Calculate Molality of Solution Given Molarity and Solvent Density

To find the molality of the given solution, we use the definitions of molarity and molality along with the given conditions.

1. Moles of Solute: The molarity of the solution is given as $M = 3.2\text{ M}$, which means $1\text{ L}$ ($1000\text{ mL}$) of the solution contains: $n_{\text{solute}} = 3.2\text{ moles}$

2. Mass of Solvent: Since it is given that there is no change in volume upon dissolution, the volume of the solvent equals the volume of the solution: $V_{\text{solvent}} = V_{\text{solution}} = 1000\text{ mL}$

Given the density of the solvent is $d_{\text{solvent}} = 0.4\text{ g mL}^{-1}$, the mass of the solvent ($w_{\text{solvent}}$) present in $1\text{ L}$ of the solution is: $w_{\text{solvent}} = V_{\text{solvent}} \times d_{\text{solvent}} = 1000\text{ mL} \times 0.4\text{ g mL}^{-1} = 400\text{ g} = 0.4\text{ kg}$

3. Molality of the Solution: Molality ($m$) is defined as the number of moles of solute per kilogram of solvent: $m = \frac{n_{\text{solute}}}{w_{\text{solvent}}\text{ (in kg)}}$

Substituting the values: $m = \frac{3.2\text{ mol}}{0.4\text{ kg}} = 8\text{ m}$

Thus, the molality of the $3.2\text{ M}$ solution is 8.