VAPOR-PRESSURE LOWERING AS A FUNCTION O F THE DEGREE O F SATURATIOS. I BY ISAAC BENCOWITZ'
Introduction Numerous functions expressing Lhe relation of the vapor pressure of binary solutions, one of the components of which is non-volatile, are found in the literature. It was known for many years that water which contained some non-volatile substance dissolved in it boiled at a higher temperature than pure water. This indicated that the presence of t,he solute had lowered the vapor pressure of the solvent. The first generalization made in this connection was that of von Babo, who, in 1848, pointed out that the relative lowering of vapor pressure is independent of the temperature, providing the solution is dilute. Wullner2 came to the conclusion that t,he lowering of the vapor pressure of water by non-volatile solutes was proportional to the concentration of the solute. Later, Tammann showed that this was not quite accurate. A great deal of apparent contradiction was removed by the experimental and theoretical work of R a ~ u l t who , ~ confirmed in certain cases both the laws of von Babo and Wullner. The most important advance made by Raoult, however, lay in the introduction of the concept of molecular lowering of vapor pressure. In 18j 8 , Kirchhoff4 obtained the formula, d P Q = R T 2 - ln-' dt P which expresses the vapor pressure as a function of the differential heat of solution. Notwithstanding the fact that this expression is theoretically sound, its validity was questioned until the experimental work of Roozeboom5 and Sholtz6 established it without doubt'. However, it must not be forgotten that this equation, as well as all relations which pure thermodynamics yields, is only an indirect relation, Le. it does not give us a picture of the intermolecular forces which exist in the solution and determine its properties. Kirchhoff's law expresse: the dependence of the vapor pressure on the differential heat of solution but does not disclose the nature of the function relating the differential heat to the National Research Fellow in Chemistry. ZPogg. Ann. 103, 529 (1858). Compt. rend. 103, 1125 (1886); 104, 1430 (1887); Z. physik. Chem. 2, 353 (1888). Pogg. Ann. 103, 177 (1858). 5 Z. physik. Chem. 4, 31 (1889). 6 Wied. Ann. 45, 193 (1892). Woitaschewsky: 2. physik. Chem. 78, I I O (1912). For the theoretical significance of this formula, see Porter: Trans. Faraday SOC.11, 19 (1915). 1
VAPOR-PRESSURE LOWERING AND SATURATION
I433
temperature1. Van Laar,2 on the basis of molecular attraction, obtained an equation for the vapor pressure of liquid-systems which can be used very satisfactorjly as an empirical equation to fit any type of vapor pressure curve.a For binary solutions, one of the components of which is nonvolatile, the equation takes the form: P =Po(I - N)C””’ where P and Po are the vapor pressures of the solution and pure solvent respectively. N is the mol-fraction and a and C are constants.4 Due to the fact that it is claimed6 that all equations connecting the vapor pressure of water with the temperature apply to saturated solutions of salts, it may be well to give a short account of the attempts made to deal with the variation of the pressure of saturated vapor (in contact with the liquid) with the temperature. These are characterized by the fact that on compression or rarefaction the volume undergoes considerable change while the pressure remains constant, until one or the other of the phases has mtirely disappeared. If the pressure of the saturated vapor depends only on the temperature, some general relation between the pressure and the temperature such as: P=f(T) must exist. The form of the function will probably depend on the nature of the substance, but no general law has as yet been found. The first attempt jn this direction was made by Dalton, who proposed the simple law that the vapor pressure increases jn a geometrical progression as the temperature increases in an arithmetical. This assumes that the relation between the pressure and temperature is of the form p = baT or log p = C‘+c. This formula, however, holds only for small limits of temperature,-near the point a t which the constants were determined.6 Young’ proposed the formula P = (a+ bT)”, the three constants of which are determined experimentally. Another equation suggested by Roche8 from theoretical considerations belongs to the type P=ba
