Composition expressed on a weight per cent basis must frequently be converted into other composition parameters which do not have a 1:l correspondence with the weights per cent. A common example of this is mole per cent. The reverse problem also exists when mixtures of specified mole percentages must be prepared by weighing operations. The present work is framed in terms of these specific conversions, although the methods are more generally applicable. Such conversions may involve a number of mathematical operations. For example, that from weight per cent to mole fraction for but one component in a ternary system requires four divisions and a sum. Conversion tables are not practical for this purpose because of the three composition variables. When the conversions must be performed repeatedly, as in continuing work with a given system, the cumulative labor is considerable. The conversion grids described below may be used to advantage under these conditions. They are easily constructed and used, and effect a significant saving in labor. Their use reduces the five operations noted just above, for example, to two sums and a division. The basis of the grids and their construction are described. Specific illustrations of conversion are given in terms of the system benzene--carbon tetrachloriden-hexadecane (B-C-H). Grid Basis
Assume the weights per cent of the i components B, C, H to be p,, p,, p,, and their molecular weights Ms,Mc, and Mn. Per 100 g of mixture there will be ns, nc, and ne moles of the components and n~ total moles. Thus n~ = ns
+ nc + n~
(1)
Consider a particular straight line in a weight per cent ternary diagram. Whether the diagram is of the equilateral or right-angled triangular type, it is possible to relate two of the weights per cent by an equation of the form, using B and H for example, PA =
mps
+b
(4)
where m and b are the constants of the slope-intercept form of the straight line equation. Substitution for each of ni in equation (1) in terns of equation (2), and use of equations (3) and (4) t o eliminate pc and p, in the resulting equation yields n~ as a function of PBI n~ = RPB
+S
(5)
where
From equations (6,7) it is evident that S and R are constant for a particular straight line, or that equation (5) is linear. Now for a set of parallel straight lines R is constant by equation (6) because m is constant. Moreover, from equations (5,7)
Thus there is a constant incremental change in n~ hetween every adjacent pair in a set of equidistant parallel lines. For the particular set where 1 m = M c [ L -M- s] - lM A
by equation (6) R reduces to zero and n~ is a constant, equal to the S fixed by equation (7). To summarize, there may be superimposed upon a weight per cent diagram a series of equidistant parallel constant nr lines, which are suitable for linear interpolation of intermediate n~ values. I n the above the interpolation has been developed in a vertical direction in the ternary diagram. Also superimposable is a scale of the various ni, t o the corresponding pi by which are equation (2). Thus a point on the weight per cent ternary diagram may also be interpreted directly in , the mole fraction by a ratio terms of niand n ~yielding determination. Reasoning parallel to that above shows that conversion from mole fraction to weight per cent is also possible by this approach. That is, a weight per cent grid may be superimposed on a mole fraction ternary diagram. Grid Construction and Use
Figure 1 shows the grid for conversion from weight per cent to mole fraction. Constant nr l'mes have been constructed a t intervals of 10 weight per cent in B along the vertical (B-C) axis. Figure 2 shows the grid for conversion from mole fraction to weight per mnt. Lines of constant total weight per mole of mixture have been constructed a t intervals of 0.10 mole fraction in B along the B-C axis. I n actual use the two grids may be combined on a single plot; they have been separated here only for clarity. Tables 1 and 2 list proportional parts useful for interpolation in Figures 1 and 2 respectively. Volume 40, Number
6,June 1963
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325
Table I. Interpolation from Weight Per Cent to Moles per 100 Grams of Mixture
Weight % increment
Moles B
Total moles
Example. Determine the mole fraction B in a mixture of weight percentages 23.2 B, 29.2 H, 47.6 C, corresponding to point x in Figure 1. moles B : 0.2560 0.0384 0.0026 0.2070
total moles: 0.7130 0.0189 0.0044 0.7363
mole fraction B: 0.404
Example. Determine the weight per cent B in a mixture of mole fractions 0.461 B, 0.342 H, 0.197 C, corresponding to point y in Figure 2. Table 2.
weight B: total weight: 31.24 138.69 4,54 4.69 0.08 0.53 36.10 143.76
lnterpolafion from Mole Fraction to Weight in Grams Per Mole of Mixture
Mole fraetion increment
Weight B
Total weight
weight per cent B: 36.01 1W -= 25.0 143.76
In each example the result obtained with the grid differs by less than 1 unit in the last place from the result obtained by the detailed fundamental calculation. It is apparent that the concentrations of the other components of the system could also be converted by use of the grids.
Figure 1.
326
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Convsrrion grid, weight per cent to mole fmdion.
Journal o f Chemical Educofion
Figure 2.
Conversion grid, mole frodion to weight per cent.
