In the Classroom
How Many Digits Should We Use in Formula or Molar Mass Calculations? Christer Svensson Department of Chemistry and Biomedical Science, University of Kalmar, 391 82 Kalmar, Sweden;
[email protected] Teachers are often asked by their students: “How many digits should I use when calculating the formula or molar mass of a substance?” A focus on the formal calculation of uncertainty may discourage students before they have had chance to learn the chemistry. Uncertainty calculations, although essential, can be intimidating and discouraging, but to ignore them is a serious error. The objective of this article is to streamline the calculations of uncertainties in the formula or molar masses of compounds. Results of uncertainty calculations are treated as estimations as they are seldom (if ever) exact quantities. Three rules of increasing complexity are proposed. These rules probably overestimate the uncertainty so there is little if any risk that the true values are missed. Significant Digits Mathematical operations with numbers do not carry any uncertainty (2 + 2 = 2 · 2 = 4.0000…), but when the uncertainty of a measurement is not declared it is conventionally accepted as one unit of the last significant digit (±0.5). A measured or calculated property such as the formula or molar mass is expressed as a number multiplied by a unit. The use of units in this case has been discussed elsewhere (1–3). The number of significant digits, or, synonymously, significant figures, can be defined as “the meaningful digits in a measured or calculated quantity” (4). These are the digits used when the quantity is expressed in scientific notation. From a strictly mathematical view, the error is not more than ±0.5 of the final digit in rounded values. In science, however, the error may be greater. For instance when weighing, if the digital display on the balance shows 3.2579 g, this is considered as five significant digits although the uncertainty may be more than ±0.0001 g. In laboratory work at secondary and college level, the number of significant digits used is rarely more than four. Elementary Propagation of Error Theory Students are introduced to elementary propagation of error theory by a simple rule: not to use more significant digits in their answer than the smallest count of significant digits in the numbers used in their calculation. This applies to multiplication and division only and is a simplification of the rule that the relative errors may be added in this case. This is, however, not the case in addition and subtraction where the absolute errors may be added. Chang explains this concept well (5). For further reading on this topic, a number of other textbooks can also be recommended (6–9). Adding two numbers each with four significant digits such as 123.4 (±0.05 at most if correctly rounded) and 5.678 (±0.0005 at most if correctly rounded) gives the rewww.JCE.DivCHED.org
•
sult 129.078 (with the sum of the errors ±0.0505 at most). With this result rounded to four significant digits and the error to one (the error should always be rounded up because otherwise it may appear to be underestimated), we get 129.1 (the error ±0.06 indicates that the result should not be given with more than four significant digits). There are the same number of significant digits in the final result as in the largest of the terms used. This simple rule does not apply to cases when there are many large terms with the same error or when errors in many small terms add up to significant values. This simple rule will be used further below. First Rule The formula of any chemical substance may be written as a standard formula. For instance acetic acid, CH3COOH, may be written C2H4O2. The mass is, of course, the same, but this form may be useful in elementary propagation of error analysis of the calculation of its formula mass or its molar mass. We will discuss molar mass calculations only because identical considerations apply to formula mass calculations. Let us study the general case in which A, D, and E are elements and k, m, and n are numbers expressing their ratio. The chemical formula is: A k Dm E n
In this general case the molar mass is calculated as:
M ( A k DmE n ) = k · M (A ) + m · M (D ) + n · M ( E ) (1) In any term of this expression, the number of significant digits is at least as great as the number of significant digits in the molar mass of that term because the numbers expressing the ratios are exact numbers. Atomic weights used in this calculation are known to four significant digits or more. Exceptions are the cases of lithium and some elements with no characteristic terrestrial isotopic composition (10). We therefore conclude that, with the exception of the few cases listed, each term has at least an accuracy corresponding to four significant digits. We now return to elementary propagation of error theory and apply it to the addition of the terms. The absolute errors of each term are summed to give the error of the final result. The largest of these terms determines the final error and if four significant digits are used we conclude that, with a few exceptions, the final result has an accuracy of at least four significant digits. Using eq 1, with molar masses rounded to four digits, the molar mass of acetic acid C2H4O2 is: M ( C 2 H 4O 2 ) = ( 2 · 12.01 + 4 · 1.008 + 2 · 16.00 ) g mol −1
Vol. 81 No. 6 June 2004
= 60.052 g mol −1 = 60.05 g mol −1
•
Journal of Chemical Education
827
In the Classroom
A goal of this article is to find a straightforward answer to the question in the introduction. From this discussion, we can conclude that in elementary chemical calculations a good rule of thumb is: Use four digits in atomic weights and consider the calculated formula or molar mass to have four significant digits.
