Slide Rule for Converting Percentages to Mole Fractions C. R. Masson, Atlantic Regional Laboratory, National Research Council, Halifax, N. S., Canada HE routine conversion of percentages T b Y weight to mole fractions can become tedious, particularly for multicomponent systems. Previous slide rules [Hermann, G., Metullwirtschuft 12, 104 (1933); Rachinger, W. A,, J . Inst. Metals 82, 38 (1953)l have been limited t o binary systems. The slide rule described has been developed specifically for the components of basic open-hearth slags. It may be constructed from any slide rule which has an L scale and a n inverted D scale on the body. Extra scales are incorporated, with a separate slide for each system of interest. These scales are linear, and no special problems in construction are involved. Two indicators are required. Certain advantages may be gained by the use of circular scales. If the linear scales on the slide represent chemical substances, and the lengths of these scales are inversely proportional to the molecular weights of the substances, concentrations by weight can be added to give numbers that are proportional to molar concentrations. These numbers can be transferred to the C and D scales for division to give mole fractions directly. Conversely, if lengths are directly proportional to molecular weights, mole fractions can be added to give numbers that are proportional to concentrations by weight. These numbers can be treated similarly to give percentages. The use of two indicators enables the number representing the first sum to be set on the D scale n-hile the second number is being determined. “Crossing” the indicators is avoided by use of the inverted D scale.
Table 1. Total Lengths of Scales for Components of Open-Hearth Slags
Total Length of
Com-
1101. Wt.
ponent FenOa PzOs
159.70 141.96 101.94 78.08 71.85 70.93 60.06 56.08 55.85 40.32 19.00
Al203
CaF2 FeO
MnO
SiOv CaO
Fe JIaO
F-
Scale, Cm.
Obverse Reverse 8.779 9.876 13.755 17.956 19.513 19.766 23.343 25.000 25,104 34.772 73.790
56.247 50.000 35.904 27.500 25.306 -. ~~. 24.982 21.154 19.752 19.671 14.209 6.692
hearth slags are listed in Table I. CaFl is listed also as F, because both methods of representation are used by workers in this field. For calculating mole fractions, the scale representing CaO has been taken as standard, because it represents the major component of most basic slags, and its total length (0 to 100%) has been arbitrarily made equal to 25 cm., the standard length of the L scale on most 10-inch slide rules. Because in open-hearth slags me are generally not concerned with the complete range of solubility of all components, only portions of the scales that adequately cover all compositions encountered in practice are placed on the slide. The 11 scales can be placed in a space normally occupied by three full-length scales. A convenient arrangement is shown in Figure l , a , which illustrates also the other features necessary for the calculations. The linear scale on the body is represented here as 1OL (graduations 0 to lo), although a common L scale (graduations 0 to 1) n-ould serve equally well. To convert mole fractions to per-
APPLICATION T O OPEN-HEARTH SLAGS
The main components of basic open-
centages, it is convenient to set the total length of the P205scale (0 to 1 mole fraction) equal to 50 cm. The total lengths of each of the other scales !vi11 then be as given in the fourth column of Table I. Only portions that cover the range encountered in practice have been utilized. A suitable arrangement is shown in Figure 1,b. OPERATION
