A mass spectrum is an intensity vs. mass-to-charge ratio plot representing a chemical analysis. Hence, the mass spectrum of a sample is a pattern representing the distribution of components (atoms or molecules) by mass (more correctly: mass-to-charge ratio) in a sample. It is usually acquired using an instrument called a mass spectrometer. Not all mass spectra are the same. For example some mass spectrometers break the analyte molecules into fragments; others observe the intact molecular masses with little fragmentation. A mass spectrum can represent many different types of information based on the type of mass spectrometer and the specific experiment applied; however, all plots of intensity vs. mass-to-charge are referred to as mass spectra.
See IUPAC MS Terms Second Draft. Since a mass spectrum x-axis represents the mass-to-charge ratio it contains mass information that may be extracted by a knowledgable mass spectrometrist. Once this is done many mass spectrometrists use dalton (Da) as the unit of mass in order to avoid the clumsy "atomic mass units".
In 1897 the mass-to-charge ratio of the electron was first measured by J.J. Thomson By doing this he showed that the electron, which was postulated before in order to explain electricity, was in fact a particle with a mass and a charge and that its mass-to-charge ratio was much smaller than the one for the hydrogen ion H+. In 1913 he measured the mass-to-charge ratio of ions with an instrument he called a parabola spectrograph [http://web.lemoyne.edu/~giunta/canal.html. Although this data was not represented as a modern mass spectrum, it was similar in meaning. Eventually there was a change to the more physically meaningful mass-to-charge ratio with some early notation as m/e giving way to the current IUPAC standard of m/z.
Early in mass spectrometry research the resolution of mass spectrometers did not allow for accurate mass determination. Francis William Aston won the nobel prize in Chemistry in 1922 "For his discovery, by means of his mass spectrograph, of isotopes, in a large number of non-radioactive elements, and for his enunciation of the whole-number rule." In which he stated that all atoms (including isotopes) follow a whole-number rule [http://web.lemoyne.edu/~giunta/aston.html. This implied that the masses of atoms were not on a scale but were quantized and could be expressed as integers. This may be an origin of the unitlessness of the representation of mass-to-charge since both mass and charge were quantized and could be expressed as unitless whole numbers. (In fact multiply charged ions were rare, so for the most part the ratio was whole as well.) Today we know this to be not true; however for the most part the nomenclature convention has held while the whole-number rule has disappeared. There have been several suggestions (e.g. the unit thomson) to change the official mass spectrometry nomenclature to be more internally consistent and compatible with the broader scientific unit system and other standards (ISO 31, IUPAC green book, IUPAP red book). Currently there is an effort to redefine the standard for x-axis notation. This effort has recently produced the IUPAC MS Terms Second Draft document.
A common way to get more quantitative information out of a mass spectrum is to create a standard curve to compare the sample to. This requires knowing what is to be quantitated ahead of time, having a standard available and designing the experiment specifically for this purpose. A more advanced variation on this the use of an internal standard which behaves very similarly to the analyte. This is often an isotopically labeled version of the analyte. There are forms of mass spectrometry, such as accelerator mass spectrometry that are designed from the bottom up to be quantitative.
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