TEXTBOOK ERRORS:'
GUEST COLUMN
XVII:
The Meaning of Mole Fraction
J. M. BINOET and A. F. PEERDEMAN University of Utrecht, The Netherlands
T H E thermodynamic treatment of solutions usually starts with the introduction of the concept of "mole fraction in solution." Most textbooks2 leave the student with the impression that it is the molecular weight of the liquid solvent that is used in the computation of this quantity so that "mole fraction" really represents the proportion of molecules as they are present in the liquid. Yet this would require that both solute and solvent have a definite molecular weight, and this certainly is not the case for associated solvents such as water. In the absence of a more precise discussion, one could therefore be led to the conclusion that the entire subsequent treatment of solutions must be ill-founded. In fact, however, we shall show that upon closer examination the molecular weight of the solvent to be used in calculating mole-fractions turns out to be either immaterial or to be that of its vapor.
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The expression for the freezing point depression is an example where the molecular weight M , of the solvent plays no role; we confine ourselves to the limiting case of extreme dilution. This expression
where dT R
TO
Q
z:
= freeaing paint depression = gas constant = absolute temperature of the freezing point = "molar" heat of fusion of the solvent = "mole fraction" of solute in solution
involves the ratio of xz and Q which are both doubled if M I is doubled. With a composition of the solution of nlmoles (a grams) of solvent and nnmoles (b grams) of solute
This example suggests that perhaps there is no need to emphasize the meaning of the molecular weight MI of the solvent. A more rigorous conclusion is that, given a solvent of definite molecular weight, the numerical value assigned to that weight is immaterial for calculating the freezing point depression. When no definite molecular weight can be given, the question arises whether or not the existence of some definite
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molecular weight is essential in the derivation of equation (1). A very different situation is presented by Raoult's law:
the thermodynamic potential of the arbitrary unit of solvent, the Gibhs-Duhem relation still reads. (6)
Consequently, where -Ap/p is the relative vapor pressure lowering and x2 again the "mole fraction" of solute. The following superficial reasoning can lead here t o an obviously wrong conclusion: Measurement of the experimental quantities on the left-hand side of the equation would yield a value for xz. Combined with the known weight fraction and a known molecular weight Mz of the solute, this measurement would be a direct determination of the molecular weight MI of the solvent even if it is an associated liquid. Yet as already mentioned, no definite molecular weight can be attrihuted to such a solvent which might even be all associated to a single giant complex. I t is striking, furthermore, that if the above calculation is made for an aqueous solution, the value calculated for MI turns out to be exactly 28. The value of MI evidently need not be related to the molecular weight of the solvent in the liquid state but seems t o be that of the vapor. In order to avoid these difficulties one has to examine the premises which lead to the assignment of meaning to the quantities MI and x2that are being introduced. The basic equations giving the concentration dependence of the thermodynamic potential of solute and solvent are N* =
1128
+ RT in z2
(4)
and pl
= PI. - RT zp
(5)
respectively. I n accordance with international convention3pl* denotes the standard thermodynamic potential of the solute and p; the thermodynamic potential of the pure liquid solvent. The first of these equations simply expresses the fact that the molar thermodynamic potential of the solute is increased by RT In 2 when the concentration is doubled. (Recall the approximation leading t o equation (2)). With the conventional value of R this defines the quantity of solute involved as one mole (Mz grams). As to equation (4) it is, however, immaterial in which units the components are expressed in calculating the fraction xz in ideal dilute solution. These concentrations are all proportional to each other under these conditions so that substituting one for the other simply changes the value of pz*. The Gibbs-Duhem relation and the first equation alone lead to the second equation as a necessary corollary. We can show this as follows, the derivation at the same time specifying the units involved: Let us choose for xz in (4) a weight fraction XZin which the weight wz of the solute is expressed in units of one mole (&I2 grams), the unit of pz in equation (4), while the weight wl of solvent is expressed in units of arbitrary number of grams M,. The value of Xz is w*). With p2 denoting the thermodythen w,/(wl namic potential of one mole of solute (Mzgrams) and
+
a "Comptes rendus de la dix-huitieme conference de I'union international de chimie pure et appliqu&,'' Zurich, 1955, p. 99.
VOLUME 35, NO. 5, MAY, 1958
or, in our first order approximation: PI
- RT XS
=
(5')
It is thus evident that-whether the solvent is associated or not--equation (50, which specifies equation (5), gives the thermodynamic potential of an arbitrary quantity (MI grams) solvent in a solution provided that the fraction X 2 is calculated using units of same weight (MI grams) and of the molecular weight of the solute (Mz grams). Let us now consider why in Raoult's law the "mole fraction" xi must he calculated using the molecular weight of the gaseous solvent. This law is derived from a consideration of the liquid-vapor equilibrium by equating the thermodynamic potentials of equal amounts of solvent, first for the pure material NI.
=
N*
+ RT In P
(7)
and then for the solution PI*
- R T z n = r a + R T 1 n ( P + AP)
(8)
Here p* is the standard thermodynamic potential of the vapor, P is the vapor pressure of the pure solvent, and - A P is the vapor pressure lowering. The right side of these equations denote the thermodynamic potential of the vapor phase. Equation (7) with the conventional value of R is true only if the quantity of solvent involved is one mole of that vapor. This must, therefore, also be the quantity (MI grams) involved in the calculation of x2 in equation (8). By subtracting (7) from (8) one obtains
+
P AP -RT z, = RT In P
(9)
which reduces to Raoult's law. AP
=
7
(10)
Thus Raoult's law is valid only if xzis calculated using the molecular weight of the solvent in the vapor phase. As the derivation makes apparent it is in fact the molecular weight of the saturated vapor at the temperature considered which is to be used. We have raised previously the question of the validity of the freezing point depression equation (I) for the case of an associated liquid. The answer requires again a re-examination of the basis of the derivation. It can be easily verified, however, that equation (1) is obtained no matter what arbitrary units are used in the basic expression (5) for the thermodynamic potential of the solvent in the liquid solution and in the pure solid as long as these units measure the same amount of solvent in the two phases. Thus it appears that t o give a deeper insight and to avoid confusion and misunderstanding, the formal character of "mole fraction in solution" should and could be clearly brought out in the early stages of the thermodynamic treatment of solutions. 241