In calculating mole fractions, the percentages of all con~ponentsare first added in accordance n-ith their lengths on the scales, so that the sum is a number on the 1OL scale. This number is transferred to the D scale by indicator 2, which remains fixed for the rest of the calculation. The number on the 1OL scale which corresponds to the percentage of each constituent (or sum of constituents) is nox placed on the C scale under indicator 2 . Mole fractions are read off on either the DI scale opposite the index of C or the C scale opposite the index of D. Two methods of reading avoid crossing the indicators. Method A is used most frequently; method B, when the number on the C scale under indicator 2 is larger than that on the D scale. ExAhfPLE 1. A hypothetical slag contains 60.2% CaO, 15.5% &On, 11.9% P205,and 12.47, FeO. Find the mole fraction of FeO and the sum of the mole fractions of SiOl and PzOc. Move hairline 2 t o 6.02 of 1OL. Draw zero of SiO, under hairline 2. More hairline 2 to 15.5 of SOz. Dram zero of Pz05under hairline 2. Move hairline 2 t o 11.9 of PZO5. Dram zero of FeO under hairline 2. Move hairline 2 to 12.4 of FeO and read value on 1OL under hairline 2 (this will be 8.90). Move hairline 2 to 8.90 of D. (This
INDICATOR 2
INDICATOR 1
a Body and slide (obverse)
L.0
‘0
s.0,
10
r.0
,o
C
3
2
L
,D I
~
,
8,
5
‘I , I
,
2,
PI>,’: Fez:,
7
1 1 0 ‘0 “ “ 0 ) I,
2
, ‘0,
3
Figure 1. ANALYTICAL CHEMISTRY
‘ 3 .1,3, ‘i,
, 1
:..,a
‘0
‘
0
0, ‘0,
I
I
b Slide (reverse)
1 122
:i 1 ,
Slide rule for open-hearth slags
‘5
9
hairline is 11oiv set, and is not further disturbed.) A. Draw zero of FeO opposite zero of 10L. Move hairline 1 to 12.4 of FeO and read value on 10L under hairline 1 (this nill be 0.9T). Draw 0.97 of C under hairline 2. = 0.109 on C opposite Read off -l-Fe~ index of D . B. Dran. zero of SiOz opposite zero of 10L. Move hairline 1 to 15.5 of SiO2. Draw zero of PzOj under hairline 1. Move lisirline 1 to 11.9 of PzOj and read value on 1OL under hairline 1 (this will be 1.02). Draw 1.92 of C under hairline 2. Read off ?;s102-p10a = 0.216 on DI opposite index of C. If the composition of the slag lies outside the range of the scales, calculations can be performed by expressing the composition in smaller units, each within the range on the rule. For example, :in experimental slag containing 22.5c0 AInO could be handled in units of 15 and T.57,, each added separately to obtain the appropriate number on the 1OL sc:ile. In coiiiputing percentages from mole fractions, the scales on the reverse side of tlie slide are used. MOLE RATIOS A N D MOLAR VOLUMES
Mole ratios can be computed siniply by dividing, on the C and D scales, the numbers on tlie 1OL scale n-hich correspond t o the percentages involved. E x a m L r : 2 . d slzag contains 407, CaO, ISc, S O 2 . and 27, P20i. Find the V ratio iPhilbrook, IT7. O., TS’ashburn, F. “Basic Open-Hearth Steelmaking.” 2nd ed.. p. 196, Am. Inst. nfining Met. Engrs., S e w York, 1951)
a Obverse
Figure 2.
of this slag, defined as (Kcao - 3Sp,O,),’ ~~
Nsiov
Move hairline 2 to 4.0 of IOL. Draw 6.0 of P,Os under hairline 2 . Move hairline-2-to zero of PzOs and read value on 1OL under hairline 2 (this will be 3.76). nIove hairline 2 to 3.76 of D. Drarr zero of SiOz opposite zero of 10L. Move hairline 1 to 18.0 of SiOz and read value on 1OL under hairline 1 (this will be 1.68). Draw 1.68 of C under hairline 2. Read off V ratio = 2.24 on D opposite index of C. If a further scale is incorporated, molar volumes can be computed. This is a linear scale, so calibrated that the percentage of any component divided by its molecular weight can be read off directly. Once this scale has been constructed for any single component, it will be valid for all other components on the same slide. By choosing the CaO scale, the required scale can be constructed by setting a 25-em. length equal to 100/56.08 = 1.7832, and subdividing linearly. This scale, designated %/M, is shown in Figure l,a, on the body of the rule. The reading on this scale corresponding to a certain percentage of any component gives the number of moles of the component in 100 grams of slag.
EXAMPLE 3. A slag has the composition given in Example 1. Find the molar volume of the slag if its density is 5.5 grams per nil.. and if 150 nil. of a solution contains 5 grams of slag. find the molar volume of the solution. Add the slag percentages as in Example 1, but take the final reading under hairline 2 on the %/AI scale instead of the 1OL scale (this nill be 1.58, the total number of moles in 100 grams of slag).
