Is there a function to approximate atomic mass from the atomic number?

The product of $2Z$ generally approximates the value of $m\_\mathrm{a}$ of atoms that comprise relatively few protons. Whereas, the product of $2.5Z$ generally approximates the value of $m\_\mathrm{a}$ of atoms that comprise relatively many protons. The product of $((2+2.5)/2)Z$ generally approximates the value of $m\_\mathrm{a}$ of atoms that comprise a moderate number of protons. 


What function from the $Z$ values produces a value that maximally approximates the $m\_\mathrm{a}$ that corresponds to $Z$?




**The short answer is that you can find a power-law fit ($1.61Z^{1.1}$) with low average error.**


I'd never really thought about it much, but after downloading the [IUPAC Atomic Weights](http://www.ciaaw.org/atomic-weights.htm), I decided to do some curve fitting.


Here's a linear fit between atomic number and atomic mass:
![linear fit of atomic mass from atomic number](https://i.stack.imgur.com/fxX6Y.png)


As you say, the fit isn't very good for small $Z$, but the overall fit isn't bad - the mean absolute error (MAE) is $2.821 \:\mathrm{u}$, and taken as a whole, the data is *surprisingly* linear. (Well, surprising to me.)


So I thought of a **quadratic** fit, requiring the intercept to be 0,0 to ensure the best fit for small $Z$:


![quadratic fit of atomic mass from atomic number](https://i.stack.imgur.com/1hNd7.png)


Looks better, right? Certainly the fit is much better for first and second row elements, but the MAE only reduces to $2.749 \:\mathrm{u}$.


So I went up to a cubic fit, again requiring 0,0 for the intercepts:


![cubic fit of atomic mass from atomic number](https://i.stack.imgur.com/47CN8.png)


Aha, now we're talking! We get the subtle nonlinearities, and the MAE is down to $1.36 \:\mathrm{u}$.


Thanks to the comment by Nicolau below, I performed a power-law fit.


![power-law fit](https://i.stack.imgur.com/pJRzo.png)


So that gives a power-law fit with MAE of $0.01 \:\mathrm{u}$ and a fairly easy-to remember function:


$m\_a \approx 1.61Z^{1.1}$


 28

Atomic mass as a function of atomic number:


(Atomic number)^2/162 + 2\*atomic number = atomic mass


It's the same as the answer given above except the final term is 2x 


The equation for the first 16 elements is obviously just 2x.


X^2/162 is needed for the rest of the elements.


Thorium has 90 protons so 90^2/162+2\*90 = 50+180=230.


The equation breaks down after thorium (140 neutrons)


After thorium 4 halo neutrons are added for every 3 protons.


<https://upload.wikimedia.org/wikipedia/commons/d/d1/Uranium_atom.svg>


 -3