# Raoult's Law

BY WILDER D. BANCROFT AND H. L. DAVIS. Using a van't Hoff cycle tve can calculate the osmotic work done when we squeeze out of a solution one gram of ...
RXOULT’S LAW* BY WILDER D . BASCROFT AND H. L. D A W S

Using a van’t Hoff cycle we can calculate the osmotic work done when we squeeze out of a solution one gram of the pure solvent by means of an ideal piston which is permeable to the solvent and impermeable to the solute. If we work with such a large mass of solution that we may ignore the change of concentration when one gram of solvent is removed reversibly, if the vapor of the solvent follows the gas law at that temperature, and if we can ignore the volume of one gram of liquid solvent relatively to the volume of one grain of solvent as vapor, we have RT PV, = - In

I , P2 where P is the osmotic pressure, Ygis the volume of one gram of the solvent in the solution,‘ >I2 is the gram-molecular weight of the solvent as vapor, p2 is the vapor pressure of the pure solvent, p’* is the partial pressure of the solvent in the solution, and T‘, is the volume of one gram-molecular weight (as occurring in the vapor) of the solvent in the solution. The first equation holds for the pressing out of one gram of the solvent and the second for the pressing out of one gram-molecular weight (as defined) of the solvent. The gram-molecular weight of the liquid solvent does not appear in the formula. If we squeeze out the amount of solvent in which one gram-molecular weight of the solute (in the solution and not as vapor) is contained, we get the equation as given by van’t Hoff M 2

p

or PT’, = RT

2

I1 where K 2 = GP/MBand nl = GliMI, G2 and GI being the masses in grams of the two components in the solution, hlt being the gram-molecular weight of the solvent as vapor, and ?VI1 the gram-molecular weight of the solute in the solution. The gram-molecular weight of the solute as vapor does not appear in the equation. V1 is the volume occupied in the solution by N grammolecules (as defined) of the solvent. It is necessary to be as meticulously explicit as this because many physical chemists do not know just what the terms mean in the van’t Hoff formula. If we assume that PV, = RT, equation I1 becomes P2 , n1 - In 7

K--Pz

I11

*This paper is part of the programme now being carried out a t Cornell University under a grant to Professor Bancroft from the Heckscher Foundation for the Advancement of Research, established by August Heckscher a t Cornell rniversity. Bancrnft and Davis: J. Phys. Chem., 32, I (1928).

WILDER D. BANCROFT A S D H. L. DAVIS

362

which is, of course, true only within the limits for which the assumption holds. For very small values of nl we can write, with a loss of accuracy, n’ - d In p2

K-

=

p2 - P’? ___ P2

An empirical study of his data led Raoult to write this last formula

“1 = kP2 - P’?

IT‘ Pz As a result of van’t Hoff’s theoretical deductions, Raoiilt made k = I . Since the resulting equation becomes an absurdity when nl = K2because we must then have p’2 = 0,the equation was changed empirically to read

x2

nl - pz - PI2 TK2 nl P? For values of n, negligible with respect to K,,Equations I11 and V are practically identical. It is only for such conditions that we can substitute (p, - p’z),’p? for ln(p21pl)without introducing a serious error. Equation 1’ has practically no theoretical basis when applied to any considerable range of concentrations, because it contains the two assumptions, cont,rary to fact, that PV1 = RT and that ln(p2‘p’?) = ( p ~- p’2),:p2. It has not been shown by anybody that introducing nl into the denominator nullifies the effect of the two errors. Speyers’ was one of the first people to try to substitute Equation F’ for Equation 111. If we define an ideal solution, for the moment, as one in which each component has the same molecular weight in the liquid phase as in the vapor phase, and as one in which the two components mix in all proportions without change of volume and without any heat effect, we find empirically that the change of partial pressures over the whole range of concentrations can be expressed accurately by the equations