This simple rule also applies to the number of significant digits used in accurate elementary laboratory work. Therefore, this study recommends that atomic weights should be given with four significant digits in students’ periodic tables. This is indeed sometimes the case, but there are many examples with either no particular number of significant digits or with a fixed number of decimal places. Second Rule The first rule is appropriate for beginners. At the university level, students should use the second rule: express the formula of a substance in the standard way and then consider the expression for the molar mass calculation. Calculate each term in this expression with the full number of significant digits. Then the number of decimal places in the final result should be no more than the number of decimal places in the term with the fewest decimal places. In this calculation, scientific notation should not be used. In rare cases, if any terms have no decimal places then the final result should have as many significant digits as the term with the fewest significant digits. Using eq 1, the molar mass of acetic acid C2H4O2 is:
certainty of the molar mass will never give a value that is too small. Using eq 1 and expressing the uncertainties as ∆, we get: ∆M ( A k DmE n ) = k · ∆M ( A ) + m · ∆M (D ) + n · ∆M (E )
This formula may be expressed in a more general way as follows: ∆M ( A k DmE n … ) = k · ∆M (A ) + m · ∆M (D ) + n · ∆M (E ) + …
M ( C 2 H 4O 2 ) = ( 2 · 12.0107 + 4 · 1.00794 + 2 · 15.9994 ) g mol −1 = 60.05196 g mol −1 ∆M ( C 2 H 4O 2 ) = ( 2 · 0.0008 + 4 · 0.00007 + 2 · 0.0003 ) g mol −1 = 0.00248 g mol −1 Then with the uncertainty rounded up to one or two significant digits and the calculated molar mass rounded to the same absolute accuracy, we get, M ( C 2 H 4O 2 ) = ( 60.052 ± 0.003 ) g mol −1
+ 2 · 15.9994 )
or
= ( 24.0214 + 4.03176 + 31.9988 ) g mol −1 = 60.05196 g mol
= 60.0520 g mol
−1
In short, this rule can be expressed as: Use all digits from the recommended atomic weights and estimate the absolute error in the calculated formula or molar mass from the term in the sum with the fewest decimal places. If any terms have no decimal places, the final result should have as many significant digits as the term with the fewest significant digits.
Third Rule In very accurate work, propagation of error theory should be applied correctly. It is then important to consider whether the uncertainties of the recommended atomic weights are random or systematic. Recommended values are derived from a process where results from many different methods are compared and combined. This is similar to the process used to establish systematic errors. The uncertainty of the recommended value may be considered random only if it is derived from a single method. In this case, it should be termed precision and expressed as a statistical term. Thus the uncertainties of the recommended values may be treated as systematic errors. Applying propagation of systematic errors theory to linear combinations (11) when calculating the un828
Journal of Chemical Education
•
(3)
The uncertainty calculated in this way is known as the absolute maximum uncertainty. It is always rounded up, usually to one or two significant digits, as it may otherwise appear to be underestimated. The molar mass should then be given with the same absolute accuracy as the rounded uncertainty. Using eqs 1 and 2, the molar mass of acetic acid C2H4O2 is:
M ( C 2 H 4O2 ) = ( 2 · 12.0107 + 4 · 1.00794
−1
(2)
M(C 2 H 4O2 ) = (60.0520 ± 0.0025) g mol −1 where the uncertainty represents the absolute maximum uncertainty of the result. The third rule can be expressed as follows: Use all the digits in recommended atomic weights and calculate the absolute maximum uncertainty of the calculated formula or molar mass. The uncertainty should be rounded up to one (two) significant digit(s) and the formula or molar mass should be rounded to the same absolute accuracy as the rounded uncertainty.
