Atomic Weights and Atomic Numbers
The integer that you find in each box of the Periodic Chart is the atomic number. The atomic number is the number of protons in the nucleus of each atom. Notice that there is one and only one integer from 1 - 110 or so in each element box, so we have found all the elements. Each element fits neatly into its niche in the Periodic Table.
Another number that you can often find in the box with the symbol of the element is usually not an integer. It is oversimplifying only a little to say that this number is the number of protons plus the average number of neutrons in that element. The number is called the atomic weight or atomic mass.
How can it be that an element must have an averaged atomic weight? The number of protons defines the type of element. If an atom has six protons, it is carbon. If it has 92 protons, it is uranium. The number of neutrons in the nucleus of an element can be different, though. Carbon 12 is the commonest type of carbon. Carbon 12 has six protons (naturally, otherwise it wouldn't be carbon) and six neutrons. The mass of the electrons is negligible. Carbon 12 has a mass of twelve. Carbon 13 has six protons and seven neutrons. Carbon 14 has six protons and eight neutrons. Carbon 14 is radioactive because, as other atoms with the wrong percentage of neutrons to protons, it is unstable. The nucleus tends to pop apart. The proper ratio of protons to neutrons is about one to one for small elements and about one proton to one and a half neutrons for the larger elements. Types of an element in which every atom has the same number of protons and the same number of neutrons are called isotopes. Carbon 14 is a radioactive isotope of carbon. Any carbon 14 that was made at the time the earth was formed is now almost all gone. Carbon 14 is continuously made from high energy electromagnetic radiation hitting nitrogen atoms in the ozone layer of the earth. This carbon 14 when taken into plants as CO2 will also be taken into animals. We can find out how much carbon 14 that normally is in a living plant or animal and from there we can find the actual amount of carbon 14 left in a plant or animal long dead. We can get a very good idea of how long ago that plant or animal was living from the amount of carbon 14 remaining in the dead body. This process is called "carbon dating." The stable, non-radioactive isotopes of carbon play no part in this. As a whole element, carbon has a more or less fixed proportion of the various carbon isotopes. For this reason, we can determine a weighted average of the isotopes for all elements. On a periodic chart you may see some atomic weights that are integers or in parentheses. These are usually on the very large or very rare or very radioactive elements. That is not really an integer atomic weight, but the atomic weight has been estimated to the nearest integer.