This paper offers a clear framework for understanding basic atomic concepts.Throughout the manuscript, mass expressions are described as scalar quantitiesthatare characterized by a symbol, a numerical value, and a unit of measurement.It defines the role of the IUPAC ratio,explains the meaning of absolute and relative atomic mass, and determines the origin and relationship among molar mass, Avogadro’s number, and the amount of substance. The approach moves from isolated isotopes to poly-isotopic elements, and from absolute atomic mass to relative atomic mass when applicable.The atomic mass ratio, , plays an important role in this work. Absolute atomic mass, a magnitude expressed in kilograms, differs from relative atomic mass, denoted by the numerical value with the unified atomic mass unit . The relationship between the ratio and the periodic table of elements is highlighted, emphasizing factors, besides the controversy between atomic mass and atomic weight, that may confuse users of the table.The concept of any mass quantity,,is introduced to clarify the meaning of the atomic mass-weighted average. Furthermore, the ideain conjunction with the ratio..is used to demonstrate the origin and relationship among molar mass, Avogadro’s number, and the amount of substance.
Some concepts remain difficult for students and even instructors to grasp. Two instances include the interpretation of atomic mass (also referred to as atomic weight) as a dimensionless value on the periodic table, as well as the justification for obtaining the molar mass of an element.
Efforts to clearly explain the mole, Avogadro’s number, the molar mass, and the amount of substance, along with ongoing discussions about potential future changes to the definition of the amount of substance "n," have been extensively documented in numerous publications 1, 2, 3, 4, 5, 6.
The International Union of Pure and Applied Chemistry (IUPAC) has designated the symbol 
to denote relative atomic mass (atomic weight), textually defined as “the ratio of the average mass of the atom to the unified atomic mass unit” 7, 8. However, the ratio 
is not commonly mentioned in chemistry textbooks and publications, resulting in atomic mass-related explanations that are usually qualitative or based only on numerical examples rather than broader scope symbolic expressions 9.
The objective of this publication is to provide a comprehensive and systematic theoretical framework that elucidates and interconnects all definitions of atomic mass concepts.
The atomic mass is a scalar quantity, and any scalar quantity is made up of a symbol, a magnitude or numerical value 
, and a unit of measurement. Figure 1 resumes the way of representing a scalar quantity.
According to the IUPAC publication 8, the atomic mass symbol for an isotope with a mass number “i” of element
is
where
is the symbol of the element. In this paper, and following IUPAC nomenclature 7, the symbol “i” changes to“A.”Therefore, the “absolute” atomic mass of an isotope is a scalar quantity with the kilogram as a unit, represented as:
 | (1) |
The unified atomic mass unit 
(also known as the Dalton “Da”) 7, defined and accepted by agreement between chemists and physicists in 1961 1, is also a scalar quantity given by:
 | (2) |
The ratio of 
to
is known as the relative atomic mass
7. In other words,
 | (3) |
The quantity 
is unknown, but the ratio 
is available through the mass spectrometer. Mass spectrometers do not provide absolute atomic masses; they only give isotopic 
valuesas well as the percentage fraction of the natural isotopic abundance. Applying equation (3) to gives
. Samples of 
are used to calibrate mass spectrometers to
obtain 
values of other elements with 
as a reference.
As demonstrated subsequently,
is derived from Avogadro's number 10, 11. As a result, since 
and
are known,
can also be expressed as a “relative” atomic mass, as shown below.
 | (4) |
Expression (4) is a scalar quantity with 
as the numerical value.
Table 1 shows the isotopic composition of element chlorine 12.
The following is an example of obtaining absolute atomic mass through the atomic mass unit 
. Let us apply equation (4) to isotope
.
 |
Unlike the extremely small numerical value of the “absolute” atomic mass, the magnitude of 
allows the ratio 
to become a whole number with up to three digits and a decimal portion with a variable number of digits, which is easier to handle than the numerical value of the “absolute” atomic mass.It is important to emphasize that expression (4) is the complete version of the relative atomic mass, whereas the ratio
is only the numerical value of that quantity. Despite that, IUPAC defines this ratio as the full representation of relative atomic mass 7.
The following text is an exact transcription of a passage found on page 267 of an IUPAC publication 8:
“Thus, the atomic mass of 12C is 12 Da, and the atomic weight of 12C is 12 exactly. All other atomic weight values are ratios to the 12C standard value and thus are dimensionless numbers.”
The revised text below is an example of how the ideas discussed in this section can be applied to enhance the original (Modifications are highlighted).
Thus, the relative atomic mass of. is 12 Da, and the numerical value of the relative atomicmass of 
is 12 exactly. This number and all other relative atomic mass numerical values are ratios to the unified atomic mass unit and thus are dimensionless numbers.
Up to this point, only isolated isotopes were considered.For elements of more than one natural isotope, the atomic mass is a weighted average 13 represented by
.
