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Question: If a substance that contains the same type of atoms is an element, why is a diamond called diamond and not carbon?
There are two types of pure substances: elements and compounds. Most pure substances are compounds, which are a mixture of elements in a fixed, constant ratio. As you point out, elements contain only one type of atom, and there are only 115 known to date, of which 83 occur naturally on Earth. For some of these elements, like carbon, they can exist in different forms, which are called "allotropes". The difference between allotropes is the layout or arrangement of the atoms, so scientists have to give a different name for each to distinguish them.
For carbon, there are several possibilities as to how the atoms bond together. One is for the carbons to connect into flat sheets or layers. The layers stack on top of each other to create the solid particle. This form of carbon is known as graphite, which is used as pencil "lead" (it is really graphite). Since the graphite layers are bound weakly together, material comes off easily when writing.
A second form of carbon is diamond, which is not layered but rather a rigid 3-D framework. It is still pure carbon, but the solid is instead a shiny crystal, versus the dull, soft graphite of pencils. Amazingly, the diamond form of carbon is less stable than graphite, so diamonds aren't forever. There is a large energy barrier to converting diamond to graphite, however, so the diamond structure is "trapped". A third allotrope of carbon was discovered less than twenty years ago, and is a cage structure known as fullerene. The layout is identical to that of a soccer ball.
So you can see that if we have a sample of a pure element, it is better to specify which form it is in. Many other elements have allotropes, such as phosphorus. Like carbon, it can have cage, layered or framework layouts. Even oxygen has allotropes. Most of it exists as O2, which we breath, but some of it exists as O3, which is called ozone. Ozone is not good to breath but exists in the upper atmosphere, protecting animal and plant life from harmful solar radiation.
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