Some Basic Concepts of Chemistry · Lesson 6 of 11

Atomic, Molecular, and Formula Masses

“Build particle masses from a common reference and interpret isotope-weighted averages.”

Learning Objectives

• Explain the carbon-12 reference and the unified atomic mass unit. • Calculate average atomic mass from isotopic abundances. • Find molecular mass by counting each kind of atom. • Distinguish molecular mass from formula mass for extended ionic substances.

A practical scale for very small masses

An atom is far too light to describe conveniently with ordinary gram values alone. A common small mass unit lets us compare atoms and build the masses of molecules. The reference is one atom of the carbon isotope carbon-12, assigned exactly 12 unified atomic mass units.

Definition

Unified atomic mass unit

The unit u is one-twelfth of the mass of one carbon-12 atom in its specified reference state. The older abbreviation amu refers to the atomic mass unit.

Carbon-12 mass referenceLaTeX

1 u = \frac{m ( ^{12} C atom )}{12} \approx 1.66054 \times 1 0^{- 24} g

m (^{12} C atom)

denotes the mass of one carbon-12 atom. The gram equivalent is approximate; carbon-12 has mass exactly 12 u by this definition.

Early relative mass scales used hydrogen as a reference because it is the lightest element. The unified carbon-12 scale makes comparisons consistent across elements. Modern mass spectrometry allows accurate determination of isotope masses; the atomic mass written in u is not the same quantity as the mass of a laboratory sample in grams.

Example — Converting a single-atom mass

Problem
An atom has mass approximately $1.6736 \times 1 0^{- 24} g$ . Express it in u.

1.Use $1 u \approx 1.66054 \times 1 0^{- 24} g$ as a conversion factor.
2.The mass in u is $(1.6736 \times 1 0^{- 24}) / (1.66054 \times 1 0^{- 24}) \approx 1.0079 u$ .
3.The ratio is close to one, consistent with a hydrogen atom being only slightly heavier than 1 u. The gram and u descriptions refer to the same atom.

Why many listed atomic masses are not whole numbers

An element can occur as isotopes: atoms of the same element with different masses. A natural sample may contain more than one isotope. Its representative atomic mass is a weighted average, so the most abundant isotope has the largest influence on the result.

Definition

Isotopes

Atoms of the same element that have different mass numbers because they differ in neutron number. Their isotope masses need not be exact whole numbers in u.

Abundance-weighted atomic massLaTeX

\overline{m} = i \sum f_{i} m_{i} with i \sum f_{i} = 1

m_{i}

is an isotope mass and

f_{i}

its fractional abundance. Divide percentage abundance by 100 before using it.

Carbon isotope	Illustrative abundance in the source	Isotope mass
$^{12} C$	98.892%	$12 u$
$^{13} C$	1.108%	$13.00335 u$
$^{14} C$	$2 \times 1 0^{- 10} %$	$14.00317 u$

Example — Carbon’s weighted average

Problem
Use the carbon isotope data to explain a value near 12.011 u.

1.The main fractional abundances are $0.98892$ and $0.01108$ . The carbon-14 fraction is about $2 \times 1 0^{- 12}$ and has negligible effect at the displayed precision.
2. $\overline{m} \approx (0.98892) (12) + (0.01108) (13.00335) = 12.011117 \dots u$ .
3.Report approximately $12.011 u$ . The mean is near 12 because carbon-12 dominates; it is not the unweighted midpoint of 12 and 13.

Example — A different isotope mixture

Problem
Chlorine has illustrative isotope masses $34.9689 u$ and $36.9659 u$ with abundances $75.77%$ and $24.23%$ . Find its average atomic mass.

1.Convert percentages to fractions: $0.7577$ and $0.2423$ . They sum to 1.
2. $\overline{m} = (0.7577) (34.9689) + (0.2423) (36.9659) = 35.4527731 u$ .
3.An appropriate rounded value is $35.45 u$ . The result is nearer the lighter isotope mass because that isotope is more abundant.

A mean is not a new kind of atom

A sample with average atomic mass 35.45 u does not require an individual atom of mass exactly 35.45 u. The average describes the isotope mixture. Use the atomic-mass values specified for a calculation consistently, since rounded tables can give slightly different final digits.

