THE APPARENT VOLUMES O F SALTS I N SOLUTION I. A Test of the Empirical Rule of hlasson BY ARTHUR F. SCOTT
The apparent molal volume cp of a salt in an aqueous solution is defined as cp=-
v
- nl v1 (1)
n2 where v is the volume of a solution containing nL mols of electrolyte and nl mols of water; v1 is the volume of one mol of pure water a t the temperature of the solution. Although there have been many studies of the apparent molal volume, very little is known regarding this characteristic property of solutions, especially with respect to its bearing on the nature of the solution state. However, the empirical relationship, recently put forward by Masson,l appears to offer some interesting possibilities in this connection. Masson’s empirical rule may be expressed in the following form: cp =
ami
+b
(2)
Here m is the molarity (gram-mols per liter); a and b are constants, characteristic of the salt involved and also dependent on temperature. Masson examined the validity of the above equation for solutions of twenty-eight salts with the following results: Of the twenty-four strong electrolytes in this group, the solutions of only two (magnesium nitrate and sodium acetate) were found not to conform to the above rule. The solutions of the four so-called weak electrolytes were found to be abnormal in that they apparently require the five-fourths power of the molarity instead of the square-root. Finally, in five instances (sulfuric acid, lithium chloride, nitric acid, and ammonium nitrate) the plotted points for a given solution exhibited an abrupt transition from the straight line, generally, a t relatively high concentrations. While the evidence in support of equation ( z ) , which we have only partially summarized above, does indicate that the linear relationship is valid for the simple type of strong electrolytes, it appears desirable to extend the examination of the equation in a more systematic manner. In this way i t should be possible not only to make a more thorough test of the equation but also to gain some insight into its significance. Therefore, in the present preliminary paper we shall examine the general applicability of equation ( 2 ) to solutions of the alkali halides ( I ) by testing its ability to represent the experimental variation of q with concentration; ( 2 ) by checking the expected additive properties of the constant b ; and (3) by comparing experimental and computed values of q in saturated solutions. The data employed in this examination are from the careful measurements of Baxter and Wallace2 and ‘Masson: Phil. Mag., (7) 8, 218 (1929). * Baxter and Wallace: J. Am. Chem. SOC.,38, 70
(1916).
2316
ARTHUR 5’. SCOTT
are well suited for the purpose. From them it is possible to compute for the halogen salts of the five alkali metals values of cp a t concentrations ranging from nearly saturated to about three-tenths molar for a t least three different temperatures. Before presenting the results of the present study it is important to call attention to certain matters affecting precision. In the first place cp is the difference of two volume terms and consequently is of a much lower order of precision than the basic quantities. Indeed, according to Baxter and Wallace, the values of cp, as determined by them, are aboLt ten times more precise for concentrated solutions than for dilute solytions, despite the fact that the basic quantities involved in the latter are the more precise. Moreover, it should be noted that the probable precision of p at a given concentration is not identical for each salt. Hence, only a very general statement regarding the precision of the p values calculated from the experimental measurements is permissible: The uncertainty in p is less than 0.01cc. in concentrated solutions but in very dilute solutions it probably amounts to several hundredths of a cubic centimeter. In considering +he applicability of equation ( 2 ) to the experinientally determined data it will be necessary to keep a second item in mind. Any error in a basic measurement of densit’y or weight of salt per unit weight of solution which would act to increase p would act t o decrease the magnitude of m*. Thus, any errors in the experimental data are magnified when represent#edby equation (2). In carrying out the first test of equation ( z ) , namely, that of determining how closely the linear relationship describe4 the variations of p with mi, it is necessary to know the values of the constants a and b for each case. These quantities, which are given in columns 3 and 4 of Table I, have been computed from the experimental points (p,mi values) by the method of “zero sum.”’ In calculating these const,ants it became apparent that, because of the limited number of experimental points available (usually five), an error in the data of one point can introduce a serious error in the values of the constants, especially in a. Since it was desired to obtain the most representative values of the constants, it was considered advisable in applying the method of “zero sum” to disregard any point which deviates very markedly from a straight line drawn through the remaining points, Information on this matter is also presented in Table I. In column j are listed the total number of concentrations (points) for which Baxter and Wallace have measurements, and in column 6 are given the number of points omitted in determining the constants. I n this same column are also the letters h, i, and 1 to indicate whether the rejected points are a t highest, int,ermediate, or lowest concentrations, respectively. We are now in position to compute the value of p a t each concentration using the values of the constants given in Table I, and thus find the difference between the cp value found experimentally and that calculated from the equation. For the sake of brevity only the average of the differences for each N. Campbell: Phil. Mag., (7) 47, 816 (1924).
