# Isotopic profiling [closed]

If a molecule contains 12 carbon atoms and 2 nitrogens, what's the probability that it contains precisely 1 carbon-13 and precisely 1 nitrogen-15?

my working: 1.1(12)(0.36)(2) = 9.5 %

where 1.1 is the natural abundance (%) of carbon-13 and 0.36 is the natural abundance (%) of nitrogen-15

This seems too high. What is the correct probability calculation and why?

• You can see the issue with your approach by considering what probability you would get of having exactly 1 carbon-13 in a chain of 100 carbon atoms. With you method, you would get over 100%, which tells you that your missing a piece that will keep your probability bounded between 0 and 1. Apr 14 '20 at 22:47

## 1 Answer

Yes, 9.5% is much to high a value.

Let's break the problem into two parts.

• $$p_\mathrm{C}$$ will be the probability of getting exactly one $$\ce{^{13}C}$$ atom in a molecule with 12 carbon atoms.

• $$p_\mathrm{N}$$ will be the probability of getting exactly one $$\ce{^{15}N}$$ atom in a molecule with 2 nitrogen atoms.

• $$p$$, the probability sought will be $$p = p_\mathrm{C}\times p_\mathrm{N}$$

For carbon

The probability of getting one $$\ce{^{13}C}$$ atom when selecting the first carbon is 1.1% or 0.011.

The probability of not getting any $$\ce{^{13}C}$$ atom when selecting eleven additional carbon atoms is $$(1-0.011)^{11} = 0.8854$$.

So the probability of having the first carbon atom be a $$\ce{^{13}C}$$ atom and the next eleven atoms not being a $$\ce{^{13}C}$$ atom is $$0.011\times 0.8854 = 0.009740$$

But the $$\ce{^{13}C}$$ atom could have been chosen in any one of the 12 positions. So

$$p_\mathrm{C} = 12\times 0.009740 = 0.1169$$

For nitrogen

The probability of getting one $$\ce{^{15}N}$$ atom when selecting the first nitrogen is 0.36 % or 0.0036.

The probability of not getting a $$\ce{^{15}N}$$ atom when selecting one additional carbon atoms is (1−0.0036)=0.9964

So the probability of having the first nitrogen atom be a $$\ce{^{15}N}$$ atom and the next atom not being a $$\ce{^{15}N}$$ atom is:

$$0.0036\times 0.9964 = ‭0.003587$$

But the $$\ce{^{15}N}$$ atom could be the first or second so the value needs to be multiplied by 2. So

$$p_N =2 \times ‭0.003587=0.007174$$

Finally $$p = p_C \times p_N = 0.1169 \times 0.007174 = 0.0008386 = 0.084\%$$

BONUS Figuring out the number of combinations with just one $$\ce{^{13}C}$$ atom and the others being non-$$\ce{^{13}C}$$ atoms was easy. However listing all the carbon possibilities when there are four $$\ce{^{13}C}$$ atoms and six non-$$\ce{^{13}C}$$ atoms can get tricky. Fortunately there a mathematical method call Combinatorics which allows the answer to be found easily.

$$\large\binom{n}{m} = \small\dfrac{n!}{(n-m)!m!}$$ is read as "n chose m."

So for for four $$\ce{^{13}C}$$ atoms and six non-$$\ce{^{13}C}$$ atoms we'd have:

$$\large\binom{10}{4} = \small\dfrac{10!}{(10-4)!4!} = 210$$ possible arrangements.

• V clear answer, but you seem to have chosen 10 C atoms instead of the 12 in the question, Apr 14 '20 at 7:54
• @porphyrin - Thanks for point out my stupid mistake!
– MaxW
Apr 14 '20 at 15:20
• no problem I make them all the time :) Apr 14 '20 at 16:08
• I should vote to close this question as off-topic because it's really about mathematics :-) Apr 14 '20 at 20:01
• @BuckThorn , maybe so, but a number of non-mathematical chemists will come to chemistry stackexchange and find this answer very useful as it's not only a common problem encountered in chemistry, but also explains the maths at a level that is easily understood by a chemist. maths stackexchange would likely either dismiss this as 'trivial' or give an overly complicated answer.
– TSA
Apr 15 '20 at 16:05