Question

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 \(=1.00783\) amu; abundance 99.9885\(\% )\) and 2H (atomic mass \(=2.01410\) amu; abundance 0.0115\(\% ) .\) (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?

Short answer

The mass spectrum of \(\mathrm{H}_{2}\) will have three peaks, corresponding to the combinations of hydrogen isotopes: \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\), \(^{1} \mathrm{H}\) - \(^{2} \mathrm{H}\), and \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\). The relative atomic masses of these peaks are \(2.01566 \ \mathrm{amu}\), \(3.02193 \ \mathrm{amu}\), and \(4.02820 \ \mathrm{amu}\), respectively. The largest peak will be the \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\) combination with an abundance of \(\approx 0.99977\), while the smallest peak will be the \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\) combination with an abundance of \(\approx 0.000000013\).

Step-by-step solution

Identify possible combinations of isotopes

First, let's identify all the possible combinations of these two hydrogen isotopes that can form \(\mathrm{H}_{2}\): 1. \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\) 2. \(^{1} \mathrm{H}\) - \(^{2} \mathrm{H}\) 3. \(^{2} \mathrm{H}\) - \(^{1} \mathrm{H}\) 4. \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\) However, we should note that combinations 2 and 3 are essentially the same, as they both consist of one \(^{1} \mathrm{H}\) isotope and one \(^{2} \mathrm{H}\) isotope.

Calculate the relative atomic masses for each combination

Now, let's calculate the relative atomic masses for each of these unique combinations (1, 2, and 4): 1. \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\): \(1.00783 \ \mathrm{amu} + 1.00783 \ \mathrm{amu} = 2.01566 \ \mathrm{amu}\) 2. \(^{1} \mathrm{H}\) - \(^{2} \mathrm{H}\): \(1.00783 \ \mathrm{amu} + 2.01410 \ \mathrm{amu} = 3.02193 \ \mathrm{amu}\) 3. \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\): \(2.01410 \ \mathrm{amu} + 2.01410 \ \mathrm{amu} = 4.02820 \ \mathrm{amu}\)

Calculate the abundance of each combination

To determine which peaks will be the largest and smallest, we'll calculate the relative abundance of each combination: 1. \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\): \(0.999885 \times 0.999885 \approx 0.99977\) 2. \(^{1} \mathrm{H}\) - \(^{2} \mathrm{H}\): \(0.999885 \times 0.000115\times 2 \approx 0.000230\) 3. \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\): \(0.000115 \times 0.000115 \approx 0.000000013\)

Answer the questions

a) How many peaks will the mass spectrum have? There will be three peaks (corresponding to the combinations \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\), \(^{1} \mathrm{H}\) - \(^{2} \mathrm{H}\), and \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\)). b) What are the relative atomic masses of each of these peaks? The relative atomic masses are as follows: 1. \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\): \(2.01566 \ \mathrm{amu}\) 2. \(^{1} \mathrm{H}\) - \(^{2} \mathrm{H}\): \(3.02193 \ \mathrm{amu}\) 3. \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\): \(4.02820 \ \mathrm{amu}\) c) Which peak will be the largest, and which the smallest? The largest peak will be the one with the highest abundance, which is the \(^{1} \mathrm{H}\) - \(^{1} \mathrm{H}\) combination with an abundance of \(\approx 0.99977\). The smallest peak will be the one with the lowest abundance, which is the \(^{2} \mathrm{H}\) - \(^{2} \mathrm{H}\) combination with an abundance of \(\approx 0.000000013\).

Key concepts

Atomic Weight

Atomic weight, often referred to as atomic mass, is a critical concept in chemistry that represents the average mass of atoms of an element, measured in atomic mass units (amu). This value takes into account the various isotopes of an element and their relative abundance in nature.
An isotope is an atom with the same number of protons but a different number of neutrons in its nucleus, leading to different mass numbers. For example, the most common isotope of hydrogen, denoted as ¹H, has an atomic weight of 1.00783 amu, which is a weighted mean that reflects the isotopic composition of hydrogen found on Earth.
Understanding the atomic weight is essential when examining mass spectrometry results, as it allows us to predict and interpret the mass spectrum's outcome.

Molecular Weight

Molecular weight, also known as molecular mass, is the sum of the atomic weights of all the atoms present in a molecule. It’s expressed in atomic mass units (amu) and is calculated by adding up the atomic weights of individual atoms within the molecule.
For instance, in our exercise with molecular hydrogen (
H₂), if we consider only the most abundant isotope of hydrogen (¹H), the molecular weight of H2 would simply be twice the atomic weight of hydrogen, approximately 2.01566 amu. Hence, the molecular weight provides indispensable information when trying to identify substances via mass spectrometry.

Isotopic Abundance

Isotopic abundance refers to the percentage of a particular isotope present in a naturally occurring sample of an element. It plays a pivotal role in the calculation of both atomic and molecular weights.
For example, hydrogen exists primarily as ¹H with an isotopic abundance of 99.9885%, along with a minor abundance of the isotope ²H (commonly known as deuterium) at 0.0115%. These figures, as mentioned in the exercise, indicate nearly all hydrogen atoms have a mass of 1.00783 amu, while a small fraction is heavier, with a mass of 2.01410 amu.
In mass spectrometry, isotopic abundance influences the intensity of observed peaks, since the higher the abundance, the more likely that isotope will create a detectable peak.

Mass Spectrum

A mass spectrum is the output resulting from mass spectrometry analysis. It graphically represents the various fragments of a sample by their mass-to-charge ratio (m/z) on the x-axis against their relative abundance on the y-axis.
In this case, the exercise shows us how the mass spectrum of H2 would have three distinct peaks corresponding to the possible combinations of hydrogen isotopes. This spectrum reveals not only the masses of these combinations but also their abundance in the sample. Thus, the tallest peak on the spectrum represents the most abundant species, in our exercise, the H₂ molecule composed of two ¹H atoms. In contrast, the smallest peak corresponds to the least abundant species, the molecule consisting of two ²H atoms.
Understanding mass spectra allows chemists to deduce the isotopic composition of a molecule and identify unknown substances by their molecular weights.