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SOLUTION:

Step 1: List the given values.

[tex]\begin{aligned} & M = 5.86 \: M \\ & MM_{\text{solute}} = \text{46.068 g/mol} \\ & d_{\text{solution}} = \text{0.927 g/mL} \end{aligned}[/tex]

Step 2: Calculate the number of moles of solute.

Assume that the volume of the solution is 1 L (1000 mL).

[tex]\begin{aligned} n_{\text{solute}} & = (M)(V_{\text{solution}}) \\ & = (5.86 \: M)(\text{1 L}) \\ & = \text{(5.86 mol/L)(1 L)} \\ & = \text{5.86 mol} \end{aligned}[/tex]

Step 3: Calculate the mass of solute.

[tex]\begin{aligned} mass_{\text{solute}} & = (n_{\text{solute}})(MM_{\text{solute}}) \\ & = \text{(5.86 mol)(46.068 g/mol)} \\ & = \text{269.96 g} \end{aligned}[/tex]

Step 4: Calculate the mass of solution.

[tex]\begin{aligned} mass_{\text{solution}} & = d_{\text{solution}}V_{\text{solution}} \\ & = \text{(0.927 g/mL)(1000 mL)} \\ & = \text{927 g} \end{aligned}[/tex]

Step 5: Calculate the mass of solvent.

[tex]\begin{aligned} mass_{\text{solvent}} & = mass_{\text{solution}} - mass_{\text{solute}} \\ & = \text{927 g} - \text{269.96 g} \\ & = \text{657.04 g} \\ & = \text{0.65704 kg} \end{aligned}[/tex]

Step 6: Calculate the molality of the solution.

[tex]\begin{aligned} m & = \frac{n_{\text{solute}}}{mass_{\text{solvent}}} \\ & = \frac{\text{5.86 mol}}{\text{0.65704 kg}} \\ & = \text{8.92 mol/kg} \\ & = \boxed{8.92 \: m} \end{aligned}[/tex]

Hence, the molality of a 5.86 M ethanol solution is 8.92 m.

[tex]\\[/tex]

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