APPhRENT VOLUMES O F SALTS I S SOLUTION
1
T.4BLE
Data bearing on the Constants of Equation
Salt
LiCl
Temp.
15.30 I
j o . 04
1.446 I .663 I ,606
16.96 16.14 15.95
6 6 6 5 5
1.673 1.1j9
21.94 24.08 24.37 23.88 32.50 35.50 36.81 36.85
0.00 25.00
50.04 7 0 . I9
NaCl
KaBr
1.42j
0,841
jO.04
0.j69
70.19
0.708
0.00 25.00
3' 2 4 0 2.153
50.04
I .804
0.00
0.00
2j.00
j o . 04 0.00 2j.00
j o , 04
70.19 100.0
KBr
I ,312
0.00
50.04 7 0 . I9
KC1
I .145
25.00
2j.00
NaI
points
1.990
100.0
UI
xo. of
I .488
7 0 . I9
LiBr
b
a
25.00
0.00
0.00 25.00
50.04
23'7
2
Average differences KO of between points expt. and rejected calc. values in C.F Ih
0.02
2h
0.01
2h
0.02
Ih
0.0j
ah
0.06
5
Ih
0.02
6 6 6
2h
0.01
2h
0.02
2h
0.04
9 9 5 5
3h 4h
0.06
2h
0.00
2h
0.02
12.36 16.40 17.Y6
5
0
0.05
5
2:
0.01
Ii
0.05
18.86 23.51 25.34
0
0.02
0
0.02
0
0.03
11
0.02
7 .OO
.
0.07
2.976 1,760 1.398 1,377
25 . 8 0
5 5 5 5
2.886 I ,346 0.826
29.34 35.10 37.68
5 5 5
0
0.02
0
0.03
0
0.05
3.291 2.327 2.087 1.982
23.00 26.5 2 27.66
7 8
11
0.05
0
0.03
5
11
0.01
27.95
I4
11, zi
0.05
2'574
26.49
4
11
0.00
3.219 1.939 I. 640
29.38 33.73 35.32
4
21
0.01
5 5
0
0.03
0
0.05
23 18
ARTHUR F. SCOTT
TABLE I (Continued) Data bearing on the Constants of Equation
Salt
KI
RbCl
RbBr
Temp.
CSCl
CsBr
CsI
h
so. of points
Average differences So. of hetween points espt. and rejected tal:. values in C.C.
0.00
3 ' 133
40.05
5
11
0.03
25.00
45.36
7
Ih
0.01
j o . 04
1'556 I . 168
4 7 . 58
5
0
0
0
0.00
3.287
28.11
4
2j.00
2.219
1s
jO.04
2.047
3',87 32-95
0.00
34.61 38.71
jo.04
3 ,I 1 2 2.038 1,757
25.00
RbI
a
2
'
'
03
003
li
0
0
0 02
4
0
0.03
3
0
40.28
LI
0
0.03 0.03 0.0j
5
04
0.00
3'158 I . 607
45.15 50.31
4 4
0
2j.00
0
0.0j
50.04
1.224
52.75
5
0
0.02
35.23 39.15 40.39 40.45
5 6 6 6
0.00
3 ' 293
25.00
2.172
50.04 70.19
2.053
0.00
3.046
1.958
21
0.02
0
0.02
0
0.02
0
0.03
>
0
0.02
6
21
0.01
6
21
0.01
11
0.02
0.00
25.00
I
j o . 04
I . 632
41.95 46.19 47 ,231
70.19
I .
