Probability of forming dihydrogen with molecular weight 3 [closed]








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During a lecture, my professor told us that the probability $P$ to form a hydrogen molecule $\ce{H2}$ with mass number 3 could be calculated out of the abundances, $\gamma$, of the isotopes of this element: $\ce{^1H}$ (normal hydrogen), $\ce{^2H}$ (deuterium), etc. So,


$$
\begin{split}
P[\ce{H2}] &= P[(\ce{^1H} \cap \ce{^2H}) \cup (\ce{^2H} \cap \ce{^1H})] \\
%
 &= P[\ce{^1H} \cap \ce{^2H}] + P[\ce{^2H} \cap \ce{^1H}] - 
%
 P[(\ce{^1H} \cap \ce{^2H}) \cap (\ce{^2H} \cap \ce{^1H})]
\end{split}
\tag{1}$$


But since


$$(\ce{^1H} \cap \ce{^2H}) \cap\ (\ce{^2H} \cap \ce{^1H}) = \ce{^1H} \cap \ce{^2H}\tag{2}$$


and


$$P[\ce{^1H} \cap \ce{^2H}] + P[\ce{^2H} \cap \ce{^1H}] = 2P[\ce{^1H} \cap \ce{^2H}]\tag{3}$$


we end up with


$$
\begin{split}
P[\ce{H2}] &= P[\ce{^1H} \cap \ce{^2H}] \\
 &= \gamma(\ce{^1H}) \cdot \gamma(\ce{^2H}) \\
 &= 0.99972\cdot 0.00028 \\
 &= 0.0002799
\end{split}
\tag{4}$$


This is my result while for my professor, it should be twice that value. It's like he's forgetting the term


$$
P[(\ce{^1H} \cap \ce{^2H})\ \cap\ (\ce{^2H} \cap \ce{^1H})]
$$


Am I right?




Extending / reformulating Karsten Theis' answer: A probabilistic approach is to draw a tree about drawing two spheres where event $A$ (normal hydrogen, $\ce{^1H}$) dominates, but event $B$ (the next abundant isotope, $\ce{^2H}$) complements, $P(\ce{^1H} \cup{} \ce{^2H}) = 1 = \Omega{}$ and eventually to multiply the probabilities for each path (e.g., $P(\ce{^1H^2H})$ as shown on the left hand side).


[![enter image description here](https://i.stack.imgur.com/OncnO.png)](https://i.stack.imgur.com/OncnO.png)


Yet because in your case *both* paths $P(\ce{^1H^2H})$ and $P(\ce{^2H^1H})$ yield a dihydrogen molecule of mass 3, these individual probabilities need to be summed up.


 3

You are wrong. There are only four cases (first atom is proton or deuteron, second atom is proton or deuteron), and you have to add up two cases to get all molecules with mass number of 3, i.e. the two light blue areas in the schematic diagram below (not to scale):


[![enter image description here](https://i.stack.imgur.com/COQQu.jpg)](https://i.stack.imgur.com/COQQu.jpg)


 2