T __
+nT
Finally, a more general form was suggested by Biot9: log p =a+b’+Cp‘ Regnault found that Young’s formula might be used to represent the results of experiments within a limited range of temperature, but that beVan Laar: Z. physik. Chem. 72, 727 (1910). Z. physik. Chom. 72, 723 (1910);82, 599 (1913). a Hildebrand and Eastman: J. Am. Chem. SOC.37, 2452 (1915). Porter: Trans. Faraday SOC. 11, 48 (1915). 6Woitaschewsky: Z. physik. Chem. 78, I I O (1912). See also Compt. rend. 176, I 5 5 2 (1923). 6 Preston: “Theory of Heat,” 3rd Ed. p. 393 (1919). 7 Nat. Phil. 11, 400. 8 See Dulong and Arago’s Memoir, MBm. de L’Institut 10, 227. Preston: “Theory of Heat,” p. 118 (1919). l
2
I434
ISAAC BENCOWITZ
yond this range it had to be abandoned. The formula of Roche, on the other hand, represents the whole series of experiments with considerable accuracy, but not quite as precisely as the more general formula of Biot. Rankinel later suggested the expression: log p = a-b/t-C/T2 where a , b and C are empirical constants, and T is the absolute temperature corresponding to the vapor pressure, p. This type of formula is in frequent use as it represents the whole series of Regnault’s experiments fairly well. A closer approximation may be obtained, however, by applying the relation due to Bertrand2 namely:
Kirchhoff3 in 1858, and Rankine in 1866, independently suggested the formula, B log p = A + - + C log T T (This equation as well as that of Young can be obtained from theoretical consideration^)^ TFe work of Kernst and others in connection with the heat theorem has drawn considerable attention to the expression first proposed by Hertz5 which has the following form:
This equation has been applied by Knudsen6 in the case of the vapor pressure of mercury, and wa,s thoroughly tested by Smith and Menzies’. A formula based partly on the theorem of corresponding stAtes was prcposed by Nernst*. This equation is based on the a,ssumption thLtt the molecular heat of the vapor at absolute zero is greater by 3.5 than the molecular heat of the liquid. Recent investigations on the specific heat a t low temperatures show thik to be not trueg. It is seen, therefore, that all attempts to express the vapor pressure of a liquid in equilibrium with its va,por as a function of the tempmature har been without recognized success. No rational law holding from the supercooled region below the triple point has been formulated. I t is true, however, that the temperature ratio law of Ramsay and Young’O holds for the whole range, from very low pressures up to the criticalll. S e w Phil. Journal Edinburgh, July ( I 849). Bertrand: “Thermodynamique,” p. 93 (188;). 3Popg. Ann. 103, 185 (1858). Callendnr: Enc. Brit. 10th Ed. p. 39;. Wied. Ann. 17, 199 (1882). 6 Ann. Physik, (4) 29, 179 (1909). 7 J. Am. Chem. Soc. 32, 1434 (1910); Lewis: Physical Chemistry, 1, 93 (1920). This formula is discussed fully in Xernst: “Thermodynamics and Chemistry,” (1907). 9 Sackur and Gibson: “Thermodynamics,” p. 218 (191;). 10 Phil. Mag. 20, 5 1 5 (1885): 21, 33 (1886); 22, 3; (1886). For a very recent application of this principle, see: Lorenz: Z. morg. allg. Chem. 138, 104 (1924); 143, 336 (1925). For its relation to Bertrand’s vapor-pressure equat,ion see Porter: Phil. Mag. 13, 724 (1907). Moss: Phil. Mag. 16,356 (1903); 25, 453 (1907).