646
48.24
6
2.885 I ,j i g
52.74
4
11
25.00
57.74
rh, I i
0.0;
50.04
1.081
60.32
0
70.19
I .
5 5 5
0.04 0.04
0.00
,901
160
61.06
21
case is included in the last column of Table I. I t should be mentioned that these average differences or deviations do not include the deviations of the rejected points which will be considered separately. In analyzing these figures it is probably best to deal first only with the unequivocal cases, those in which no i or 1 points have been rejected. *Iltogether 41 out of the 56 cases tabulated belong in this class and involve over two hundred points: the mean of their average differences is just a little more than 0 . 0 2 cc. In view of the difficulty in determining q experimentally the magnitude of this uncertainty
APPARENT VOLUMES OF SALTS IN SOLUTIOS
2319
is all that can be expected. Furthermore, in those cases where i or 1 points were rejected in determining the constants the mean of the deviations of the accepted points is essentially the same as in the other cases, 0 . 0 2 cc. The deviations of the rejected points will be considered in some detail, for it is important to ascertain as far as possible whether the deviations in question result from esperimental error or from the inapplicability of equation ( 2 ) : h points: The marked deviations of the calculated from observed values of p at high concentrations occur, with two exceptions, with a11 the lithium salts. They resemble the case of lithium chloride described by Masson in that they deviate very definitely and uniformly from the linear relationship. Therefore, since these deviations occur at concentrations where the linear relationship is presumably not valid, they have no bearing on the applicability of equation ( 2 ) . The deviation b e h e e n calculated and observed values of p is 0.15 cc. for the high concentration points rejected in the cases of K I and CsI. Since the deviation is in the same direction in both instances and opposite to that found with the lithium salts, the cause cannot be the same R S in the latter cape. I t is not unreasonable to suppose that the extreme deriation arises from an uncertainty in the basic measurements because it is evident a t only one temperature. i points: The average deviation of the five points at intermediate concentrations which deviate most decidedly from the linear relationship is 0.26 cc. The deviations in these cases are not only irregular in direction but appear only a t one, high temperature. From these facts it may be concluded that here also the basic measurements involve a slight but definite error. 1 points: An analysis of these rejected points is especially important because an incorrectness of equation might be expected to make itself most evident at low Concentrations. I t is found, however, that the mean of the deviations of the twenty points in this cl is 0.24 cc. and that the deviations are not consistently in one direction. Since this mean figure is approximately ten times that obtained for the other points, it may be concluded that thcse deviations arc no greater than what might result from the greater uncertainty in determining p at low concentrations. Indeed, practically all that \vas gained by the omission of these points in the determination of the equ:ttion constants was to give more weight to the points at higher concentrations which are the more accurate. In summarizing the results of the foregoing test we may state that with few esceptions the linear relationship describes within the limits of espcrimental error the variation of p with concentration. h scrutiny of the few exceptions reveals no clear-cut evidence of the failure of the equation. On the contrary, there is evidence to indicate that most of the unusual deviations arise from exceptionally large errors in the basic measurements. An objection to the above test is that, since it covers only a limited range of concentration, a discrepancy between t,hc calculated and experimental variation of cp is not likely to be very evident. K e can, however, test the equation further at the extreme limit where the molal concentration is zero.
