Atomic weight refers to the mass of an average atom of a particular element. This weight is calculated by adding together the number of protons and neutrons in the average nucleus (electrons, which also form an atom, are of extremely low mass and are therefore usually omitted for the sake of convenience). In chemical formula shorthand, the atomic weight for an element or isotope is usually written directly after the element name, such as Carbon-12 (C-12) or Uranium-235 (U-235). Atomic weight is important not only for identifying what types of atoms may make up a sample based on the sample's mass, but also for differentiating isotopes of a particular element, some of which are stable, but others of which are radioactive.
HOW TO CALCULATE THE ATOMIC WEIGHT
Atomic weight is calculated both for the average atom of an element (by taking the weighted average of the atomic weights of all isotopes based on their normal frequency within a sample), or, for a sample of a single isotope, the number of neutrons and protons in the nucleus of that particular isotope. Chemists are now able to identify with complete accuracy the number of neutrons and protons in the nuclei of stable compounds; the more difficult task, for the moment, is identifying isotope frequency. For example, there are three isotopes of hydrogen, two of which are stable: hydrogen-1 and hydrogen-2 (also known as deuterium). Deuterium makes up only a very small amount of naturally occurring hydrogen, so the atomic weight, or the average atomic mass of all hydrogen in a sample is 1.008 units.
In chemistry, atomic weight is usually expressed in terms of the mole - that is, the mass of about about 602 sextillion, or 6.02 x 10^23, atoms. This number is special because 602 sextillion protons or neutrons weighs 1 gram - and, therefore, one mole of any atom will have a mass equal to its atomic number.
ATOMIC WEIGHT vs. ATOMIC MASS
In physics and chemistry, weight and mass are technically different properties: that is, there is a difference between mass and weight. Note that the same is true of the terms atomic weight and atomic mass, although, very confusingly, the difference between mass and weight is different than the difference between atomic weight and atomic mass.
In this case, atomic mass is the mass of a particular atom at rest, which must be of a specific isotope (since any particular atom cannot be of multiple isotopes at once). Whereas atomic weight refers to the average mass of all of the atoms of an element in the sample, weighted according to the expected frequency of isotopes, atomic mass refers to the specific mass of a single atom of a particular isotope.