GUEST AUTHOR M. 1. MeGIashan
Textbook Errors,
The University Reoding, England
Deviations from Raoult's Law
I t is often assumed, and has sometimes been stated explicitly in text books,' that there are only two thermodynamically possible ways in which a binary mixture can deviate from Raoult's law, namely either that the deviations shall be everywhere positive for both components, or that the deviations shall be everywhere negative for both components. Certainly no such restriction is imposed by thermodynamics and there are in fact several contrary examples. The only thermodynamic restriction on the partial pressures (strictly fugacities) pl and p, of the components of a binary mixture is that they must satisfy the-Duhem-Margules relation (1
- z)(d In fi/dz)
+ z(d In j d d z )
=
0 ( T , P conatant)
Here x is the mole fract,ion of component 2, and and fi are the activity coefficientsdefined by j =1
-p
f,
=
p*lzp2"
(1) fi
which has often been used (1, 2, 3) to test liquid-vapor equilibrium data for thermodynamic consistency. Since it follows from (I), equation (3) is a necessary condition for thermodynamic consistency, but since it is less restrictive than (1) it is not a su&ient condition. For example, the equations in fl = Az and In ft = 9 ( 1 - x) satisfy (3) but fail to satisfy the Dnhem-Margules relation (1) and so are thermodynamically impossible. Equation (3) is, however, sufficient immediately to prove that if one component of a binary mixture shows positive deviations (In ,fi > 0) over the whole range of empositions then the other component cannot show negative deviations (In ,f2 < 0) over Suggestions of material suitable for this column and guest columns suitable for publication directly are eagerly solicited. They should be sent with as many details as possible, and particularly with references to modern textbooks, to Karol J. Mysels, Department of Chemistry, University of Southern California, Los Angeles 7, California. Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts. the source of errors discussed will not be cited. In order to an error must occur in s t least two independent rehe cent standard books.
/
Journal of Chemical Education
the whole range of compositions, and conversely. No other general restriction on the signs of In fi and In fi can be derived from equation (1). There remain, in addition to the well-known cases of deviations everywhere positive (or negative) for both components, possible cases in which In f changes sign for one or both components between x = 0 and z = 1. For brevity we shall confine this discussion to kinds of behavior which are not only thermodynamically possible but have actually been observed. In none of these does In f change sign more than once for each component. The excess Gibbs function G" is defined as the excess of the increase of the Gibbs function on mixing over that for an ideal mixture, and is related (4) to the activity coefficients by the equations
(2)
where p,O and pz0 are the vapor pressures of the pure substances. Does equation (1) imply any general restrictions on the signs of In fi and In f,, that is on the signs of the deviations from Raoult's law? One such restriction is readily derived but it is a much less severe restriction than that challenged in the first paragraph. Iutegration by parts of equation (1) between the limits 0 and l leads to the equation
516
46
Whereas the composition dependence of In f, and in fa must he such as to satisfy equation (1) the only restriction on the composition dependence of GB is that GE shall be zero a t x = 0 and a t z = 1. Subject to that restriction any expression for G Pas a function of x mill lead to thermodynamically possible activity coefficients by use of equations (5) and (6) since these equations satisfy the Dnhem-Margules relation identically. For the purpose of illustrating unusual kinds of deviations from Raoult's law by specific examples we shall use as a general expression for G Rthe series OP/RT = z(1 - z ) I A + B ( l - 25) + C(1 - 2z)l ...1 (7)
+
~vhichhas often been used (.5, Z, 6 ) to fit experimental results, and it will be sufficient for our purpose if we retain only the first three terms. Using equations (5) and (6) with (7) we then obtain for the activity coefficients
We shall now- consider some special cases of equations
(7), (8), and (9), first dealing briefly with the wellknown cases of ideal and of regular or simple (4) mixtures, and t,hen going on to give illustrative examples chosen for their mathematical simplicity, of more unusual but no less thermodynamically possible kinds of behavior. Case I
The equations for an ideal mixture can be obtained if we put. A = B = C = 0 in equations (7) to (9) so that
(An ideal mixture is thus defined by the equation GB = 0. A mixture which obeys Raoult's law is strictly defined by the relations p, = (1
- x)p? and pa
=
zpso
We remind the reader that we have ignored any distinction between partial pressure ratios and fugacity ratios, and that this is equivalent to ignoring the distinotion between an ideal mixture and one which obeys Raoult's lau. Our discussion therefore applies strictly to deviations from ideality rather than to deviations from Raoult's Ism, hut the distinction will usually be unimportant.)