2320
ARTHUR F. SCOTT
At this concentration the apparent molal volume is, of course, equal to b. Furthermore, if the constant a is assumed to be correct, a measure of the uncertainty attached to b in each case is the average deviation given in the last column of Table I. In infinitely dilute solution the apparent molal volume becomes identical with the partial molal volume of the salt. K e are therefore able to compare the values of b (Table I) with the corresponding pa.rtia1 molal volumes, calculated by Lanler and Gronwalll who, it may be remarked, also based their computations on the measurements of Baxter and Kallace. A comparison of these two quantities shows, however, that the partial molal volumes are consistently larger than the values of b, the difference being greater than the estimated uncertainty of eithe:. quantity. As a matter of fact, the difference appears to be directly proportional to the constant a, the proportionality factor being roughly 0.3. Since this discrepancy may appear at first glance to be evidence of a defect in hlasson's equation, it is important to show that its origin does not necessarily reside entirely in equation ( 2 ) . The apparent molal volume as well as the partial molal volume at infinite dilution can be regarded as the sum of the corresponding volumes of the ions. Since the latter quantities are presumably independent of the other ions present, we may, on the principle of additivity, expect the differences between any two ions to be the same regardless of the other ion present in the salt. Some of the possible combinations whereby this expected additivity can be tested are given in Table 11. A scrutiny of Table I1 shows that with exception of those differences which involve KCl a t 2 j"and soo, and CsI at oo, the values of various combinations check very well with each other. The average deviation of the values of a given combination from the mean is given in brackets and is of the order of magnitude to be expected from the estimated uncertainty of the different values of b. For the purpose of comparison there are included in the above table the corresponding mean values and average deviations obtained by La Mer and Gronwall. If the two sets of average deviations are compared, it is seen a t once that the checks obtained for the different combinations are appreciably better with the b values than with the partial molal volumes: the mean of all the average deviations in the first case is 0.07 cc. and in the second, 0.1j cc. Moreover, it should be noted that), in addition to those differences (bracketed in Table IT) which exhibit a somewhat marked divergence from the mean value of a given combination, La Mer and Gronwall found others which involve KCI a t oo and RbCl at 0'. The average amount of these extreme divergences is only 0 . 2 j cc. in the case of the b values but is I , I I cc. in the case of the partial molal volumes. These facts may be taken to indicate that the method employed by La Mer and Gronaall to find the partial molal volumes a t infinite dilution is rather more susceptible to slight experimental errors in the basic measurements than is equation ( 2 ) . On the other 1
LaMer and Gronwall : J. Phys. Chem., 31, 393 (1927).
APPAREST VOLUBIES OF SALTS IS SOLCTION
I
I
r *
-
r.c
I
I
I
--
h-
-
N
0
0
A -
cz 0
0
2321
ARTHUR F. SCOTT
2322
hand, the actual difference between the two sets of mean va!ues of the various combinations is not very great. The average difference is 0 . 2 7 cc.; but, if the two combinations which show the greatest divergence are omitted, this difference drops to 0 . 2 0 cc., a figure not much greater than the average deviations of the partial molal volume differenem Since the foregoing discussion suggests that the disagreement between the b values and the partial molal volumes a t infinite dilution may arise from an uncertainty inherent in the procedure employed by La Mer and Gronwall as well as from a defect in equation ( z ) , it is interesting to note corroborating evidence. From very precise measurements Lamb and Lee1 have determined the apparent molal volumes of a number of salts in extremely dilute solutions at 2 o T . The apparent molal volumes of three salts a t infinite dilution, derived by these authors from their data, are given below in column 2. These figures, it should be remarked, are quite reliable because of the short range of extrapolation. The corresponding figures for the partial molal volumes and the b values are given in columns I and 3 , respectively. These latter quantities are all for 20' and were obtained by graphical interpolation. I.
LiCl KaC1 KCl
Diff. -0.4
16.7
-1.0
26.6 (27.3)
-0.
2.
17.13
1 7 . j
j
15.71
26.10
Diff. 0.2
-0.1 0.0
3.