VAPOR-PRESSURE LOWERING AND SATURATION
I43 5
Many of the above equations derived for pure liquids in equilibrium with their own vapor have been applied to saturated solutions with varying success. Speranskil applied Bertrand’s equation and found that it expresses accurately the experimental vapor pressure of saturated solutions of several salts. He also showed2that it holds for solutions of chloroform, benzene and liquid ammonia. The vapor pressure of potassium chloride, however, i s more accurately expressed by the equation of Hertz3. Relations connecting the vapor pressure with the solubility are rather scarce. The only one that came to my attention is that of Speranski4 which has the form, log p = a log C+b where a and b are empirical constants and C is the weight of the salt in gms. which is required to saturate I O O gms. cf water. It holds for C u S 0 4 . gHzO and Na2S04within the experimental error, but fails completely in the case of potassium chloride and potassium nitrate.4. Thus, all formulae suggested connecting the vapor-pressure and the temperature hold only for saturated solutions and then only for a few salts and within a short range of temperature; the equation connecting the vapor-pressure and the solubility is valid only in the case of a few salts. No function has been formulated as yet which expresses the vapor-pressure of aqueous solutions of non-volatile solutes, as a function of the temperature and the solubility, an equation which would express the experimental data within reasonable precision of most, if not all salts. Degree of Saturation Various methods are used to express the concentration of solutions. Each one of the methods has its field of usefulness. All of them, however, are arbitrary. The “volume-normal” system jn expressing concentration while advantageous and correct for analytical purposes is both disadvantageous and illogical whenever any phenomenon is to be studied in which the influence of the solvent on the solute is involved. The only theoretical justification for the use of this system was based on van’t Hoff’s discovery5 of the analogy between the osmotic pressure in dilute solutions and the gas law. Later, in order to bring the experimental values of the osmotic pressure in a closer coilformity with the gas laws, the “weight-normal” system was introduced.6 Since then, however, the analogy between osmotic and gas pressures was definitely established to be a myth.’ The theoretical foundation on which ‘Z. physjk. Chem. 70, 519 (1910); J. Russ. Phys. Soc. 41,91 (1910). Z. physik. Chem. 78, 86 (1912). Pavlovich: Z. physik. Chem. 84, 169 (1913). * Z. hvsik. Chem. 78,86 (1912); Another equation was recently proposed by Moudain-
Mouvaf Compt. rend. 178, I164 (1924). 52. physik. Chem. 1, 481 (1887). Morse and Frazer: Am. Chem. J. 34, I (190j); 37,324, 425, 458 (1906); 38, 175 (1907); Morse: “The Osmotic Pressure of Aqueous Solutions” p. 97 (1914); Findlay, “Osmotic Pressure” (1913); Bancroft: J. Phys. Chem. 10,320 (1910). ’Lewis: J. Am. Chem. SOC. 30, 660 (1910); Kendall: 43, 1391 (1921); Hildebrand; “Solubility,” p. 24 (1924).
1436
ISAAC BENCOWITZ
these two systems were based was, thus shown to be groundless. Of the two, however, the (‘weight-normal” is to be preferred for the following reasons: In the first place, the atomic weight is subject to constant revision, and therefore, varies from year to year. In the second place, the true molar weight except in a few cases, is not known, and the formula which is employed expresses only a surmise as to the composition of the ultimate molecule, or more frequently is merely the simplest expression which represents the stoichjometric proportions of the elements involved1. In the third place, the volumenormality changes with the temperature, so that to change from one temperature to another requires the knowledge of the density of the solution, which is seldom accurately known. The mol fraction method of expressing concen trabion, because of Raoult’s law assumes a theoretical significance. This law, however, is valid only in a few cases, for the so-called “ideal solutions.” But even the ideal solution cannot be accurately defined by Raoult’s law2. One thing is obvious and that is that the mol fraction is not a functim of tbe intermolecular forces which exist in the solution and determine its properties. Whereas, when two properties of a solution are expressed as a function of each other with the object in view of gaining some knowledge about these intermolecular forces, both properties must be functions of bhese forces in so far as it is possible to determine. The similarity between the process of solution and evaporation has been frequently pointed out. NernsV went so far as to derive an expression for the lowering of the solubility similar to that of Raoult for vapor-pressure lowering. Whatever the theoretical basis for this and similar relations may be, it is plausible to assume that both the vapor pressure lowering and solubility are functions of the same solution forces. The composition expressed in terms of the solubility will, therefore, be a function of the solution, as much as the vapor pressure lowering. And all relations expressing any property of the solution as a function of the concentration expressed in this manner will be essentially fundamental. The degree of saturation is the composition of a solution expressed in terms of the solubility. Thus, if the concentration of a solution is given as n gms. (or mols) per ’\Ii gms. (or mols) of solvent and the concentration of a saturated solution as N gms. (or mols) per W. gms. (or mols) of solvent, the degree of saturation, S, is then given as, S = n/N The ratio of two degrees of saturation SI,and SZat a given temperature will be : S ~ S=Znl/n2 This ratio is, obviously, identical with the ratio of two concentrations expressed in gms. (or mols) per 1000 gms. of water. It is however, not identical with the ratio of two mol-fractions. Lewis and Randall: “Thermodynamics,” p. 33 (1923). “Solubility,” p. 27 (1924). Nernst. Z physik. Chem. 6, 19 (1890).
