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Matter isn't a term that is clearly defined, however I have found it most commonly used distinctively different than mass by using the assumption that an atom is a quantized form of matter. Ergo, the number of atoms is the amount of matter in an object.

The unit for a number of atoms is Moles. A mole is Avagadro's Number of atoms, which is independent of the mass of the matter. Thus, knowing the mass of a discrete number of moles of a particular substance, and knowing the mass of a given sample, you can calculate how much "matter" is actually present in the sample.

Also, don't confuse any of this with volume. This was all independent of volume.

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