17.5 Determining the Number of Carbon Atoms using Mass Spectrometry

This section outlines a method used in mass spectrometry to determine the number of carbon atoms in a molecule by analyzing the relative intensities of the molecular ion peak ($M^{+}$) and the $M+1$ peak.

The Molecular Ion ($M^{+}$) and $M+1$ Peaks

In mass spectrometry, the molecular ion peak ($M^{+}$) represents the molecule with its most common isotopes (e.g., ${}^{12}C$, ${}^{1}H$, ${}^{16}O$). The $M+1$ peak is a smaller peak found at one mass unit higher than the molecular ion peak. Its existence is primarily due to the natural abundance of the heavy isotope of carbon, carbon-13 (${}^{13}C$), which is present at approximately 1.1% in nature.

The intensity (or abundance) of the $M+1$ peak relative to the $M^{+}$ peak is proportional to the number of carbon atoms in the molecule. This relationship allows us to calculate the number of carbons directly from the mass spectrum.

Formula for Calculating Carbon Atoms

The number of carbon atoms in a molecule can be calculated using the following formula, which compares the relative abundances of the $M+1$ and $M^{+}$ peaks:

$\text{Number of C atoms}=\frac{100\times (\text{Abundance of }M+1\text{ peak})}{1.1\times (\text{Abundance of }{M}^{+}\text{ peak})}$

• 1.1 in the denominator represents the natural abundance percentage of the ${}^{13}C$ isotope.
• Multiplying by 100 converts the ratio into a usable number.

Worked Example

Problem: Determine the number of carbon atoms in a molecule with a molecular ion peak ($M^{+}$) of relative abundance 27.32% and an $M+1$ peak with a relative intensity of 2.10%.

Solution:

1. Write the given values:

   • Abundance of $M^{+}$ peak = 27.32%
   • Abundance of $M+1$ peak = 2.10%

2. Apply the formula:

   $\text{Number of C atoms}=\frac{100\times (\text{Abundance of }M+1\text{ peak})}{1.1\times (\text{Abundance of }{M}^{+}\text{ peak})}$

3. Show the calculation:

   $\text{Number of C atoms}=\frac{100\times 2.10}{1.1\times 27.32}=\frac{210}{30.052}\approx 6.99$

4. Interpret the result: The calculated value is approximately 6.99, which rounds to the nearest whole number.

   Therefore, there are 7 carbon atoms in the molecule.

Possible Questions/Answers

• Q: Why is the factor 1.1 used in the denominator of the formula?

• A: The factor 1.1 corresponds to the natural abundance of the ${}^{13}C$ isotope, which is approximately 1.1%. The M+1 peak's intensity is directly related to this abundance.

• Q: What would happen to the intensity of the M+1 peak if a molecule had more carbon atoms?

• A: The relative intensity of the M+1 peak would increase. A molecule with more carbon atoms has a higher probability of containing at least one ${}^{13}C$ atom.

Summary

• The number of carbon atoms in an organic molecule can be determined from its mass spectrum.
• This method relies on comparing the relative intensities of the molecular ion peak ($M^{+}$) and the $M+1$ peak.
• The $M+1$ peak's existence is due to the natural abundance of the ${}^{13}C$ isotope (1.1%).
• The key formula is:

$\text{Number of C atoms}=\frac{100\times [M+1{]}_{\text{abundance}}}{1.1\times [M^{+}{]}_{\text{abundance}}}$

• Significance: This is a powerful and rapid technique in analytical chemistry for elucidating the structure of unknown organic compounds, especially in fields like drug discovery, forensics, and environmental analysis.