* Hildebrand
'437
VAPOR-PRESSURE LOWERING AND SATURATION
This method of expressing concentration is not advocated for practical or for theoretical use. The disadvantages of this system are self-evident. In the first place, its accuracy is limited to that of the available data for the solubility. In most cases, this is very poor or not available at all. Furthermore, it cannot be applied to solutions the components of which are miscible in all proportions. Kevertheless the theoretical importance of this methDd, especially in the study of the theory of solutions, is not invalidated by these shortcomings. It is hoped that the relation between the vapor pressure lowering and the degree of saturation derived in this paper will prove this point. Vapor Pressure Lowering and the Degree of Saturation Using the concept of the degree of saturation, the following relation is Dbtained : logAP=K[$-a
(I
-'!!)]
where AP is the vapor pressure lowering, T, the absolute temperature, S, the degree of saturation, i.e. S =n/N, where n js the number of gms. or mols in a given weight of solvent and N is the number of gms or mole necessary to saturate the same weight of solvent, i.e. the solubility, and K, a,and b are constants. This equation is derived on the basis of the following three postulates and is mathematically exact if the postulates are experimentally true. Postulate I : The coefficient,
[ a]':::
=constant
= K,
i. e. when log
S
of the vapor pressure lowering, AP at a given degree of saturation, S, is plotted against the reciprocal of the corresponding absolute temperacure a straight line is obtained. The slope of this line is K. Postulate 2 : The value of the coefficient is independent of the degree of saturation. ___
is constant, i.e. when the log of [ the degree of saturation is plotted against the reciprocal of the absolute Postulate 3 : The coefficient
; ; :a
AP = I m m
temperature a t which the vapor pressure lowering is I mm. a straight line is obtained. Assuming for the moment the validity of these postulates, relation ( I ) is derived as follows : According to postulate I
[a 'Oag,;
'1,
=K
which on integration gives log*,=,(+) where I is an integration constant.
+I
1438.
ISAAC BENCOWITZ
When A P = Imm. I =K / T (4) where T, is the absolute temperature at which the solution of the given degree of saturation has a vapor pressure lowering of I mm. According to postulate 3, when log S is plotted as ordinates and I/T as abscissa, where T is the absolute temperature at which AP = I mm. we obtain a straight line, the equation of which is, log s (A) -+ 2 . = 1 b a where a and b are the intercepts with log S snd the r/T axis respectively Equat'ion (5) can be written as r / T = a ( I - log s
T)
substituting this in (4) we get,
I = -Ka
(
I--
This substitution is possible becausg 3f postulate for I ia (3) the final function is obtained. log A P = K [
$ -a
2.
