2.3: Calculating Atomic Masses

Skills to Develop

• Define the atomic mass unit and average atomic mass
• Calculate average atomic mass and isotopic abundance
• Define the amount unit mole and the related quantity Avogadro’s number
• Explain the relation between mass, moles, and numbers of atoms or molecules, and perform calculations deriving these quantities from one another

Video \(\PageIndex{1}\): A review of counting subatomic particles and a preview of isotopes and relative atomic mass.

 

Isotopes

The symbol for a specific isotope of any element is written by placing the mass number as a superscript to the left of the element symbol (Figure \(\PageIndex{4}\)). The atomic number is sometimes written as a subscript preceding the symbol, but since this number defines the element’s identity, as does its symbol, it is often omitted. For example, magnesium exists as a mixture of three isotopes, each with an atomic number of 12 and with mass numbers of 24, 25, and 26, respectively. These isotopes can be identified as ²⁴Mg, ²⁵Mg, and ²⁶Mg. These isotope symbols are read as “element, mass number” and can be symbolized consistent with this reading. For instance, ²⁴Mg is read as “magnesium 24,” and can be written as “magnesium-24” or “Mg-24.” ²⁵Mg is read as “magnesium 25,” and can be written as “magnesium-25” or “Mg-25.” All magnesium atoms have 12 protons in their nucleus. They differ only because a ²⁴Mg atom has 12 neutrons in its nucleus, a ²⁵Mg atom has 13 neutrons, and a ²⁶Mg has 14 neutrons.

Figure \(\PageIndex{1}\): The symbol for an atom indicates the element via its usual two-letter symbol, the mass number as a left superscript, the atomic number as a left subscript (sometimes omitted), and the charge as a right superscript.

Information about the naturally occurring isotopes of elements with atomic numbers 1 through 10 is given in Table \(\PageIndex{2}\). Note that in addition to standard names and symbols, the isotopes of hydrogen are often referred to using common names and accompanying symbols. Hydrogen-2, symbolized ²H, is also called deuterium and sometimes symbolized D. Hydrogen-3, symbolized ³H, is also called tritium and sometimes symbolized T.

Table \(\PageIndex{1}\): Nuclear Compositions of Atoms of the Very Light Elements

Element	Symbol	Atomic Number	Number of Protons	Number of Neutrons	Mass (amu)	% Natural Abundance
hydrogen	\(\ce{^1_1H}\) (protium)	1	1	0	1.0078	99.989
	\(\ce{^2_1H}\) (deuterium)	1	1	1	2.0141	0.0115
	\(\ce{^3_1H}\) (tritium)	1	1	2	3.01605	— (trace)
helium	\(\ce{^3_2He}\)	2	2	1	3.01603	0.00013
	\(\ce{^4_2He}\)	2	2	2	4.0026	100
lithium	\(\ce{^6_3Li}\)	3	3	3	6.0151	7.59
	\(\ce{^7_3Li}\)	3	3	4	7.0160	92.41
beryllium	\(\ce{^9_4Be}\)	4	4	5	9.0122	100
boron	\(\ce{^{10}_5B}\)	5	5	5	10.0129	19.9
	\(\ce{^{11}_5B}\)	5	5	6	11.0093	80.1
carbon	\(\ce{^{12}_6C}\)	6	6	6	12.0000	98.89
	\(\ce{^{13}_6C}\)	6	6	7	13.0034	1.11
	\(\ce{^{14}_6C}\)	6	6	8	14.0032	— (trace)
nitrogen	\(\ce{^{14}_7N}\)	7	7	7	14.0031	99.63
	\(\ce{^{15}_7N}\)	7	7	8	15.0001	0.37
oxygen	\(\ce{^{16}_8O}\)	8	8	8	15.9949	99.757
	\(\ce{^{17}_8O}\)	8	8	9	16.9991	0.038
	\(\ce{^{18}_8O}\)	8	8	10	17.9992	0.205
fluorine	\(\ce{^{19}_9F}\)	9	9	10	18.9984	100
neon	\(\ce{^{20}_{10}Ne}\)	10	10	10	19.9924	90.48
	\(\ce{^{21}_{10}Ne}\)	10	10	11	20.9938	0.27
	\(\ce{^{22}_{10}Ne}\)	10	10	12	21.9914	9.25

Use this Build an Atom simulator to build atoms of the first 10 elements, see which isotopes exist, check nuclear stability, and gain experience with isotope symbols.

