# How to find the mass spectrum of Cl3+?

The setting of this problem is on the moon of Jupiter: Io.

Given that chlorine exists in two isotopes: $\ce{{}^{35}_{17}Cl}~(33.3\%)$ and $\ce{{}^{37}_{17}Cl}~(66.7\%)$, what is the mass spectrum of $\ce{Cl3+}$?

What I know now is that there are 3 peaks in the mass spectrum of $\ce{Cl2+}$ because there are 3 possible combinations ($\ce{{}^{35}_{17}Cl}$ + $\ce{{}^{35}_{17}Cl}$ / $\ce{{}^{35}_{17}Cl}$ + $\ce{{}^{37}_{17}Cl}$ / $\ce{{}^{37}_{17}Cl}$ + $\ce{{}^{37}_{17}Cl}$). Therefore, there are 4 possible combinations and 4 peaks for $\ce{Cl3+}$.

However, I do not know how to make a mass spectrum graph with only this much information.

• Consider Pascal's Triangle. – Abel Friedman Nov 18 '14 at 19:42

Okay, let's tackle this. First of all you worked out correctly that there are four different combinations, resulting in four different peaks. You have also given the probability for finding a $\ce{{}^{35}_{17}Cl}$ is $a=33.3\%\approx\frac13$ and the probability for finding a $\ce{{}^{37}_{17}Cl}$ is $b=66.7\%\approx\frac23$. Since you have three positions to fill, this will result in a binomial formula of third order, $(a+b)^3$. You can solve this easily with Pascal's triangle, $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. You spectrum should therefore look like this table: \begin{array}{lrr}\hline & \text{m/z} &\text{abundance}/\%\\\hline \ce{[{}^{35}_{17}Cl_3]^{+}} & 105 & 3.7 \\ \ce{[{}^{35}_{17}Cl_2{}^{37}_{17}Cl]^{+}} & 107 & 22.2 \\ \ce{[{}^{35}_{17}Cl{}^{37}_{17}Cl_2]^{+}} & 109 & 44.4 \\ \ce{[{}^{37}_{17}Cl_3]^{+}} & 111 & 29.7 \\\hline \end{array}
• Aren't there eight possible combinations of isotopes? (as I see from the bottom row of your graph, and as is also easily observed from the fact there are three atoms in $\ce{Cl3+}$ each having two choices) – Gaurang Tandon Mar 9 '18 at 6:31