. For (b) a difference between room temperature and thermostat temperature of 15" corresponds to a volume change of 5 parts in lo8 for a change in meniscus height of 10 ern (0.5 cm) ; a correction was applied. Temperature of Maximum Density. The temperature of maximum density of each dilute sucrose solution left in the dilatometer after mixing at 5" was found by measuring the change in meniscus height with temperature. The temperature of the thermostat was regulated to within 10.001" ; temperature differences were determined to 10.01" using a 6" Beckmann thermometer calibrated at 5°C against the N.P.L. certified platinum resistance thermometer. Measurements ol meniscus height were made with a cathetometer to *0.001 cm at 0.2" intervals with +1.5" of the temperature corresponding to the minimum in meniscus height, @,(observed). The dilatometer attained thermal equilibrium with the thermostat within 0.5 hr of a temperature change; readings of meniscus height were taken every few minutes until two consecutive readings agreed to *0.001 cm. The plot of meniscus height us. temperature was approximately symmetrical (parabolic) about @,(observed) for each solution. The temperature, &>(observed), corresponding to the minimum in meniscus height is not the true temperature of maximum density of that solution because of the thermal expansion of the dilatometer itself. Each dilatometer was calibrated by doing a run using pure water (0, = 3.98') and the true temperature of maximum density of each solution determined as explained below. The values of 0, are estimated to be accurate to .t0.04". The fact that the emergent stem of the dilatometer was at room temperature was calculated to affect 0, by less than 0.001"(0.02"). Kow the measured volume change is equal to the expansion of the liquid of volume V less the expansion of the Pyrex glass dilatometer, so that Adh
N
d l i - dli,
365
and
v
=
v* [I + a(@-
Om)2]
(6)
where 0 is the temperature, 0, is the true temperature of maximum density of the solution, V* is the volume of solution at Om, ag is the coefficient of expansion of the glass vessel, and a the coefficient of expansion of the solution, then dh/dO
=
K
+ 2V* a(@- @,)/A
(7)
where K = -Vgoag/A and is a constant for the particular vessel. Accordingly, if Ah/AO is plotted against 0 for pure water then K is equal to the value of Ah/AO at 0 = 3.98". For each solution Ah/AO is then plotted against 0 and when Ah/AO is equal to K then 0 = 0, for that solution. The temperature of maximum density could have been obtained by plotting h against 0 and then correcting @,(observed) by the difference [@,(observed) e,] of pure water for that dilatometer; however, this assumes that the coefficient of expansion of the solution is equal to that of water.
-
Results Determinations of the densities of the concentrated solutions were made at 5, 15, and 25". The pycnometers were calibrated with conductance water immediately after a density measurement; buoyancy corrections were applied to the masses. Densities for the concentrated solutions are given in Table I. Values for the density of water were taken from the compilation of Tilton and Taylor.6 Values for the apparent molar volumes were calculated at these molalities; these values were plotted together with values at other molalities calculated from data in the literature and the results smoothed graphically to give the apparent molar volumes recorded in Table I as 'Vinitial smoothed. Values for the apparent molar volume were calculated at 25" and 15" from the density data of SchneiderI2at 25" from the data of Mantovani and Indelli,' and at 5" from extrapolation of Plato's d a h 8 The values of the apparent molar volume and molality obtained on diluting the concentrated solution in the dilatometer are recorded in Table I as 'VFinal and final molality. The smoothed values of the initial apparent molar volumes were used in calculating the apparent molar volumes of the dilute solutions. The apparent molar volumes at each temperature
(4)
where A is the cross-sectional area of the capillary, V , is the volume of the vessel, and V is the volume of solution. If we write
(6) L. W. Tilton and J. K. Taylor, J . Res. Nat. Bur. Stand. 18, 205 (1937). ( 7 ) G. Mantovani and A. Indelli, International Sugar Journal, 6 8 , 104 (1966). ( 8 ) F. Plato, Kaiserlichen-Normal-Eichungs Kommision, Wiss. Abh. 2, 153 (1900). F. T. Bates, "Polarimetry and Saccharimetry of the sugars," National Bureau of Standards Circular 440, U. S. Government Printing Office, Washington, D. C.
