Problem 40 - Chemistry The Central Science In Si Units

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 \(=1.00783 \mathrm{u}\); abundance \(\left.99.9885 \%\right)\) and \({ }^{2} \mathrm{H}(\) atomic mass \(=2.01410 \mathrm{u} ;\) 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?

Step-by-Step Solution

Answer

The mass spectrum will have 3 peaks: at 2.01566 u, 3.02193 u, and 4.02820 u. The largest peak is at 2.01566 u (\(^{1}\mathrm{H}-^{1}\mathrm{H}\)), and the smallest is at 4.02820 u (\(^{2}\mathrm{H}-^{2}\mathrm{H}\)).

Step 1: Identify Possible Combinations

First, we need to determine the possible combinations of isotopes in the hydrogen molecule \(\mathrm{H}_2\). Since \(\mathrm{H}_2\) is made of two hydrogen atoms, the possible combinations using the isotopes \(^{1}\mathrm{H}\) and \(^{2}\mathrm{H}\) are:1. \(^{1}\mathrm{H} - ^{1}\mathrm{H}\)2. \(^{1}\mathrm{H} - ^{2}\mathrm{H}\)3. \(^{2}\mathrm{H} - ^{2}\mathrm{H}\).This analysis shows that there will be 3 different isotopic combinations, and therefore, 3 peaks in the mass spectrum.

Step 2: Calculate the Molecular Masses

Next, calculate the molecular mass for each of these combinations:1. For \(^{1}\mathrm{H} - ^{1}\mathrm{H}\): \[ \text{Mass} = (1.00783 + 1.00783) \text{ u} = 2.01566 \text{ u}. \]2. For \(^{1}\mathrm{H} - ^{2}\mathrm{H}\): \[ \text{Mass} = (1.00783 + 2.01410) \text{ u} = 3.02193 \text{ u}. \]3. For \(^{2}\mathrm{H} - ^{2}\mathrm{H}\): \[ \text{Mass} = (2.01410 + 2.01410) \text{ u} = 4.02820 \text{ u}. \]These are the atomic masses relative to the molecular composition.

Step 3: Determine the Relative Abundance

Calculate the relative abundance of each combination:1. For \(^{1}\mathrm{H} - ^{1}\mathrm{H}\): \[ \text{Abundance} = (0.999885 \times 0.999885) = 99.977\%. \]2. For \(^{1}\mathrm{H} - ^{2}\mathrm{H}\): \[ \text{Abundance} = (2 \times 0.999885 \times 0.000115) = 0.023\%. \]3. For \(^{2}\mathrm{H} - ^{2}\mathrm{H}\): \[ \text{Abundance} = (0.000115 \times 0.000115) = 0.000013\%. \]The factor of 2 in the second case comes from the fact that \(^{1}\mathrm{H} - ^{2}\mathrm{H}\) and \(^{2}\mathrm{H} - ^{1}\mathrm{H}\) are identical and hence add up their probabilities.

Step 4: Analyze the Peaks

Based on the relative abundances:- The peak corresponding to \(^{1}\mathrm{H} - ^{1}\mathrm{H}\) (2.01566 u) will be the largest at 99.977%.- The peak corresponding to \(^{1}\mathrm{H} - ^{2}\mathrm{H}\) (3.02193 u) will be smaller at 0.023%.- The peak corresponding to \(^{2}\mathrm{H} - ^{2}\mathrm{H}\) (4.02820 u) will be the smallest at 0.000013%.

Key Concepts

Isotopes

In the study of chemistry, isotopes are variants of a particular chemical element that share the same number of protons but differ in neutron count. This difference in the number of neutrons gives isotopes their unique masses. For instance, hydrogen has two naturally occurring isotopes, often denoted as \(^1\mathrm{H}\) and \(^2\mathrm{H}\). The first isotope, \(^1\mathrm{H}\), has an atomic mass of 1.00783 u, while the second, often called deuterium \(^2\mathrm{H}\), has an atomic mass of 2.01410 u. Although the name and chemical behavior of the element remain the same regardless of the isotope, the mass can significantly affect physical properties and reactions.
This uniqueness in mass is critical for techniques like mass spectrometry, where isotopic variation leads to different mass spectrum peaks. Essentially, isotopes allow us to explore the finer details of molecular makeup and unravel complex chemical identities.

Molecular Weight

Molecular weight, also referred to as molecular mass, is the sum of the atomic weights of all the atoms in a molecule. In mass spectrometry, the molecular weight is an essential concept, as it determines the position of the peaks on the mass spectrum. For a molecule like \( \mathrm{H}_2 \), its molecular weight is calculated by combining the atomic masses of its constituent atoms.
For example, depending on the isotopic composition, the molecular weight of \( \mathrm{H}_2 \) could differ. With two \(^1\mathrm{H}\) atoms, the molecular weight would be \[ 2.01566 \text{ u} \]. For a molecule consisting of one \(^1\mathrm{H}\) and one \(^2\mathrm{H}\), the molecular weight becomes \[ 3.02193 \text{ u} \]. Finally, with two \(^2\mathrm{H}\) atoms, the molecular weight is calculated to be \[ 4.02820 \text{ u} \].
Understanding molecular weight helps in predicting the mass spectrum peaks and deducing molecular structures.

Relative Abundance

Relative abundance is a concept that describes how common an isotope is in a natural sample of an element. It is expressed as a percentage and indicates the proportion of each isotope compared to the total. This metric is crucial in mass spectrometry as it influences the intensity of the mass spectrum peaks.
Taking our example of hydrogen isotopes: \(^1\mathrm{H}\) occurs in nature with a relative abundance of 99.9885%, while \(^2\mathrm{H}\) has an abundance of 0.0115%. When forming \( \mathrm{H}_2 \) molecules, these abundances help determine the likelihood of each isotopic combination. Consequently, the most prominent peak in the mass spectrum is from \(^1\mathrm{H} - ^1\mathrm{H}\), with a high abundance of 99.977%.
Understanding relative abundance enables scientists to interpret spectral data accurately and is fundamental for analyzing unknown compounds.

Mass Spectrum Peaks

Mass spectrum peaks are the graphical representation of molecules detected in mass spectrometry. Each peak on a mass spectrum corresponds to an ion of a particular mass-to-charge ratio (\( m/z \)). The position and height of these peaks provide insights into the composition and abundance of isotopes within the sample.
In the context of \( \mathrm{H}_2 \), mass spectrum peaks resulted from distinct isotopic compositions of hydrogen. For instance, use of \(^1\mathrm{H} - ^1\mathrm{H}\) leads to a peak at 2.01566 u, which is also the most intense due to its high relative abundance. The \(^1\mathrm{H} - ^2\mathrm{H}\) combination gives a peak at 3.02193 u and is less intense, while \(^2\mathrm{H} - ^2\mathrm{H}\) manifests as a faint peak at 4.02820 u due to its rarity.
Interpreting these peaks allows scientists to determine molecular structures and quantify isotopic distributions within a sample, making it an invaluable tool in analytical chemistry.