Empirical Formulas from Atom Ratios A Simple Method To Obtain the Integer Factors of a Rational Number E. Weltin University of Vermont, Burlington, VT 05405
The atom ratios in a n empirical formula of a molecule are rational numbers.
2 Starting with the initial pair Ira; sol88 long ass; + 0, define the next riel = Usi. For the given example, we get
In many cases the integersp and q can be obtained from experience. If, on the other hand, a very careful chemical analysis of 5.53101 g of a hydrocarbon CP% shows that it contains 4.68429 g C, the carbon:hydrogen ratio is found to be 0.4642857.In such a case we might not be able to immediately write down what the coefficients p and q should be. The Procedure I n a similar problem, we may be given a balanced chemical equation with noninteger mefficients. Then we must find a multiplying factor to bring the equation into the standard form with integer stoichiometric coefficients. Instead of just trial and error, which is the method used in virtually all chemistry textbooks, the following simple procedure is an effective way to find the integer factors.'
where sg is rounded to zero. 3. Form the product of the r:s, excluding the first term, ro. The result is the number q. Finally, caleulatep = r ~ . In the example, q = 28 a n d p = 13,that is, 0.4642857 = 13/28.The empirical formula of the hydrocarbon is C13H28. The fraction obtained is always in reduced form. In other words, numerator and denominator have no wmmon factors. Using Fractions The e x a m ~ l eclearlv shows how the method works: i t is much less informati& about why the method works.'with perfect hindsight, that is, the knowledge that
1 Associate with a given rational number r a simplified rational numbers using the following procedure.
a. If r 2 1, dmp the integer part. b. If the remaining fractional part f is < 0.5, sets = f. Otherwise, sets = 1- f .
the sequence above maybe recognized as the decimal equivalent of the sequence of fractions given below.?
Obviously, if r is an integer, then s = 0; s is always 5 0.5. Some examples using the notation (r;s) are 17; 01, 12.367; 0.3671, 11.811; 0.1891, etc
'Tne standaro proced~re,multlpllcallon of rwlth two d~lferentpowers of 10 to ahgn tne repeatang oeclma afterthe dec~malpo~nt-liO~ - 104 r = i, integeris, in general, inconvenient. It cannot even be applied to this example because, at the given precision of r, it is not clear what the reoeatina decimals are. 'The numerator p o i s = plqcan be related to the numerator p in r using the following steps. (a,) Subtract qfrom p as many times as possible until the remainder t is between 0 and (q- I)inclusive, that is, t = p modulus q. (b.) Then p' is the smaller'of t and (q- 0.
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Journal of Chemical Education
Forming the products of the r{s from last to first, one obtains
Evaluating the Method
Of course, the algorithm also applies when we generally know the answers, as shown below in two examples.
Does this method always work? From the way the sequence is constructed, the numerators in the simplified rational numbers q ,which are 13,2,1,0 in the first example, yield a sequence in which each integer is less than or equal to one half of the preceding integer. Thus, sooner or later, we must reach 0, the terminator of the process. Rounding
You must be careful when rounding off the numbers. In the first example, it was quite obvious that r2 might be 6.5 and r, = 2. In particular, the partial products, working from the bottom up, should all be integers. The initial rational number ro must be given to a fairly large number of significant digits. In some respect, the method works too well: If the initial term is rounded off, the algorithm determines the factors of the rounded number, not those of the desired value. As chemists we probably would accept all three values 0.32,0.33, and 0.34 as representing the ratio 1:3. Using the present method, the ratios 8125,331100, and 17/50, respectively, are obtained. On chemical grounds, we can reasonably guess how large q might be. Except for ro, the r;'s are necessarily greater than or equal to 2. By keeping track ofthe rnnning product of the r/s starting with rl, we can see when the product exceeds the expected magnitude of q. By examining the previous steps one might round a t some earlier point. For example,
Rounding r , to 3.0, we get q = 3 and consequently p = 1. In other words, 1 3 is not an exact re~resentationof 0.34. but it is quite close. A Rational Approximation to an Irrational Number: n and e
The same type of argument may be used to obtain a rational approximation to an irrational number. An irrational number, such as n = 3.141592654, leads to a nonterminating sequence.
If we observe a terms; that is rather small, then we can obtain a rational approximation to the irrational number. For example, if we truncate a t the second line and round off r to 7, we get the approximation 2217 for n. In the days before electronic calculators, this very ratio was used quite frequently Truncating at the third line, the rational approximation 3551113 = 3.141592920 is obtained, which is an exceptionally close match. The number e = 2.71828 is a little less accommodating; 1917 = 2.71428 is about the best one can do among relatively small values ofp and q. Conclusion
In the age of hand-held electronic calculators, the method discussed here is one of the most convenient and fastest ways to find the integer factors of a rational number. Given its simplicity, the algorithm is bound to have been rediscovered repeatedly. Typical mathematics texts defme the rational numbers and point out that they have repeating decimals? Neither they nor books on numerical methods discnss systematic methods for finding the integer factors from the decimal representations, and it is unlikely that a typical chemistry student has been exposed to such a method. More to the point, 1 have not seen this algorithm mentioned in any chemistry text, particularly for general chemistry, where it would do the most good educationally The only caution is that effects due to rounding off and truncation must be carefully considered. %ee, forexample, Kolman 6.;ShaDirO A. Alaebra for Colleoe Students. 2nd ed.; ~arcourt,Brace, ~ovanovich:tiew York, 1986;
Volume 70 Number 4 April 1993
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