Volume ?4, Number .9 Januaru 2.9,1,970
JOHN E. GARROD AND THELMA M.
366
HERRINQTON
~~
Table I : Densities and Apparent Molar Volumes of Sucrose Solutions 10e
x ma88 fraction
Denjity/ g om-8
of aucrose
d vInitial . . smoothEd/ om3 mol-1
Initial molality/ mol kg -1
IO2 X Final molality/ mol kg -1
VFinal/
cma mol-1
6 , ooo Water
0.999964 1.10693 1.18078 1,22555 1.25100 1.26231
24.836 39 I748 48.117 52 683 64.623 I
0.9653 1.9272 2.7093 3.2527 3.6166
209.16 210.52 211.41 211.93 212.18
0.759 1.296 6.465 1.820 7.858
207.60 207 64 207.72 207.68 207.80
211.19 211.36 212.02 212.10 213.01 213.02 213.69
0.747 3.306 1.167 4.775 6.433 1.628 7.572
209.96 210.04 209 98 210.05 210.03 209.97 210.10
212.04 212.16 212.23 212.66 212.89 214.60 214.97
0.379 0 458 0.497 0.750 0.875 6.955 1.920
211.43 211.50 211 48 211.49 211.52 211.61 211.54
I
15.00'
Water
0,999101 1.10310 1.11484 1.16019 1.16513 1.22141 1.22263 1.25968
24.536 27.074 36.437 37.412 48.091 48.265 54.900
0.9498 1.0846 1,6747 1.7462 2.7065 2.7254 3.5563
I
25.00'
Water
0.997047 1.04961 1.10606 1.06589 1 10047 1,11757 1.23516 1.25726
13.082 15.670 16.913 24.713 28.404 51.433 55.372
0.4397 0.5428 0 * 5947 0.9589 1,1590 3.0938 3.6247
I
I
Table I1 : Sucrose: Coefficients of the Polynomial
T/OC
P P om8 mol-'
5.00 15.00 25.00
207 62 209.97 211.49
A' x 1041 kg mol-1 atm-1
I
1.752 1.166 1.107
were fitted by least squares to a polynomial in the molality. The best fitting polynomial was found using the null hypothesis, Le., the polynomial is assumed to be the best fit which gives the minimum value of 8pz/(n-p-l), where is the sum of the squares of the deviations, n the number of points, and p the degree of the polynomial. If we write (see later) @V= V2*
+ RT
A'm 1
-
4
Ct'
v2*
+ -51 Of' m4 + . . .
3
where is the partial molar volume of sucrose at infinite dilution, T the absolute temperature, and m the molality, then the coefficients of the best fitting polynomial are given in Table 11. The pycnometric data were given twice the weighing of the dilatometric The Journal of Physical Chemistry
-5.61 -1.13 -1.64
Ct' x 1061 mol-a atm-1
kg3
Dt' X loa/ kgr mol-4 atm-1
20.18 -0.20 1.15
-3.16
data. (The results were compared with calculations giving the data equal weighting: the value of V2* was not affected; A' was altered by 2% and the higher coefficients by rather more). The degree of the polynomial did not affect V2*, but A' was affected by 5%. Table 111: Temperature of Maximum Density of Sucrose Solutions
+ -31 Bt' m2 + m3
Bt' X l O S / kga mol-* atm-*
10% X Molality/ mol kg -1
Water 0.759 1.296 3.464 4.965 6.465 7.558
em/ oc
ABm/ OK
(3.98) 3.84 3.70 3.46 3.16 2.92 2.68
-0.14 -0.28 -0.52 -0.82 -1.06 -1.30
n/IOLAR VOLUMES AND
TEMPERATURES O F SUCROSE
367
SOLUTIONS
-1.4
-
-1.2
-
where gz0 is the partial molecular volume of the solute a t infinite dilution. Thus the total volume, V , of the solution is given by
-1.0
-
V / N l = 4"
-0.8
-
-0.6
-
-0.4
-
AOfK
0 A
1 - B'222m3 3 THIS
3
4
5
6 102m
a
;7
+
- AZ2'm2
+ 41- C2222'm4+ 51 D'22222m6 + . . .}
(13)
where the dash represents differentiation with respect to pressure, and the apparent molar volume is given by
AND
+V = V2*
I 2
{:
WGRK
WADA
UMEDA
1
+ mi&€+ kT
+ RT
A'm
9
+ 31- B'm2 +
mol hg-'
Figure 2. Lowering of the temperature of maximum density of water by sucrose a t low values of the molality.