16.9 15.9 26.1
(-1.2)
The second, brackcted value given for the partial molal volume of KC1 is a corrected value, calculated on the basis of the additivity principle. From the above data it appears that equation ( 2 ) yields reRults which are in better agreement with the measurements of Lamb and Lee than are those obtained by the method of La RIer and Gronwall. We may now summarize the conclusions of this second way of testing equation ( 2 ) as follows. The b values satisfy excellently the requirements of the principle of additivity. They do not, however, agree very well with the partial molal volumes calculated from the same basic data by La Mer and Gronwall. Although this disagreement may be the consequence of a defect in equation ( z ) , there is some evidence which points to the possibility that the method employed in the determination of the partial molal volumes at infinite dilution is at fault. As the third and final means of test,ing the equation under discussion we can calculate the apparent molal volumes of salts in saturated solutions and compare them with the values obtained experimentally. The essential data for such a comparison are assembled in Table 111. In the second, fourth, and fifth columns are given the temperatures, molarity, and (a values of the experimental measurements. The figures contained in column 6 are the cp values computed by use of the constants given in Table I and the molarity 'Lamb and Lee: J. Am. Chem. SOC.,35, 1683 (1913).
APPAREST VOLUMES OF SALTS I N SOLUTIO3
2323
in column 4. It will be noted that some of the measurements are not exactly a t 50.04' for which temperature the constants are valid. The difference, however, is practically negligible. I n order to show to what extent the calculated values lie beyond the concentration range employed in determining the constants of equation ( 2 ) , the maximum value of m' involved in each determination is given in column 3. Even though the concentrations of the saturated solutions are not much greater than these maximum values, the excellent agreement between calculated and observed values of cp is additional evidence in support of the applicability of equation ( 2 ) . In order to reduce
TABLE I11 Comparison of Observed and Calculated Values of Salt
Temp.
KaC1
0 . ool 25.00'
50,OOl
XaBr
0,002 25.003
jO.3l2
Sa1
o.oo? 2 5 . 003
50,022
KC1
0 . 004
25.00~
5 0 . OZ'!
IiBr
2.268 2.268 2.267
2.332 2.331 2.336
2.341 2.344 2.339
2
0.1;
0.0j
29.4
0.08
2.634 2.633 2.633
764 2.879 3.056
37.2 38.8
37.3 39.0
0.16
40.2
40.2
1.831 I . 830 I .jrq
,840
29.3 31.3 32.2
29.1 31.3 32.2
0 . IO
2.043
35.6 37.8 39.1
0.15
2.
I
2.190
0.08
-
-
2.313
36.0 37.8 39.5
2.360 2.360 2.359
2,375 2.489
47.7 49.3
47.5
0.10
49.2
0.06
2 . j81
j0.8
j0.6
0.10
I ,805
36.I 37.7 38.3
35.7 37.4 38 . o
0.20
43.9 45 I 46 . o
43.8
0.06 -
'
'44
2.301 2.474
5 0 .004
2'554
2,582
0 .004
2.076
2.603
j .004
2.744
2
2
,743
,741
2.843
'
45.1
46.0
Tschernaj : Taken from Comey: "Dictionary of Chemical Solubility." J. Phys. Chem., 34, 1424 (1930). Scott and Frazier: J. Phys. Chem., 31, 459 (1927). International Critical Tables; z j" values are interpolated.
* Scott and Durham:
0.0;
29.3
2.468
5 0 . 004
0.10
22.2
19.9
2.902
2 j ,004
2
21.4
0.07
2
CsCl
19.8 21.6 21.9
28.3
j0.212
0. 004
calc.
26.1
1,975
RbCl
Percent Difference
obs.
28.2
542
2.119
50.132
in Saturated Solutions
26.2
'
I.jIj
0.002
LS
2.699
2.14j
0,002
2 j ,003
'
mi
25.002
IiI
I
ml max.
