Question

How would you prepare \(1.0 \mathrm{~L}\) of an aqueous solution of sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) having an osmotic pressure of \(15 \mathrm{~atm}\) at a temperature of \(22^{\circ} \mathrm{C} ?\) Sucrose is a nonelectrolyte.

Short answer

To prepare 1.0 L of an aqueous solution of sucrose with an osmotic pressure of 15 atm at 22°C, first, calculate the molar concentration using the osmotic pressure formula, which gives \(c ≈ 0.6595 \mathrm{ ~M}\). Then, calculate the mass of sucrose required by multiplying the molar concentration, molar mass of sucrose, and the volume of the solution, resulting in \(m ≈ 225.63 \mathrm{ ~g}\). Finally, dissolve approximately 225.63 g of sucrose in water and adjust the total volume to be 1.0 L.

Step-by-step solution

1. Convert temperature to Kelvin

To convert the given temperature from Celsius to Kelvin, add 273.15: $$T(K) = T(^\circ C) + 273.15$$ $$T = 22 ^\circ C + 273.15 = 295.15 K$$

2. Calculate molar concentration from the osmotic pressure formula

Using the osmotic pressure formula, \(\Pi = cRT\), we can isolate the molar concentration (c) and solve for it: $$c = \frac{\Pi}{RT}$$ Plug in the values: $$c = \frac{15 \mathrm{~atm}}{(0.0821 \mathrm{ ~L\cdot atm/mol \cdot K})(295.15 \mathrm{ ~K})}$$ Calculate the molar concentration (c): $$c ≈ 0.6595 \mathrm{ ~M}$$

3. Calculate the mass of sucrose needed

To calculate the mass of sucrose needed, multiply the molar concentration by the molar mass of sucrose and the volume of the solution: Molar mass of sucrose (C12H22O11) = 12(12.01) + 22(1.008) + 11(16.00) = 342.3 g/mol Mass of sucrose needed (m) = Molar concentration (c) × Molar mass × Volume (V): $$m = (0.6595 \mathrm{ ~M})(342.3 \mathrm{ ~g/mol})(1.0 \mathrm{ ~L})$$ $$m ≈ 225.63 \mathrm{ ~g}$$

4. Prepare the solution

To prepare 1.0 L of the aqueous sucrose solution with osmotic pressure of 15 atm at 22°C, dissolve approximately 225.63 g of sucrose in water, and adjust the total volume to be 1.0 L.