Is phosphorus a rare earth or a family name?

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Neither the term "rare earth", nor "family name", apply to phosphorus.

The rare earths refer specifically to a series of about 17 elements, predominately in the lanthanide series, which are considered rare because they tend not to appear in significant amounts in any one location or ore. Phosphorus is not located anywhere near the lanthanide series on the periodic table, and is in fact abundant throughout the earth.

The term "family name" is just another word for each column, or group, of the periodic table. The noble gases and halogens are one example. The group in which phosphorus is located, Group 15 (or occasionally Group 5A), are called the pnictogens, or the nitrogen family. Phosphorus is otherwise generally referred to as a nonmetal, rather than referring specifically to its family name.

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