Relative Atomic and Molecular Mass
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You should be comfortable interpreting chemical formulae and distinguishing between elements, compounds, and ions. Familiarity with the periodic table and the concept of atomic number is assumed.
Why this matters
We cannot count atoms directly in the laboratory, so we must weigh them.
In practical chemistry, we measure mass, not particle count. To bridge the gap between the microscopic world of atoms and the macroscopic world of scales and beakers, we use relative masses. This lecture establishes the definitions of relative atomic mass and relative molecular mass, which serve as the fundamental conversion factors for all subsequent stoichiometric calculations. Without a precise understanding of how these values are derived from isotopic abundance and standard reference scales, any calculation involving moles, concentrations, or reaction yields will be fundamentally flawed.
The Scale of Atoms
Imagine you are asked to count a specific number of grains of sand. If the pile contains ten billion grains, counting them one by one is impossible. Instead, you might weigh a single grain, determine its average mass, and then weigh a large pile to estimate the total count. Chemistry operates on the same principle, but with atoms. Atoms are incredibly small, with masses measured in kilograms that are so tiny they are impractical for daily use. For example, a single carbon atom has a mass of approximately \( 2 \times 10^{-26} \) kg. Writing and calculating with such small numbers is cumbersome and prone to error. To solve this, chemists use a relative scale. We do not measure the absolute mass of an atom in kilograms; instead, we measure how many times heavier it is compared to a standard reference. This reference is one-twelfth of the mass of a carbon-12 atom. By defining this standard, we create a dimensionless number called the relative atomic mass. This number allows us to compare the "heaviness" of different elements without dealing with exponents. It is a ratio, not a weight. When we say the relative atomic mass of oxygen is 16, we mean an oxygen atom is sixteen times heavier than one-twelfth of a carbon-12 atom. This relative scale is the foundation of the mole, which we will explore in the next lecture. For now, focus on the idea that we are comparing masses against a fixed benchmark to create a manageable numerical system for counting particles by weighing them.
Relative Atomic Mass and Isotopes
The relative atomic mass (\( A_r \)) of an element is not simply the mass of a single isotope. Most elements exist as a mixture of isotopes, which are atoms of the same element with different numbers of neutrons. For example, chlorine exists primarily as chlorine-35 and chlorine-37. The value listed on the periodic table is a weighted average of these isotopes, based on their natural abundance. To calculate the relative atomic mass, we multiply the mass of each isotope by its fractional abundance and sum the results. Consider chlorine: chlorine-35 has a mass of approximately 35 and an abundance of 75%, while chlorine-37 has a mass of approximately 37 and an abundance of 25%. The calculation is \( (35 \times 0.75) + (37 \times 0.25) = 26.25 + 9.25 = 35.5 \). This explains why the relative atomic mass of chlorine is 35.5, a non-integer value. It is crucial to understand that this value is an average. If you isolate a single atom of chlorine, its mass will be either 35 or 37 (ignoring small mass defects), never 35.5. The 35.5 value represents the average mass of a large sample of naturally occurring chlorine. This concept is vital for accurate stoichiometry because laboratory samples contain natural isotopic mixtures, not pure isotopes. When performing calculations, always use the weighted average value from the periodic table unless a specific isotope is explicitly stated.
Relative Molecular Mass
While relative atomic mass applies to single atoms, relative molecular mass (\( M_r \)) applies to molecules. It is calculated by summing the relative atomic masses of all atoms in the molecular formula. For example, consider water, \( \mathrm{H_2O} \). Hydrogen has a relative atomic mass of 1, and oxygen has a relative atomic mass of 16. The relative molecular mass of water is \( (2 \times 1) + (1 \times 16) = 18 \). This means a water molecule is 18 times heavier than one-twelfth of a carbon-12 atom. For more complex compounds, such as sulfuric acid, \( \mathrm{H_2SO_4} \), the calculation involves multiple elements. Hydrogen contributes \( 2 \times 1 = 2 \), sulfur contributes \( 1 \times 32 = 32 \), and oxygen contributes \( 4 \times 16 = 64 \). The total relative molecular mass is \( 2 + 32 + 64 = 98 \). It is important to distinguish between relative molecular mass and relative formula mass. The term "molecular" is strictly used for covalent compounds that exist as discrete molecules, like water or carbon dioxide. For ionic compounds, such as sodium chloride, \( \mathrm{NaCl} \), which form giant lattices rather than discrete molecules, we use the term relative formula mass. The calculation method is identical: sum the relative atomic masses of the ions in the formula unit. For \( \mathrm{NaCl} \), this is \( 23 + 35.5 = 58.5 \). Using the correct terminology is a matter of precision, but the mathematical process remains consistent across all chemical species.