When ( 7 ) is substituted
(I-'%')]
It is readily seen from the above simple mathematical derivation that relation (I) is exact if the three postulates are experimentally valid. Experimental Proof of the Three Postulates The experimental data used in testing the above three postulates were taken from Landolt' and Bornstein's Tables and from Seidell's Solubilities. The degrees of saturation corresponding to each temperature and vapor pressure lowzring were obtained by dividing the concentration given in gms. per IOO gms. of water, by the proper solubiljties similarly expressed. Smooth curves were drawn through the experimental points obtained by plotting the log of the degree of saturation, S, againat the log of the vapor pressure lowering AP. The log A P for any given degree of saturation was read off from this series of isotherms and plotted against the reciprocal of the corresponding absolute temperature. The curves thus obtained are shown in Pigs. I and 2 . It will be seen from these figures thst the curve of each substance is a straight line which is a striking corroboration of Postulate I , in view of the fact bhat 3 I salts of various types are considered. It may be of interest to point out that there is a break of the log A P - I/T curve at the reciprocal of the absolute temperature corresponding t o the transition point of two hydratek. In order to prove Postulate 2,j.e. that the slopes of the logAP- ~ / T c u r v e s are independent of the degree of saturation, a family of curves were obtained by plotting the values of log A P a t different degrees of saturation against the reciprocal of the corresponding absolute temperature. I n all cases, the slope of these curves is independent of the degree of saturation. The values of K,
VAPOR-PRESSURE LOWERING AND SATURATION
I
1
I
I
I
2.9 7.I 3.s TC/PPOCAf ABSOLUT€ TLMP‘RATURf x /O
I43 9
I
3.5
FIG.I Abscissas, reciprocal of absolute temperature x 103. Ordinates, log of the vapor pressure lowering in mm; there is a different origin of this axis for each curve.
LEGEND OF FIGS.I No.of
Curve I 2
Name of Substance
Sa,SO,
ka.iCO;. HzO 3 MgClz.6H20 4 B a B r z . 2HpO j BaC12.2H2O 6 CaCIp. z H Z O 7 (KHq)pS04 8 KapS03 9 KzCO3 IO K2Cr04 I1 CsCl 12 RbCl 13 K a I . 2Hz0 14 NaCl I 5 CoSOq. 7 H 2 0 16 ”A21 17 L i c i
log S
log A P No. of scale curve
0 . 4 S;btr. 0.4
4.8 3.9 3.3 3.2 3.2 2.8 2.7
0.5
”
:;;:: O.9
0.5 0.5
0.5 0.5 0.8 O.5
z:: O.4
0.4 0.9
”
>, I’
:: ,, ,> ”
,I ”
2.1
1.9 2.2
1.8 1.3 0.5
0.7
0.2 0.2
O.K.
AND 2
Name of Substance
18 I9 20 21 22
23 24a 24b 25 a 25 b 26a 26 b
27 28 29 30 31
KR; NaBr . H ~ O NaBr .2H20 ZnSOl. H 2 0 ZnS04. 6 H z 0 LiBr . HzO LiBr . 2 H 2 0 BeSol. 4H20 SrBrp.6 H 2 0 SrClp.6 H z 0 NaC108 CuSOi. jHpO
log AP scale
0. K . Subtr. 0 . 2 Add 0 . 6 ” 0.4 Subtr. 2 . 5 ” 2.6 ’I
1.6
”
2.1
”
0.8
”
1.2
” ”
0.3 0.3 0.3 0.4
” ’I
I440
ISAAC BENCOWITZ
the slope of the line log4P- I/T, for each substance were obtained graphical13 from such curves and are given in columns 2 and 6 of Table I. According to postulate 3, when log S is plotted against the reciprocal of the absolute temperature at which a solution corresponding to the degree of saturation, S, has a vapor pressure lowering of I mm., the curve obtained is a straight line. To prove this statement, values of ( I / T ) A ~ =were ~ ~ ob~,
FIG.2 Abscissas. reciprocsl of absolute temperature x 103. Ordinates, log of the vapor pressure lowering in mm; the origin of this axis is different for each curve.
