Mass spectrometry is more often applied to molecules than to atoms. We will see in Chapter 3 that the molecular weight of a molecule is the sum of the atomic weights of the atoms in the molecule. The mass spectrum of \(\mathrm{H}_{2}\) is taken under conditions that prevent decomposition into \(\mathrm{H}\) atoms. The two naturally occurring isotopes of hydrogen are ${ }^{1} \mathrm{H}\( (atomic mass \)=1.00783 \mathrm{u}\(; abundance \)\left.99.9885 \%\right)\( and \){ }^{2} \mathrm{H}\( (atomic mass \)=2.01410 \mathrm{u}$; abundance \(\left.0.0115 \%\right)\). (a) How many peaks will the mass spectrum have? (b) Give the relative atomic masses of each of these peaks. (c) Which peak will be the largest, and which the smallest?

To identify the number of peaks in the mass spectrum, we need to consider all possible combinations of hydrogen isotopes that may form \(\mathrm{H}_{2}\) molecules. There are two hydrogen isotopes, so we could have three possible combinations: \({ }^{1} \mathrm{H}-{ }^{1} \mathrm{H}\), \({ }^{1} \mathrm{H}-{ }^{2} \mathrm{H}\), or \({ }^{2} \mathrm{H}-{ }^{2} \mathrm{H}\). These three combinations would give rise to three diferent mass peaks in the mass spectrum.

For each peak, we can find the relative atomic mass by summing the atomic masses of the isotopes forming the \(\mathrm{H}_{2}\) molecule in each combination: 1. For \({ }^{1} \mathrm{H}-{ }^{1} \mathrm{H}\): \(1.00783 \mathrm{u} + 1.00783 \mathrm{u} = 2.01566 \mathrm{u}\) 2. For \({ }^{1} \mathrm{H}-{ }^{2} \mathrm{H}\): \(1.00783 \mathrm{u} + 2.01410 \mathrm{u} = 3.02193 \mathrm{u}\) 3. For \({ }^{2} \mathrm{H}-{ }^{2} \mathrm{H}\): \(2.01410 \mathrm{u} + 2.01410 \mathrm{u} = 4.02820 \mathrm{u}\)

In order to find the largest and smallest peaks, we must consider the abundance of each isotope combination. The abundance for a specific combination of isotopes can be calculated by multiplying the individual isotope abundances: 1. For \({ }^{1} \mathrm{H}-{ }^{1} \mathrm{H}\): abundance = \(0.999885 \times 0.999885 = 0.999770\) 2. For \({ }^{1} \mathrm{H}-{ }^{2} \mathrm{H}\): abundance = \(0.999885 \times 0.000115 = 0.000115\) 3. For \({ }^{2} \mathrm{H}-{ }^{2} \mathrm{H}\): abundance = \(0.000115 \times 0.000115 = 0.000000013\) From these abundances, we can observe that the largest peak corresponds to \({ }^{1} \mathrm{H}-{ }^{1} \mathrm{H}\) (abundance = \(0.999770\)) with a relative atomic mass of \(2.01566 \mathrm{u}\), while the smallest peak corresponds to \({ }^{2} \mathrm{H}-{ }^{2} \mathrm{H}\) (abundance = \(0.000000013\)) with a relative atomic mass of \(4.02820 \mathrm{u}\).