T H E CALCULATION OF PARTIAL MOLAL SOLUTE QUANTITIES AS FUNCTIONS OF T H E VOLUME CONCENTRATION, WITH SPECIAL REFERENCE TO T H E APPARENT MOLAL VOLUME1 FRANK T. GUCKER, JR.
Department of Chemistry, Northwestern University, Evanston, Illinois Received J u l y 6 , 1933
The partial molal solute quantity is important and frequently used in thermodynamic calculations. Many methods have been developed for calculating from a particular extensive property, usually through the intermediate apparent molal quantity (Gz), but most of these are based upon the weight concentration. Lewis and Randall (7) outlined various methods, both analytical and graphical. To mention only a few of the other methods most recently used: La Mer and Gronwall ( 6 ) analytically derived a fourth order equation for the partial molal volume as a function of the molality. Adams and Gibson (1) calculated the partial specific volume from the apparent specific volume plotted against the square root of $he weight fraction. Randall and Rossini (9) developed an equation for calculating the partial from the apparent molal heat capacity as a function of the square root of the molality. Young and Vogel(l4) discussed at length methods of calculating partial molal quantities in general, but dealt only with equations based on weight concentration. Within recent years, however, experimental evidence has shown, in the case of most electrolytes, a linear relationship between many apparent molal properties and the square root of the volume concentration, which holds even in concentrated solutions. This relationship in the case of +(V2),discovered by Masson (8), has been checked by the very careful measurements of Geffcken (2, 3) and also by Scott (13). Moreover, Redlich and Rosenfeld (10) derived from the Debye-Huckel theory an and hence also @(Vz), should be linear functions equation showing that VZ, of c+ a t very low concentrations. The same relationship was found in the case of the apparent molal compressibility by the present author, who discussed the experimental and theoretical aspects of this problem in a recent article (4). Such a simple and general relationship seems an ideal starting
+
1 Some conclusions in this article were briefly discussed in the Symposium on Electrolytes held a t the Washington meeting of the American Chemical Society, March, 1933. 3G7
308
FRANK T. GUCKER, JR.
point in calculating partial molal quantities. such calculations may be made.
This paper indicates how
DEFINITIONS
G is any extensive property of a solution containing nl and n2 moles of solvent and solute of molecular weights MI and MO,respectively. The corresponding apparent molal solute property is defined thus : a (GI)= (G
- nl@)/np
(1)
and the partial molal properties of solute and solvent are given by the equations
and
The superscript O is used at zero concentration. d and d l are the densities of solution and of solvent, c is the concentration (moles of solute per liter of solution), and m is the molality (moles of solute per 1000 g. of water). The apparent molal volume is defined by any of the following equations : (3)
1 [lo,, - M,] di
1000
aJ(V212)= - - c
a (VJ
=
1000
-
1000
(4) (5)
CALCULATION O F THE APPARENT MOLAL VOLUME
The usual method is to calculate @(V,) directly from density measurements, using equation 4. These values are then plotted against c? and the best curve fitted to them. The following method is, however, much simpler. Rearrange equation 4 as follows:
If we now plot F
sz
’)
1000 ( d l against ct we can calculate
function of c*, for the intercept FO
=
d18
- MP
as a
CALCULATION OF PARTIAL MOLAL SOLUTE QUANTITIES
309
and the slope
hence
and
34
32
30
28
d
26
32
ft 24
34
0
.-
c \ 0
U e8
22
1-
E
0
88
20
0 P
111
IE
90
LL
32
Y4 FIG.1. 0
AND
F
FOR
CALCIUM CHLORIDEAT 25°C.