Partial pressures for an ideal mixture are shown as broken straight lines in all the following diagrams of the partial pressures of nonideal mixtures. Case II
The equations for regular or simple (4) mixtures can be obtained if we put B = C = 0 in equations (7) to (9). We then obtain
Examples of this kind of behavior are shown for A = 1in Figure 3 and for A = - 1in Figure 4. Mixtures of ethanol and chloroform a t 35OC (7) give deviations from Raoult's law like those of case I I I a (Fig. 3). So do solid solutions of silver chloride and sodium chloride a t 150°C (S), the deviations for this pair being large enough to cause a region of partial miscibility. Mixtures of pyridine and water, which according to von Zawidski (9) behave like case IIIa, have since been shown (IO), however, to give deviations which are everywhere positive like case IIa (Fig. 1). Mixtures of nitromethane and acetone a t 45°C have recently been shown (11) to behave like case I I I b (Fig. 4). Liquid mixtures of thallium and mercury at 325'C (12), cadmium and antimony a t 480°C (13), zinc and antimony at 550°C ( I d ) , and potassium and mercury a t 275'C and 325°C (15), and solid solutions of lead bromide and lead chloride at 200°C (16),have also been shown to behave like case I I I b (Fig. 4).
As examples of the behavior of regular mixtures plots of GE/RT, In fl, and in fz, and of p1/pio and pz/pyo are shown for A = 1 (positive deviations from Raoult's law) in Figure 1, and for A = -1 (negative deviations from Raoult's law) in Figure 2.
Figure 1.
Care Ilo;
Figure3.
Carelllo;
A=l,B=-i,C=O.
Figure 4.
Case Illb: A = -1,
A = 1, B = C = 0.
B = 1, C = 0.
Case IV
Another kind of unusual behavior can be illustrated if we put A = 0 and C = O in equations (7) t o (9). We then obtain
~i~~~~ 2. Core ilb:
A = - 1, B = C = 0.
Case Ill
One kind of unusual behavior can he illustrated if we put B = - A and C = 0 in equations (7) to (9). We then obtain GB/RT= 2Aze(l
- 2) In j, = 2Az2(2z- 1) In fx = 4A(1 - z ) %
An example of this kind of behavior is s h o ~ mfor B = 1 in Figure 5. Solid solutions of p-dichlorobenzene and p-dihromobenzene at 50°C (17) have recently been shown to behave like case IV. The author knom of no other example. Case V
Finally, yet another kind of unusual behavior can he illustrated if we put B = 0 and C = - A in equations (7) to (9). We then obtain Volume 40, Number 10, October 1963
/
517
Examples of this kind of behavior are shown for A = 1 in Figure 6 and for A = - 1 in Figure 7. Mixtures of benzene and bromohenzene a t 80°C (18) are nearly ideal but have nevertheless been shown to behave like case Va (Fig. 6). Liquid mixtures of cadmium and potassium at 475°C (19) also behave like case '\'a (Fig. 6), the deviations for this pair being large enough to cause a region of partial miscibility. Liquid mixtures of bismuth and cadmium a t 431°, 477', and 533°C (80) and a t 500°C (21), and of lead and thallium a t 4 3 8 T (28) have been shown to behave like case Vb (Fig. 7). I t was shown by Dolezalek (83) tbat deviations from Raonlt's law which are everywhere negative (case I I b (Fig. 2); for example chloroform and ether) can be ascribed to the formation of a 1: 1 compound in equilibrium with the parent substances, if it is assumed that the three species together form a ternary ideal mixture. Since then it has been commonly but wrongly supposed that "compound formation always leads to negative deviations from Raoult's law." I n fact it has been
shown (24, 25) that while tbat statement is true for 1: 1 compounds, the formation of any compound other than 1 : l always leads to deviations which change sign for one or both of the components. For example the formation of a 1 : 2 compound leads to behavior like case IIIb (Fig. 4). The formation of both 1:l and 1:2 compounds in the same mixture can lead either to behavior like case IIb (Fig. 2) (deviations everywhere negative), or to behavior like case IIIb (Fig. 4) (deviations change sign for component I), according to the relative amounts formed of the two compounds. Mitures of chloroform and dioxane which probably form both 1:1 and 1 : 2 compounds turned out (26) to behave like case I I b (Fig. 2), in spite of the author's hope that they might behave like case IIIb (Fig. 4) and so be another example of unusual behavior. Apart from the advantages of not telling lies in the lecture room the author has found that these examples of unusual behavior provide a useful way of familiarizing the student with the Duhem-Margules relation and its consequences and applications. Calculations, with various chosen values for the equilibrium constants, of the properties of mixtures in which one or two compounds are formed, have proved a useful exercise for advanced students. Literature Cited (1) HERINGTON, E. F. G , Nature, 160, 610 (1947): COULSON ,
E. A,,
AND
HERINGTON, E. F. G., Trans. Faraday Sor.,
44,629 (1948). ( 2 ) REDLICH, O., (1948).
