<p>How can I calculate the expression of the abundance of a given element Xi in mass fraction from its abundance by number Ni/NH, where log(NH) = 12 is the abundance by number of hydrogen?</p>
<p>I've tried the following: suppose I have a star with, among others, an element Xi and hydrogen, then the mass fraction would be:</p>
<p><span class="math-container">$X_{i} = \frac{m_{i}}{m_{tot}}$</span></p>
<p>and as the number of atoms is given by:</p>
<p><span class="math-container">$N_{i} = \frac{m_{i}N_{A}}{M_{i}}$</span>     (((<span class="math-container">$M_{i}$</span> is the molar mass)))</p>
<p>you can get:</p>
<p><span class="math-container">$\frac{N_{i}}{N_{H}} = \frac{m_{i}M_{H}}{M_{i}m_{H}} = \frac{X_{i}M_{H}}{X_{H}M_{i}}$</span></p>
<p>so</p>
<p><span class="math-container">$X_{i} = \frac{N_{i}M_{i}}{N_{H}M_{H}}X_{H}$</span></p>
<p>Now, let's take into account that <span class="math-container">$A = \log\left(\frac{N_{i}}{N_{H}}\right)+12$</span></p>
<p>so, finally:</p>
<p><span class="math-container">$X_{i} = 10^{A-12}\frac{M_{i}}{M_{H}}X_{H}$</span></p>
<p>However, here: <a href="https://astronomy.stackexchange.com/questions/36976/difference-in-stellar-abundance-numbers">Difference in stellar abundance numbers</a> they obtain this:</p>
<p><span class="math-container">$X_{i} = \frac{N_{i}}{N_{H}}\frac{M_{i}}{M_{H}}\frac{1}{1-\frac{N_{i}}{N_{H}}}X_{H} = \frac{N_{i}}{N_{H}-N_{i}}\frac{M_{i}}{M_{H}}X_{H}$</span></p>
<p>the only difference is that I get <span class="math-container">$N_{H}$</span> and they get <span class="math-container">$N_{H}-N_{i}$</span>.</p>
<p>Why do they get that and which is the correct way?</p>

CC BY-SA 4.0
2023-02-18T18:37:45.277Z
But, then there are more steps in that other calculation to compensate for Z.  I think you have the simpler solution.
eshaya
CC BY-SA 4.0
2023-02-18T19:33:20.423Z
@eshaya I don't understand... Suppose we have a start with many elements, Xi is one of them. My calculation is right?
Elena
CC BY-SA 4.0
2023-02-18T19:59:52.110Z
Yes.  The other is overly complicated and specific for Helium.
eshaya
<astrophysics>
Abundance of an element Xi in mass fraction from its abundance by number
Elena
# Abundance of an element Xi in mass fraction from its abundance by number

(Asked by: username_0 on 2023-02-17)

How can I calculate the expression of the abundance of a given element Xi in mass fraction from its abundance by number Ni/NH, where log(NH) = 12 is the abundance by number of hydrogen?

I've tried the following: suppose I have a star with, among others, an element Xi and hydrogen, then the mass fraction would be:

$X\_{i} = \frac{m\_{i}}{m\_{tot}}$

and as the number of atoms is given by:

$N\_{i} = \frac{m\_{i}N\_{A}}{M\_{i}}$ ((($M\_{i}$ is the molar mass)))

you can get:

$\frac{N\_{i}}{N\_{H}} = \frac{m\_{i}M\_{H}}{M\_{i}m\_{H}} = \frac{X\_{i}M\_{H}}{X\_{H}M\_{i}}$

so

$X\_{i} = \frac{N\_{i}M\_{i}}{N\_{H}M\_{H}}X\_{H}$

Now, let's take into account that $A = \log\left(\frac{N\_{i}}{N\_{H}}\right)+12$

so, finally:

$X\_{i} = 10^{A-12}\frac{M\_{i}}{M\_{H}}X\_{H}$

However, here: [Difference in stellar abundance numbers](https://astronomy.stackexchange.com/questions/36976/difference-in-stellar-abundance-numbers) they obtain this:

$X\_{i} = \frac{N\_{i}}{N\_{H}}\frac{M\_{i}}{M\_{H}}\frac{1}{1-\frac{N\_{i}}{N\_{H}}}X\_{H} = \frac{N\_{i}}{N\_{H}-N\_{i}}\frac{M\_{i}}{M\_{H}}X\_{H}$

the only difference is that I get $N\_{H}$ and they get $N\_{H}-N\_{i}$.

Why do they get that and which is the correct way?

**Tags:** astrophysics
> **Comments:**
> **username_1**: But, then there are more steps in that other calculation to compensate for Z.  I think you have the simpler solution.  
> **username_0**: @eshaya I don't understand... Suppose we have a start with many elements, Xi is one of them. My calculation is right?  
> **username_1**: Yes.  The other is overly complicated and specific for Helium.  

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