Isotopes, Atomic Mass, and Mass Spectra
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You should be comfortable with the basic subatomic particles (protons, neutrons, electrons) and understand that the atomic number defines the element while the mass number defines a specific nucleus.
Why this matters
The periodic table lists average masses, not the masses of individual atoms, because nature rarely produces a single isotope in isolation.
When you look up the atomic mass of chlorine, you see 35.45 u, yet no single chlorine atom has this mass. This discrepancy arises because natural chlorine is a mixture of two distinct isotopes: chlorine-35 and chlorine-37. Understanding isotopes and how we measure their relative abundances is fundamental to interpreting the periodic table, calculating molar masses for stoichiometry, and utilising techniques like mass spectrometry in analytical chemistry. This lecture bridges the gap between the discrete nuclear world and the continuous values found in standard reference data.
The Isotope Concept in Plain Language
Elements are defined by their proton count. If an atom has 6 protons, it is carbon, regardless of how many neutrons it possesses. However, the number of neutrons can vary. These different versions of the same element are called isotopes. Think of isotopes as siblings: they share the same identity (protons) and therefore have identical chemical behaviour, but they differ in their physical weight (neutrons). For example, carbon-12 has 6 neutrons, while carbon-14 has 8. Because electrons determine chemical bonding, isotopes react almost identically in chemical reactions. The difference is purely nuclear and mass-based. This distinction is crucial because while chemistry cares about electron configuration, physics and nuclear chemistry care about the nucleus. The atomic mass listed in the periodic table is a weighted average of all naturally occurring isotopes, reflecting the proportion of each isotope found in nature.
Mass Number vs. Atomic Mass
We must distinguish between the mass number (\(A\)) and the atomic mass. The mass number is an integer representing the total count of protons and neutrons in a specific isotope. It is always a whole number. In contrast, the atomic mass is the actual mass of a single atom of that isotope, measured in unified atomic mass units (\(\mathrm{u}\)). Because of the binding energy that holds the nucleus together, the actual mass is slightly less than the sum of its individual protons and neutrons (mass defect). Therefore, the atomic mass of carbon-12 is defined as exactly \(12\ \mathrm{u}\), but the atomic mass of carbon-13 is approximately \(13.003\ \mathrm{u}\), not exactly 13. When we speak of the "atomic mass" of an element in the periodic table, we refer to the weighted average of the atomic masses of its isotopes, weighted by their natural abundances.
Calculating Average Atomic Mass
To determine the average atomic mass of an element, we use the relative abundance of each isotope. The formula is a weighted sum. If an element has isotopes with masses \(m_1, m_2, \dots\) and fractional abundances \(f_1, f_2, \dots\) (where the sum of fractions equals 1), the average atomic mass \(M_{\mathrm{avg}}\) is calculated as:
For instance, if an element has two isotopes, one with mass \(10\ \mathrm{u}\) and abundance \(20\%\) (\(0.20\)), and another with mass \(11\ \mathrm{u}\) and abundance \(80\%\) (\(0.80\)), the average mass is \((0.20 \times 10) + (0.80 \times 11) = 2.0 + 8.8 = 10.8\ \mathrm{u}\). This value would appear in the periodic table. It is critical to use fractional abundances (decimals) rather than percentages in the calculation to ensure the units remain in \(\mathrm{u}\).
Mass Spectrometry Basics
How do we know the masses and abundances of isotopes? We use mass spectrometry. In a mass spectrometer, atoms are ionised to form cations. These ions are accelerated by an electric field and then deflected by a magnetic field. The degree of deflection depends on the mass-to-charge ratio (\(m/z\)) of the ion. Heavier ions deflect less than lighter ions. The instrument detects the ions and records a spectrum, which is a plot of signal intensity (abundance) versus mass-to-charge ratio. Each peak corresponds to a specific isotope. The height of the peak is proportional to the relative abundance of that isotope. By analysing the positions and heights of these peaks, chemists can determine the isotopic composition of an unknown sample. The technique consumes the sample, but it needs only tiny amounts and is highly precise, making it a cornerstone of modern analytical chemistry.
A common misconception is that the atomic mass in the periodic table represents the mass of the most abundant isotope. This is incorrect. The value is a weighted average. For example, chlorine's average mass is \(35.45\ \mathrm{u}\), but the most abundant isotope is chlorine-35 (\(\approx 75\%\)) and the less abundant is chlorine-37 (\(\approx 25\%\)). If you simply round the average mass to the nearest whole number, you get 35, which happens to match the most abundant isotope here, but this is not a general rule. For copper, the average mass is \(63.55\ \mathrm{u}\), which rounds to 64, but the most abundant isotope is copper-63. Always calculate the weighted average using the specific isotopic masses and abundances provided in a problem; do not assume the rounded integer is the dominant isotope. C
Worked example
Problem. Boron has two naturally occurring isotopes: boron-10 and boron-11. The atomic mass of boron-10 is \(10.013\ \mathrm{u}\) and its natural abundance is \(19.9\%\). The atomic mass of boron-11 is \(11.009\ \mathrm{u}\). Calculate the average atomic mass of boron and determine the abundance of boron-11.
- Convert the abundance of boron-10 to a decimal fraction: \(19.9\% = 0.199\).
- Determine the abundance of boron-11. Since the total abundance must be \(100\%\), the abundance of boron-11 is \(100\% - 19.9\% = 80.1\%\). As a decimal, this is \(0.801\).
- Apply the weighted average formula: \(M_{\mathrm{avg}} = (f_{10} \times m_{10}) + (f_{11} \times m_{11})\).
- Substitute the values: \(M_{\mathrm{avg}} = (0.199 \times 10.013) + (0.801 \times 11.009)\).
- Calculate the contributions: \(0.199 \times 10.013 = 1.992587\) and \(0.801 \times 11.009 = 8.818209\).
- Sum the contributions: \(1.992587 + 8.818209 = 10.810796\ \mathrm{u}\).
- Round to an appropriate number of significant figures (three decimal places is standard for atomic masses): \(10.811\ \mathrm{u}\).
Units check. The calculation multiplies dimensionless fractions by masses in \(\mathrm{u}\), resulting in a final value in \(\mathrm{u}\), which is consistent with the definition of atomic mass.
Three quick tests
- What is the difference between mass number and atomic mass?
Answer
Mass number is the integer count of protons and neutrons in a specific isotope. Atomic mass is the actual measured mass of that isotope in \(\mathrm{u}\), which is not an integer due to mass defect. - If an element has two isotopes, one with mass \(20\ \mathrm{u}\) (25% abundance) and one with mass \(22\ \mathrm{u}\) (75% abundance), what is the average atomic mass?
Answer
\((0.25 \times 20) + (0.75 \times 22) = 5 + 16.5 = 21.5\ \mathrm{u}\). - In a mass spectrum, what does the height of a peak represent?
Answer
The relative abundance of that specific isotope in the sample.
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Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.