Boiling Temperature vs. Composition An Almost-Exact Explicit Equation for a Binary Mixture Following Raoult's Law Mario Emilio Cardinali and Claud10 Glomini "La Sapienza" University, 00161 Rome, Italy For binarv mixtures with Raoultian behavior1. the boiline temperature as a function of composition IS usually treated bv textbooks in the context of fract~onaldlst~llation. The following exact relation applies2: x,. expl-(A;JR). [(llTe) - (l/T;b)ll
+ (1- x,).
exp 1-(h;llR). [(l/T$) - ( ~ / T $ ~= ) ]1J (1)
where x~ and 1 - x~ = xc are the mole fractions of the components B and C of the mixture, Tib and TE: are the boiling temperatures of B and C as pure liquids, XB and A; are the standard molar enthalpies of evaporation of the same liquids, assuming that ,their values remain constant and equal to XB(Tib) and Xc(TEb), respectively, over the temperature range Tib TEb,R is the perfect gas constant, and T$ is the boiling temperature of the mixture (i.e., the equilibrium parameter). A drawback of eq 1 is that i t cannot be written as an explicit expression for T$ a s a functionof x ~however, ; it can be used to draw the curve of T 2 vs. xg by numerical inversion, or according to the method suggested by Silverman2. In the present paper, we propose a simple method based on an expansion of the exponential terms of eq 1in a Taylor series, which leads directly to an explicit relation of Te_bvs. XB, that works almost as well as the exact, but implicit, eq 1 throughout the whole composition range of the mixture, as we will show by applying it to the case of the benzenetoluene mixture.
+
Theory
T o obtain a notation less cumbersome than in eq 1, let us define
Equation 1can then be rewritten in terms of XB as x,
.KB. exp (-OBIT,)
+ (1- x8). KC. exp (-BclTm) = 1
or, in terms of xc, as
Taking the ratio of the two above forms of eq 1,we obtain X~ x~ -=-=-
zc
I - r,
1-Kc. exp (-OclT,J 1- KB exp (-OBIT,)
.
(5)
for which i t would be KB = KC,if Trouton's rule were exactly valid. The two exponentialterms in eq 5 are analytical functions of the variable T,,, in the range of interest TB-Tc, so that they can be expanded in a convergent Taylor series for any sufficiently small neighborhood of each point To of the range3. Setting
'
McGlashan. M. L. Chemical Thermodynamics; Academic: London, 1979; p 250. 2Silverman,M. P. J. Chem. Educ. 1985,62,112-114. Aleksandrov, A. D.; Kolmogorov. A. N.; Lavrent'ev. M. A. Mathe matics, Its Content. Methods and Meaning, 2nd ed.; translated from Russian; M.I.T.: Cambridge, MA. 1969; Vol. 1, p 175.
Volume 66
Number 7 July 1969
549
