# How do you calculate the atomic mass of carbon?

Jan 3, 2014

The term "atomic mass" refers to the mass of a single atom. The mass of a single atom of carbon-12 is defined as exactly 12 u.

The term atomic mass is also often used (though technically, incorrectly) to refer to the average atomic mass of all of the isotopes of an element.

This second definition is actually the relative atomic mass of an element — a single average value of the element's mass based on the masses of its isotopes.

Carbon has 15 known isotopes, of which only two ($\text{^12"C"color(white)(l) "and"color(white)(l) ""^13"C}$) are stable. $\text{^14"C}$ exists only in trace amounts, so it makes no measurable contribution to the average atomic mass.

Carbon consists of 98.93% $\text{^12"C}$ (12 u) and 1.07% $\text{^13"C}$ (13.003 u). What is the average atomic mass of these isotopes?

METHOD 1

To find the average atomic mass, you take a certain number of atoms, find the total mass of each isotope, and then divide the total mass of all the atoms by the total number of atoms.

Assume that you have, say, 10 000 atoms of carbon. Then you have 9893 atoms of $\text{^12"C}$ and 107 atoms of $\text{^13"C}$.

$\text{Mass of"color(white)(l) ""^12"C" = 9893 color(red)(cancel(color(black)("atoms"))) × "12 u"/(1 color(red)(cancel(color(black)("atom")))) = "118 716 u}$

"Mass of"color(white)(l) ""^13"C" = 107color(red)(cancel(color(black)("atoms"))) × "13.003 u"/ (1 color(red)(cancel(color(black)("atom")))) =color(white)(ml) "1 390 u"

$\text{Total mass"color(white)(l)color(white)(mmmmmmmmmmmmm) = "120 110 u}$

$\text{Average mass" = "120 110 u"/"10 000 atoms" = "12.011 u}$

METHOD 2

Another way of determining the average mass is to multiply the atomic mass of each isotope by its percentage and then add the numbers.

The two methods are mathematically equivalent. Thus,

$\text{Average mass" = "(0.9893 × 12 + 0.0107 × 13.003) u" = "(11.872 + 0.139) u" = "12.011 u}$

Method 2 is probably mathematically simpler, but Method 1 makes it clear that you are determining an average mass.

Choose the method that you prefer.