Image from “Russell’s Blog” which you can access by clicking on the image
Last time we talked about the important parts of atoms – protons and electrons – and how they come together to define the unique properties of every element. Now let’s tackle the one remaining part, which is how neutrons fit into the mix. Remember, neutrons and protons are practically the same thing in terms of mass, except the proton carries a positive charge. The neutron even says it in the name. It is neutral, meaning no charge, just mass.
That mass is what the neutron brings to an atom. The mass of an atom is dictated by the number of protons and neutrons. The identity of an atom is defined by the number of protons, so a given element will always have the same number of protons. Lithium always has three protons, carbon has six protons, iron has 26 and so on. The number of protons is so important, it is called the atomic number and you will find it on every periodic table you will see. But the number of neutrons in an atom does not change the identity of the atom at all. In fact, anywhere you go and scoop up a handful of carbon atoms, every atom will have 6 protons but you find a mixture of the number of neutrons in each atom. Some will have 6 neutrons, some will have 7 neutrons and others will have 8 neutrons. These are all carbon, but this is an example of atoms existing as isotopes. An isotope of a given atom – say carbon – will always have the number of protons described by the atomic number, but one atom may possess 6 atoms (which we call “carbon 12” – 6 protons plus 6 neutrons), another has 7 neutrons (carbon 13) and yet another will have 8 neutrons (carbon 14).
An individual atom will have a single number of neutrons but any time a macroscopic amount of atoms is used, a mixture of isotopes exist unless a difficult purification process has already occurred. All atoms are created with what is known as the “natural abundance” of isotopes of that atom. Some atoms have only one or two isotopes, while others have 10 or more isotopes. But keep in mind, it is the number of protons that sets the identity and the major thing that changes with isotopes is the mass. Some isotopes are also radioactive, and this will be a topic of another time. But for now, the important details are that an atom’s identity is set by the number of protons, given by the atomic number. The number of electrons, when compared to the number of protons in an atom, defines the net charge of the atom, and the number of neutrons changes only the mass. However, any given atom cannot have any given number of neutrons. The range in the number of neutrons in an atom is set by the natural isotopic abundance. Anywhere you look in the known universe that is cooler than 10,000 degrees, you will find the same natural abundance of isotopes for each element.
One last thing. Because of the natural abundance of isotopes, it is rarely necessary to talk about individual isotopes of an atom and that’s why you see no mention of it on most periodic tables. 
You do see the evidence for isotopes when you look at the mass of an atom on a periodic table. In rough numbers, 1 proton has a mass of 1 atomic mass unit, or amu, and the same is true for a neutron. Yet if you look at the mass of carbon on a periodic table, you will see mass given as 12.01 amu. That’s because a handful of carbon atoms will be made up of mostly carbon 12 – 6 protons and 6 neutrons. But a few will be carbon 13 – 6 protons and 7 neutrons – and a few will be carbon 14. Carbon 14 still has 6 protons but has 8 neutrons.
This is the isotopic notation for copper 63.
What is reported on a periodic table is the weighted average of a handful of atoms and that is why the reported mass of carbon is a tad higher than 12 amu. Look around the periodic table and you will see none of the atoms have a nice neat mass because all are weighted higher or lower than the most abundant isotope. If you want to specify a certain isotope of an atom you would use “isotopic notation” to do so. The image at left is isotopic notation for the copper 63 isotope. Notice on a periodic table, copper has an atomic number of 29 but a mass of 63.546 amu. This means that a random sample of copper must contain copper atoms with 29 protons and (63 – 29 = 34) 34 neutrons, but also some fraction of the atoms must have greater than 24 neutrons, because the average mass is greater than 63 amu. There is no way to determine how many isotopes are in each element from most periodic tables. That information must be found by doing a web search, or looking at a periodic table that gives that specific information. Remember however, it is usually not relevant to the vast majority of chemical problems one will encounter. Atoms are so very very small, any sample that can be weighed with standard equipment or seen even by a microscope contains millions and trillions of atoms, so we use average properties of things like mass to describe the sample. That is why it is the average mass for an element that is displayed on the vast majority of periodic tables you will see.
To summarize, the identity of an atom is fixed by the number of protons, identified by the atomic number. The net charge is set by the number of electrons in an atom relative to the number of protons. The mass of an element is fixed by the number of nucleons, that is, the number of protons plus the number of neutrons. And finally, the number of neutrons in a given element can vary over a range. All are still the specific identity set by the atomic number but the variation in the number of neutrons are called isotopes.
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