# 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.

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