Atomic Mass

Because each proton and each neutron contribute approximately one amu to the mass of an atom, and each electron contributes far less, the atomic mass of a single atom is approximately equal to its mass number (a whole number). However, the average masses of atoms of most elements are not whole numbers because most elements exist naturally as mixtures of two or more isotopes.

The mass of an element shown in a periodic table or listed in a table of atomic masses is a weighted, average mass of all the isotopes present in a naturally occurring sample of that element. This is equal to the sum of each individual isotope’s mass multiplied by its fractional abundance.

\[\mathrm{average\: mass}=\sum_{i}(\mathrm{fractional\: abundance\times isotopic\: mass})_i\]

For example, the element boron is composed of two isotopes: About 19.9% of all boron atoms are ¹⁰B with a mass of 10.0129 amu, and the remaining 80.1% are ¹¹B with a mass of 11.0093 amu. The average atomic mass for boron is calculated to be:

\[\begin{align*}
\textrm{boron average mass} &=\mathrm{(0.199\times10.0129\: amu)+(0.801\times11.0093\: amu)}\\
&=\mathrm{1.99\: amu+8.82\: amu}\\
&=\mathrm{10.81\: amu}
\end{align*}\]

It is important to understand that no single boron atom weighs exactly 10.8 amu; 10.8 amu is the average mass of all boron atoms, and individual boron atoms weigh either approximately 10 amu or 11 amu.

Example \(\PageIndex{1}\): Calculation of Average Atomic Mass

A meteorite found in central Indiana contains traces of the noble gas neon picked up from the solar wind during the meteorite’s trip through the solar system. Analysis of a sample of the gas showed that it consisted of 91.84% ²⁰Ne (mass 19.9924 amu), 0.47% ²¹Ne (mass 20.9940 amu), and 7.69% ²²Ne (mass 21.9914 amu). What is the average mass of the neon in the solar wind?

Solution

\[\begin{align*}
\mathrm{average\: mass} &=\mathrm{(0.9184\times19.9924\: amu)+(0.0047\times20.9940\: amu)+(0.0769\times21.9914\: amu)}\\
&=\mathrm{(18.36+0.099+1.69)\:amu}\\
&=\mathrm{20.15\: amu}
\end{align*}\]

The average mass of a neon atom in the solar wind is 20.15 amu. (The average mass of a terrestrial neon atom is 20.1796 amu. This result demonstrates that we may find slight differences in the natural abundance of isotopes, depending on their origin.)

Exercise \(\PageIndex{1}\)

A sample of magnesium is found to contain 78.70% of ²⁴Mg atoms (mass 23.98 amu), 10.13% of ²⁵Mg atoms (mass 24.99 amu), and 11.17% of ²⁶Mg atoms (mass 25.98 amu). Calculate the average mass of a Mg atom.

Answer

24.31 amu

We can also do variations of this type of calculation, as shown in the next example.

Example \(\PageIndex{2}\): Calculation of Percent Abundance

Naturally occurring chlorine consists of ³⁵Cl (mass 34.96885 amu) and ³⁷Cl (mass 36.96590 amu), with an average mass of 35.453 amu. What is the percent composition of Cl in terms of these two isotopes?

Solution

The average mass of chlorine is the fraction that is ³⁵Cl times the mass of ³⁵Cl plus the fraction that is ³⁷Cl times the mass of ³⁷Cl.

\[\mathrm{average\: mass=(fraction\: of\: ^{35}Cl\times mass\: of\: ^{35}Cl)+(fraction\: of\: ^{37}Cl\times mass\: of\: ^{37}Cl)}\]

If we let x represent the fraction that is ³⁵Cl, then the fraction that is ³⁷Cl is represented by 1.00 − x.

(The fraction that is ³⁵Cl + the fraction that is ³⁷Cl must add up to 1, so the fraction of ³⁷Cl must equal 1.00 − the fraction of ³⁵Cl.)

Substituting this into the average mass equation, we have:

\[\begin{align*}
\mathrm{35.453\: amu} &=(x\times 34.96885\: \ce{amu})+[(1.00-x)\times 36.96590\: \ce{amu}]\\
35.453 &=34.96885x+36.96590-36.96590x\\
1.99705x &=1.513\\
x&=\dfrac{1.513}{1.99705}=0.7576
\end{align*}\]

So solving yields: x = 0.7576, which means that 1.00 − 0.7576 = 0.2424. Therefore, chlorine consists of 75.76% ³⁵Cl and 24.24% ³⁷Cl.