The experimental points are thus smoothed by the F plot, and the extra calculations, involving the d l and Mz terms, are required only once, t o obtain the final values of the coefficients. I n order to illustrate the method, the density data for calcium chloride
310
FRANK T. GUCKER, JR.
at 25’C., taken from the International Critical Tables ( 5 ) , have been used to calculate F and a. Both are linear functions2 of c+ up to 3 c as may be seen in figure 1. The equation for F was found to be P = -92.81
whence, @ =
+ 5.97~4
+ 5.99ci
18.24
A direct plot of CP values gives the almost identical result, @ =
18.26
+ 5.99~4
The method of calculating a from F is not limited to the special case in which each is a linear function of c+. If F is anl~known function of c?, the corresponding equation for CP can easily be calculated. Thus, if F
=
FO
+F
(ct)
@ = -
di
W. C. Root (12) has derived a useful equation for the density of a solution obeying Masson’s law. Using molal concentration this becomes d = d i f AC 4- B c ~
This equation may be generalized as follows: FOC
c
1000
1000
d=d1-----
(10)
F (c9
can be expressed as a power series in terms of c*, the In particular, if corresponding equation for the density will be d = dl
+ AC + B c ~f
CC*
+--
where the coefficients A , B, C are calculable from those of the
(11’)
equation.
@(Gz) The equation connecting these two functions is derived as follows: Differentiating equation 1 gives T H E CALCULATION O F
2
FROM
The relationship does not hold a t higher concentrations.
CALCULATION OF PARTIAL MOLAL SOLUTE QUANTITIES
311
Now 1000 d
c =
Eliminating d between equations 4 and 13'and rearranging terms gives NV2) =
1000
, .- Mdl 11.1
n2
Differentiating equation 14 with respect to In gives
n2
and rearranging terms
Combining equations 12 and 15 gives the desired result: u 2
= @(GJ
+c
Whence13
L
If 9 is expressed as a function of cf instead of c, equation 16 becomes
and
This equation is perfectly general, and does not impose any particular functional relation on @ and c+. I n the particular case where, as with most 3 This equation was given by Geffcken (reference 2, equation 4). H e discussed the relationship between @ (V,) and v2 in the dilute solution, but did not use this equation in actual calculation of values of Vz.
THE JOURNAL OF PHYBICAL CHEJIISTRY, VOL. XXXYIII, KO.
3
312
FRANK T. GUCKER, JR.
electrolytes, a(G2) and CP(V2)are linear functions of e*, equation 17 can be further simplified to
If we are concerned with V2, this equation becomes
At infinite dilution, G2'
=
+O(Gz)
At low concentrations, ca0(V2)and d-
dci
are small compared with
3000 and 2000, respectively, and, as Geffcken pointed out,
The limiting form of the equation at low concentration is similar to that developed by Randall and Rossini (9) for calculating the partial from the apparent molal heat capacity expressed as a function of the square root of the molality, namely,
WEIGHT AND VOLUME CONCENTRATION
Plots against c: do not differ very much from those against mt at very low concentrations, since at the limit m is proportional to c. However, the difference may become appreciable at higher concentration. In general, rearranging equation 5 gives the relationship between c and m, namely4
c and m may be almost identical if a(V,) ,is small or negative; or very different, if CP is large. To illustrate this strikingly specific character of c/m we have chosen two univalent electrolytes as different as possible, The ratio is given by Geffcken (2) without the d l . c and m are proportional and
not equal, although the difference is small.
313
CALCULATION O F PARTIAL MOLAL SOLUTE QUANTITIES
namely, lithium hydroxide and cesium iodide. The corresponding equations for @ ( V 2are6 )
-
O(VJ =
6.00
@(V&= 47.9
+ 3.00d
+ 1.47d
The values of c/m and c+/micalculated from equation 20 are given in table 1 a t several values of ct and are plotted in figure 2. TABLE 1 V a l u e s of c/m and ct/m+calculated from equation 10 LiOH
d
CsI
1
0.9986 0.9991 1.0000 1.0002 0.9985
0,9971 0.9982 1.0001 1.0005 0.9971
0.0 0.5 1.0 1.5 2.0
.9s
1
c%/mi
0.9971 0.9849 0.9478 0,8847 0.7943
e+/,?
0.9985 0.9924 0.9736 0.9406 0.8800
.+ FIQ.2. DEPENDENCE OF c/m RATIO o s @(V,) CALCULATION O F
7,
As an application of equation 18' we have taken the case of calcium chloride which was previously mentioned. The equation for the partial molal volume is -
V2 = 18.25 -$
3000 - 18.25~ 333.89 c3'2 cb
+
1
The first was calculated from the density data of the International Critical Tables, 1st edition, Vol. 111,p. 76; the second was taken from Geffcken (2).
