Mass spectrometry application: relative atomic mass

As mention in an earlier article, the relative masses of isotopes can be determined by comparing the charge-to-mass ratio of an isotope of interest with that of carbon-12. For example, mass spectrometric data for the ratio of the mass-to-charge ratio (u/z) of ²H to that of ¹²C is 0.167842. Thus, the relative mass of ²H on the carbon-12 unified atomic mass unit scale is:

$u\; of \; ^{2}H=\frac{u/z\; of\;\; ^2H}{u/z\; of\; \; ^{12}C}\; \times \; u\; of\; ^{12}C$

$u\; of \; ^{2}H=0.167842\times 12\, u$

$u\; of \; ^{2}H=2.014104\, u$

Consequently, the relative atomic mass of an element can be calculated from its isotopic abundance spectrum.

With reference to the spectrum above, the relative atomic mass of chlorine is:

$A_{r,Cl}=\frac{100\times 34.969}{100+31.96}+\frac{31.96\times 36.966}{100+31.96}=35.45$

Question

With reference to the table below, deduce the spectrum for the chlorine gas ion, Cl₂⁺.

Isotope	Relative isotopic mass	Relative abundance, %
³⁵Cl	35	75.78
³⁷Cl	37	24.22

Answer

Assuming a random distribution of isotopes in the sample of chlorine gas used and that no fragmentation occurred, the possible permutations of chlorine isotopes in a chlorine gas ion are:

Permutation, ^XCl^YCl⁺	Relative isotopic mass	No. of permutations
³⁵Cl³⁵Cl	70	1
³⁵Cl³⁷Cl or ³⁷Cl³⁵Cl	72	2
³⁷Cl³⁷Cl	74	1

The probability of observing each permutation is:

Permutation, ^XCl^YCl⁺	Probability = abundance X x abundance Y x no. of permutations
³⁵Cl³⁵Cl	0.7578 x 0.7578 x 1 = 0.5742
³⁵Cl³⁷Cl or ³⁷Cl³⁵Cl	0.7578 x 0.2422 x 2 = 0.3671
³⁷Cl³⁷Cl	0.2422 x 0.2422 x 1 = 0.0587

Hence, the spectrum has the following characteristics:

Permutation	Relative isotopic mass	Relative abundance (%)	% of predominant peak
³⁵Cl³⁵Cl	70	57.42	100.00
³⁵Cl³⁷Cl or ³⁷Cl³⁵Cl	72	36.71	63.93
³⁷Cl³⁷Cl	74	5.87	10.22

The values for ‘% of predominant peak’ are calculated relative to the abundance of ³⁵Cl³⁵Cl.

Here’s another way to explain the permutation of isotopes in chlorine gas:

Chlorine gas is formed by the collision of two chlorine atoms. Consider an infinitesimal two-dimensional space in a vessel that is filled with chlorine atoms, with the space large enough to accommodate two chlorine atoms (see diagram below).

If we further consider a single axis of collision through that space, and that chlorine atoms are approaching the space along that axis from the left and from the right, we end up with four possible products: ³⁵Cl³⁵Cl, ³⁵Cl³⁷Cl,³⁷Cl³⁵Cl and ³⁷Cl³⁷Cl. If the abundances of ³⁵Cl and ³⁷Cl are equal, we have twice as many chlorine molecules with mixed isotopes than ³⁵Cl³⁵Cl or ³⁷Cl³⁷Cl, i.e. in the ratio of 2:1:1. If the abundances of ³⁵Cl and ³⁷Cl are unequal, we multiply the numbers in the ratio with the respective abundances of isotopes to get the probabilities of occurrence of the three molecular species. The same result is obtained if we rotate the axis of collision in any direction in three dimensions, or if the collision happens at an angle.

Mass spectrometry application: relative atomic mass

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