The change in the temperature of maximum density of water produced by a solute is defined by AB,/"C
=
Bm/"C - 3.98
(9)
The observed values of 0, and ABm for dilute sucrose solutions are given in Table 111. The data are plotted in Figure 2 together with data obtained by Wada and Umeda.3
Discussion (i) Partial Molar Volumes. Let us write for the Gibbs energy of a solution of mole ratio
+
G/NtkT = ptD/kT %p2"/k/'
where m is now the molality, V2* the partial molar volume of the solute at infinite dilution, and A' = (1000/M1)2A22',B' = (1000/1V1)3B222'etc. ( M t is the molar mass of solvent.) According to the theory of RlcMillan and Mayer," for a solution of a solute in a solvent the osmotic pressure, II, is given by
II/kT = p
- f i + In @z
+
+
(10)
where m = NP/N1(Nt and Nz are the number of molecules of solvent and solute, respectively) and the coefficients Azzetc. are functions of temperature and pressure only. Then for the partial molecular volume of the solvent
A22gI0 =
2B22*'
- i&*
+ bito
(16)
where B Z 2 * O = bozo; b02' is the cluster integral for two molecules of solute in pure solvent and bllo is the cluster integral for one molecule of solute and one of solvent in pure solvent. In relating coefficients higher than AZ2( i e . , B222 etc.) to the coefficients B*O, it must be remembered that these equations are only applicable in dilute solutions when y 1 N In y ; in more concentrated solutions
-
+ B?m2 + . . .
(17)
B t = B222 - + A d etc. Thus, instead of eq 14 in general we write
(18)
In y = Az2m
$V
is the molecular volume of pure solvent, and for the solute
(15)
where p is the number density of the solute. Hillt2 has shown that the coefficients A22 etc., may be related to the coefficients BE* etc. For example
m9*lo
1 - D22222~j~~ . . . 5
where
+ B2z*p2 + B2z2*pa + . . .
=
Ir,"
+ RT
A'm
+ -31 Bt'm2 +
It is preferable to determine the coefficients A' etc., from the functional dependence of the apparent molar (9) T. L. Hill, J . Amer. Chern. Sac., 79,4885 (1957). (10) J. E. Garrod and T. M. Herrington, J . Phys. Chem., 73, 1877 (1969). (11) W. G. MoMillan and J. E. Mayer, J . Chem. Phys., 13, 276 (1945). (12) T. L. Hill, J . Chem. Phys., 30,93 (1959).
