Relation between molarity (M) and mole fraction is ___?

Let ‘M’ be the molarity of solution and ${\chi }_A,{\chi }_B$be mole fractions of solvent and solute, respectively.

Suppose $n_A and n_B$moles of solvent and solute are mixed to form solution.

Mass of solution = $n_A\times M_A+n_B\times M_B$

where $M_A and M_B$are molar masses of solvent and solute, respectively.

$V_{solution}=\frac{mass}{density}=\frac{n_A{M}_B+n_B{M}_B}{d}$

$Molarity, M=\frac{No.ofmolesofsolute\times 1000}{volumeofsolutioninml}$

$$\begin{gathered} \therefore M=\frac{n_B\times 1000\times d}{n_A{M}_B+n_B{M}_B} \\ dividebothnumeratorand\text{ denominator }with\left(n_A+n_B\right) \end{gathered}$$

$M=\frac{\frac{n_B}{n_A+n_B}\times 1000\times d}{\frac{n_A{m}_A}{n_A+n_B}+\frac{n_B{m}_B}{n_A+n_B}}=\frac{{\chi }_B\times 1000\times d}{{\chi }_A.M_A+{\chi }_B{M}_B}$

$M=\frac{{\chi }_B\times 1000\times d}{{\chi }_A.M_A+{\chi }_B{M}_B}$