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Anleitung zum Gebrauch des Rechenschiebers für Chemiker
Instructions for the use of the Chemical Slide Rule
Verlag Chemie, Weinheim
Instructions for the Use of the Chemical Slide Rule
The chemical slide rule is designed to facilitate calculations in organic and inorganic analyses and also to meet all the demands made of a normal slide rule.
A. Description
The scales designed for rapid evaluation of analytical data are to be found on the front side of the rule. The accuracy achievable with this pocketsize rule exceeds that usually obtained in practical analysis. The remaining scales correspond to those found on conventional slide rules.
I. Front side (see Figure)
The following main scales are to be found on the front side of the rule
(from bottom to top):
1. x Scale on the fixed rule (below)
2. x Scale on the slide (below).
This scale is marked with "Molvol." for the molar volume of gases under standard conditions (22.41 l mole^{1}), with "N_{L}" for the Loschmidt (Avagadro) {sic} number (6.023 x 10^{23} mole^{1}), and with "R/lg e" for the product of the logarithmic modulus and the gas constant (2.303 x 1.98 = 4.574); the latter quantity frequently occurs in physicochemical equations, e.g. in the calculation of vapor pressure.
3. 10:x Scale on the slide (center), in red.
4. (10:x)^{2} Scale on the slide (center), in red.
The righthand half of this scale is marked with a series of atomic weight ratios).
5. x^{2} Scale on the slide (above).
This scale is also marked with the % sign.
6. 10x^{2} Scale on the fixed rule (above).
This scale has the additional markings "H" and "MG" for the atomic and molecular mass (molecular weight).
In addition to the 10x^{2} scale there are further scales on the upper and lower edges of the fixed rule which carry markings for the masses of atoms and groups (atomic and group weight) and multiples thereof. The use of two colors (red and black) permits interpolation of the scales for the two atoms or group of atoms. All markings are based on the IUPAC Table of Atomic Weights (1961), i.e. refer to the carbon isotope ^{12}C = 12.0000.
The scales are designed so as to facilitate most of the calculations required in inorganic and organic analyses. The upper limit for molecular weight is 1300. Apart from hydrogen, the compounds may contain up to 80 carbon atoms, up to 12 nitrogen or oxygen atoms, up to 10 chlorine atoms, up to 6 fluorine, bromine, sulfur, phosphorus, or silicon atoms, or methoxy groups, up to 3 iodine or boron atoms or 2 ethoxy groups (EtO), and one atom of each of the other more important elements.
II. Reverse side
The scales (both black and red) on the reverse side of the rule have been compiled and designed with due consideration to specific requirements of the chemist and physical chemist. They are connected to the front side of the rule via the x scale so that a calculation begun on one side may be continued on the other. The px scale simplifies area and volume calculations and will prove useful in all calculations involving formulas which contain the constant p (angular frequency, polar coordinates, etc.). It also facilitates, in a rather elegant way, the continuation of a calculation started on the lower C and D scales, in which adjustment of the slide would normally be necessary, without the slide having to be moved. For example: (3:8) x 2 = 0.75
Fixed rule (above)  
Cubic scale 
K 
x^{3} 
Tangent scale 
T_{1} 
tan 0.1x (cot) 
Tangent scale 
T_{2} 
tan x (cot) 
Solid p scale 
DF 
px 
Slide  
Movable p scale  CF  px 
Mantissa scale  L  lgX 
Reciprocal scale  CI  10x 
Movable base scale 
C 
x 
Fixed rule (below)  
Base scale 
D 
x 
Exponential scale for positive 
LL_{3} 
e^{x} 
Exponential scale for negative 
LL_{03} 
e^{x} 
ditto 
LL_{02} 
e^{0.1x} 
III. Cursor
On the front of the cursor there is a line marked "J" to the left of the central hairline and a line marked "cal" to the right. Both lines cross the 10x^{2} scale and facilitate conversion from the chemist's unit of heat energy, the (kilo)calorie, into the unit required under the SI system, the Joule, and vice versa (1 Kcal » 4.184 x 10^{3} J; 1 J » 0.239 cal). A second pair of lines marked at the same height as the x^{2} scale are of particular importance to the electrochemist and the spectroscopist (1 kcal/mole » 4.34 x 10^{2} eV; 1 eV » 23.05 kcal/mole).
The lines marked T and E on the reverse side of the cursor can be used with the scales LL_{02}/LL_{03} and DF for the conversion of transmissions into decadic extinctions which are related to each other by the expression
E = log 1/T
For example, in order to find the value of E corresponding to T = 1 %, the line marked T on the cursor is placed over the marking 0.01 on the LL_{03} scale and the value E = 2 is read off on the DF scale. The value of 0.60 on LL_{02} belonging to T = 60% corresponds to an extinction of E = 0.221 (reading 2.21 on DF; take care with orders of magnitude).
B. Calculation of analyses
1.
Calculation of empirical formula from analytical values and molecular weight
Example: A substance has the analysis
(found) 