+

VI where the M I and N, concealrd in X1and X2 refer to the gram-molecular weights of the two components respectively as vapors. Equation VI is called Raoult’s law by the California school,2following the lead of Speyers, because Equation T’ reduces t o it for the case, and only for the case, that the molecular weight of the solute is the same in the liquid as in the vapor, in which case nl = SI. I n fact Hildebrand puts the cart before the horse by saying, p. 59: ‘ W e may, therefore, follow G. 5 .Lews in defining an ideal solution as one which obeys Raoult’s law

J. Am. Chem. Soc., 21, 72j (1897). “Solubility,” 24 (1924).

* Cf. Hildebrand:

3 63

RAOULT’S LAW

a t all concentrations and pressures, a definition which has some important consequences. It follows from it, as Lewis has shown, that the formation of such a solution will take place from its component liquids without any heat of mixing and without any change in volume.” It seems to us that the identity between Equation VI and the special case of Equation T’ is purely formal and not real. I n Equation V, it is certain that one molecular weight refers to one substance in solution and the other molecular weight to the other substance as vapor. We believe, and expect to show in this paper, that the molecular weights for both components in Equation 1-1refer to the vapor phase and not to the solution phase. Of course this makes no difference in the ideal case, where there is no polymerization in the liquid phase. The difference between tweedledum and tweedledee becomes very important when one wishes to include other binary systems. Hildebrand did not have any very striking success in discussing non-ideal solutions on the assumption that what he calls Raoult’s law has to do with molecular weights in solution. Consequently, i t will be wise to find out what can be done on the assumption that both the molecular weights in Equation VI refer theoretically to the vapor phase. By doing algebraic transformations we can change Equation VI into

VI I TABLE I Ethyl Alcohol and Water at

Gram c7c alcohol in liquid 0

12.36 20.j I

28.40 33.90 39.32 j0.46 56.50. 71.09 78.0; 90.12 100.

log Kl calc.

0.8030 0,684; 0.6653 0.6697 0,6636 0.6791 0.6721 0,6678 0.6663 0.4979 ~

p’ water

found mm

calc. mm

23.i3 23.ij* 22.67 22.39 21.78 2 1 . 7 0 21.15 21.13 20.j9 2 0 . 7 7 20.36 20.38 19.60 1 9 , j o 19.01 18.9j 1 7 . 3 1 17.30 1 6 . 1 8 16.19 10.68 12.98 0.0

o.o*

2 jo

log K P calc.

-__

p’ alcohol

found mm 0.0

calc. mm

o.o*

I , 8034

1.918j

10.50

16.66

19.0j

0.003 j

22.27

0.0028 0.0260 0.0048 0.0160 1.9948 0,0037 0,0829 __

24.90 26.8j 30.73 32.16 36.64 39.53 47.40 j9.01

22.26 24.25 27.3 30.68 31.73 36.89 39.51 45.50 j9.01*

Ij.1

*These values have no significance because they are given by every equation of this type.

3 64

WILDER D. BASCROFT AND

n. L.

DAVIS

I n Equation VI1 each side of the equation varies from zero to infinity as we change from Gl = o to G z = 0. We can tell whether this equation holds by plotting the data on logarithmic co-ordinates and seeing whether we get a straight line. Since >I?'111is a constant when the vapor of each liquid follows the gas law and when the mixtures follow Dalton's law, it is immaterial whether we plot the concentrations in grams or in molecular weights. We have found empirically that a number of systems can be represented with an unexpected degree of accuracy by the modified equation VI11 In Table I are the data by Dobson' for ethyl alcohol and water. The data for the partial pressures of water agree with the calculated values within the experimental error through ;8% alcohol. There is quite a discrepancy between the calculated and found values for ninety percent alcohol. Special experiments will have to be made to discover the cause of the discrepancy. In any event the equation describes the facts at least up to a mol fraction of 0.6, which is a long way beyond the orthodox limit of tenth-normal. The partial pressures of alcohol agree very well except for the two dilute solutions and for the most concentrated solution. I n view of the good agreement for water vapor in the two dilute solutions, it is possible that the discrepancy with the alcohol pressures for these two solutions may be due to experimental error. TABLE I1 Methyl Alcohol and JTater a t 30.90' pl = partial pressure and XI = number of mols of alcohol p2 = partial pressure and SZ= number of mols of water 0.9 log

