Question

An aqueous solution is \(1.00 \% \mathrm{NaCl}\) by mass and has a density of \(1.071 \mathrm{~g} / \mathrm{cm}^{3}\) at \(25^{\circ} \mathrm{C}\). The observed osmotic pressure of this solution is \(7.83\) atm at \(25^{\circ} \mathrm{C}\).

Short answer

The i-factor (Van't Hoff factor) for the given NaCl aqueous solution is approximately 1.923. This was calculated using the mass percent, density, and osmotic pressure data provided, along with the van't Hoff equation.

Step-by-step solution

Calculate the mass of NaCl in 1 L solution.

To find the mass of NaCl in 1 L of solution, we need to use the density of the solution and the mass percentage of NaCl. One liter of the solution has a mass of 1071 grams, and 1.00% of this mass is NaCl. Mass of NaCl in 1 L solution = \((1.071 \times 1000) \times \frac{1.00}{100} = 10.71\) g

Calculate the moles of NaCl in 1 L solution.

Using the molecular weight of NaCl (58.44 g/mol), we can calculate the moles of NaCl in 1 L of solution. Moles of NaCl = \(\frac{10.71 \,\text{g}}{58.44 \,\text{g/mol}} = 0.1833 \,\text{mol}\)

Calculate the molar concentration of NaCl in 1 L solution.

Now we can use the moles of NaCl and the volume of the solution to find the molar concentration of NaCl. Molar concentration of NaCl = \(\frac{0.1833 \,\text{mol}}{1 \,\text{L}} = 0.1833 \,\text{M}\)

Apply the van't Hoff equation to find the i-factor.

The van't Hoff equation relates the osmotic pressure, molar concentration, temperature, and i-factor (Van't Hoff factor) for a solution: Osmotic Pressure (π) = i × Concentration (C) × Ideal Gas Constant (R) × Temperature (T) Given the osmotic pressure (7.83 atm), concentration (0.1833 M), and temperature (25°C = 298.15 K) the only missing parameter is i-factor. Rearranging the formula, we get, i = \(\frac{\text{Osmotic Pressure}}{(\text{Concentration})(\text{Ideal Gas Constant})(\text{Temperature})}\) i = \(\frac{7.83 \,\text{atm}}{(0.1833 \,\text{M})(0.0821 \,\frac{\text{L.atm}}{\text{mol.K}})(298.15 \,\text{K})}\) i = 1.923 The calculated i-factor for the NaCl solution is approximately 1.923, meaning that each molecule of NaCl dissociates into approximately 1.923 particles in the solution.

Key concepts

Aqueous Solution Concentration

To explore the concept of aqueous solution concentration, let's take a closer look at the notion of solute concentration in a solvent, such as water. In chemistry, whenever we dissolve a compound, like sodium chloride (NaCl), in a solvent, we create a solution. The extent to which the solute is dissolved—an indicator of the solution's 'strength'—is what we refer to as its concentration.

The concentration can be expressed mathematically in different ways, but in the context of our exercise, we are particularly interested in molarity. Molarity (M) is the measure of the number of moles of solute per liter of solution. Calculating the molarity involves determining the amount of solute—here 10.71 grams of NaCl—and converting this mass to moles using the substance's molecular weight. Then, we divide by the volume of the solution to find the molar concentration. It's important because it directly relates to the physical properties of the solution, such as osmotic pressure, freezing point, and boiling point.

Mass Percentage

When we talk about mass percentage, we are illustrating how to express the concentration of a component in a mixture or a solution. It is a way of articulating how much of a substance is present, by mass, compared to the entire mixture or solution. For example, a 1.00% NaCl solution means that for every 100 grams of the solution, there is 1 gram of NaCl.

To find the mass of a solute in a particular volume of solution, we need to know the density of the solution, which provides us with its mass for a given volume. Then, we apply the mass percentage to determine the mass of the solute. This amount is crucial for calculations in chemistry, especially when it comes to preparing solutions for reactions or for calculating properties like osmotic pressure, as seen in our exercise.

Van't Hoff Factor

The Van't Hoff factor (i) plays a significant role in physical chemistry, particularly in colligative properties—properties depending on the number of particles in the solution but not their nature. It indicates the number of particles a solute yields when dissolved. For instance, in our exercise with NaCl, this salt typically dissociates completely into 2 ions, and thus we would expect the i-factor to be 2.

However, there could be instances where the i-factor deviates from the textbook ideal because of ion associations or incomplete dissociation, leading to the necessity for experimental determination as we did in the solution steps provided. Applying the Van't Hoff equation for osmotic pressure and rearranging it allows us to solve for the i-factor, giving insight into the behavior of the solute in the solution. Understanding the Van't Hoff factor helps in predicting the solution's behavior and is crucial for designing processes in industry and science.