Volume 74,Number 2
January $2, 19'70
JOHNE. CARROD AND THELMA 34, HERRINGTON
368 volume on the molality rather than that of the molar volume as it can be seen from eq 8 and 13 that the polynomial is of lower degree. S t i g t e P has used eq 16 together with activity coeffi~ient'~and osmotic pressure15 data on sucrose solutjons to evaluate b11'. He finds N0bl1' = -257 cm3 mol-l. S o w , according to Garrod and Herringtonlo eq 69 b11'
= -fize
+ kTx
(19)
where 7~ is the isothermal compressibility of the solvent. Values for NaBn*" for sucrose and water calculated from our values for aye are given in Table IV. Compressibility data for water were taken froni PeGa and 31cGlashan.16 The magnitude of NOBll*' increases slightly with temperature. We can consider Bll*' as being compoeed of an attractive and a repulsive contribution from the intermolecular forces. Now
Bll*'
=
4n
Lrn (1
e-o"/kT
) r 2 dr
(20)
where oll is the potential of average force between one molecule of solute and one of solvent in the pure solvent, including averaging of the force over all rotational coordinates. Let Rlz be the distance of closest approach between the centers of two molecules; then
and a hard prolate ellipsoid with long axis 2ba and short axis 2az Bll*' = 4 - ru13 3
+ 4 razz b2 + 2nalbz az (1 -
[{
-
3
e} + a1 (1 + E
+
E ~ ) " ~
I)'"
___ - " In
2E
(24)
1 - s
where e 2 = (bz2 - a2z)/ba2. A model of sucrose was constructed using CoreyPaulirig models and taking into account crystallographic datal9 which showed that there are two H bond linkages between the pyranose and furanose rings. The molecule may be approximated by a prolate ellipsoid with semiaxes 5.9 A and 3.5 Lk, The water molecule was assumed to be a sphere of radius 1.52 8. It was found that for a rigid sucrose and water molecule
S
=
588 =t 5 cm3 mol-l
This may be compared with Stigter'slz values of 655 om3 mol-'; he used values of fi2* for sucrose to calculate values of u2 and bz. Thus at 25', the attractive contribution to Bll*' is given by Nab, = N*B11*' - NaS = -378 em3 mol-1
A very approximate estimate of the effect of hydrogen bonding on the attractive part of Bll*' may be made by considering the sucrose molecule to be a right angled parallelepiped surrounded by a square-well potential. Then b,=
where S is the repulsive and b, the attractive contribution. The signs of the integrals can be obtained from considerations of the magnitude of wll. In the region 0 < r < R,,, (311 is mostly positive and very much greater than IcT, so that X is positive. For Rlz < r < a , all is negative and comparable with 1cT so that b, is negative. Thus the sign and magnitude of Bll*' depend on the relative magnitudes of S and 4. For two hard spheres BIZ*' is given l7 by
where R1 and Rz are the diameters of the spheres. Isiharal6 has evaluated B22*' for noninteracting rigid ellipsoids; he finds that (23) where vz denotes the particle volume and f is equal to unity for a sphere but increases when the particles become more asymmetrical. If we take formulas 2, 4, and 21 from his paper and remember that Bll*' must reduce to the case Of two hard spheres when the eccentricity, e = 0, then for a hard sphere of radius a1 The Journal of Physical Chemistry