5101 percent alcohol in liquid 0

N1

- log

found

10

39.8 74.4

50

60 70 80 90 IO0

101.I

122.8 143.7 165.3 188.0 210.4 235.0 260.;

J. Chem. Soc., 27, 2866 (1925).

log K1 =

0.100

p' water

calc. mm

found mm

calc. nim

55.0

0.0 20.2

20

=

PI

p' aicohol

mm

5

30 40

l

K2 -

21.3 38.7 69.2 96.4 121.5

145.I 167,9 190. I 212,I

234.7

52.8 50.7 45.1

41.9 37.7 32.8 27.5

52.7 50.7 46.5 42.2 40.3 32.8 27.4

21.0

21.7

14.8 7.8

12.1

0.0

6.7

RAOULT'S LAW

365

In Table I1 are given the data by Ferguson and Funnell' for methyl alcohol and water. The equations fit the data very well over the whole range with the exception of the alcohol values for the twenty and thirty molecular percents alcohol solutions. When plotted on logarithmic co-ordinates, the values for these two concentrations lie well off the curve. Professor Ferguson says that the curve does have a hump in it over this range. I t remains to be seen to what this is due. The value for the pressure of water vapor in equilibrium with forty molecular percent of alcohol is also not xhat it should be; but this might, well be experimental error, though Professor Ferguson does not concede this. Some preliminary data by Morton2 for methyl alcohol and acetone, Table 111, came out pretty well except for the most dilute solutions, where the formula exaggerates any experimental error very much. Similar data by ?*lorton for m-ater and acetone gave moderately good results for t'he partial pressures of acetone; but the corresponding figures for water vapor were not satisfactory enough to be worth giving. As a matter of fact the logarithmic data in Table IV could be represented better by a curve than by a straight line. It is probable, however, that more accurate data will give a really good straight line. JVe had hoped to use the data of Wrewsky3 on ethyl alcohol and water; but inspection shows that they are too inaccurate to be of any real value. At 39.76' for percentage concentrations of ethyl alcohol of 15.92, 18.25, 3 0 . 2 5 , 31.88, 36.42, 42.0, 43.75, and 47.54, Wrewsky found 44.3, 44.6, 45.95, 1 5 . 5 , 45.4, 44.7, 49.3, 41.8, and 43.8 respectively for the partial pressures of water vapor expressed in millimeters. This is quite impossible, since alcohol and water do not form two liquid layers a t this or any other temperature. I n Tables V and VI are given the data by Sameshima* for acetone and ethyl ether a t 30' and a t zoo. The calculated values for acetone in ether do not agree any too well with the experimental values and there is the painful possibility that the logarithmic graph is really a flat curve which has been made to fit fairly well by a judicious placing of the straight line. The values for the partial pressures of acetone, however, are a joy to the soul. At each of the two temperatures the equation represents the facts with considerable accuracy over the whole range of concentrations. Since methyl alcohol, ethyl alcohol, and water are highly associated liquids with normal vapors, the fact that Equation TI11 appears to hold very well for these liquids and also for acetone, which is a slightly associated liquid shows that it is not the molecular weight in the liquid phase that count,s. Any polymerization in the liquid phase will show in the exponent. If the exponent is unity and if K = AII,/M~,Equation VI11 reduces to Equation VI, the so-called Raoult law. All ideal binary solutions therefore form a special case under Equation VIII, and we have at least made a start towards a J. I'hys. Chem., 33, I (1929). J. Phys. Chem., 33, 384 (1929). 3 2. physik. Chem., 81, I (1912). ' J. Am. Chem. SOC.,40, 1482 (1918).

WILDER D. BASCROFT AND H . L. DAVIS

366

TABLE I11 Methyl Alcohol and Acetone a t 20' Galc 0.83log - - log 'lAc = log K1 = 0.11 GAC

0.62

PIAc

G.4C

log - - log GAk

% acetone in liquid 0.0

P

,I l C

= log Kz = 0.03

p' acetone

log I