2.5 Mass Spectrometry
29
2.5.4 Calculation of Isotope Distributions
The characteristic abundance patterns resulting from the combination of more
than one polyisotopic element can be calculated from the relative abundances
of the different isotopes. The following polynomial expression gives the isotope
distribution of a polyisotopic molecule:
{p
i1
Α
0
+ p
i2
Α
(m i2 - m i1 )
+ p
i3
Α
(m i3 - m i1 )
+ …}
n i ×
{p
j1
Α
0
+ p
j2
Α
(m j2 - m j1 )
+ p
j3
Α
(m j3 - m j1 )
+ …}
n j × {…
where p ix is the relative abundance of the xth isotope of element i, m ix is the mass of
the xth isotope of the element i, and the exponent n i stands for the number of atoms
of the element i in the molecule. The expansion of this polynomial expression after
inserting the p ix and m ix values for all the isotopes 1, 2, 3, … of the elements i, j,
… of a given molecule yields an expression that represents the isotope distribution:
w
0
Α
0
+ w r
Α
r
+ w s
Α
s
+ w t
Α
t
+ …
where the values of w 0 , w r , w s , w t , … are the relative abundances of M +· , [M + r] +· ,
[M + s] +· , [M + t] +· , … , respectively. The use of Α
(m ix - m i1 )
allows to determine the
values of r, s, t, … simply by expanding the general polynomial. A numerical value
for A, which has no intrinsic meaning, is never needed.
For example, for CBr 2 Cl 2 , the above equation gives rise to the following
expression:
{p
12 C
Α
0
+ p
13 C
Α
(m 13 C
- m 12 C
)
} ×
{p
79 Br
Α
0
+ p
81 Br
Α
(m 81 Br
- m 79 Br
)
}
2
×
{p
35 Cl
Α
0
+ p
37 Cl
Α
(m 37 Cl
- m 35 Cl
)
}
2
For sufficient resolution, (m ix - m i1 ) and (m jx - m j1 ) differ from one another. This
results in very complex isotope patterns even for very small molecules. Thus, owing
to the occurrence of 12 C, 13 C, 79 Br, 81 Br, 35 Cl, and 37 Cl, there are 18 signals for
CBr 2 Cl 2 . However, the limited resolution of many real life experiments can make
many pairs of (m ix - m i1 ) and (m jx - m j1 ) indistinguishable within experimental error,
thereby reducing the number of separate peaks. For example, at unit resolution, one
obtains (m 13 C
- m 12 C
) = 1 and (m 81 Br
- m 79 Br
) = (m 37 Cl
- m 35 Cl
) = 2. Consequently,
the expression for CBr 2 Cl 2 becomes:
29
2.5.4 Calculation of Isotope Distributions
The characteristic abundance patterns resulting from the combination of more
than one polyisotopic element can be calculated from the relative abundances
of the different isotopes. The following polynomial expression gives the isotope
distribution of a polyisotopic molecule:
{p
i1
Α
0
+ p
i2
Α
(m i2 - m i1 )
+ p
i3
Α
(m i3 - m i1 )
+ …}
n i ×
{p
j1
Α
0
+ p
j2
Α
(m j2 - m j1 )
+ p
j3
Α
(m j3 - m j1 )
+ …}
n j × {…
where p ix is the relative abundance of the xth isotope of element i, m ix is the mass of
the xth isotope of the element i, and the exponent n i stands for the number of atoms
of the element i in the molecule. The expansion of this polynomial expression after
inserting the p ix and m ix values for all the isotopes 1, 2, 3, … of the elements i, j,
… of a given molecule yields an expression that represents the isotope distribution:
w
0
Α
0
+ w r
Α
r
+ w s
Α
s
+ w t
Α
t
+ …
where the values of w 0 , w r , w s , w t , … are the relative abundances of M +· , [M + r] +· ,
[M + s] +· , [M + t] +· , … , respectively. The use of Α
(m ix - m i1 )
allows to determine the
values of r, s, t, … simply by expanding the general polynomial. A numerical value
for A, which has no intrinsic meaning, is never needed.
For example, for CBr 2 Cl 2 , the above equation gives rise to the following
expression:
{p
12 C
Α
0
+ p
13 C
Α
(m 13 C
- m 12 C
)
} ×
{p
79 Br
Α
0
+ p
81 Br
Α
(m 81 Br
- m 79 Br
)
}
2
×
{p
35 Cl
Α
0
+ p
37 Cl
Α
(m 37 Cl
- m 35 Cl
)
}
2
For sufficient resolution, (m ix - m i1 ) and (m jx - m j1 ) differ from one another. This
results in very complex isotope patterns even for very small molecules. Thus, owing
to the occurrence of 12 C, 13 C, 79 Br, 81 Br, 35 Cl, and 37 Cl, there are 18 signals for
CBr 2 Cl 2 . However, the limited resolution of many real life experiments can make
many pairs of (m ix - m i1 ) and (m jx - m j1 ) indistinguishable within experimental error,
thereby reducing the number of separate peaks. For example, at unit resolution, one
obtains (m 13 C
- m 12 C
) = 1 and (m 81 Br
- m 79 Br
) = (m 37 Cl
- m 35 Cl
) = 2. Consequently,
the expression for CBr 2 Cl 2 becomes: