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 ^{2}H to that of ^{12}C is 0.167842. Thus, the relative mass of ^{2}H on the carbon-12 unified atomic mass unit scale is:
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:
Question
With reference to the table below, deduce the spectrum for the chlorine gas ion, Cl_{2}^{+}.
Isotope |
Relative isotopic mass |
Relative abundance, % |
^{35}Cl |
35 |
75.78 |
^{37}Cl |
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, ^{X}Cl^{Y}Cl^{+} |
Relative isotopic mass |
No. of permutations |
^{35}Cl^{35}Cl |
70 |
1 |
^{35}Cl^{37}Cl or ^{37}Cl^{35}Cl |
72 |
2 |
^{37}Cl^{37}Cl |
74 |
1 |
The probability of observing each permutation is:
Permutation, ^{X}Cl^{Y}Cl^{+} |
Probability = abundance X x abundance Y x no. of permutations |
^{35}Cl^{35}Cl |
0.7578 x 0.7578 x 1 = 0.5742 |
^{35}Cl^{37}Cl or ^{37}Cl^{35}Cl |
0.7578 x 0.2422 x 2 = 0.3671 |
^{37}Cl^{37}Cl |
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 |
^{35}Cl^{35}Cl |
70 | 57.42 |
100.00 |
^{35}Cl^{37}Cl or ^{37}Cl^{35}Cl |
72 | 36.71 |
63.93 |
^{37}Cl^{37}Cl |
74 | 5.87 |
10.22 |
The values for ‘% of predominant peak’ are calculated relative to the abundance of ^{35}Cl^{35}Cl.
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: ^{35}Cl^{35}Cl, ^{35}Cl^{37}Cl,^{ 37}Cl^{35}Cl and ^{37}Cl^{37}Cl. If the abundances of ^{35}Cl and ^{37}Cl are equal, we have twice as many chlorine molecules with mixed isotopes than ^{35}Cl^{35}Cl or ^{37}Cl^{37}Cl, i.e. in the ratio of 2:1:1. If the abundances of ^{35}Cl and ^{37}Cl 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.