Carbon has two stable isotopes, \({ }_{6}^{12} \mathrm{C}\) and \({ }_{6}^{13} \mathrm{C},\) and fluorine has only one stable isotope, \({ }_{9}^{19} \mathrm{~F}\). How many peaks would you observe in the mass spectrum of the positive ion of \(\mathrm{CF}_{4}^{+} ?\) Assume that the ion does not break up into smaller fragments.

Understanding the role of stable isotopes is crucial when studying chemical elements, as it influences the interpretation of mass spectrometric data. Isotopes are variants of a particular chemical element that differ in neutron number, and thus in mass. However, the protons are the same in every isotope of an element, which means the chemical properties remain consistent.

Stable isotopes do not undergo radioactive decay, making them permanent fixtures in the periodic table. Carbon, for instance, has two stable isotopes, \( ^{12}C \) and \( ^{13}C \). These isotopes allow us to identify the presence of carbon in molecular compounds through techniques such as mass spectrometry. Since stable isotopes have a consistent presence over time, they also serve as important tracers in environmental investigations, medical diagnostics, and forensic science.

The differing masses of stable isotopes are key to molecular identification. In our exercise, the recognition that carbon has two stable isotopes allows us to predict the mass spectrum of the \( CF_4^+ \) ion accurately.

'Mass spectrometry' is an analytical technique that measures the mass-to-charge ratio of ions to identify the amount and type of chemicals in a sample. It is extremely sensitive and can detect even trace amounts of substances, which is why it's widely used in chemistry, biotechnology, and forensic analysis.

When a molecule is ionized in a mass spectrometer, it may break into smaller fragments, each of which generates a peak in the mass spectrum. These peaks represent the molecular mass of the ion or fragment. The number of peaks in a mass spectrum provides insights into the isotopic composition of the molecule as well as its structure. In our textbook exercise, we assume the \( CF_4^+ \) ion does not fragment, meaning the spectrum will show peaks corresponding to the whole ions based on the isotopic composition of carbon, the only variable element in the ion.

The 'molecular mass calculation' involves adding the atomic masses of all atoms in a molecule. When isotopes are involved, the calculation takes into account the different masses of these isotopes. Because different isotopes have different masses, molecules containing different isotopes will have varying molecular masses, even if they are of the same type.

In the case of \( CF_4^+ \), a molecular ion composed of carbon and fluorine, we use the atomic masses of \( ^{12}C \) and \( ^{13}C \) isotopes and the single stable isotope of fluorine, \( ^{19}F \), to calculate the potential molecular masses of the ion. This results in two different molecular masses, one for each carbon isotope, leading to the prediction of two distinct peaks in the mass spectrum. This process demonstrates how mass spectrometry can distinguish between molecules that differ only in isotopic composition.