Question

What is the density of an aqueous solution of potassium nitrate that has a normal boiling point of \(103.0^{\circ} \mathrm{C}\) and an osmotic pressure of 122 atm at \(25^{\circ} \mathrm{C}^{2}\)

Short answer

Answer: The density of the aqueous solution of potassium nitrate is 101.7785 g/L.

Step-by-step solution

Determine the boiling point elevation of the solution

To determine the boiling point elevation (∆Tb) of the solution, subtract the normal boiling point of water (100°C) from the given boiling point of the solution (103.0°C): ∆Tb = 103.0°C - 100°C = 3.0°C

Calculate the molality of the solution using the boiling point elevation

Using the boiling point elevation formula, we can find the molality (m) of the solution: ∆Tb = Kb * m, where Kb is the molal boiling-point elevation constant for water, which is 0.512°C/molal. Rearranging the formula to solve for molality (m): m = ∆Tb / Kb = 3.0°C / 0.512°C/molal = 5.8594 molal.

Determine the molar concentration of the solution using the osmotic pressure

Using the osmotic pressure formula, we can find the molar concentration (C) of the solution: π = CRT, where π is the osmotic pressure (122 atm), C is the molar concentration, R is the ideal gas constant (0.08206 L atm/mol K), and T is the temperature (25°C = 298.15 K). Rearranging the formula to solve for molar concentration (C): C = π / (RT) = 122 atm / (0.08206 L atm/mol K * 298.15 K) = 5.0266 mol/L.

Find the mass of potassium nitrate in the solution

Using the molality (m) found in step 2 and the molar concentration (C) found in step 3, we can calculate the mass of potassium nitrate (KNO3) in the solution: mass of KNO3 = C * molar mass of KNO3 / m, where the molar mass of KNO3 is 101 g/mol. mass of KNO3 = 5.0266 mol/L * 101 g/mol / 5.8594 molal = 86.9385 g/L.

Calculate the mass of water in the solution

Since we have the molality (m) of the solution, we can calculate the mass of water in the solution: mass of water = mass of KNO3 / m = 86.9385 g/L / 5.8594 molal = 14.84 g/L.

Calculate the density of the solution

Using the mass of potassium nitrate and mass of water in the solution, we can calculate the density (ρ) of the solution, which is the mass per unit volume: ρ = (mass of KNO3 + mass of water) / volume = (86.9385 g/L + 14.84 g/L) / 1 L = 101.7785 g/L. Therefore, the density of the aqueous solution of potassium nitrate is 101.7785 g/L.

Key concepts

Boiling Point Elevation

Boiling point elevation occurs when a solute is dissolved in a solvent, causing the boiling point of the resulting solution to be higher than that of the pure solvent. In our exercise, potassium nitrate is dissolved in water, elevating the boiling point of water from its normal 100°C to 103°C. This difference of 3°C is the boiling point elevation (ΔTb). The phenomenon is dependent on the number of solute particles in the solution and not on their identity. Understanding this concept is crucial for students as it applies to daily life scenarios, such as adding salt to water when cooking to raise its boiling point.

Molality

Molality (m) is a measure of the concentration of a solute in a solvent and is defined as the number of moles of solute per kilogram of solvent. This value does not change with temperature, which is why it's particularly useful in boiling point elevation and freezing point depression calculations. To calculate molality, as seen in the solution, we use the formula ΔTb = Kb * m, where Kb is a constant unique to each solvent. Converting the boiling point elevation to molality gives students a means to link qualitative observations, such as boiling point change, to quantitative measurements involving moles and mass.

Osmotic Pressure

Osmotic pressure (π) is a property of solutions that describes the pressure needed to prevent the inward flow of its pure solvent across a semipermeable membrane. It is a critical concept which finds extensive use in fields such as biology and medicine, for example, when determining the properties of blood or plant cells. The osmotic pressure equation, π = CRT, involves the ideal gas constant (R) and the temperature (T) in Kelvin, showcasing the relationship between the physical parameters of gases and solute concentration in solutions.

Molar Concentration

Molar concentration (C) refers to the moles of solute per liter of solution, often expressed in mol/L, and is also known as molarity. It's a pivotal concept for students to grasp because it allows for direct comparison of solute concentrations across different solutions. By understanding molarity, students can solve for various properties of solutions, such as osmotic pressure or the total amount of a substance in a given volume, which is essential in chemical reactions and stoichiometry. In the exercise, knowing the osmotic pressure and using the ideal gas equation facilitates the calculation of molar concentration.

Ideal Gas Constant

The ideal gas constant (R) is a fundamental value in chemistry that appears in the universal gas equation, PV = nRT. It's a bridge between the macroscopic properties of gases—pressure (P), volume (V), and temperature (T)—and the microscopic aspect of moles of gas (n). This constant enables us to apply the principles of gaseous behavior to solutions, as seen in the calculation of osmotic pressure. The value of R is 0.08206 L atm/mol K, a number that students should be familiar with to solve a variety of chemical problems involving gases and solutions.