0.20
0.18 0.20
-
2324
ARTHCR F. SCOTT
the differences between observed and calculated values of cp to a common basis, the calculated values of 9 have been assumed to be correct and the error in the experimental determination of cp, necessary to cause the difference, has been computed. This error expressed in per cent is given in the last column of Table 111, and is well within the known limits of experimental error. Incidentally, the values of mt are expressed in four significant figures to indicate the experimental precision required in order to calculate cp to three significant figures. We have completed the examination of the applicability of equation (2). The conclusion to be drawn from the evidence considered is that the equation is valid within the limit’s of experimental error. This conclusion, however, cannot be generalized because the data which have been employed in testing t’he equation are not sufficiently complete or varied. On the other hand, it must not be forgotten that Masson found the equation to be applicable to solutions of electrolytes which are much more diverse in type than those which we have considered, a fact which strengthens our present conclusion. Whether the square-root function of m is the correct function or whether it is merely a good approximation’ of this function is a question which possibly cannot be decided by the method of analysis which we have employed so far. A partial decision can be reached, however, by a consideration of the theoretical implications of the equation, a matter which will be taken up in the following paragraphs. Attempts t o interpret the variation of apparent molal volume with concentration have been based, as rule, on the assumption that the change in the ionization of the solute with concentration is the primary factor. The difficulty encountered in developing a theory along these lines is that the concept of apparent molal volume is by no means clear. According to one view the change in cp is a change in the volume of the solute itself. The following extracts from blasson’s paper will suffice to illustrate this interpretation. ‘‘TThere water is mixed with a soluble substance the change in volume that occurs is the sum of two changes. One is that due to the formation of a hydrate from its constituent molecules; the other is caused by the dilution of that hydrate with excess of water or with excess of the other component. The former may be called the ‘chemical effect’ . . . the latter may be distinguished as the ‘physical effect’.” Regarding the change in cp with mi in . . and the suggestion is as inevitable terms of hydrates, Masson states: that the hydrate exists in two forms, a greater and a smaller, and that cp as measured, belongs always to a mixture of the two in a proportion determined by concentration. The smaller form could exist by itself only if it were possible for the hydrate to exist at all in the complete absence of solute, Le., at infinite dilution. The larger form could exist by itself only if it were possible for the hydrate to exist all in the complete absence of water.” Apart from the change in volume which the water undergoes in forming the hydrate For example, Jaiilczynski [Roczniski Chemji, 3, 362 (1923).] has employed the cuberoot of the molarity which holds very well for the solutions he deals with but is obviously incorrect when tested by the data employed in this paper.
APPAREST VOLUMES O F SALTS I N SOLUTION
23-25
.
it is assumed “that any excess of water which remains as solvent . , retains its original specific volume practically unchanged.” The alternative concept attributes the change in cp to a change in the volume of the solvent molecules rather than to a change in the volume of the solute. Webb’ offers an adequate picture of this mechanism in his mathematical theory of electrostriction. In very dilute solutions the ions are considered to be within a cavity which is free from water molecules and which represents the effective volume of the ion. Since the radius of the cavity is considerably larger than the radius of the ion proper, it is assumed that the cavity approximates a sphere with considerable exactness. About this spherical cavity the water molecules are compressed, the amount of contraction depending on the field strength of the ion involved, Le., on the radius and charge of the ion. In terms of this mechanism the apparent molecular volume of an ion becomes equal to its effective volume (the spherical cavity) minus the total contraction suffered by the solvent. Neither of the two concepts of cp, outlined above, offers obvious grounds for an interpretation of the square-root function. Nevertheless, by means of certain postulates it appears possible to indicate a connection between Masson’s equation and the electrostriction concept of p. According to this theory the change in cp with concentration is the direct result of a change in the amount of total contraction of the solvent molecules and probably also a change in the dimensions of the ion cavities. Even though the factors which control these changes are exceedingly complex, it may be assumed that they have their ultimate origin in the interaction of the ions of the solute. Consequently, if solutions of electrolytes of the same concentrations are compared, it may be supposed that differences in the rates of change of q with concentration arise from differences in the effect of the inter-ionic forces involved. This very general assumption would lead us to expect the relative magnitude of the constant a (equation 2 ) to be a measure of the inter-ionic forces because this constant is a relative measure of the rate of change of cp with concentration when solutions are compared under conditions of identical concentration. This argument can be supported only indirectly by showing that the constant a is related to other properties of elect#rolyteswhich are dependent on the inter-ionic forces. One factor which appears in the treatment of ionic interaction problems’ is the so-called valence factor which is determined by the number and valence charges of the ions constituting the electrolyte. That this valence factor affects the magnitude of the constant a can be seen from the following values for uni-bivalent and bi-bivalent salts, taken from Masson’s paper : Salt: MgClz a : 6.17
CaCL 7.10
CdCb (NH4)*S04 K2CO3 5.72
10.13
13.60
MgS04
ZnS04
11.12
10.71
Webb: J. Am. Chem. SOC., 48, 2589 (1926). example, the Debye-Hiickel theorv of solutions. I t is of interest to note that in the mathematical elaboration of this theori a quantity is obtained which is defined as the “characteristic or probability” distance between ions in solution and which varies inversely as the square-root of the molal concentration.