314
FRANK T. GUCKER, JR.
The graph of this function is included in figure 1. Notice that v2 is nearly linear up to about 1 c. It is only above this concentration that the curve for 7 2 deviates appreciably from the dotted line which represents the limiting slope of the tangent. In order to test the more general case to which equation 17’ is applicable we have studied the very precise specific volume data for ammonium nitrate solutions from 0.1 to 11 c at 25°C. which were obtained by Adams and Gibson (1). These particular data were chosen because the authors had made very careful computations on a weight basis with which we 53
52
51
ci s o 0
.-c
-pE49 3
48
47 1
0
0
04
08
1.2
2 4
28
3.2
:
FIG. 3. AXMOXIUMNITRATEAT 25°C.
could compare the results obtained from our equation. Moreover, in this case @ ( V 2is ) large, so that the weight and volume scales diverge sharply at high concentration. We calculated values of @ ( V 2 )from their data a t fourteen points and found that they could be represented fairly well b y the linear equation @ = 47.280 4- 1.2049~1 However, appreciably better agreement was given by a second order equation in et, namely, 0 = 47.558
+ 0.966ct + 0 . 0 4 7 4 ~
The average deviation in d calculated from the linear equation is 4.6 X The improved agreement and from the quadratic curve 3.2 X
CALCULATION O F PARTIAL MOLAL SOLUTE QUANTITIES
315
was particularly noticeable in the lower concentrations, and for this reason the extrapolation to zero concentration is much better with the quadratic equation. This is shown by the heavy line in figure 3. The lighter line represents the linear equation which extrapolates to an appreciably lower point. The two curves are practically identical above about 0.25 c. From the second order equation for @ we calculated the corresponding equation for d, of the form indicated by equation 11', namely: d = 0.997077
+ 0.032628~- 9.63 X
C*
- 4.73 X
10-j c2
In calculating the partial molal volume, equation 17' is changed to the form
where
a@ ( VZ) = 0.966 dC*
+ 0.948d
Our equation for 7 2 - a(V2) is plotted a t the bottom of figure 3. The equation for 7 2 is likewise included. Calculations from the linear and quadratic equations for @ differ appreciably only below 0.25 c. Adams and Gibson calculated cp (apparent volume per gram) and 7 2 (partial volume per gram). Plotting cp against the square root of the weight fraction, 2 2 , two equations were required to cover the whole range of concentration, namely: 'p 'p
= =
+
0.59318 0.0491612z* (0 0.594118 f 0.043431~z* 0.0205242~:
+
< 2 2 < 0.10) ($2 > 0.10)
They used the first equation in extrapolating to zero concentration. Our molal properties, ~ ( V Zand ) 7 2 are obtained by multiplying their values by the molecular weight (80.047). Their extrapolated value for (Po(V2), indicated by the black triangle in figure 3, lies between our values ext,rapolated by the quadratic and linear equations. It is considerably nearer the former and agrees very well with our conclusions. They calculated the partial volume by substituting in the equation
aP
They evaluated the tangent - by tabular difference plots of cp and also axz checked the values of 2U2 analytically from their interpolation equations. A comparison of the results obtained by the weight and volume plots,
316
F R A N K T. GUCKER, JR.
given in table 2, illustrates the extremely good agreement in almost all cases. In figure 3, the values of Adams and Gibson for VZ- ip(Vz)are indicated by circles on the plot of our equation. The only appreciable deviations are shown a t the two highest concentrations. In comparing the two different methods of calculation the volume concentration plot is simpler, since one equation will cover the whole range of concentration. The weight fraction method requires two equations with different values for cpo as well as for the coefficient of xzt. Moreover, the volume plot smoothes the 7, curve in accordance with our present knowlTABLE 2
Ammonium nitrate at 25"C.-co
52
c
*
103 A
Q
=
(Gucker) "obsd."
- oalcd.*
vz - Q vz - Q Gucker
(A.&G.)
