Calculation of Isotopic Patterns

The calculation of isotopic patterns as just shown for the carbon-only molecule Qo can be done analogously for any X-rl element. Furthermore, the application of this scheme is not restricted to molecular ions, but can also be used for fragment ions. Nevertheless, care has be taken to assure that the presumed isotopic peak is not partially or even completely due to a different fragment ion, e.g., an ion containing one hydrogen more than the presumed X-rl composition. 

The calculation of isotopic patterns of molecules of several 10 u is not a trivial task, because slight variations in the relative abundances of the isotopes encountered gain relevance and may shift the most abundant mass and the average mass up or down by 1 u. In a similar fashion the algorithm and the number of iterations employed to perform the actual calculation affect the final result. [16]... 

Calculation of Isotopic Patterns 12.3.1 The Origin of Isotopic Patterns... 

Note Mass spectrometers usually are delivered with the software for calculating isotopic distributions. Such programs are also offered as internet-based or shareware solutions. While such software is freely accessible, it is still necessary to obtain a thorough understanding of isotopic patterns as a prerequisite for the interpreting mass spectra. 

The treatment of polyisotopic elements does not require other techniques as far as calculation or constmction of isotopic patterns are concerned. However, the appearance of isotopic patterns can differ largely from what has been considered so far and it is worth mentioning their peculiarities. 

Mass spectra of isotopic patterns of two alkanes having the molecular formulae C20H42 and C100H202, respectively. The monoisotopic mass is the lighter mass of the isotopic pattern whereas the average mass, used by chemists in stoichiometric calculations, is the balanced mean value of all the observed masses. 

Fig. 5.8. ESI FTICR mass spectrum of the supramolecular rhomb s (S,S,S,S) enantiomer. The intense signal at m/z 662 represents the doubly-charged complex [M—2 HNO3-2 NOs]. The insets show the experimental (top) and calculated (bottom) isotope patterns of the doubly-charged ion [M—2 HNO3-2 NO3]2 (right inset) and the triply-charged ion [M—I—IN O3—3 NO3] + (left inset). The inset of the doubly-charged species at m/z 662 is superimposed by another isotope pattern of a triply-charged 3 3 complex, that is also included in the calculation.

Deterministic permeability models. Application of the above principles to high temperature stable isotopes was pioneered by Norton and Taylor (1979) in their models of isotopic alteration of the Skaergaard layered intrusion and its host rocks. They used discreet zones and layers to which they assigned individual permeability values. Cartwright (1997) presented two-dimensional cases in which he modeled individual high permeability networks (fractures). Cook et al. (1997) used multiple, constant permeability zones to model the distribution of lithologies in the Alta stock area (see detailed discussion below). The advantage of this approach is that the calculated stable isotope patterns can be compared directly with measured patterns provided the permeability structure is adequately known. Permeability is also a function of time. Bolton et al. 

The basic technique, how to calculate relative abundance of isotope clusters, has been given in the literature [4], Computer programs that calculate the isotopic patterns of molecules are available [5-7]. 

From the abundance generating functions that are used to calculate the isotopic pattern, also the average molar mass of the molecular can be calculated. If the abundance generating function, e.g., for carbon tetrachloride, is as given in Eq. (12.12), then the average molar mass is given as... 

Proving the identity of isotopic patterns requires careful comparison with calculated patterns. The mass differences must be consistent with the mass of the presumed neutral losses. In order to hold true, a pattern can only be assigned to signals at or above the mass given by the sum of all contributing atoms. 

Fig. 11.17. Positive-ion LDI-FT-ICR mass spectmm of a fuUerene soot. The insets show expanded views of the experimental upperparts) and calculated lowerparts) isotopic patterns of Cgo , Cvo .and Ci2o "- Sample courtesy of W. Kratschmer, Max Planck Institute for Nuclear Physics, Heidelberg. Reproduced from Ref. [142] by permission. Wiley-VCH, Weinheim, 2009.

Figure 13. Comparison of the measured (bottom) and calculated (top) isotope pattern of the ion peak [179-PFg] in the ESI-MS spectrum of 179 [64,101].

Applications With the current use of soft ionisation techniques in LC-MS, i.e. ESI and APCI, the application of MS/MS is almost obligatory for confirmatory purposes. However, an alternative mass-spectrometric strategy may be based on the use of oaToF-MS, which enables accurate mass determination at 5 ppm. This allows calculation of the elemental composition of an unknown analyte. In combination with retention time data, UV spectra and the isotope pattern in the mass spectrum, this should permit straightforward identification of unknown analytes. Hogenboom et al. [132] used such an approach for identification and confirmation of analytes by means of on-line SPE-LC-ESI-oaToFMS. Off-line SPE-LC-APCI-MS has been used to determine fluorescence whitening agents (FWAs) in surface waters of a Catalan industrialised area [138]. Similarly, Alonso et al. [139] used off-line SPE-LC-DAD-ISP-MS for the analysis of industrial textile waters. SPE functions here mainly as a preconcentration device. 