* For
2326
ARTHUR F. SCOTT
These values are all considerably larger than the values of a for the uniunivalent electrolytes given in Table I and, with the exceptions of the values of (?;Ha)&Oa and &COB, fall into two distinct groups according to valence type. This valence factor is, of course, not the only one which determines the value a as can be seen at a glance from the dat'a in Table I. Since the positions of the water molecules about each ion and also the positions of ions about the oppositely charged ions are determined by electrostatic forces of attraction and repulsion, and since these latter forces depend on the ionic dimensions' it may be surmised that the ionic dimensions are of considerable importance in determining the rate of change of p with concentration (a). One property which, on the basis of electrostatic theory, is also a function of ionic dimensions is the interionic distance in a crystal lattice. In a preliminary trial to see whether the differences in the a values corresponded in any way with the differences in lattice distances, it was observed for the alkali halides that, when log a is plotted against lattice distance, the plotted points exhibit distinct regularities. These regularities become more definite, if, instead of the lattice distances, the inter-ionic distances corrected for the radius ratio effect2are used, and they may be summarized briefly: The plotted points for the three halogen salts of each cation fall closely on straight lines which are sensibly parallel; analogous lines cannot be drawn through the points of salts with a common anion apparently because of a fairly uniform displacement of the points of the lithium and cesium salts. The above-mentioned graphs are not presented because the possibility of a relat>ionshipbetween the constant a and the forces involved in determining the lattice distances is evidenced more clearly in a slightly different form. The actual inter-ionic distances in crystals, according to Pauling, can be represented as the consequence of two factors: ( I ) the radius sum r + r - and ( 2 ) the radius ratio p = r-'r.-, The corrected inter-ionic distances It, are those of hypothetical substances in which the radius ratio effect has a standard value but whose other ionic properties are unchanged. Therefore, the fact that the log a relationships, described above, are more regular with K,than with the actual lattice distances may mean that the effect of the radius ratio on lattice distance is quite different from its effect on the complex forces which determine p. An empirical function of p which appears to represent approximately the effect of the radius ratio so far as log a is concerned is ( I I ' p l , To show this possibility Fig. I has been constructed by plotting values of log a against the values of the expression R, ( I 1,'~). The former quantities are obtained from the data in Table I and the latter are computed from the figures given by Pauling. To avoid misunderstanding it should be stated that the rclationships portrayed in Fig. I are important only insofar as they call attention to a possible
+
+
+
Garrick: Phil. Mas., ( 7 ) 9 , 1.30 (1930); 10, ;6, ; i (19301, has treated the problem of co-ordination numhers of ion hydrates and amino-(.ompounds from this viewpoinr. and has ohtained results in agreement with Yidgwirk's co-valency rule. Pauling: 2. Krystallographie, 67,37; (1928).