This section introduces the concept of anymass quantity
to help understand the origin of the atomic mass-weighted average.For a sample with any number “N” of atoms of an element 
with
natural isotopes,any mass quantity is given by,
 | (5) |
Where 
is the percentage fraction of the natural isotopic abundance, this factor equals one for elements with only one natural isotope. Dividing by
on both sides of equation (5), it is obtained the atomic mass-weighted average,
 | (6) |
Therefore,
 | (7) |
In this way, a single theoretical quantity 
is equivalent to the atomic mass of natural isotopes of element
, and the element can be treated as if it only has one isotope. Dividing by 
on both sides of equation (7), the result is the atomic mass-weighted average in terms of the numerical value of the relative atomic masses,
 | (8) |
Be careful not to confuse
with the atomic mass number 
.
The following example shows how to compute the value
. From Table 1 and applying equation (8):
 |
Hence, the percentage contribution of each isotope in creating 
is: 74.72% 
and 25.20%
.Most cells in the periodic table of elements display a decimal number named atomic mass or atomic weight. Regardless of the name, this quantity is just the ratio 
as given in equation (8) and shown in Figure 2.
The remaining cells display whole numbers in parentheses, corresponding to the mass number of the most stable isotope among the unstable isotopes of radioactive elements 13.
Like isolated isotopes, the formula for the relative atomic mass of poly-isotopic elements is,
 | (9) |
Therefore, and as mentioned before, referring to the ratio 
as the relative atomic mass or, even worse, atomic weight is confusing. Due to a lack of units, this ratio is only part of a scalar quantity, but not the whole scalar quantity as such. Even in the IUPAC Periodic Table of the Isotopes 8, which is a very complete source of information, neither the ratio 
nor the unified atomic mass 
are mentioned or related to the atomic weight (in this table, the term atomic weight is used instead ofatomic mass).
All the factors above can lead to confusion about the meaning of atomic mass or atomic weight in the periodic table. The following is a summary of them:
• A number without units is called atomic mass or atomic weight. Most periodic tables omit the unit
.
• Given that the terms weight and mass are used indistinctly, it could seem that both concepts are equivalent at the atomic level.
• The ratio 
and the word “atomic weight” are incompatible. This number is the numerical value of the "relative" atomic mass.
The idea of “any mass quantity”
already mentioned, togetherwith the ratio 
is used to demonstrate through equations the origin as well as the relationship among the molar mass
, the Avogadro´s number
, and the amount of substance 
[14]. This approach also applies to molecules and ionic compounds.
Any mass quantity of any element 
can be expressed as:
 | (10) |
Where 
is the number of atoms.
Substituting 
from equation (2) it is obtained,
 | (11) |
If the expression (11) is given in terms of grams, it will be equal to:
 | (12) |
Let denote the numerical factor within square brackets as
. Hence,
 | (13) |
Where 
could be any positive number.
Combining expressions (12) and (13) yields:
 | (14) |
and if 
the expression (14) becomes the molar mass
, formerly known as “gram atomic weight” 13. Therefore, expression (14) can be rewritten as
 | (15) |
Expression (15) shows that the molar mass is a scalar quantity with the ratio 
as the numerical value. The relative atomic mass (9) also has 
as the numerical value. For that reason, there is a wrong tendency to think that the molar mass comes from an arbitrary change of units, 
for in expression (9) 2, 3, 4, 5, 6, 7, 8, 9, without considering the numerical factor
.
Also, if 
equals 1, the number of atoms 
in equation (13) is equal to Avogadro's number NA which, once cleared, gives
 | (16) |
Since 
is always equal to one for the molar mass, it means that the molar mass of any element always has 
atoms.
Regarding expression (16) and contrary to what might be assumed, 
is derived from experimental work, and relation (16) is used only to compute the numerical value of 
4, 10.
Figure 3 summarizes the relationship between different mass amounts and the corresponding number of atoms for element (E).
From equations (13) and (16), it is found that 
is just the ratio of 
to
, which shows that 
is precisely the amount of substance; therefore,
 | (17) |
Where 
From the ratio (17), it is deduced that 
represents the number of moles that fit in
, which suggests that "amount of moles" would be a better name for 
than "amount of substance"when the entities are atoms.As an alternative, equation (17) can be rearranged into a scalar quantity as shown below
 | (18) |
where 
is the numerical value and 
the measurement unit.
Table 2 shows the scalar quantities covered in this article, emphasizing the units of measurement. Except for the unified atomic mass unit 
, all other units are SI base units. However, 
is accepted for use with SI units [14].
Remembering that mass is a scalar quantity is the first step in understanding atomic mass-related basic concepts.
