Expressing the mass of an atom

The mass of a hydrogen atom is 1.6727$\times$10$^{-27}$ kg. To avoid having to work with such small numbers all the time, the tradion of using a relative atomic mass scale was started. Although such a relative scale has undergone several incarnations, in 1961, by international agreement, a scale was settled upon.

A relative scale can be devised by assuming that one of the elements has exactly one mass unit for each of its nucleons. In 1961, it was decided that $^{12}$C would be the element that defined the scale. Thus, the scale of relative atomic mass of $^{12}$C is taken to be exactly 12 (and no units are assigned to this number). Then, the masses of all other elements are defined with respect to this scale. Thus, knowing the actual mass of a $^{12}$C atom (1.9927$\times$10$^{-26}$ kg), and the actual mass of a hydrogen atom, we can determine the relative mass of hydrogen on this scale from

$${{\rm mass\ of\ an\ atom\ of\ H} \over {\rm mass\ of\ an\ atom\ of\ C}} = {{\rm relative\ mass\ of\ an\ atom\ of H} \over {\rm relative\ mass\ of\ an\ atom\ of\ }^{12}{\rm C}}$$

$${1.6727\times 10^{-27} kg \over 1.9927\times 10^{-26} kg} = {{\rm rel.\ mass\ of\ H} \over 12}$$

$${\rm re.\ mass\ of\ H} = 1.0073$$

A complete list of relative mass of atoms is given in the table in the inside back cover of the book.

Notice, however, that the number we computed above does not correspond to what is in the table under hydrogen. Since many elements have isotopes, it is customary to express the relative mass of an atom as an average over the different isotopes. Thus, is an element has isotopes $^{A_1}$X, $^{A_2}$X, $^{A_3}$X, ... with relative masses $A_1$, $A_2$, $A_3$,... then the relative mass $A$ associated with an atom of X is given by

$$A = A_1 p_1 + A_2 p_2 + A_3 p_3 + \cdots = \sum_{i=1}^n A_i p_i$$

where $p_i$ is the fractional abundances of the $i$th isotope.

Example: $^{12}$C and $^{13}$C are stable isotopes of carbon with fractional abundances of 98.934% and 1.0664%, respectively. The relative mass of $^{13}$C is 13.003354. What is the average relative mass of carbon?

$$A = 12.00000\times 0.98934 + 13.00354\times 0.010664 = 12.00107$$

In order to establish a link between the mass of macroscopic amounts of matter that we ordinarily deal with and the masses of individual atoms, a connection exists through a number known as Avogadro's number ($N_0$), which is defined to be the number of atoms in exactly 12 g of carbon. The currently accepted value of $N_0$ is 6.0221420$\times$10$^{23}$. Then, the mass of any atom can be determined by dividing its relative mass $A$ by $N_0$:

$$m_{X} = {A \over N_0}$$

For example, the mass of one atom of $^{12}$C is

$$m_{^{12}C} = {12.000g \over 6.022142\times 10^{23}} = 1.9926465\times 10^{-23}g$$