where D is the width of the well. w l l is taken arbitrarily to be -4.0 kcal mol-' at 25'; this is the mean of the values given for the enthalpy of formation of the H bond in methanol and in water by Pimentel and ;\fcClellan.20 D is found to be 0.72 A, which is a reasonable figure for atomic vibrations in the 0-H . . . 0 bond. The absolute magnitude of 4 decreases slightly with increasing temperature (Table IVA) , assuming that the rigid particle contribution to Bll*' is unchanged. If we assume that D is constant then all decreases with decreasing temperature, (13) D. Stigter, J . Phys. Chem., 64, 118 (1960). (14) G. Scatchard, W. J . Hamer, and S. E. Wood, J . Amer. Chem. Soe.. 6 0 , 3061 (1938); R. A. Robinson and D. A. Sinclair, ibid., 56, 1830 (1934). (16) H. N. Morse, Publ. CarnegieTnst. Wash., 198 (1914). (16) M. Diaz Peza and M.L. LfcGlashan, Trans. Faraday SOC.,55, 2018(1959). (17) B. H. Zimm, J . Chem. P ~ U S . , 14, 164 (1946). (18) A. Isihara, ibid., 18, 1446 (1950). (19) G. A. Jeffrey and R. D. Rosenstein, Adzan. Carbohyd. Chem., 19, 11 (1964). (20) G. C. Piinentel and A. L. RlcClellan, "The Hydrogen Bond," w. H. Freeman and y e w Pork, 1960. 3
co.,
MOLARVOLUNESAND TEMPERATURES OF SUCROSE SOLUTIONS Table I V A. Sucrose. Attractive Contribution to the Solute-Solvent Interaction Coefficient R Tx/ T/CO
om3 mol-1
NABI*O/ cma mol-1
5.0 15.0 25.0
1.14 1.13 1.13
206.48 208.84 210.36
NA#/ om8 mol-1
- 382 - 379
Ts4/
RTx/
om2 mol-'
om8 mol-1
88.0a 87.9 87.8
0.5 5.0 25.0
kcal mol-1
-3.74 -3.87 (-4.00)
-378
N~Bii*'l NA#/ om3mol-1 om3 mol-'
86.4 86.2 85.5
1.6* 1.7 2.3
a
loll/
-146.6 -146.8 -147.5
-3.67 -3.73 (-4.00)
as shown in Table IVa. On the other hand, if we assume that M i l is independent of temperature, then D increases with temperature (Table VI). (ii) The Temperature of Maximum Density. Let us rewrite eq 13 in the form = 55.51v1"
+ mv2* + A'm2
+ -31 Bt' rn3 . ,} ,
(26)
and for the temperature dependence of the molar volume of water in the neighborhood of its temperature of maximum density 0,
VIo = VIo*[I +
=
-
- em)2]
(27)
1 [bV2*/bT]em m2 55.51 al'(;;*o
1 [b(TA')/dT]e, -R m2 4 55.51 a 1 V 1 * O
+ ...
(28)
The first term on the right-hand side of eq 28 is the "ideal dilute" contribution to A@,, AO,(ideal dilute), and the second term is the nonideal dilute contribution, AO,(nonideal dilute). S o t e that the first term is linear in the molality whereas the second term is a function of the square of the molality. Now from (16) and (19) A,zq"
- 2bo2"
+ 2b1i0 - kTlt
(29) Thus, AO(nonidea1 dilute) is a measure of solutesolute and solute-solvent interaction, but from =
Wada and Umeda, ref 3.
E/QK* mol-$ kga
X/QK. mol-a kga
4.2 1.5 -3.2 -4.1 -18.3 - 18 - 16
-10.3 -2.0 -0.4 -1.5 2 X 10' 6 2
3.8 0.3
* Our data.
(19), AO(idea1 dilute) is a measure of solute-solvent interaction alone. If we assume that [bV2'z"/bT]s, and [d(TA')bTls, are constant for small values of AO, then we can rewrite eq 28 as As, = [m 4-fm2
+ xm3 + . . .
(30)
so that if Ae,/m is plotted against m, the intercept will be { and the coefficient of m, [. In Table V our values of { and f , obtained by a least-squares fit for sucrose are compared with those of Wada and Umeda.7 Also given in the table are values for tertiary butanol, ethanol, ethylene glycol, and glycerol obtained by appropriate least-squares fits to their data. Also given in Table V are values of { and [ for sucrose calculated using our experimental and A' as functions of temperature; leastdata for squares fits for these latter two quantities as functions of temperature gave
vz*
V2*/cm3 mol-1 = 206.13
+ 3.18 lo-'
(t/"C)
4.15
Then, substituting (27) into (26) and differentiating with respect to temperature gives for the solution at its temperature of maximum density, 8,
ae,
t-Butyl alcohol& Ethanola Ethylene glycola Glycerol@ Sucroseb (experimental) Sucrose* (calculated) Sucrosea
t/"K' mol-1 kg
kcal mol-'
x calculated from a value at 20' using the a Reference 21. same temperature coefficient as for 1-propanol. Int. Crit. Tables, Vol. I11 (1928); W. Brsostowski and T. M. Hardman, Bulletin de 1'Academie Polonaise des Sciences, Vol XI, 447 (1963).
