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 atm at a temperature of \(22^{\circ} \mathrm{C} ?\) Sucrose is a nonelectrolyte.

Short answer

To prepare 1.0 L of the aqueous sucrose solution having an osmotic pressure of 15 atm at a temperature of 22°C, first calculate the molarity (M) using the formula: Osmotic Pressure = Molarity × Gas Constant × Temperature. Find the molarity to be 0.685 mol/L. Next, calculate the mass of sucrose needed, which is approximately 234.27 g. Finally, dissolve 234.27 g of sucrose in water and adjust the volume to 1.0 L.

Step-by-step solution

Identify the given information

We are given: - Osmotic pressure = 15 atm - Temperature = 22°C (which we will convert to Kelvin) - Molecular formula of sucrose = C12H22O11

Convert the temperature to Kelvin

To convert the temperature from Celsius to Kelvin, add 273.15: Temperature = 22 + 273.15 = 295.15 K

Determine the osmotic pressure formula and solve for molarity

We will use the formula: Osmotic Pressure = Molarity × Gas Constant × Temperature Where the gas constant (R) has a value of 0.08206 L atm/mol K. Now we will solve for Molarity (M): 15 atm = M × 0.08206 L atm/mol K × 295.15 K M = (15 atm) / (0.08206 L atm/mol K × 295.15 K) M = 0.685 mol/L

Calculate the mass of sucrose needed

To prepare 1.0 L of the solution, we need to determine the mass of sucrose needed for a molarity of 0.685 mol/L. First, calculate the molar mass of sucrose: Molar mass of sucrose = (12 × 12) + (22 × 1) + (11 × 16) = 342 g/mol Now, calculate the mass of sucrose needed: Mass of sucrose = Molarity × Volume × Molar mass of sucrose Mass of sucrose = 0.685 mol/L × 1.0 L × 342 g/mol Mass of sucrose ≈ 234.27 g

Prepare the solution

To prepare 1.0 L of the aqueous sucrose solution having an osmotic pressure of 15 atm at a temperature of 22°C, dissolve approximately 234.27 g of sucrose in water and adjust the volume to 1.0 L.

Key concepts

Sucrose Solution

A sucrose solution is created when sucrose, a type of sugar, is dissolved in water. This is a popular experiment to understand properties like osmotic pressure. Sucrose is a common carbohydrate found in plants, and it is usually extracted from sugarcane or sugar beets. In a laboratory setting, creating a sucrose solution helps students explore concepts of concentration and solution dynamics. Since sucrose is a nonelectrolyte, it doesn’t dissociate into ions when dissolved, which means every molecule of sucrose stays intact. This property is important when calculating osmotic pressure because it affects the number of solute particles in the solution.

When working with sucrose solutions, it's crucial to accurately measure the amount of sucrose and water to ensure the desired concentration. This precision is important when performing experiments like determining osmotic pressure, where the concentration of the solution directly impacts the calculated results.

Molarity Calculation

Molarity is a common way to express the concentration of a solution, and it is defined as the number of moles of solute per liter of solution. In our exercise, we need to calculate the molarity to find out how much sucrose is required for the solution to have a specific osmotic pressure.

The formula for osmotic pressure (\( \pi \)) is given as:

\[\pi = M \times R \times T\]where:

• \( \pi \) = Osmotic pressure
• \( M \) = Molarity
• \( R \) = Gas constant (0.08206 L atm/mol K)
• \( T \) = Temperature in Kelvin

To find molarity, rearrange the formula:

\[M = \frac{\pi}{R \times T}\]Using the given osmotic pressure of 15 atm and a temperature of 295.15 K (converted from Celsius), we calculate the molarity to be 0.685 mol/L. This tells us how much sucrose is needed per liter to achieve the desired osmotic pressure.

Temperature Conversion

Temperature conversion is an essential step in this calculation because scientific equations often require temperatures to be in Kelvin rather than Celsius. The Kelvin scale is preferred in scientific calculations because it is an absolute scale, starting from absolute zero, where molecular motion theoretically stops.

To convert the temperature from Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example, the temperature given in the problem is 22°C, so the conversion is:

\[22 + 273.15 = 295.15 \, \text{K}\]This accurate conversion is crucial in ensuring that subsequent calculations, such as those involving the gas constant, are correct.

Always double-check your conversion, as errors can significantly impact results, especially in experiments dealing with pressure or concentrations.

Molar Mass Determination

To determine the mass of sucrose needed for the solution, it's essential to calculate its molar mass accurately. The molar mass of a compound is the sum of the masses of all the atoms in its formula. For sucrose (\(\text{C}_{12}\text{H}_{22}\text{O}_{11}\)), this involves:

• 12 carbon atoms each with an atomic mass of 12 g/mol,
• 22 hydrogen atoms each with an atomic mass of 1 g/mol, and
• 11 oxygen atoms each with an atomic mass of 16 g/mol.

The total molar mass is calculated as:

\[(12 \times 12) + (22 \times 1) + (11 \times 16) = 342 \, \text{g/mol}\]Once you have the molar mass, you can determine how many grams of sucrose you need by using the previously calculated molarity and desired volume of the solution.

In this exercise, for a molarity of 0.685 mol/L in 1.0 L of solution, about 234.27 g of sucrose is required. Accurate molar mass determination is pivotal in preparing solutions with precise concentrations, necessary for reliable experimental outcomes.