# Derive the Relation Between Molarity and Molality

The relation between molarity and molality is:

Consider,

• W to be the mass of the given solute
• W’ to be the weight of the solvent
• V to be the volume of the solution
• m to be the molality
• M to be the molarity
• M’ to be the molar mass of the solute

Molarity, M = $frac{W}{M’}times frac{1000}{V}$ (equ.1)

Molality, m = $frac{W}{M’}times frac{1000}{W’}$ (equ.2)

Density, d = $frac{M}{V}=frac{W+W’}{V}$

From equ.1

$V=frac{Wtimes 1000}{MM’}$ (equ.3)

From equ.2

$W’=frac{Wtimes 1000}{mM’}$

Therefore,

w+W’ = W + $frac{Wtimes 1000}{mM’}$ $frac{1000}{M’}[frac{mM’}{1000m}+frac{1}{m}]W'$ (equ.4)

Dividing equ.4 by equ.3,

$frac{d}{M}=frac{1}{m}+frac{M’}{1000}$ $frac{1}{m}=frac{d}{M}-frac{M’}{1000}$ $m=frac{Mtimes 1000}{(dtimes 1000)-M(M’)}$

Therefore, above is the relation between molality and molarity.