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A problem from Introduction to Spectroscopy by Pavia et al. [1, p. 137]:

*4. An unknown hydrocarbon has a molecular ion peak at $m/z = 84,$ with a relative intensity of $31.3.$ The $\ce{M + 1}$ peak has a relative intensity of $2.06,$ and the $\ce{M + 2}$ peak has a relative intensity of $0.08.$ What is the molecular formula for this substance?

I normalized $31.3$ to $100\%,$ so the relative intensity $I$ was calculated as follows:

$$ \begin{array}{lr} \hline \text{Peak} & I/\% \\ \hline \ce{M^+} & 100.00 \\ \ce{(M + 1)+} & 6.58 \\ \ce{(M + 2)+} & 0.25 \\ \hline \end{array} $$

The textbook is saying the molecular formula is $\ce{C6H12}.$ To me the answer doesn't make sense because if I calculate the $\ce{(M + 1)+}$ intensity using this formula from my textbook given for $\ce{C_xH_y}$:

$$[\ce{M + 1}] = 1.1x + 0.015y$$

Thus for $\ce{C6H12}$:

$$[\ce{M + 1}] = 1.1 × 6 + 0.015 × 12 = 6.78\%$$

However, $6.78\%$ is off from $6.58\%$ by about $0.2\%$. Am I missing something or is there some sort of acceptable tolerance for peak intensity differences?

Reference

  1. Pavia, D. L.; Lampman, G. M.; Kriz, G. S.; Vyvyan, J. R. Introduction to Spectroscopy, 5th ed.; Cengage Learning: Stamford, CT, 2015. ISBN 978-1-285-46012-3.
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  • $\begingroup$ Mass spectrometers can be slightly off due to calibration. $\endgroup$
    – M. Farooq
    Sep 13 '20 at 23:41
  • $\begingroup$ Here, the experimental errors are such that $6.78$ can be considered as equal to $6.58$. $\endgroup$
    – Maurice
    Sep 14 '20 at 11:22
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Such problems in chemistry don't work out to infinite precision like pure math problems. You have to apply some chemical knowledge to the problem too.

The book formula is obviously an approximation to the exact solution. To find the exact values expected you'd need to use combinatorics.

Considering the chemistry:

  • 7 carbons have a mass of 84, so a hydrocarbon molecule can't have that many carbons since there wouldn't be any hydrogen atoms.

  • 5 carbons have a mass of 60, but that would mean at most 12 hydrogen atoms for an aliphatic molecule, so the mass could be at most 72.

Thus there must be 6 carbons in the molecule.

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  • $\begingroup$ But then why did the problem add the peak intensity information if I all need to know is the mass of the molecular ion? $\endgroup$
    – Mohamed
    Sep 14 '20 at 0:02
  • $\begingroup$ @Mohamed Why not ? In real cases, you need not to use all info that is available. In study cases, extra info may be included, if it would come naturally, to train people to think what data to use. $\endgroup$
    – Poutnik
    Sep 14 '20 at 7:04

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