V
Table V : Analysis of the Effect of the Solute-Solvent Interaction on the Temperature of Maximum Density
loll/
B. Tertiary butanol: Attractive Contribution to the SoluteSolvent Interaction Coefficient T/CO
369
A'/kg mol-1 atm-1 = 2.244 lo-* 1.11 10-5 (t/"c) - 2.6 10-7
-
(t/"C)z (31)
( t / O c ) z
(32)
The values used for al and PI*" were 8.0 10-60K-2 and 18.02 em3 mol-', respectively. The agreement obtained for { is good; the poor agreement for f is probably due to the rapid variation of A' with temperature in the neighborhood of 5". It can be seen from the variation of the sign and magnitude of Bt', Ct', and Dt' with temperature that a quantative analysis is not justified. The sign and magnitude of { which gives the "ideal dilute" or "solute-solvent" contribution to A8, is determined by [bVz*/dT]e,. { varies from small and positive for tertiary butanol to large and negative for sucrose. For sucrose [bV2*/bT]smis positive so that bq5/bT is positive and -dwl1/bT is positive as shown in Table IVA; for tertiary butanol on the other hand [bV20/bT]s,is negativez1: if we assume (21) F. Franks and H. T. Smith, Trans. Faraday. SOC.,64,2862 (1968).
Volume 74, Number I January II,1970
370
JOHNE. GARROD AND THELMA M. HIERRINGTON
Table VI: Variation of D with Temperature, ( d l kcal mol-1)
=
-4.0
D/A T/OC
Sucrose
0.6 5.0 15.0 25.0
0.61 0.66 0.72
(-Butyl alcohol
0.43 0.45
Ils
xsw = xsxw
XSW, = XsXw2, etc.
(33)
where A is the absolute activity. Equilibrium constants are then given by
Kl = asw/asaw, Kz = usw2/asaw2etc.,
+ + qsw + Ilw
+ ... = 1
gswz
(35)
from the definition of mole fraction. Then, substituting (34) into (35)
0.52
the tertiary butanol molecule to be a sphere of radius 3.0 A, then the rigid particle contribution to Bn*” is 233 cms mol-’. Thus for tertiary butanol b+/bT is negative but, as shown in Table IVB, -d&/bT is positive and almost identical with that for sucrose. On the other hand, if we consider wll to be independent of temperature then thewidth D of the attractive squarewell potential for tertiary butanol increases with temperature in a very similar manner to that for sucrose (Table VI). The sign and magnitude of f determines the sign of At),,, in dilute solutions and, on our model, although A&, is different in both sign and magnitude for tertiary butanol and sucrose, the temperature dependence of the depth or width of the attractive square well potential is almost the same in each case. “Xemiideal” Xolutions. We investigated the application of the concept of a “semiideal dilute” solution in the analysis of our results. The concept of a “semiideal” solution was first introduced by Scatchard22and refined by Robinson and Stokes.23 A “semiideal” solution is defined as one in which all the departures from ideal behavior are attributed to chemical reactions, and the activity of each actual species in the solution is equal to its actual mole fraction when the chemical reactions have reached equilibrium. We define a “semiideal dilute” solution as one in which the activity of the solvent on the Raoult’s law scale is equal to its actual mole fraction and the activity of each solute species on the Henry’s law scale is equal to its actual mole fraction when the chemical reactions have reached equilibrium. However, using our concept of a “semiideal dilute” solution, a certain inequality (eq 40) must be satisfied by the activity coefficients of solute and solvent and we found that this criterion was not obeyed for solutions of sucrose and water. Let us assume that in a solution of sucrose and water, we have the species S, SW, SW, etc. in equilibrium. Then
The Journal of Physical Chemistrg
where uw is the Raoult’s law activity of the water and as, asw etc. are the Henry’s law activities of the solute species on the mole fraction scale. Let qs, qw etc. represent the actual mole fractions of each species present at equilibrium, then
(34)
1 - 7s =
Thus 1
- rw
= qsw
+
qswz
+ ...
+ Kzasuw2 + . . .
Klasaw
(36)
- qs - qw must be positive, i.e. 17s
+
Ilw
