Question

In using a mass spectrometer, a chemist sees a peak at a mass of \(30.0106 .\) Of the choices \(^{12} \mathrm{C}_{2}\) \(^{1} \mathrm{H}_{6},\) \(^{12} \mathrm{C}\) \(^{1} \mathrm{H}_{2}\) \(^{16} \mathrm{O},\) and \(^{14} \mathrm{N}^{16} \mathrm{O},\) which is responsible for this peak? Pertinent masses are \(^{1} \mathrm{H}\), \(1.007825 ;^{16} \mathrm{O}, 15.994915 ;\) and \(^{14} \mathrm{N}, 14.003074.\)

Short answer

The molecular isotope responsible for the peak at a mass of 30.0106 is \(^{12}C^1H_2^{16}O\), since its calculated mass (30.010565) is the closest value to the observed peak mass value.

Step-by-step solution

Calculate the mass of the first molecular isotope, \(^{12}C_2$$^1H_6\)

To calculate the mass of this molecule, we first need the mass of the constituent isotopes. The molecular formula has 2 carbon-12 atoms and 6 hydrogen atoms. The mass of carbon-12 is 12 by definition and the mass of hydrogen is provided in the problem: mass of \(^1H = 1.007825\) Now, we calculate the mass of the molecule: Mass of \(^{12}C_2$$^1H_6 = 2\times(12) + 6\times(1.007825) = 24 + 6.04695 \approx 30.04695\)

Calculate the mass of the second molecular isotope, \(^{12}C$$^1H_2$$^{16}O\)

The molecular formula of this molecule contains 1 carbon-12 atom, 2 hydrogen atoms, and 1 oxygen-16 atom. The mass of oxygen-16 is provided in the problem: mass of \(^{16}O = 15.994915\) Now, we calculate the mass of the molecule: Mass of \(^{12}C$$^1H_2$$^{16}O = 12 + 2\times(1.007825) + 15.994915 = 12 + 2.01565 + 15.994915 \approx 30.010565\)

Calculate the mass of the third molecular isotope, \(^{14}N$$^{16}O\)

The molecular formula of this molecule contains 1 nitrogen-14 atom and 1 oxygen-16 atom. The mass of nitrogen-14 is provided in the problem: mass of \(^{14}N = 14.003074\) Now, we calculate the mass of the molecule: Mass of \(^{14}N$$^{16}O = 14.003074 + 15.994915 \approx 30.007989\)

Compare the calculated masses with the observed peak value

We will now compare the calculated masses of the three molecular isotopes with the observed peak value of 30.0106: 1. Mass of \(^{12}C_2$$^1H_6 \approx 30.04695\) 2. Mass of \(^{12}C$$^1H_2$$^{16}O \approx 30.010565\) 3. Mass of \(^{14}N$$^{16}O \approx 30.007989\) From the comparison, we can observe that the mass of the second molecular isotope \(^{12}C$$^1H_2$$^{16}O\) is the closest to the observed peak value of 30.0106.

Identify the molecular isotope responsible for the peak

Since the mass of \(^{12}C$$^1H_2$$^{16}O\) (30.010565) is the closest value to the observed peak mass value (30.0106), we can conclude that the molecular isotope responsible for the peak is \(^{12}C$$^1H_2$$^{16}O\).

Key concepts

Isotopic Mass Calculation

Isotopic mass calculation is a fundamental concept in chemistry, especially in the realm of mass spectrometry. It involves determining the mass of a molecule by summing the masses of the individual isotopes that compose it.

To calculate isotopic mass, we must first understand what isotopes are. Isotopes are atoms of the same element that have the same number of protons but a different number of neutrons, leading to different mass numbers. For example, the most common isotope of carbon is Carbon-12 or \( ^{12}C \), which has 6 protons and 6 neutrons.

In our exercise, the chemist is trying to find out which molecule corresponds to a specific peak in the mass spectrometry data. For each molecule option given, the isotopic mass is calculated by adding up the mass of each atom in the molecule. It's important to note that the mass of isotopes like Carbon-12 is often set by definition, while the masses of other isotopes, like Hydrogen-1 ( \( ^{1}H \)) and Oxygen-16 ( \( ^{16}O \)), are measured experimentally and provided.

Due to the nature of isotopic masses being an average weighted by abundance, the exact mass of a molecule might be a decimal value rather than a whole number. This precision is crucial when matching an experimental peak with the calculated mass of a potential molecule.

Molecular Formula Determination

Molecular formula determination is a process by which chemists deduce the actual number of atoms of each element present in a molecule. It's a step beyond the empirical formula that provides the simplest whole-number ratio of the elements.

For instance, in our problem, the exercise requires us to consider different molecular formulas, like \( ^{12}C_2^{1}H_6 \) and \( ^{12}C^{1}H_2^{16}O \) , to find which one is responsible for a given mass spectrometry peak. The accurate molecular formula is essential to determine because it gives complete information regarding the molecular structure, which is invaluable in chemical analysis and synthesis.

Determining the molecular formula requires a clear understanding of not only the compound's mass but also the relative masses of the constituent atoms. By considering the molecular weight from isotopic masses and the chemical laws – such as the Law of Multiple Proportions – chemists can ascertain the molecular formula. This will, in turn, help in identifying the substance and understanding its properties and potential reactions.

Mass Spectrometry Analysis

Mass spectrometry analysis is an analytical technique used to measure the mass-to-charge ratio of ions. It provides detailed information about the molecular weight and structure of a compound, which is crucial in identifying unknown substances and understanding their properties.

In a mass spectrometer, the substance being analyzed is ionized, and the resulting ions are then separated based on their mass-to-charge ratio. This separation process results in the creation of a spectrum that displays peaks corresponding to the various fragments of the molecule. Each peak represents an ion, with its position relative to the mass and its height indicating the relative abundance.

In the given exercise, the mass spectrometer reveals a peak with a specific mass, and our goal is to identify which molecule corresponds to this peak from the list of possibilities. By employing mass spectrometry analysis, we are able to pinpoint the exact mass and therefore determine that the molecule \( ^{12}C^{1}H_2^{16}O \) is responsible for the observed peak. This ability to correlate mass spectrometric data with molecular structures makes mass spectrometry an indispensable tool in modern chemistry for both qualitative and quantitative analysis.