0.00 0.18 0.40 0.57 0.70 0.80 0.88 0.98 1.03 1.01 0.98 0.93 0.89
0.00 0.19 0.40 0.55 0.68 0.78 0.87 1.01 1.05 1.01 0.95 0.86 0.79
parison of methods 102 A
t
(G - A.&G.)
-vz
(Guoker)
102 A v z
(G - A. & G.)
I _ _ _ _ _ -
0.00 0 . 0 47.558 (-80) 0.01 0.125 47.906 -30 -6 0.05 0.635 48.358 1.297 48.720 6 0.10 -7 0.15 1.984 49.013 -7 0.20 2.700' 49.273 0.25 3.444 49.514 -13 -5 0.35 5,024 49.961 0.45 6.733 50.384 6 11 0.55 8.584 50.795 10 0.60 9.567 50.999 -1 0.65 10.590 51.204 0.68 11.224 51.326 -11
0 -1 0 2 2 2 1 -3 -2 0 3 7 10
Av. = 3
47.56 48.08 48.76 49.29 49.71 50.07 50.39 50.94 51.41 51.80 51.98 52.13 52.22
8 1 1 1 2 2 2 -3 -3 -1 2 7 11
4v.
=
3
* "obsd." values were slightly corrected to even concentrations by Adams and Gibson. Calcd. values from our equation. t A = (V,- @) (G) - (V, - @) (A.$G.) edge of simple functional relationships. The advantage of using a function which has theoretical justification rather than an empirical equation is particularly evident in the extrapolation to zero concentration. Redlich and Rosenfeld (11) illustrate strikingly the difference between a linear plot of T2against ci and the fourth order empirical equation of La Mer and Gronwall, which passes through the same points and yet extrapolates to a very different value. SUMMARY
We have described a simple method of calculating the apparent molal volume, +(Vz),as a function of cb, which is shown to give the same results as the usual more laborious method.
CALCULATION O F PARTIAL MOLAL SOLUTE QUAKTITIES
317
We have compared volume and weight concentration. The ratio c/m is a specific property of the individual solute which depends on (a( V,) and may vary from 1 to less than 0.8. We have developed a general equation for calculating any partial molal from the corresponding apparent property Q,(Gz) as a solute quantity, function of c, and a special equation applicable when Q, is a linear function of e$. Since many apparent molal properties of most electrolytes and some non-electrolytes show such a linear relationship, this equation has a very wide applicability. We have shown the application of these equations in calculating partial molal volumes. I n the case of ammonium nitrate we have compared our calculated from a single equation for @(V,)with those which values of 7, Adams and Gibson calculated from two weight fraction plots. Our results are in substantial agreement with theirs, but theory and numerical simplicity seem to favor the volume concentration method.
a,,
REFERENCES (1) ADAMSAND GIBSOS: J. Am. Chem. Soc. 64,4522 (1932). (2) GEFFCKEN: Z. physik. Chem. A166,l (1931). BECKMANS, A N D KRUIS: Z. physik. Chem. B20, 398 (1933). (3) GEFFCKEN, (4) GUCKER: Chem. Rev. 13,111 (1933). (5) International Critical Tables, 1st ed., Vol. 111, p. 72. McGraw-Hill Book Co., New York (1928). (6) LAMERA N D GRONWALL: J. Phys. Chem. 31, 393 (1927). Thermodynamics, 1st ed., pp. 36-41. McGraw-Hill Book (7) LEWISAND RANDALL: Co., Yew York (1923). (8) MASSON:Phil. Mag. 8,218 (1928). (9) RANDALL A N D ROSSIKI:J. Am. Chem. Soc. 61,323 (1929). (10) REDLICH AKD ROSENFELD: Z. physik. Chem. A166, 65 (1931). (11) REDLICH AND ROSEKFELD: Z. Elektrochem 27, 705 (1931). (12) ROOT:J. Am. Chem. Soc. 66, 850 (1933). (13) SCOTT:J. Phys. Chem. 36, 2315 (1931). (14) YOUKG AND VOGEL:J. Am. Chem. SOC.64, 3025 (1932).
