## 2 Critical Analytical Parameters
### Mass Resolution of the Mass Spectrometer Instrument  
Mass resolution is defined as the mass of an ion divided by the mass width at half height of the spectral peak (Equation (2)) and is a measure of how well two similar masses are separated in a spectrum. Accurate mass instruments such as Orbitraps and TOFs provide much higher mass accuracies than unit mass quadrupoles due to their higher mass resolutions.  
\[R=m/\Delta m \tag{2}\]  
Equation (2) is a formula for determining mass resolution of a mass spectrometer, where m is the mass of an ion, and \(\Delta m\) is the mass width at half height.  
Mass resolution for TOF instruments using the formula in Equation (2) is defined when two peaks have equal height and width. When the peak height ratio is 100:1, the resolution may need to be 10\(\times\) higher than calculated, since the mass at lower abundance will otherwise be included in the base of the more abundant ion [29]. High-resolution accurate mass (HRAM) instruments such as FT-ICR, TOF, and Orbitrap are, therefore, better suited to chemical characterization of NAFCs than unit mass quadrupoles. Their relatively high cost of purchase, maintenance, and operation preclude them from use in many labs. They are, however, required for elemental speciation and separation of OSPW into compound classes due to their ability to produce accurate mass assignments to the fourth or fifth decimal place. Mass accuracy at this level is required for unambiguous assignment of chemical formulae to unknown compounds. Correct formula assignments typically require mass accuracies of \(\pm\)5 ppm when restricting the formula to C, H, O, N, and S in the molecules. This forward calculation between two chemical formulae provides straightforward mass error calculations. However, since mass spectrometers measure accurate masses, the calculation must be carried out in reverse, i.e., start with an accurate mass and use it to determine the chemical formula. Constraints for the type and number of elements that are included in the calculation are used to limit the number of possible formulae. For example, if the mass of an ion was measured to be 281.1763 (mass error \(-\)1.7 ppm for C\({}_{16}\)H\({}_{26}\)O\({}_{4}\)), there are six possible formulae when C, H, O, N, and S are allowed, with mass error up to \(\pm\)10 ppm, (Table 3). Using an MS with mass accuracy known to be \(\pm\)5 ppm, only the top three formulae would be possible. The third ion formula (C\({}_{17}\)H\({}_{21}\)N\({}_{4}\)) would not be forbidden but is unlikely to be an anion. If mass accuracy was known to be \(\pm\)2 ppm, neither of the first two formulae could be dismissed but the third would be eliminated. Adding an LC separation to the analysis and using model compounds with known formulae and retention times may separate isobaric compounds (i.e., compounds with the same unit mass) and add more confidence to otherwise ambiguous compound assignments.  
\begin{table}
\begin{tabular}{l c c c} \hline
**Ion Formula** & \(m/z\) **(Calc)** & **Diff (ppm)** & **Resolution** \\ \hline C\({}_{9}\)H\({}_{25}\)N\({}_{6}\)O\({}_{2}\)S & 281.1765 & 0.8 & 1,406,000 \\ \hline C\({}_{16}\)H\({}_{25}\)O\({}_{4}\) & 281.1758 & \(-\)1.7 & 562,000 \\ \hline C\({}_{17}\)H\({}_{21}\)N\({}_{4}\) & 281.1772 & 3.1 & 312,000 \\ \hline C\({}_{11}\)H\({}_{27}\)N\({}_{3}\)O\({}_{3}\)S & 281.1779 & 5.5 & 176,000 \\ \hline C\({}_{14}\)H\({}_{23}\)N\({}_{3}\)O\({}_{3}\) & 281.1745 & \(-\)6.4 & 156,000 \\ \hline C\({}_{19}\)H\({}_{23}\)N O & 281.1785 & 7.8 & 128,000 \\ \hline \end{tabular}
\end{table}
Table 3: Mass accuracies of chemical formulae of ions within 10 ppm of \(m/z\) measured at 281.1763 and the mass resolution that would be required to separate that chemical species from the observed mass.