Adding masses within a molecule

A molecular formula states the number of atoms of each element in one molecule. To find molecular mass, multiply each atomic mass by its atom count and add the contributions. A missing subscript means one atom, and parentheses in a formula multiply the enclosed group when a subscript follows them.

Definition

Molecular mass

The sum of the atomic masses of all atoms in a molecule, expressed in u.

Example — Methane and water

Problem
Find the molecular masses of $C H_{4}$ and $H_{2} O$ using C = 12.011 u, H = 1.008 u, and O = 16.00 u.

1.Methane has one carbon and four hydrogens: $12.011 + 4 (1.008) = 16.043 u$ .
2.Water has two hydrogens and one oxygen: $2 (1.008) + 16.00 = 18.016 u$ , commonly reported as $18.02 u$ with these inputs.
3.The subscripts determine how many times each atomic mass contributes. Neither formula means a mixture of separate elemental samples.

Example — Glucose

Problem
Calculate the molecular mass of $C_{6} H_{12} O_{6}$ with C = 12.011 u, H = 1.008 u, and O = 16.00 u.

1.Carbon contributes $6 (12.011) = 72.066 u$ . Hydrogen contributes $12 (1.008) = 12.096 u$ .
2.Oxygen contributes $6 (16.00) = 96.00 u$ . Add the unrounded contributions: $72.066 + 12.096 + 96.00 = 180.162 u$ .
3.The result is about $180.16 u$ at a precision consistent with the supplied atomic masses. Retaining 180.162 in working avoids early rounding.

Formula mass when there are no separate molecules

Solid sodium chloride is an extended arrangement of sodium and chloride ions. It does not consist of independent NaCl molecules sitting next to one another. The formula NaCl expresses the one-to-one ratio of the ions in the structure.

An ionic formula describes a repeating ratio— This is a two-dimensional slice, not a full coordination model. In the three-dimensional rock-salt structure, each ion has six nearest neighbours of the opposite charge.

Definition

Formula mass

The sum of the atomic masses represented by one formula unit. It is useful for ionic substances and other substances not described as discrete molecules.

Example — Sodium chloride formula mass

Problem
Calculate the formula mass of NaCl using Na = 23.0 u and Cl = 35.5 u.

1.One formula unit represents one sodium ion for each chloride ion.
2.Add one contribution of each element: $23.0 + 35.5 = 58.5 u$ .
3.Call this formula mass, rather than the mass of a discrete NaCl molecule in the solid. The calculation follows the formula ratio.

Quick check

One unified atomic mass unit is defined using:

Quick check

Which factor determines how much an isotope contributes to average atomic mass?

Quick check

A mean atomic mass lies closer to the lighter of two isotope masses. What is usually implied?

Quick check

Using C = 12 and O = 16, the molecular mass of $C O_{2}$ is:

Quick check

Why is formula mass preferred for solid sodium chloride?

Practice Problems

Find the mass in grams of one $^{12} C$ atom using the approximate gram equivalent of u.
Explain why a two-isotope average must lie between the two isotope masses.
Calculate a weighted mean for isotope masses 10.0 u and 11.0 u with abundances 20.0% and 80.0%.
Calculate molecular masses of $C O_{2}$ and $N H_{3}$ using C = 12.01, O = 16.00, N = 14.01, H = 1.008.
Calculate the formula mass of $N a_{2} S O_{4}$ using Na = 23.0, S = 32.0, O = 16.0.
Explain why 52 u of helium is a much smaller sample than 52 g of helium.
Draw a small alternating-ion pattern and explain what its simplest formula represents.

Key Takeaways

The unified mass unit uses one-twelfth of a carbon-12 atom mass. Average atomic mass is weighted by isotope abundance, not an ordinary unweighted mean. Molecular mass adds the masses of all atoms in one molecule. Formula mass applies the formula ratio without implying a discrete molecule. Particle masses in u and bulk masses in grams must be distinguished.

Previous · Lesson 5

Laws of Chemical Combination and Atomic Theory

Next · Lesson 7

The Mole and Molar Mass

A practical scale for very small masses

Why many listed atomic masses are not whole numbers

Adding masses within a molecule

Formula mass when there are no separate molecules

Quiz

Practice Problems

Key Takeaways