tained by plotting l o g 4 p against I/T for differentvalues of log S. The intersections of these lines with the I/T axis are the required ( I / T ) A ~ = ~values ~~~, corresponding to log S of the particular log Ap- I/T curve. When these intersections are projected downward a distance equal to log S of the corresponding log4p - I/T curve, as shown in Fig. 3, a curve is obtained which is the desired log S- I/T curve. Such curves for 3 I salts are shown in Fig. 4. It will be seen that with the exception of CuSO4. jHzO, these curves are straight lines. The instances where there is a deviation from this rule are so few that it is plausible to assume that the experim.enta1data in such cases are in error. This is conclusive proof of the validity cf Postulate 3'. 1 The experixnental data for ALz(S04)3 and LiI are not in agreement with this postulate. The deviations, however, are so irregular that it makes one doubt the accuracy of the experimental data. We hope, however, in the near future, to publish new experimental data on the vapor pressure lowering of these salts and a few others.
VAPOR-PRESSURE LOWERING AND SATURATION
1441
The intersections of log S - I/T curve with the I/T and log S axes give the values of the constants a and b respectively. These values are given jn columns 3, 4, 7 and 8 of Table I. The values of the three constants, K, a and b given in Table I are not final. They are very sensitive to slight changes in the experimental data and
FIG.3 Abscissas, reciprocal of absolute temperature x 19.Ordinates, upper part, log of the vapor pressure lowering of BaBra . zHz0; the lower part, the log of the corresponding degree of saturation.
will no doubt change as mare accurate data should became available. It may even be predicted that more accurate data will bring out regular deviations from the three postulates; deviations which a t present are masked by large experimental error. The degree of saturation as expressed in this paper is probably only the “apparent” degree of saturation and should be corrected for such effects as “self-saltjng-out.” However, this, as well as other factors involved, are probably very small as evidenced by the fact that the values of A P calculated with equation ( I ) are in very close agreement with the experimental data. The comparative values of the experimental and calculated APs are given in Table 11. In order to save space, these values are given only
ISAAC BENCOWITZ
I442
FIG.4 Abscissas, reciprocal of absolute temperature x 103. Ordinates, log of the degree of saturation; the origin of this axis is different, for each curie.
LEGENDOF FIG.4 F a m e of Curve Substance
No.of I 2
KC1 RrC12,2Ha0
4
KBr RI NH4Br
7
IdC1 LiBr . HzO xis04. 6Pz0 IiBr . zF90
3
2 8
9
log
s
Scale
0. K. 0. K. 0. K. 0. I(. Add 2 0. R. 0. K. Add I . 2 0
!’
2.0
N o . Of
Curve
17
18
I9 20 21 22
23 24 25 26 27
10 I1 I2
”
0.6
”
1.6
”
13 14
”
1.2 1.5
28 29
2.4
33 31 32
15 16
:: ”
2 .O 2.2
Name of Substance
NaCl SrBr, . 6H20 SrC12. 6H20 RbCl Sa2S04 S R ~ C OHa0 ~.
Log s Soal e
Add 2 . 4 1, 2.6
2.9
3 .o 3.4 3.4 3.3 3.7 3.8 4.4 4.4 4.7 j.0
4.8 4.8 4.8
VAPOR-PRESSURE LOWERING AND SATURATION
I443
for three concentrations and a few temperatures. The agreement, however, is of the same order throughout. Columns I and 2 are self explanatory. Column 3 gives the solubility expressed in gms. per I O O gms. of water. In columns 4, 7 and IO are given the degrees of saturation obtained by dividing the concentration by the corres-
TABLE I Values of the Three Constants Kame of Substance
LiCl LiBr . H 20 LiBr .2HzO KCl KBr
KI NHdC1 KH4Br NaCl NaBrzHzO NaBrHzO Na12Hz0 RbCl CsCl KN03 KZCrO4
KZC032H:O Na &O aH2 0 NaN03 Na2S04 "(4)
zS0 4
NaC103 CaClzzHzO BaClz.2H2O B a B r z .2HzO SrClz. 6 H z 0 SrBrz.6 H z 0 MgC12.6HzO BeS04.4Hz0 XiSO,. 6Hz0 CoS04.7Hz0 ZnS04.6H20 ZnS04.HzO C U S O ~5Hz0 .
