Topic 2 · page 6 of 15

Predicting spectra of diatomic molecules

A molecule made of two atoms of an element with two isotopes gives three molecular ion peaks. Their heights come from the chance of each pairing, which for chlorine gives the ratio 9 to 6 to 1.

12 min read2.7

Read first: Relative atomic mass from mass spectra, Molecules and 2+ ions

Why chlorine gives three molecular ion peaks 2.7

Chlorine atoms come in two kinds, ³⁵Cl and ³⁷Cl, in a ratio of about 3 : 1. A molecule of chlorine, Cl₂, contains two atoms, and each atom can be either kind. So there are three different molecules:

Molecule Relative mass m/z of Cl₂⁺
³⁵Cl–³⁵Cl 70 70
³⁵Cl–³⁷Cl 72 72
³⁷Cl–³⁷Cl 74 74

Which molecule is more common? Treat the atoms as joining at random. The chance of an atom being ³⁵Cl is 0.75 and the chance of it being ³⁷Cl is 0.25. For two atoms you multiply the chances.

³⁵Cl (0.75) ³⁷Cl (0.25)
³⁵Cl (0.75) 70: 0.75 × 0.75 = 0.5625 72: 0.75 × 0.25 = 0.1875
³⁷Cl (0.25) 72: 0.25 × 0.75 = 0.1875 74: 0.25 × 0.25 = 0.0625

The mixed molecule can form in two ways (³⁵Cl first or ³⁷Cl first), so its total chance is 0.1875 + 0.1875 = 0.375. The totals are:

m/z Chance Ratio (divide by 0.0625)
70 0.5625 9
72 0.375 6
74 0.0625 1

The chances add up to 1, as they must.

The electron beam also breaks some molecules into atoms. These form Cl⁺ ions, which appear at m/z 35 and 37 in the ratio 3 : 1, the same as the atoms themselves. How tall those peaks are compared with the molecular ions depends on the instrument, so questions ask you to predict only the ratios.

Predicted mass spectrum of chlorine. Fragment peaks at m/z 35 and 37 in the ratio 3 to 1, and molecular ion peaks at m/z 70, 72 and 74 in the ratio 9 to 6 to 1.
Chlorine, Cl₂. The heights of the Cl⁺ peaks relative to the Cl₂⁺ peaks are illustrative; only the ratios 3 : 1 and 9 : 6 : 1 are predictions.

A method for any diatomic molecule 2.7

MethodPredicting the molecular ion peaks of X₂

  1. List the isotopes of X and the fraction of each (75% is 0.75).
  2. Draw a grid with the isotopes along both sides. Each cell is one pairing.
  3. In each cell, add the two masses (for m/z) and multiply the two fractions (for the chance).
  4. Add together cells with the same m/z.
  5. Write the heights as a ratio of whole numbers.

With two isotopes of fractions p and q (so p + q = 1), the three peaks are always in the ratio p² : 2pq : q². For chlorine, p = 0.75 and q = 0.25 give 0.5625 : 0.375 : 0.0625, which is 9 : 6 : 1.

Worked exampleBromine, Br₂

Bromine is about 50% ⁷⁹Br and 50% ⁸¹Br.

  1. Fractions: 0.5 and 0.5.
  2. ⁷⁹Br–⁷⁹Br: m/z 158, chance 0.5 × 0.5 = 0.25. ⁷⁹Br–⁸¹Br (two ways): m/z 160, chance 2 × 0.25 = 0.5. ⁸¹Br–⁸¹Br: m/z 162, chance 0.25.
  3. The peaks are at 158, 160 and 162 in the ratio 1 : 2 : 1.

Worked exampleAn element X₂ with unequal isotopes (illustrative)

Suppose ¹⁰X is 20% and ¹¹X is 80%, so p = 0.2 and q = 0.8.

  1. ¹⁰X–¹⁰X: m/z 20, chance 0.2 × 0.2 = 0.04.
  2. ¹⁰X–¹¹X: m/z 21, chance 2 × 0.2 × 0.8 = 0.32.
  3. ¹¹X–¹¹X: m/z 22, chance 0.8 × 0.8 = 0.64.
  4. Ratio 0.04 : 0.32 : 0.64 = 1 : 8 : 16.

A molecule made of two different elements is simpler. Hydrogen chloride, HCl, contains hydrogen, which is almost all ¹H, so its molecular ion peaks come from the two chlorine isotopes alone: m/z 36 (¹H³⁵Cl) and 38 (¹H³⁷Cl) in the ratio 3 : 1.

Reading a spectrum back to the isotopes 2.7

The same pattern works in reverse. If a diatomic element shows three molecular ion peaks in the ratio 9 : 6 : 1, the outer peaks are p² and q², so take square roots: √9 : √1 = 3 : 1. The two isotopes are in the ratio 3 : 1, with the lighter one more abundant (it is at the tall end).

Check yourself

  1. Predict the m/z values and the relative heights of the molecular ion peaks in the spectrum of bromine, Br₂, if ⁷⁹Br and ⁸¹Br are present in equal amounts.
  2. What peaks do you expect for HCl, if ¹H is the only hydrogen isotope present?
  3. A diatomic element Z₂ has two isotopes, ⁶³Z and ⁶⁵Z, in the ratio 7 : 3 (illustrative). Predict the m/z values and the relative heights of its molecular ion peaks.
  4. A spectrum of a diatomic element X₂ shows three molecular ion peaks in the ratio 1 : 2 : 1. What does this tell you about the isotopes of X?
  5. Explain why the peak at m/z 72 for Cl₂ is smaller than the peak at m/z 70 but much taller than the peak at m/z 74.
  6. What fraction of all Cl₂ molecules has a relative mass of 72?

Answers cover these until you have tried the questions

  1. ⁷⁹Br₂⁺ at 158, ⁷⁹Br⁸¹Br⁺ at 160, ⁸¹Br₂⁺ at 162, in the ratio 1 : 2 : 1 (0.25 : 0.5 : 0.25).
  2. m/z 36 (¹H³⁵Cl) and m/z 38 (¹H³⁷Cl), in the ratio 3 : 1, the same as the ratio of the chlorine isotopes.
  3. The fractions are 0.7 and 0.3. m/z 126 (⁶³Z–⁶³Z): 0.7² = 0.49. m/z 128 (mixed): 2 × 0.7 × 0.3 = 0.42. m/z 130 (⁶⁵Z–⁶⁵Z): 0.3² = 0.09. The peaks are at 126, 128 and 130 in the ratio 49 : 42 : 9.
  4. The outer peaks are p² and q², and they are equal, so p = q. X has two isotopes in equal amounts (50% each).
  5. The chance of two ³⁵Cl atoms is 0.75 × 0.75 = 0.5625. The chance of the mixed pair is 2 × 0.75 × 0.25 = 0.375, because it can form in two ways. The chance of two ³⁷Cl atoms is 0.25 × 0.25 = 0.0625. So 72 is lower than 70 and much higher than 74.
  6. 0.375, which is 3/8.