Lesson 1.2 · Unit 1 · CED topic 1.2
Mass spectra of elements
The periodic table says chlorine has a mass of 35.45 amu. Yet no chlorine atom weighs 35.45 amu. Every chlorine atom has a whole number of protons and neutrons, so its mass is very close to 35 or to 37, never in between. The reconciling fact is that natural chlorine is a mixture of two kinds of atoms, and 35.45 is their average. The instrument that reveals this, and measures how much of each kind there is, is the mass spectrometer.
Explain the quantitative relationship between the mass spectrum of an element and the masses of the element’s isotopes.
- The mass spectrum of a sample containing a single element can be used to determine the identity of the isotopes of that element and the relative abundance of each isotope in nature.
- The average atomic mass of an element can be estimated from the weighted average of the isotopic masses using the mass of each isotope and its relative abundance.
- Exclusion: interpreting spectra of samples with several elements, or peaks from anything other than singly charged monatomic ions, is not assessed.
Isotopes
The number of protons fixes the element; the number of neutrons can vary. Isotopes are atoms of the same element with different numbers of neutrons, and therefore different masses. Chlorine-35 has 17 protons and 18 neutrons; chlorine-37 has 17 protons and 20 neutrons. They have the same electron arrangement and so behave identically in reactions, but they differ in mass. The mass number (protons plus neutrons) is the whole number in the name. The actual isotopic mass in amu is very close to the mass number but not exactly equal (34.969 amu for chlorine-35).
How a mass spectrometer works
A sample is vaporized, and a beam of electrons knocks one electron off each atom, producing singly charged positive ions. The ions are accelerated by an electric field so that all of them gain the same energy, then pass through a magnetic field that bends their path. Heavier ions have more inertia and bend less; lighter ions bend more. Each isotope lands at its own position on a detector, and the strength of the signal is proportional to how many ions arrive there.
The result is a mass spectrum: a bar graph with mass (in amu, since the ions carry a charge of +1) on the horizontal axis and relative abundance on the vertical axis. One bar means one isotope. The position tells you the isotope’s mass; the height tells you the percentage of atoms in nature that are that isotope.
Read the two spectra above. Chlorine has two bars, at 35 and 37 amu, with the 35 bar about three times as tall: about 75.8% of chlorine atoms are chlorine-35. Magnesium has three bars, at 24, 25 and 26, and the tallest, at 24, accounts for about 79% of atoms. Because the bars show the share of atoms, the heights in a complete spectrum add up to 100%.
Average atomic mass
The mass in the periodic table is a weighted average: each isotope’s mass is multiplied by its fractional abundance, and the products are added. It is weighted because an isotope that makes up 75% of the atoms contributes three times as much as one that makes up 25%.
\[ \text{average atomic mass} = \sum (\text{isotopic mass}) \times (\text{fractional abundance}) \] Use fractions (0.7577) rather than percentages (75.77%), or divide the final sum by 100.
Chlorine-35 has a mass of 34.969 amu and an abundance of 75.77%. Chlorine-37 has a mass of 36.966 amu and an abundance of 24.23%. Find the average atomic mass.
Step 1. \(34.969 \times 0.7577 = 26.496\) amu from chlorine-35.
Step 2. \(36.966 \times 0.2423 = 8.957\) amu from chlorine-37.
Step 3. Add: \(26.496 + 8.957 = 35.453\), which rounds to 35.45 amu, the periodic table value.
Reasonableness. The average must lie between 35 and 37, and closer to 35 because that isotope is more abundant. It does.
On the AP exam you are often asked only to estimate, using mass numbers. For magnesium: \(24(0.7899) + 25(0.1000) + 26(0.1101) = 18.96 + 2.50 + 2.86 = 24.32\), so the average is a little over 24.3 amu. The accurate value using exact isotopic masses is 24.305 amu.
Copper has two isotopes, copper-63 (62.930 amu) and copper-65 (64.928 amu). The average atomic mass is 63.55 amu. What is the abundance of each?
Let \(x\) be the fraction of copper-63. Then copper-65 is \(1 - x\): \(62.930x + 64.928(1-x) = 63.55\), so \(64.928 - 1.998x = 63.55\) and \(x = \dfrac{1.378}{1.998} = 0.690\). Copper-63 is 69.0% and copper-65 is 31.0%. Check. The average (63.55) is nearer 63 than 65, so copper-63 must be the more abundant. It is.
A quick check that always works: the average atomic mass is pulled toward the isotope with the tallest bar. If the periodic-table mass is 35.45, the spectrum must show a taller peak at 35 than at 37. Also notice what the table value is not: it is not the mass of any real atom, and it is not simply the midpoint of the isotopes.
- Peak position is mass; peak height is abundance. Do not swap them.
- The average is weighted. Averaging 35 and 37 to get 36 is the classic error.
- The answer must fall between the lightest and heaviest isotope, and lean toward the more abundant one. Use this to catch arithmetic slips.
- Same element, same number of protons. Isotopes differ only in neutrons, so they have the same chemical behavior.
- Stay within the scope. The exam uses spectra of one element with singly charged ions, so the horizontal axis is the isotope mass in amu.
- Boron has two isotopes: boron-10 (10.013 amu, 19.9%) and boron-11 (11.009 amu, 80.1%). Estimate the average atomic mass using mass numbers.
Show answer
\(10(0.199) + 11(0.801) = 1.99 + 8.81 = 10.80\) amu. This is close to the accepted 10.81. The answer lies nearer 11 because boron-11 is much more abundant.
- A mass spectrum of an element shows peaks at 24 amu (height 79), 25 amu (height 10) and 26 amu (height 11). How many isotopes does the element have, and which is the most abundant?
Show answer
Three isotopes, one per peak. The most abundant is the 24 amu isotope, because its peak is the tallest (79%).
- Why is the atomic mass of chlorine on the periodic table not a whole number even though every chlorine atom has a whole number of protons and neutrons?
Show answer
Natural chlorine is a mixture of isotopes, about 76% chlorine-35 and 24% chlorine-37. The table value is the weighted average of the isotopic masses, so it falls between 35 and 37 and is not the mass of any single atom.
- An element has two isotopes with masses 79 amu and 81 amu. The average atomic mass is 79.90 amu. Estimate which isotope is more abundant and find the percentages.
Show answer
The average is nearer 79 than 81, so the 79 amu isotope is more abundant. If \(x\) is the fraction at 79: \(79x + 81(1-x) = 79.90\), so \(81 - 2x = 79.90\) and \(x = 0.55\). The 79 amu isotope is 55% and the 81 amu isotope is 45%.
- A student reports that the average atomic mass of an element with isotopes of mass 20 and 22 is 21, because 21 is the middle of the two. What information would show that this is probably wrong?
Show answer
The average is weighted by abundance, so it equals 21 only if the two isotopes are present in exactly equal amounts. The spectrum’s peak heights would show whether that is true; if one peak is taller, the average lies nearer to that isotope.
- A spectrum of a sample of element X shows two peaks of equal height at 69 amu and 71 amu. Estimate the average atomic mass and describe the percent abundances.
Show answer
Equal heights means 50% each, so the average is \(0.50(69) + 0.50(71) = 70\) amu. Each isotope makes up half of the atoms.