K
2570 2405 2430 2 500
2595 2610 2625 2 500 2308 2610
2393 2750
2487 2400 3200 23 50 242 j 2025
2590 2090 2 400 2650 2315 2575 2425 2770 2750 2460 2775 2610 3073 2575 208 j 2685
a x 103
b
3.945 4.040 4.012 3.594 3.598 3.688 3.625 3.682 3.690 3.736 3.790 3.780 3.696
--
3.495 3.593 3.817 3.708 3.670 3.585 3.631 3.690 4.022 3.480 3.690 3.664 3.724 3.836 3.551 3.447 3.417 3.543 3.622 3.263
1 2 .52
7.315 7.394 7.242 11.36 7 ' 370 8.720 10.21
7.071 8.078 7.216
7 770 7 770 '
'
6,573 5 ,803 5.771 6.287 9.032 7 ' 592 16.80
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VAPOR-PRESSURE LOWERING AND SATURATION
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VAPOR-PRESSURE LOWERING A N D SATURATION
I449
ISAAC BENCOWITZ
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VAPOR-PRESSURE LOWERING AND SATGRATION
14.51
pondjng solubility given in column 3. In columns 5 , 8 and I I the experimental vapor pressure lowerings are given, and the corresponding calculated values are given in columns 6, 9 and 12.
Conclusion I n conclusion, jt is of interest to point out that the constant, a, in equation ( I ) is the reciprocal of the absolute temperature at which a saturated solution has a vapor pressure lowering of I mm. This is seen from Figs. z and can be readily proven theoretically. This is important, because if the vapor pressure lowering of a saturated solution of 2 (or better 3) temperatures is known, then the line l o g n P I -/T can be drawn. The slope of this line is K and its intersection with t,he I/T axis is a. The third constant of the equation can be obtained if only one other value of AP and the corresponding degree of saturation are known. Two generalizations follow directly from equation ( I ) , which can be written in the form, logAP
g l o g S+K b For a saturated solution, since S = I and log S = 0, this equation reduces to, =
logAP,=K( $-a)
(9)
Substract,ing (9) from (8), We obtain, AP Ka log - = - log s AP, b A P or dropping the logs -AT, This equation can be stated in words in the form of two gsneralizations. I . The ratio of the vapor pressure lowering at any degree of saturation over the vapor pressure lowering of a saturated solution at a given temperature is independent of the temperature. 2 . The value of this ratio is a simple function of the degree of saturation. These two generalizations will be more fully discussed and experimentally proven in an early publication.
SF
Summary ( I ) It was suggested that the degree of saturation is a fundamental method of expressing concentration and should be used in the study of the theory of solution wherever possible. ( 2 ) . Using this method of expressing concentrations, a relation is arrived at which gives the vapor pressure lowering, A P , as a function of the absolute temperature, T, and the degree of saturation, S, or indirectly the solubility.
This equation has the form, log A P
=
I ;.
K--a
( -lotS)] I
where K, a and b are constants, and holds for 31 salts.
ISAAC BENCOWITZ
1452
(3) The values of the three constants for 3 I substances are given. (4) Two generalizations are directly deducible from the equation: Generalization 1 : The value of the ratio of the vapor pressure lowering at
any degree of saturation over that of a saturated solution at a given tempera-
AP
t,ureisindependent of the temperature, i.e. -is the same for all temperatures. Ape Generalization 2 : The value of this ratio is a very simple function of the degree of saturation, i.e.
E
=
AP, constants in the above equation. Haverneyer Chemical Laboratory New York Unioersitu New York Cify.
8 where K, a
and b are the three
