Question

Dinitrogen monoxide \(\left(\mathrm{N}_{2} \mathrm{O} ;\right.\) nitrous oxide \()\) is a greenhouse gas that enters the atmosphere principally from natural fertilizer breakdown. Some studies have shown that the isotope ratios of \({ }^{15} \mathrm{~N}\) to \({ }^{14} \mathrm{~N}\) and of \({ }^{18} \mathrm{O}\) to \({ }^{16} \mathrm{O}\) in \(\mathrm{N}_{2} \mathrm{O}\) depend on the source, which can thus be determined by measuring the relative abundances of molecular masses in a sample of \(\mathrm{N}_{2} \mathrm{O}\). (a) What different molecular masses are possible for \(\mathrm{N}_{2} \mathrm{O}\) ? (b) The percent abundance of \({ }^{14} \mathrm{~N}\) is \(99.6 \%,\) and that of \({ }^{16} \mathrm{O}\) is 99.8\%. Which molecular mass of \(\mathrm{N}_{2} \mathrm{O}\) is least common, and which is most common?

Short answer

The least common molecular mass is 48 u and the most common is 44 u.

Step-by-step solution

Identify Isotopes and their Mass Numbers

Identify the isotopes involved: \ For nitrogen: \ - \ \(^{14}N\) with a mass of 14 \ - \ \(^{15}N\) with a mass of 15 \ \ For oxygen: \ - \ \(^{16}O\) with a mass of 16 \ - \ \(^{18}O\) with a mass of 18

Calculate Possible Molecular Masses of N\(_2\)O

Calculate the different molecular masses: \ \ \(14N - 14N - 16O = 44\) \ \(14N - 14N - 18O = 46\) \ \(14N - 15N - 16O = 45\) \ \(14N - 15N - 18O = 47\) \ \(15N - 15N - 16O = 46\) \ \(15N - 15N - 18O = 48\)

Determine the Abundance of Each Isotope

Given the percent abundances: \ \(^{14}N\) is 99.6% \ \(^{15}N\) is 0.4% \ \ \(^{16}O\) is 99.8% \ \(^{18}O\) is 0.2%

Calculate the Relative Abundance of each Molecular Mass

Calculate the abundance of each combination using the given abundances: \ For the most common mass: \ \(14N - 14N - 16O = (0.996) * (0.996) * (0.998) \ \approx 0.990\) \ For the least common mass: \ \(15N-15N-18O=(0.004)*(0.004)*(0.002) \ \approx 3.2*10^{-8}\)

Key concepts

molecular masses

Understanding molecular masses is crucial for isotopic analysis. Nitrous oxide, \(\text{N}_2\text{O}\), is composed of two nitrogen atoms and one oxygen atom. To determine the different possible molecular masses, we need to consider the isotopes of nitrogen and oxygen. Nitrogen has two stable isotopes, \(^{14}N\) and \(^{15}N\). Oxygen has two stable isotopes as well, \(^{16}O\) and \(^{18}O\). By combining these isotopes, we calculate the molecular masses of nitrous oxide as follows:
- \(^{14}N \text{-} ^{14}N \text{-} ^{16}O = 44\)
- \(^{14}N \text{-} ^{14}N \text{-} ^{18}O = 46\)
- \(^{14}N \text{-} ^{15}N \text{-} ^{16}O = 45\)
- \(^{14}N \text{-} ^{15}N \text{-} ^{18}O = 47\)
- \(^{15}N \text{-} ^{15}N \text{-} ^{16}O = 46\)
- \(^{15}N \text{-} ^{15}N \text{-} ^{18}O = 48\).
By knowing these molecular masses, scientists can measure and analyze the isotopic composition of nitrous oxide to derive meaningful insights.

isotope ratios

Isotope ratios are essential in determining the sources and history of a compound like nitrous oxide \(N_2O\). The ratio of \(^{15}N\) to \(^{14}N\) and \(^{18}O\) to \(^{16}O\) can reveal various environmental and biological processes. In isotopic analysis, the ratio of heavier to lighter isotopes in a sample is compared. This is often expressed in terms like \(\frac{^{15}N}{^{14}N}\) or \(\frac{^{18}O}{^{16}O}\). Variations in these ratios can indicate different sources or pathways of formation. For instance, a higher ratio of \(^{15}N\) could indicate a biological source, while lower ratios might suggest different origins. These measurements rely on mass spectrometry, which can precisely detect small differences in atomic masses. Understanding isotope ratios helps in tracing the origins of gases in climate studies and pollution science.

relative abundances

Relative abundances refer to the proportions of different isotopes within a sample. For nitrogen atoms in nitrous oxide, the relative abundances are:
- \(^{14}N = 99.6\text{\text{\text{%} \(\text{\)}}\).
- \(^{15}N = 0.4\text{\text{\text{%} \(\text{\)}}\).
And for oxygen atoms:
- \(^{16}O = 99.8\text{\text{\text{%} \(\text{\)}}\).
- \(^{18}O = 0.2\text{\text{\text{%} \(\text{\)}}\).
These percentages help calculate the likelihood of each molecular mass occurring. For example, the most common molecular mass \(^{14}N-^{14}N-^{16}O\) has a high probability as calculated by:
\((0.996) \times (0.996) \times (0.998) \approx 0.990\), meaning it's about 99% abundant. In contrast, the least common mass \(^{15}N-^{15}N-^{18}O\) calculates as
\((0.004) \times (0.004) \times (0.002) \approx 3.2 \times 10^{-8}\), indicating a very low abundance. Knowing relative abundances is pivotal in isotopic analysis as it guides us on which isotopes are dominant and which are rare.

nitrous oxide

Nitrous oxide \(N_2O\) is an important greenhouse gas that has significant environmental impacts. It is commonly known as 'laughing gas' and is used as an anesthetic. However, its environmental role is much more critical. \(N_2O\) enters the atmosphere from various sources, especially from soils due to the breakdown of fertilizers in agricultural practices. This gas is a key focus in climate studies because of its potent greenhouse effect, contributing significantly to global warming. Analyzing its isotopic composition helps in identifying the specific sources and processes responsible for its emissions. By understanding the molecular masses, isotope ratios, and relative abundances of \(N_2O\), scientists can better track and manage its impact on the environment.