The ratio 
is not popular as a symbol for educational purposes. It is uncommon to find it in chemistry textbooks, even though there is no other symbol equivalent to 
. The use of the IUPAC ratio 
should be expanded throughout the field of science education to standardize nomenclature and enhance the teaching and learning processes.
Recognizing the decimal number in some cells of the periodic table of the elements as the ratio 
reveals that it is wrong to refer to this number as the relative atomic mass or atomic weight. Without the unit 
this number is not a complete scalar quantity. To be accurate, it is only the numerical value of the relative atomic mass and not the whole scalar quantity as erroneously stated by IUPAC 7. Those mentioned above,besides the controversy between mass and weight,are a main source of confusion among students and even teachers.
Introducing the idea of any mass quantity,
besides being useful in defining the weighted average atomic mass, has been fundamental in providing a sound demonstration that explains the origin and relationship among the molar mass, Avogadro´s number, and the amount of substance. It eliminates the need to use or seek out qualitative definitions, official or not, that could help describe these concepts.
Substituting 
for grams in the expression for relative atomic mass to obtain molarmass is only a rule of thumb that disguises the true origin of molar mass.Choosing grams as a unit for molar mass is a matter of convenience. If given in kilograms, the only change is that Avogadro’s number would increase by a factor of one thousand.
| [1] | De Bièvre, P., Peiser, H. S. Pure & Appl. Chem., Vol. 64, No. 10, pp. 1535-1543. 1992. | ||
| In article | View Article | ||
| [2] | Balocchi, E., Modak, B., Martínez, M., Padilla, K., Reyes C. F. andGarritz, A., Aprendizaje cooperativo del concepto ‘cantidad de sustancia’ con base en la teoría atómica de Dalton y la reacción química,Educación Química, 16(1), 14-21. 2006. Vilches, A., Gil Pérez, D., Algunas consideraciones clave, pero generalmente olvidadas, para lograr la comprensión del concepto de cantidad de sustancia,Educación Química, 21(3), 207-211. 2010. ISSN 0187-893-X. | ||
| In article | View Article | ||
| [3] | Barański, A., The atomic mass unit, the Avogadro constant, and the mole: a way to understanding, J. Chem. Educ., 89, 97-102. 2012. | ||
| In article | View Article | ||
| [4] | Giunta, C.J., The mole and amount of substance in chemistry and Education: beyond official definitions, J. Chem. Educ., 92, 1593-1597. 2015. | ||
| In article | View Article | ||
| [5] | Kacker, R.N., Irikura, K. k., The SI unit mole and Avogadro constant, Measurement: Sensors, Vol. 38, Supplement, May 2025, 101767. | ||
| In article | View Article | ||
| [6] | IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019) created by S. J. Chalk. ISBN 0-9678550-9-8. | ||
| In article | |||
| [7] | Meija, J., Coplen, T. B., Berglund, M., Brand, W. A., De Bièvre, P., Gröning, M., Holden, N. E., Irrgeher, J., Loss, R. D., Walczyk, T. and Prohaska, T. (2016)."Atomic weights of the elements 2013 (IUPAC Technical Report)" Pure and Applied Chemistry, vol. 88, No. 3, pp. 265-291. | ||
| In article | View Article | ||
| [8] | Brown, T. E.; LeMay, H. E.; Bursten, B. E.; Murphy, C.; Woodward, P.; Stoltzfus, M. E,Química. La ciencia central, 1era ed.; Pearson Educación de México, S.A. de C.V. 2021. | ||
| In article | |||
| [9] | Umaña, C.E.,"The Atomic Mass Unit and the Avogadro Constant: An Approach to Introduce these Concepts", National Educators Workshop: Update 2001. Standard Experiments in Engineering, Materials Science, and Technology, NASA/CP-2002-211735, 351-359. 1 Jun. 2002. | ||
| In article | |||
| [10] | CODATA Recommended Values of the Fundamental Physical Constants: 2014. https:// codata.org/blog / 2015/ 08/04/codata-recomm ended-values -of-the- fundamental- physical-constants-2014/. | ||
| In article | |||
| [11] | NIST Atomic Weights and Isotopic Composition for all elements. https://physics.nist.gov/cgi-bin/ Compositions/ stand_alone. plDevelopers and Contributors:J. S. Coursey, D. J. Schwab, J. J. Tsai, and R. A. Dragoset. NIST Physical Measurement Laboratory.The atomic weights are available for elements 1 through 118 and isotopic compositions or abundances are given when appropriate. The atomic weights data were published by J. Meija et al in Atomic Weights of the Elements 2013, and the isotopic compositions data were published by M. Berglund and M.E. Wieser in Isotopic Compositions of the Elements 2009. The relative atomic masses of the isotopes data were published by M. Wang, G. Audi, A.H. Wapstra, F.G. Kondev, M. MacCormick, X. Xu1, and B. Pfeiffer in The AME2012 Atomic Mass Evaluation. These data have been compiled from the above sources for the user's convenience and do not represent a critical evaluation by the NIST Physical Measurement Laboratory. | ||