Move hairline 1 to 1.58 of D. Draw index of C under hairline 1. A. Read off molar volume of slag = 11.5 ml. on DZ opposite 5.50 of C. B. Move hairline 1 to 5.00 of C. Draw 1.50 of C under hairline 1. Read off molar volume of solution = 1.90 liters on DZ opposite index of C. CIRCULAR SCALES
Circular scales offer a number of advantages (Figure 2 ) . If arranged concentrically, the zero of each can be made to coincide without the slide rule’s becoming too unwieldy, and a larger range can be covered on each scale. The accuracy of readings can be improved by placing the smaller scales near the periphery. Use of a common zero setting for all scales decreases the number of operations required for the calculations; if a second 1OL scale is fixed in relation to the “component” scales, these scales need not be brought into coincidence each time a reading on the 1OL scale is required. The inverted D scale can be omitted, as crossing of the indicators causes no problems. On the obverse side, the CaO scale is again taken as standard, so that a revolution of 360” corresponds to 100% of this component. On the reverse side, a full revolution corresponds to 0.5 mole fraction of Pz06. The scales are arranged, for highest accuracy, in order of increasing molecular weight toward the periphery for the obverse side; in the opposite sense for the reverse side. ACCURACY
The scales are arranged for good accuracy consistent 153th ease of manipulation. Compromise is possible for most
b Reverse
Slide rule with circular scales for open-hearth slags VOL. 31, NO. 6, JUNE 1959
1123
applications. Even with a 10-inch linear rule, components of highest molecular weight can be read directly to 0.570. This corresponds to a n absolute displacement of about 0.5 mm., the smallest interval between graduations on the 1OL scale. Estimation to 0.25% is, therefore, possible. For most components, estimation to 0.1% is feasible. Such accuracy is possible because
all possible ranges of composition need not be covered. Where all possible compositions must be allowed, the largest scale must be set equal to the full length of the rule. If the components exhibit a wide range of molecular weights, the smaller scales (representing components of high molecular weight) might be badly compressed. This difficulty can be overcome in the circular
rule by choosing some intermediate scale as standard and handling by parts compositions that ,cannot be incorporated on the longer scales. It may be necessary to “overrun” the zero setting in some summations, but this presents no difficulty. Issued as N.R.C. No. 5091.
Concentration of Impurities from Organic Compounds by Progessive Freezing Joseph S. Matthews and Norman
D. Coggeshall, Gulf
it is necessary to determine 0 trace amounts of impurities in organic compounds. Mass spectrometry FTEN
and ultraviolet or infrared spectroscopy have limits of sensitivity and are not always applicable. The usual procedure, when concentrations are lorn, is to concentrate the impurities by distillation or extraction prior to instrumental analysis. These methods have certain limitations. Crystallization by fractional freezing, fractional melting, or zone melting techniques has come to the fore as a means of obtaining ultrahigh purity. Starting with Schm-ab and Kichers (9, 10) various workers have shown that such methods may be used to purify organic compounds (1, 8, 4-6, 8 ) . Goodman (3) suggested that they might also be used to concentrate impurities that occur in high dilution. I n 1957 Schildknecht and Mannl (7) concentrated small quantities of biological material from aqueous solution by zone melting the frozen solution. Progressive freezing can conveniently be used t o concentrate impurities from organic liquids.
Research & Development Co., Pittsburgh, Pa.
frozen is lowered into the freezing bath, TI hile the unfrozen part is stirred by a motor-driven glass stirrer. The tube is lonered until 1 ml. or less of liquid remains unfrozen a t the top of the tube. If all the liquid solidifies, a eniall amount is melted. The liquid is lvithdran n with a long dropper pipet, transferred to a vial, weighed, and analyzed. After removal of the fraction, the tube contents are melted and the cycle is repeated if desired.
wire frame which holds the sample tube and allows it to be stirred while being lowered into the cooling bath. The stirrer motor remains in a fixed position. The top of the sample tube is closed by a rubber stopper fitted with a Teflon sleeve t o serve as a bearing for the stirrer shaft. The cooling bath consists of a copper pipe, 2 inches in outside diameter by 14 inches long, insulated on the outside with glass fiber pipe insulation, and cooled by circulating acetone cooled by a dry ice-acetone bath. The temperature is regulated by adjusting the coolant f l o ~rate. The coolant is maintained a t a constant level by an overflow through which the circulating acetone returns to a reservoir.
Several parameters n-ere studied: effect of stirring, lowering speed, temperature of cooling, and size of fraction removed, and effect of impurity concentration on efficiency of separation of impurities.
EXPERIMENTAL
RESULTS AND DISCUSSION
The tube containing the liquid to be
Stirring was necessary for increasing
Figure 1. Freezing apparatus
LOWERING APPARATUS
The apparatus consists of a lowering mechanism and a cooler. The lowering mechanism is a small synchronous motor with a speed of 1800 r.p.m., geared down through a worm drive which passes the torque on to a ball and disk integrator that serves as a variable speed reducer. The ball cage of the integrator is attached to a calibrated micrometer screw, so that the ball can be moved across the face of the disk to vary the output speed. The torque from the speed reducer is transferred through a worm drive to the output shaft, which can be regulated between 0.25 and 2.87 revolutions per hour. A spirally grooved drum attached to the output shaft carries a very thin metal cable for lowering the sample tube into the cooling bath. The range of speeds at which the cable is unwound can be controlled by the size of the cable drum. With a drum of 1-inch diameter, the speed of the cable can be regulated between 4 and 16 em. per hour. Attached to the end of the cable is a 1 124
ANALYTICAL CHEMISTRY
STIRRER
MOTOR
N
DRY ICE COOLING
ACETONE
PbMP
