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The IUPAC Gold Book defines

Amount concentration

Amount of a constituent divided by the volume of the mixture. Also called amount-of-substance concentration, substance concentration (in clinical chemistry) and in older literature molarity. For entities B it is often denoted by B. The common unit is mole per cubic decimetre or mole per litre sometimes denoted by M.

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Number concentration

Number of entities of a constituent in a mixture divided by the volume of the mixture.

I can't see a distinction between the two terms since it seems to me that "Amount of a constituent" is the same as "Number of entities of a constituent".

Could you please explain the difference between them? Is it a conceptual difference or just a difference in the context in which they are most commonly used?

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The "amount" of a substance is the proper term for the "number of moles" (Wikipedia; Gold Book). So, the two quantities are related by a factor of the Avogadro constant, $\pu{6.022\times 10^23 mol-1}$.

If you have $\pu{1 mol}$ of helium in a $\pu{1 m3}$ box, the amount concentration is $\pu{1 mol m-3}$, or $\pu{0.001 mol dm-3}$ if you prefer those units.

The corresponding number concentration is $\pu{6.022\times 10^23 m-3}$.

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  • $\begingroup$ Isn't mol just a specific unit for amount? According to the International System of Units, amount of substance is a quantity, while mol is a unit. I think one can use other units for amount if one wishes: tens, thousands, pi, dozens, etc. $\endgroup$ – toliveira Jul 26 '18 at 11:15

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