The molecular ion of Ci2H27SnCl is expected to produce a very complex pattern, because of the combination of the characteristic natural abundances of the isotopes of Sn, Cl and C. The theoretical calculation of the intensity pattern has taken into account all the ten isotopes of Sn, the two isotopes of Cl, three distinct contributions due to the 12 carbons (144, 145, 146), and the two significant contributions due to the 27 hydrogens (27 and 28). Of the 120 combinations, many overlap. 

Fig. 3.2. Calculated isotopic patterns for carbon. Note the steadily expanding width of the pattern as X+2, X+3, X+4,... become visible. At about C90 the X-i-1 peak reaches the same intensity as the X peak. At higher carbon number it becomes the base peak of the pattern.

Again, we obtain w+1 terms for the isotopic pattern of w atoms. The binomial approach works for any di-isotopic element, regardless of whether it belongs to X+1, X+2 or X-1 type. However, as the number of atoms increases above 4 it is also no longer suitable for manual calculations. 

Example The isotopic pattern of CI2 is calculated from Eq. 3.9 with the abundances a = 100 and = 31.96 as (100 + 31.96) = 10000 -1- 6392 + 1019. After normalization we obtain 100 63.9 10.2 as the relative intensities of the three peaks. Any other normalization for the isotopic abundances would give the same result, e.g., a = 0.7578, b = 0.2422. The calculated isotopic pattern of CI2 can be understood from the following practical consideration The two isotopes Cl and Cl can be combined in three different ways i) Cl2 giving rise to the monoisotopic composition, ii) Cl Cl yielding the first isotopic peak which is here X-i-2, and finally iii) Cl2 giving the second isotopic peak X+4. The combinations with a higher number of chlorine atoms can be explained accordingly. 

Note For a rapid estimation of the isotopic patterns of chlorine and bromine the approximate isotope ratios Cl/ Cl = 3 1 and Br/ Br =1 1 yield good results. Visual comparison to calculated patterns is also well suited (Fig. 3.3). 

Fig. 3.3. Calculated isotopic patterns for combinations of bromine and chlorine. The peak shown at zero position corresponds to the monoisotopic ion at m/z X. The isotopic peaks are then located at m/z = X+2, 4, 6,. .. The numerical value of X is given by the mass number of the monoisotopic combination, e.g., 70 u for CI2.

Fig. 3.6. Calculated isotopic pattern of the molecular ion of ethyl propyl thioether, C5H12S with the respective contributions of and C to the M+1 and of and C2 to the M+2 signal indicated.

Fig. 3.7. Calculated isotopic pattern of tetrabutyltin, CigHsgSn, with labels to indicate major isotopic contributions.

Fig. 3.10. Calculated and experimental (FD-MS, cf. Chap. 8.5.4) isotopic pattern of a ruthenium carbonyl porphyrin complex. The isotopic pattern supports the presumed molecular composition. The label is attached to the peak corresponding to the ° Ru-contaming ion. Adapted fromRef. [17] with permission. IM Publications, 1997.

Isotopic patterns provide a prime source of such additional information. Combining the information from accurate mass data and experimental peak intensities with calculated isotopic patterns allows to significantly reduce the number of potential elemental compositions of a particular ion. [31] Otherwise, even at an extremely high mass accuracy of 1 ppm the elemental composition of peptides, for example, can only be uniquely identified up to about 800 u, i.e., an error of less than 0.8 mmu is required even if only C, H, N, O and S are allowed. [27,32,33]... 

Example The [M-Cl]" ion, [CHCl2], represents the base peak in the El spectrum of chloroform. The results of three subsequent determinations for the major peaks of the isotopic pattern are listed below (Fig. 3.15). The typical printout of a mass spectrometer data system provides experimental accurate mass and relative intensity of the signal and an error as compared to the calculated exact mass of possible compositions. For the [ CH Cl2] ion, the experimental accurate mass values yield an average of 82.9442 0.0006 u. The comparatively small absolute error of 0.6 mmu corresponds to a relative error of 7.5 ppm. 

Note The assignment of empirical formulae from accurate mass measurements always must be in accordance with the experimentally observed and the calculated isotopic pattern. Contradictions strongly point towards erroneous interpretation of the mass spectrum. 

Fig. 3.26. Calculated isotopic patterns of large polystyrene ions. Adapted from Ref. [38] with permission. American Chemical Society, 1983.

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