Exercise \(\PageIndex{2}\)

Naturally occurring copper consists of ⁶³Cu (mass 62.9296 amu) and ⁶⁵Cu (mass 64.9278 amu), with an average mass of 63.546 amu. What is the percent composition of Cu in terms of these two isotopes?

Answer

69.15% Cu-63 and 30.85% Cu-65

 

 

Summary

Video \(\PageIndex{6}\): Watch this video for a review of relative atomic mass and isotopes.

An atom consists of a small, positively charged nucleus surrounded by electrons. The nucleus contains protons and neutrons; its diameter is about 100,000 times smaller than that of the atom. The mass of one atom is usually expressed in atomic mass units (amu), which is referred to as the atomic mass. An amu is defined as exactly \(1/12\) of the mass of a carbon-12 atom and is equal to 1.6605 \(\times\) 10⁻²⁴ g.

Protons are relatively heavy particles with a charge of 1+ and a mass of 1.0073 amu. Neutrons are relatively heavy particles with no charge and a mass of 1.0087 amu. Electrons are light particles with a charge of 1− and a mass of 0.00055 amu. The number of protons in the nucleus is called the atomic number (Z) and is the property that defines an atom’s elemental identity. The sum of the numbers of protons and neutrons in the nucleus is called the mass number and, expressed in amu, is approximately equal to the mass of the atom. An atom is neutral when it contains equal numbers of electrons and protons.

Isotopes of an element are atoms with the same atomic number but different mass numbers; isotopes of an element, therefore, differ from each other only in the number of neutrons within the nucleus. When a naturally occurring element is composed of several isotopes, the atomic mass of the element represents the average of the masses of the isotopes involved. A chemical symbol identifies the atoms in a substance using symbols, which are one-, two-, or three-letter abbreviations for the atoms.

Looking Beyond

Video \(\PageIndex{7}\): Remember our exploration into the size of an atom last week? This video goes deeper into investigating the size of the subatomic particles we just discussed.

Footnotes

1. Read more about the redefinition of SI units including the kilogram here (Laura Howe, CE&N, Nov. 16, 2018).

Key Equations

• \(\mathrm{average\: mass}=\sum_{i}(\mathrm{fractional\: abundance \times isotopic\: mass})_i\)

Glossary

anion

negatively charged atom or molecule (contains more electrons than protons)

atomic mass

average mass of atoms of an element, expressed in amu

atomic mass unit (amu)

(also, unified atomic mass unit, u, or Dalton, Da) unit of mass equal to \(\dfrac{1}{12}\) of the mass of a ¹²C atom

atomic number (Z)

number of protons in the nucleus of an atom

cation

positively charged atom or molecule (contains fewer electrons than protons)

chemical symbol

one-, two-, or three-letter abbreviation used to represent an element or its atoms

Dalton (Da)

alternative unit equivalent to the atomic mass unit

fundamental unit of charge

(also called the elementary charge) equals the magnitude of the charge of an electron (e) with e = 1.602 \(\times\) 10⁻¹⁹ C

ion

electrically charged atom or molecule (contains unequal numbers of protons and electrons)

mass number (A)

sum of the numbers of neutrons and protons in the nucleus of an atom

mole

amount of substance containing the same number of atoms, molecules, ions, or other entities as the number of atoms in exactly 12 grams of ¹²C

molar mass

mass in grams of 1 mole of a substance

unified atomic mass unit (u)

alternative unit equivalent to the atomic mass unit

Contributors

• Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110).

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• Adelaide Clark, Oregon Institute of Technology
• Fuse School, Open Educational Resource free of charge, under a Creative Commons License: Attribution-NonCommercial CC BY-NC (View License Deed: https://creativecommons.org/licenses/by-nc/4.0/)
• Crash Course Chemistry, Crash Course is a division of Complexly and videos are free to stream for educational purposes.
• TED-Ed’s commitment to creating lessons worth sharing is an extension of TED’s mission of spreading great ideas. Within TED-Ed’s growing library of TED-Ed animations, you will find carefully curated educational videos, many of which represent collaborations between talented educators and animators nominated through the TED-Ed website.