Units and Dimensionless Nature
A common source of confusion is the unit of relative mass. Relative atomic mass and relative molecular mass are dimensionless numbers. They are ratios of masses, so the units cancel out. We do not say "18 grams" for the relative molecular mass of water; we say "18". However, when we move to the next lecture on the mole, we will introduce the molar mass, which has units of \( \mathrm{g\,mol^{-1}} \). The numerical value of the relative molecular mass is identical to the molar mass in \( \mathrm{g\,mol^{-1}} \). For example, the relative molecular mass of water is 18, and the molar mass of water is \( 18\ \mathrm{g\,mol^{-1}} \). This numerical equivalence is a convenient feature of the SI system, but it is critical not to conflate the two concepts. Relative mass is a comparative number used for ratios within a molecule or between elements. Molar mass is a physical quantity that relates the mass of a substance to the number of particles it contains. Confusing these two leads to significant errors in unit analysis. Always check whether your calculation requires a dimensionless ratio or a mass per amount of substance. In this lecture, we focus on the dimensionless relative values, which serve as the building blocks for the molar mass calculations that follow.
A frequent misconception is that the relative atomic mass of an element must be an integer because protons and neutrons have integer mass numbers. This is incorrect. The relative atomic mass is a weighted average based on natural abundance, and because isotopic abundances are rarely exact integers, the resulting average is often a decimal. For instance, the relative atomic mass of boron is 10.8, reflecting a mixture of boron-10 and boron-11. Assuming integer values will lead to incorrect stoichiometric ratios and yield predictions. Always use the precise values provided in the periodic table for calculations. C
Worked example
Problem. Calculate the relative molecular mass of copper(II) sulfate pentahydrate, \( \mathrm{CuSO_4 \cdot 5H_2O} \). Use the following relative atomic masses: \( \mathrm{Cu} = 63.5 \), \( \mathrm{S} = 32.1 \), \( \mathrm{O} = 16.0 \), \( \mathrm{H} = 1.0 \).
- Identify the number of atoms of each element in the formula. There is 1 copper atom, 1 sulfur atom, 4 oxygen atoms in the sulfate ion, 5 water molecules (each containing 2 hydrogen and 1 oxygen), so total hydrogen is \( 5 \times 2 = 10 \) and total oxygen is \( 4 + 5 = 9 \).
- Calculate the mass contribution of copper: \( 1 \times 63.5 = 63.5 \).
- Calculate the mass contribution of sulfur: \( 1 \times 32.1 = 32.1 \).
- Calculate the mass contribution of oxygen: \( 9 \times 16.0 = 144.0 \).
- Calculate the mass contribution of hydrogen: \( 10 \times 1.0 = 10.0 \).
- Sum all contributions: \( 63.5 + 32.1 + 144.0 + 10.0 = 249.6 \).
Units check. The result is dimensionless, as expected for a relative mass.
Three quick tests
- What is the relative molecular mass of carbon dioxide, \( \mathrm{CO_2} \), given \( \mathrm{C} = 12.0 \) and \( \mathrm{O} = 16.0 \)?
Answer
44.0 - Why is the relative atomic mass of chlorine 35.5 rather than 35 or 37?
Answer
It is a weighted average of chlorine-35 and chlorine-37 based on their natural abundances. - Is the relative molecular mass of sodium chloride, \( \mathrm{NaCl} \), a molecular mass or a formula mass?
Answer
It is a relative formula mass, as sodium chloride is an ionic compound.
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