Question

Consider an aqueous solution containing sodium chloride that has a density of 1.01 \(\mathrm{g} / \mathrm{mL}\) . Assume the solution behaves ideally. The freezing point of this solution at 1.0 \(\mathrm{atm}\) is \(-1.28^{\circ} \mathrm{C}\) . Calculate the percent composition of this solution (by mass).

Short answer

The percent composition of the sodium chloride solution (by mass) is approximately 21.2%.

Step-by-step solution

Calculate the molality of the solution using freezing point depression formula

We are given the freezing point depression of the solution (ΔTf = 1.28°C). Using the freezing point depression formula, we can calculate the molality (m) of the solution: ΔTf = Kf × m where Kf is the cryoscopic constant, which is 1.86°C kg/mol for water. Now, we can calculate the molality: 1.28°C = 1.86°C kg/mol × m m = \( \frac{1.28}{1.86} \) mol/kg

Calculate the mass of solute using molality and density

Now that we have the molality, we can use the given density of the solution to find the mass of solute in 1 liter (or 1,000mL) of the solution: Density = \( \frac{mass_{solution}}{volume_{solution}} \) 1.01 g/mL = \( \frac{mass_{solution}}{1000\,\mathrm{mL}} \) mass_solution = 1010 g We know that molality (m) = \( \frac{moles_{solute}}{mass_{solvent_{(kg)}}} \) Rearranging this formula, we get: mass_solute = moles_solute × molar_mass_solute mass_solvent = mass_solution - mass_solute We now have a system of two equations with two variables (mass_solute and mass_solvent): \( \frac{mass_{solute}}{mass_{solvent}/1000} \) = \(\frac{1.28}{1.86}\) mass_solution = mass_solute + mass_solvent

Solve for mass_solute and mass_solvent

By solving the two equations, we can find the mass_solute and mass_solvent: mass_solute = \( \frac{mass_{solution}}{1 + \frac{molar\_mass_{solute} \times 1.86}{1.28 \times 1000}} \) For sodium chloride (NaCl), the molar_mass_solute = 58.44 g/mol. Plugging the values: mass_solute = \( \frac{1010}{1 + \frac{58.44 \times 1.86}{1.28 \times 1000}} \) = 214.16 g (approximately) mass_solvent = mass_solution - mass_solute = 1010 - 214.16 = 795.84 g (approximately)

Calculate the percent composition by mass

Now we can calculate the mass percent of sodium chloride in the solution: Percent composition (NaCl) = \( \frac{mass_{solute}}{mass_{solution}} \times 100\% \) Percent composition (NaCl) = \( \frac{214.16}{1010} \times 100\% \) ≈ 21.2% The percent composition of the sodium chloride solution (by mass) is approximately 21.2%.

Key concepts

Molality

Molality is a way of expressing the concentration of a solute in a solution. Unlike molarity, which is based on the volume of the solution, molality (\( m \)) refers to the number of moles of solute per kilogram of solvent. This makes molality particularly useful for situations involving temperature changes, as its value remains constant even when there is a change in volume due to temperature fluctuations.
To find molality, the equation used is:

• \[ m = \frac{\Delta T_f}{K_f} \]
• Where \( \Delta T_f \) is the freezing point depression, and \( K_f \) is the cryoscopic constant of the solvent.

In our example, \( \Delta T_f \) is 1.28°C and with water's \( K_f \) of 1.86°C kg/mol, the calculated molality is \( \frac{1.28}{1.86} \) mol/kg. Understanding how to calculate and interpret molality is crucial for solving problems related to colligative properties like freezing point depression.

Cryoscopic Constant

The cryoscopic constant (\( K_f \)) is a property specific to every solvent and plays a fundamental role in calculating the freezing point depression of a solution. This constant primarily depends on the nature of the solvent and represents how much the freezing point of the solvent will lower with the addition of 1 molal of a solute.
For water, the widely recognized cryoscopic constant is 1.86°C kg/mol. This means for each mole of solute added per kilogram of water, the freezing point decreases by 1.86°C.
This constant is crucial for calculating molecular weights and concentrations in solution chemistry. When dealing with solutions, particularly when wanting to understand the change in freezing points, always remember to identify the correct cryoscopic constant for the solvent used.

Percent Composition

Percent composition by mass is an important concept that helps us understand the proportion of each component in a solution. It is expressed as the mass of the solute divided by the total mass of the solution, multiplied by 100 to convert it into a percentage.
The formula is:

• \[ \text{Percent Composition} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 100\%\]

In the given example, the mass of sodium chloride (solute) is approximately 214.16 g, and the total mass of the solution is 1010 g. Thus, the percent composition of NaCl in the solution is approximately 21.2%.
Understanding percent composition is crucial for formulating solutions with precise concentrations, necessary in both industrial applications and everyday chemistry tasks.