# Whats the dimension of no of molecules per unit volume

I know its trivial but still i am confused,i know that no of molecules per unit volume (n) is dimentionless but my confusion is what if we describe it by total number of molecules by total volume(n=N/V) it gets a dimention.I know it isnt possible but i cant find a satisfactory reason.

I would guess you're thinking that unit volume is the number $1$ so molecules per unit volume is just a number divided by a number i.e. dimensionless.

But unit volume is not just the number $1$ because it could be one cubic metre or one cubic furlong or one cubic parsec etc etc. The unit volume has dimensions that tell you what that unit is. So if we are using SI units the number of molecules per unit volume has dimensions of $m^{-3}$.

If the number of molecules per unit volume is $n$ you might ask what is the number of molecules in a volume $V$.

The answer is $nV$ and it is a number ie dimensionless.

The volume $V$ has dimensions $\rm L^{3}$ so the dimensions of $n$ when multiplied by the dimensions of $V$ must be $1$.

$[n] \times \rm L^{3} =1 \Rightarrow [n] = \rm L^{-3}$

An SI unit of the the number of molecules per unit volume is $\rm m^{-3}$

The SI unit should be mole m^(-3).And the dimension should be L^(-3).