AND
KISTER,A. T.,Ind. Eng. Chem., 40, 345
(3) ROWLINSON, J. S., "Liquids and liquid mixtures," Rutterworths, London, 1959, pp. 132-7. (4) GUGGENHEIM, E. A,, ''Therm~dyn~mic~," 3rd ed., North Holland Publ. Co., Amsterdam, 1957, Sect. 5.34. et sep. (5) GUGGENHEIM, E. A,, Trans.Famday Soc ,33,151(1937). (6) SCATCHARD, G., Chem. Revs., 44,9 (1949). (7) G.. A N D RAYMOND. C. L.. J . 4 m . Chem. 8oe.. . . SCATCHARD.
60, i2mii93k). (8) WACHTER, A., J. Am. Chem.Soc., 54,919(1932). J., Z. phvSik. Chem., 35, 129 (1900). (9) VON ZAWIDSKI, (10) ANDON,R. J. L., COX,J. D., AND HERINGTON, E. F. G., Tmns. FamdavSoe., 53,410 (1957). (11) BROWN, I., AND SMITH,F.,A u s ~J ~. Chem., . 13,30 (1960). (12) HILDERHAND, J. H., AND EASTMAN, E. D., J. Am. Chem. Soe., 37,2452 (1915). (13) SELTZ, H., d N D DEWIT? B., J. Am. Chem. Soc., 60, 1305 (1938). (14) DEWITT,B., A N D SELTZ,H.,J. Am. Chem. Soc., 61, 3170 11030>. ~.~ ~
Figure 6. Care Va: A =
0, B = 1 , C =
-1
~
(15) LANTRATOV, M. F., AND TSARENKO, E. V., Z ~ W priklad. . Khim., 33.1.579 (1960). (16) WACHTER, A., J. Am. Chem. Soc., 54,2276 (1932). (17) Wn1.s~ P. N., AND SMITH,N. O., J. Phvs. Chew., 65, 718 (1961). (18) MCGLASHAN, M. L., AND WINGROVE, R. J., Trans. Faradaj, Soc., 52,470 (1956). (19) LANTRATOY, M. F., AND TSAHENKO, E. V., Zhur. priklad. Khzm., 33,1116 (1960). (20) TAYLOR, N. W., J . Am. Chem. Soe., 45,2865(1923). (21) ELLIOTT,J. F.. AND CHIPMAN, J., T ~ MFaraday . Sac., 47, 138(1951). (22) HILDEBIMND, J. H., AND SHARMA, J. N., J . Am. Chem. Soe. 51,462 (1929). (23) DOLEZALEK, F., Z. physik. Chem. 64,727 (1908). (24) METZGER, G., AND SAUERWALD, F., Z. anwg. Chem., 263, 324 (1950). (25) E., Arkiv. Kemi, 7,315(1954); Ree. Trav. Chim., 75,790(1956). (26) McG~,asnrN,M. L., AND RASTGGI, R. P., Trans. Faraday Sm., 54, 496 (1958).
H~~ELDT,
Figure 7.
5 18
/
C..e
Vb: A =
- 1, B
= 0, C = 1.
journol of Chemical Education