APPAREKT VOLUMES OF SALTS IN SOLUTION
2327
functional dependence of the constant a on the same ionic factors which are involved in determining the lattice distances. The linearity of these relationships, indicated by the drawn straight lines, is, t o he sure, open to question because of the uncertainty in the basic data but it will be a convenient
FIG.I T h e Variation of log3 with the Function R,
(I
+ k)
assuniption in discussing a number of interesting points. I n the first plnce there is a marked difference in the linear constants for the three different temperatures, a difference which most probably originates in the changes produced by temperature in the electrical properties of thr water molecule. Prohably the most striking feature of the graph is the existence of at least two set: of points at each temperature with practically a constant differencp of 0.0:
2328
ARTHUR F. SCOTT
in the values of log a. Thus, the a values of the lower set of points a t any tcmperature are approximately 0.93 of the values necessary to agree with the upper set of points. An examination of the figure discloses that the points for the cesium salts are the ones which occur most consistently in this “lower” group. Since the R, values employed in plotting these points are those of the non-existent rock-salt structure, the cause of the above division into two groups which suggests itself is the co-ordination number of the ions. In this connection it is interesting t o note the following co-ordination numbers found by Garrick for ion-hydrates: Li = 1 or 6 ; Ka and K = 6 ; R b and Cs = 8. Finally, we may call attention to the fact that the plotted points of the lithium salts show little in common with those of the other salts except at 2 5 ’ . In eonsidering these exceptional points it should be recalled that the values of the constant a are for dilute solutions only because, as has already been pointed out, the linear relationships for these salts exhibit marked discontinuities a t moderately high concentrations. In view of our present argument that the relative values of t,he constant a are dependent on inter-ionic forces, such a discontinuity suggests that a marked change in these forces takes place in solutions of lithium salts. We could account for these changes by postulating the existence of a “chemical” hydrate’ of the lithium ion whose stability is dependent on concentration and possibly also on temperature; for the properties of such an aggregate would be quite different from those of the simple ion. This same postulate, furthermore, would be sufficimt to account for the irregularities of the points plotted for the lithium salts in Fig. I because a hydrated ion would not be comparable to the other simple ions. In the foregoing discussion it is shown that the constant, a is dependent on the valence factor and that it is also related to the interionic distmces in crystals. Although we are unable to derive any direct evidence in substantiation of equation (2) from these connections, we have implied a theoretical basis for the empirical rule. X further confirmation of this argument is found in the consideration of the maximum value of p, that is, the hypothetical value when no water is present. This problem will be dealt with in a subsequent article.
Summary
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The present paper is a study of the linear relationship p = am’ b, put forward by Masson. In the first part) of the paper the applicability of the equation is tested ( I ) by examining its ability to reproduce the experimental data of Baxter and Wallace; ( 2 ) by examining the values of b on the basis of the principle of additivity; and (3) by computing p values in saturated solutions and comparing them with experimental values. Although the test is not conclusive, the evidence, with few exceptions, tends to support the validity of the empirical relationship. ‘ A rather suggestive fact bearing on this matter is that, according to Masson, with lithium chloride solutions at I jo, the abrupt break in the linear relationship occurs when the composition of the solution is LiCl I O H20.
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APPARENT VOLUNES O F SALTS IN SOLUTION
2329
In the second part of the paper the theoretical significance of the equation is considered briefly. The fact that the constant a is dependent on the valence factor and is likewise related t o inter-ionic distances affords grounds for the suggestion that a basis for the equation may be found in the interaction of ionic forces. Since the completion of this article two notes bearing on the present subject have been published. Redlich [Naturwissenchaften, 15, 2 5 1 ( r p , ~ )finds ] that, according to the Debye-Huckel theory, the partial molal volumes of salts in solution should vary linearly with the square-root of the concentration. Geffcken [Naturwissenschaften, 15, 32 I (1931)] states that in connection with a study of the refractivity of solutions he has established a linear relationship between the apparent molal volumes of salts in solution and the square-root of the concentration. This relationship is identical with that proposed by Masson. Since Geffcken’s conclusion is based chiefly on new measurements in very dilute solutions (less than 0.3iLI), it may be taken as additional evidence in support of the empirical rule of Masson. The Rice Institute, Houston. Texas.