| In article | |||
| [12] | Kolb, D., But if atoms are so tiny, Journal of Chemical Education, Vol. 54, No. 9, 543-547.September 1977. | ||
| In article | View Article | ||
| [13] | The International System of Units (PDF), V3.01 (9th ed.), International Bureau of Weights and Measures, Aug 2024, ISBN 978-92-822-2272-0. | ||
| In article | |||
Published with license by Science and Education Publishing, Copyright © 2025 Carlos E. Umaña
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
http://creativecommons.org/licenses/by/4.0/
| [1] | De Bièvre, P., Peiser, H. S. Pure & Appl. Chem., Vol. 64, No. 10, pp. 1535-1543. 1992. | ||
| In article | View Article | ||
| [2] | Balocchi, E., Modak, B., Martínez, M., Padilla, K., Reyes C. F. andGarritz, A., Aprendizaje cooperativo del concepto ‘cantidad de sustancia’ con base en la teoría atómica de Dalton y la reacción química,Educación Química, 16(1), 14-21. 2006. Vilches, A., Gil Pérez, D., Algunas consideraciones clave, pero generalmente olvidadas, para lograr la comprensión del concepto de cantidad de sustancia,Educación Química, 21(3), 207-211. 2010. ISSN 0187-893-X. | ||
| In article | View Article | ||
| [3] | Barański, A., The atomic mass unit, the Avogadro constant, and the mole: a way to understanding, J. Chem. Educ., 89, 97-102. 2012. | ||
| In article | View Article | ||
| [4] | Giunta, C.J., The mole and amount of substance in chemistry and Education: beyond official definitions, J. Chem. Educ., 92, 1593-1597. 2015. | ||
| In article | View Article | ||
| [5] | Kacker, R.N., Irikura, K. k., The SI unit mole and Avogadro constant, Measurement: Sensors, Vol. 38, Supplement, May 2025, 101767. | ||
| In article | View Article | ||
| [6] | IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019) created by S. J. Chalk. ISBN 0-9678550-9-8. | ||
| In article | |||
| [7] | Meija, J., Coplen, T. B., Berglund, M., Brand, W. A., De Bièvre, P., Gröning, M., Holden, N. E., Irrgeher, J., Loss, R. D., Walczyk, T. and Prohaska, T. (2016)."Atomic weights of the elements 2013 (IUPAC Technical Report)" Pure and Applied Chemistry, vol. 88, No. 3, pp. 265-291. | ||
| In article | View Article | ||
| [8] | Brown, T. E.; LeMay, H. E.; Bursten, B. E.; Murphy, C.; Woodward, P.; Stoltzfus, M. E,Química. La ciencia central, 1era ed.; Pearson Educación de México, S.A. de C.V. 2021. | ||
| In article | |||
| [9] | Umaña, C.E.,"The Atomic Mass Unit and the Avogadro Constant: An Approach to Introduce these Concepts", National Educators Workshop: Update 2001. Standard Experiments in Engineering, Materials Science, and Technology, NASA/CP-2002-211735, 351-359. 1 Jun. 2002. | ||
| In article | |||
| [10] | CODATA Recommended Values of the Fundamental Physical Constants: 2014. https:// codata.org/blog / 2015/ 08/04/codata-recomm ended-values -of-the- fundamental- physical-constants-2014/. | ||
| In article | |||
| [11] | NIST Atomic Weights and Isotopic Composition for all elements. https://physics.nist.gov/cgi-bin/ Compositions/ stand_alone. plDevelopers and Contributors:J. S. Coursey, D. J. Schwab, J. J. Tsai, and R. A. Dragoset. NIST Physical Measurement Laboratory.The atomic weights are available for elements 1 through 118 and isotopic compositions or abundances are given when appropriate. The atomic weights data were published by J. Meija et al in Atomic Weights of the Elements 2013, and the isotopic compositions data were published by M. Berglund and M.E. Wieser in Isotopic Compositions of the Elements 2009. The relative atomic masses of the isotopes data were published by M. Wang, G. Audi, A.H. Wapstra, F.G. Kondev, M. MacCormick, X. Xu1, and B. Pfeiffer in The AME2012 Atomic Mass Evaluation. These data have been compiled from the above sources for the user's convenience and do not represent a critical evaluation by the NIST Physical Measurement Laboratory. | ||
| In article | |||
| [12] | Kolb, D., But if atoms are so tiny, Journal of Chemical Education, Vol. 54, No. 9, 543-547.September 1977. | ||
| In article | View Article | ||
| [13] | The International System of Units (PDF), V3.01 (9th ed.), International Bureau of Weights and Measures, Aug 2024, ISBN 978-92-822-2272-0. | ||
| In article | |||
