difficulty in that the matrix containing the rate constants is not a symmetric matrix. It is, however, a "banded" matrixi.e., one in which all elements are zero except those on the main diagonal and the two diagonals adjacent to it. It can be shown that the eigenvalues (but not the eigenfunctions) of this matrix are the same as those of a symmetric matrix with the same main diagonal elements, but off-diagonal elements lii = The eigenvalues are therefore obtained by solving this symmetric matrix and the coefficients are calculated by solving the five simultaneous linear equations with five unknowns.A mass spectrometric procedure far determining up to 21 compound types in petroleum aromatic fractions is described. The entire composition of any sample is accounted for in terms of 12 hydrocarbon types, 3 thiopheno types, and 6 unidentified groups. Inclusion of the unidentified components avoids the difficulties encountered in earlier methods which described composition in terms of a fixed number of named types. Reasonable results have been obtained for a wide variety of samples.
MASSSPECTROMETRIC methods for determining compound types in petroleum have been used routinely for more than 15 years. Since the first application to the analysis of gasolines ( I ) , such methods have been extended to the heaviest volatile fractions of petroleum. However, a method to determine compound types in the wide variety of heavy aromatic fractions encountered in the petroleum industry has not been reported. The absence of such a procedure undoubtedly reflects the uncertain knowledge of the composition of petroleum aromatics and also reflects the complexity of the problem of obtaining adequate group type analyses. Hastings et al. (2) determined 12 compound types, including three thiopheno types, by using summations of the most intense (M-I)+ peaks for each type. (If a series of molecular ions are designated (M)+ the series of ions at each of the next lower masses may be expressed as (M-l>+.) When applied to very light or very heavy fractions, the results are difficult to interpret. Orkin et al. (3) used monoisotopic peaks to determine seven compound types. Summations over a limited mass range contain both (M)+ and (M-l)+ peaks, and equal sensitivities are assumed for each type. No sulfur types are included. This forces a large number of compound types into the seven groups and fails to further resolve the types. Gallegos et al. (4) developed a high resolution mass (1) R. A. Brown, ANAL. CHEM., 23,430(1951).