Comparison of the Rules For acetic acid C2H4O2, the molar mass calculated using these rules gives the following results. With the first rule, using eq 1, M ( C 2 H 4O 2 ) = 60.05 g mol −1
with the second rule, using eq 1, M ( C 2 H 4O 2 ) = 60.0520 g mol −1
and with the third rule, using eqs 1 and 2,
M ( C 2 H 4O 2 ) = ( 60.052 ± 0.003 ) g mol −1
Vol. 81 No. 6 June 2004
•
www.JCE.DivCHED.org
In the Classroom
or −1
M ( C 2 H 4O 2 ) = ( 60.0520 ± 0.0025 ) g mol where the uncertainty represents the absolute maximum uncertainty of the result. In this case, there is good agreement between the results of the calculations based on the different rules. This is, however, not always the case, so great care must be taken when using the first two rules in accurate work. The molar mass of lead is given to one decimal place only. In the case of PbS, there is also good agreement between the results of the calculations based on the different rules. Using eqs 1 and 3 the results are 239.3 g mol᎑1 from both the first and the second rule and (239.3 ± 0.2) g mol᎑1 from the third rule. With computers the third rule can easily be implemented. This is demonstrated in the McDalton Learning Edition (12), a freeware computer program for the calculation of formula or molar masses. Some Guidance for Use As mentioned above, the first rule is appropriate for beginners. For the majority of school-age students, the concept of significant digits is not easy to understand and a simple but reliable method for molar mass calculations is necessary. With this approach, sufficiently accurate results are obtained for the highest standard of school laboratory work. In periodic tables in some textbooks, atomic weights are given to just one decimal place, perhaps because the atomic weight of lead is not known with better accuracy. Presumably, the intention of the authors of these textbooks is not that these values should be used in students’ calculations. However, if these quantities are used then the molar mass of substances such as H2 or LiH will be known to just two significant digits. It is proposed here that if atomic weights are printed in periodic tables for students’ calculations, they should be given with four significant digits. Students should use the first rule, and if they do not have to round the atomic weights to four significant digits in every calculation, they are able to focus instead on the chemistry. Calculations to two decimal places cannot be recommended either. This is because Pb is not known to that accuracy, and to use 1.01 g mol᎑1 for H and 14.01 g mol᎑1 for N in a molecule such as NH3 gives the result 17.04 g mol᎑1, when a better answer is 17.03 g mol᎑1. An error in the final significant digit is introduced unnecessarily. This may be the case for other small molecules. When, at university level, analytical balances are introduced, it is necessary to estimate the uncertainties of the calculated molar masses. This is the case, for instance, when 12.3412 g of ZnO (molar mass 81.39 g mol᎑1) is calculated to 0.1516 mol. At this time the second rule should be introduced to students. The third rule is appropriate for advanced use in higher education as well as in research. It is not the only way to handle molar masses with assigned uncertainties, at this level there are also other ways to treat the concept (13). The atomic weights are assigned uncertainties that are estimated by combining experimental uncertainties and terrestrial variabilities (10). The Commission on Atomic Weights and Isotopic Abundances uses the term precision when the ranges in the isotopic composition of normal terrestrial material are diswww.JCE.DivCHED.org
•
cussed and accuracy when discussing the uncertainties of the standard atomic weights. These terms are used in the same sense in this article. For many elements, however, the limited accuracy of measurements is insignificant in comparison with terrestrial variation (10). In these cases, especially as the uncertainties estimated by calculations with the third rule are overestimated, the formula 2
2
∆M ( A k DmE n … ) = k · ∆M ( A )
(4) 2 2 + m · ∆M ( D ) + n · ∆M ( E ) + …
may be used. The uncertainty calculated this way is known as the absolute probable uncertainty. It is, however, important to note that it is likely to give a result that is too small and that the uncertainties of molar masses calculated this way may not be good estimations of the true values. This is not likely when the third rule is used and it is therefore recommended. Most atomic masses are known with sufficient accuracy that although the third rule overestimates the uncertainties, this is seldom the limiting aspect in further propagation of error analysis, and their contribution can often be disregarded. Acknowledgments I gratefully acknowledge my students and colleagues past and present for initiating this work and for valuable discussions. I also gratefully acknowledge the reviewers of this Journal for their helpful suggestions. Literature Cited 1. Paolini, M.; Cercignani, G.; Bauer, C. J. Chem. Educ. 2000, 77, 1438. 2. Gorin, G. J. Chem. Educ. 2002, 79, 163. 3. Paolini, M.; Cercignani, G.; Bauer, C. J. Chem. Educ. 2002, 79, 163. 4. Chang, R. Chemistry, 7th ed.; McGraw-Hill: New York, 2002; p 21. 5. Chang, R. Chemistry, 7th ed.; McGraw-Hill: New York, 2002; p 22–24. 6. Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed.; Ellis Horwood: Chichester, U.K., 1993. 7. Taylor, J. R. An Introduction to Error Analysis—The Study of Uncertainties in Physical Measurements, 2nd ed.; University Science Books: Sausalito, CA, 1997. 8. Clarke, G. M.; Kempson, R. E. Introduction to the Design and Analysis of Experiments; Arnold: London, 1997. 9. Hart, H.; Lotze, W.; Woschni, E.-G. Messgenauigkeit, 3rd ed.; Oldenburg, 1997. 10. Commission on Atomic Weights and Isotopic Abundances, Inorganic Chemistry Division, International Union of Pure and Applied Chemistry. Pure Appl. Chem. 2001, 73, 667. 11. Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed.; Ellis Horwood: Chichester, U.K., 1993; p 50, Eq. 2.20. 12. Svensson, C. McDalton–Freeware Calculator for Formula Masses. http://www.chem4free.info/mcdalton (accessed Mar 2004). 13. National Bureau of Standards. Precision Measurement and Calibration, Selected NBS Papers on Statistical Concepts and Procedures; NBS Special Publication 300: Washington DC, 1969; Vol. 1.
Vol. 81 No. 6 June 2004
•
Journal of Chemical Education
829
