Answer :
To find the molality of the methanol ([tex]\(CH_3OH\)[/tex]) solution, we need to use the relationship between molarity, density, and molality. Here's a step-by-step guide on how to calculate the molality:
1. Understand the Information Given:
- Molarity of the solution is [tex]\(20.3 \, M\)[/tex], which means there are 20.3 moles of methanol per liter of solution.
- The density of the solution is [tex]\(0.858 \, \text{g/mL}\)[/tex]. Converting to [tex]\(g/L: 0.858 \, \text{g/mL} \times 1000 \, \text{mL/L} = 858 \, \text{g/L}\)[/tex].
2. Calculate the Mass of Methanol in One Liter of Solution:
- The molar mass of methanol ([tex]\(CH_3OH\)[/tex]) is approximately [tex]\(12.01 + (3 \times 1.01) + 16.00 + 1.01 = 32.04 \, \text{g/mol}\)[/tex].
- The mass of methanol in the solution: [tex]\(20.3 \, \text{moles/L} \times 32.04 \, \text{g/mol} = 650.615 \, \text{g}\)[/tex].
3. Calculate the Mass of the Solvent (Water) in One Liter of Solution:
- Total mass of 1 liter of solution is 858 g (from its density).
- Therefore, the mass of water (solvent) is: [tex]\(858 \, \text{g} - 650.615 \, \text{g} = 207.385 \, \text{g}\)[/tex].
4. Convert the Mass of the Solvent to Kilograms:
- Mass of water = [tex]\(207.385 \, \text{g} = 0.207385 \, \text{kg}\)[/tex].
5. Calculate the Molality of the Solution:
- Molality is the number of moles of solute per kilogram of solvent.
- Molality [tex]\(m = \frac{20.3 \, \text{moles of CH}_3\text{OH}}{0.207385 \, \text{kg of water}} \approx 97.9 \, \text{m}\)[/tex].
Therefore, the molality of the solution is approximately [tex]\(97.9 \, m\)[/tex], which closely matches option C.
1. Understand the Information Given:
- Molarity of the solution is [tex]\(20.3 \, M\)[/tex], which means there are 20.3 moles of methanol per liter of solution.
- The density of the solution is [tex]\(0.858 \, \text{g/mL}\)[/tex]. Converting to [tex]\(g/L: 0.858 \, \text{g/mL} \times 1000 \, \text{mL/L} = 858 \, \text{g/L}\)[/tex].
2. Calculate the Mass of Methanol in One Liter of Solution:
- The molar mass of methanol ([tex]\(CH_3OH\)[/tex]) is approximately [tex]\(12.01 + (3 \times 1.01) + 16.00 + 1.01 = 32.04 \, \text{g/mol}\)[/tex].
- The mass of methanol in the solution: [tex]\(20.3 \, \text{moles/L} \times 32.04 \, \text{g/mol} = 650.615 \, \text{g}\)[/tex].
3. Calculate the Mass of the Solvent (Water) in One Liter of Solution:
- Total mass of 1 liter of solution is 858 g (from its density).
- Therefore, the mass of water (solvent) is: [tex]\(858 \, \text{g} - 650.615 \, \text{g} = 207.385 \, \text{g}\)[/tex].
4. Convert the Mass of the Solvent to Kilograms:
- Mass of water = [tex]\(207.385 \, \text{g} = 0.207385 \, \text{kg}\)[/tex].
5. Calculate the Molality of the Solution:
- Molality is the number of moles of solute per kilogram of solvent.
- Molality [tex]\(m = \frac{20.3 \, \text{moles of CH}_3\text{OH}}{0.207385 \, \text{kg of water}} \approx 97.9 \, \text{m}\)[/tex].
Therefore, the molality of the solution is approximately [tex]\(97.9 \, m\)[/tex], which closely matches option C.