C 
34.2 
H 
2.1 
Br 
38.0 
N 
6.0 
O 
19.8 
and a molecular weight of 216
1.1.
The Ù sign on the x^{2} scale of the slide (which corresponds to the value 100%) is brought to coincide with the experimentally found molecular weight (216) on the righthand half of the 10x^{2} scale of the fixed rule (MG).
1.2.
The number of atoms of the individual elements may now  with the aid of the cursor  be found, one after the other, as the integer on the relevant scale of atomic masses corresponding most closely to the analytical value on the x^{2} scale (% scale), due care being taken of the colors in the atomic scales. For hydrogen the number of atoms is read off on the 10x^{2} scale; however, the tenfold % value must be used on the x^{2} scale (e.g. 21.0 instead of 2.1).
Thus in this particular example, the best values found are 6 for carbon, 4.52 (i.e. 4 or 5) for hydrogen, 1 for bromine, 1 for nitrogen, and 3 for oxygen. Hence the formula is either C_{6}H_{4}BrNO_{3} or C_{6}H_{5}BrNO_{3}, of which the latter can be rejected on the grounds of valency. The actual composition of the former (2bromo3nitrophenol) is
(calculated) 

C 
33.06 
H 
1.85 
Br 
36.66 
N 
6.43 
O 
22.02 
with a molecular weight of 218.01. The large deviations in this example have been chosen purposely. Normally such an uncertainty in the number of hydrogen atoms seldom occurs. The influence of analytical errors on the result can be estimated immediately.
2.
Calculation of molecular weight from a percentage value.
Analysis of a compound containing one bromine atom (Br_{1}) yields a value of 38 % bromine. This value (38.0) on the x^{2} scale of the slide is brought under the scale value Br_{1} (red markings, third scale from top) and the corresponding molecular weight can be read off from the 10x^{2} scale of the fixed rule above the Ù sign (210).
3.
Calculation of percentage composition from empirical formula and molecular weight.
Once again, 2bromo3nitrophenol will be used as example.
as example.
3.1.
As in 1.1 (under 218)
3.2.
The cursor is now placed over the requisite multiple on the element scale and can be used to read off directly the desired percentage composition on the x^{2} scale of the slide, e.g. 33.0 for C_{6}. Note: The element scales on the lower part of the fixed rule are used in the same way as those on the upper part. They are, of course, to be used in conjunction with the 10x^{2} scale on the upper part of the fixed rule and not with the neighboring x scale. Lithium and beryllium have been included with a 10fold atomic weight; it is therefore necessary to use the 10fold %content values in these cases. The scales for O and N respectively are to be used for S and Si along with a factor of 1/2 since the atomic weights of the latter differ by almost exactly a factor of 2 from those of the former.
4.
Calculation of empirical formula from analytical values when the molecular weight is unknown.
Previously the calculation of the empirical formula from analytical data in those cases where the molecular weight was unknown was usually carried out by trial and error. The atomic ratios, referred to the element of lowest number in the substance (e.g. N), were first determined and scaled to the composition that best fitted the analytical values. Here, the use of the slide rule with its (10:x)^{2} scale markings for atomic ratios of the more important combinations of elements greatly simplifies this problem.
The atomic weight ratios are (except S/Cl) quoted with the heavier element as numerator, i.e. they are > 1.0. Accordingly, determination of the atomic ratio always starts with the heavier atom (except in the pair S, Cl). If the correct order of magnitude is calculated then the resulting atomic ratio also has the correct order of magnitude. On the other hand, if, for the sake of better readingoff an incorrect order of magnitude is calculated (in which, e.g. the number 10 is brought under the cursor hairline instead of the sign Ù ) then only the numerical result is correct and one must work out in the head whether the ratio is e.g. 1:6 or 10:6.
Example:
A substance containing C, H, N, O gives on analysis
(found) 

C 
54.50 
H 
8.63 
N 
26.76 
O 
10.11 
(The O value is calculated by difference).
One determines first the ratio of N to C atoms.
1. Place the hairline of the cursor over the value 26.8 on the 10x^{2} scale of the fixed rule (left half).
2. Place the value 54.5 on the x^{2} scale of the slide (right half) under the hairline of the cursor (= division of the wt.% values).
3. Set the cursor hair line over the N/C marking on the slide.
4. Bring the marking Ù under the cursor hairline.
5. Read off the atomic number ratio N/C on the 10x^{2} and x^{2} scales. The nearest whole number is taken, e.g. N/C = 3/7. The value 5/12 may be a valid alternative.
The ratio O/C = 1/7 (and if necessary O/N =1/3) is determined in the same way.
In calculations involving hydrogen, operation 3 above is even simpler, since, instead of using the special marking on the (10:x)^{2} scale, the ratio of the atomic weight of the element relative to that of hydrogen (i.e. 12 for C/H, 14 for N/H, etc.) can be used. In the above example the ratio of carbon to hydrogen is » 1:2. Since at least 7 carbon atoms must be present in the molecule (cf. oxygen!), the nearest whole number opposite 7 on the x^{2} scale is read off, i. e. C/H = 7/13; consequently the desired empirical formula is
C_{7}H_{13}N_{3}O
the exact composition of which is:
(calculated) 

C 
54.17 
H 
8.44 
N 
27.08 
O 
10.31 
It is recommended that all the likely values for each of the pairs of
elements be noted and the "most probable" ones underlined.
Subsequent inspection of the individual values then leads to correct choice of the valid
empirical formula.
The use of the basic scales not mentioned explicitly in these instructions is discussed in a leaflet on CastellDuplex precision slide rules.
If you are interested, please write:
A. W FaberCastell
8504 Stein bei Nürnberg, Germany
Scanned and formatted by Dr. Jon Iza, 2000.09.09