Question

A solution contains \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) and an unknown number of moles of sodium chloride. The vapor pressure of the solution at \(29^{\circ} \mathrm{C}\) is \(3.85 \mathrm{kPa}\). The vapor pressure of pure water at this temperature is \(4.05 \mathrm{kPa}\). Calculate the number of grams of sodium chloride in the solution. (Hint: Remember that sodium chloride is a strong electrolyte.)

Short answer

There are 1.44 grams of sodium chloride in the solution.

Step-by-step solution

Find the mole fraction of water using Raoult's Law

Raoult's Law states that the vapor pressure of a component in an ideal solution is proportional to the mole fraction of that component in the solution. Mathematically, this can be represented as: \[P_A = X_A \times P^*_A\] where: \(P_A\) = vapor pressure of component A in the solution \(X_A\) = mole fraction of component A in the solution \(P^*_A\) = vapor pressure of pure component A We can rearrange this equation to solve for the mole fraction of water: \[X_{H_2O} = \frac{P_{H_2O}}{P^*_{H_2O}}\] Given the vapor pressure of the solution, \(P_{H_2O} = 3.85\, kPa\), and the vapor pressure of pure water, \(P^*_{H_2O} = 4.05 \, kPa\), we can find the mole fraction of water: \[X_{H_2O} = \frac{3.85}{4.05} = 0.9506\]

Calculate the mole fraction of sodium chloride

Since there are only two components in the solution, water and sodium chloride, the sum of their mole fractions must be equal to 1. Therefore, the mole fraction of sodium chloride, \(X_{NaCl}\), can be calculated as: \[X_{NaCl} = 1 - X_{H_2O} = 1 - 0.9506 = 0.0494\]

Consider sodium chloride as a strong electrolyte

Sodium chloride is a strong electrolyte, which means it completely dissociates into its constituent ions in the solution. This results in two moles of ions (Na+ and Cl-) for each mole of sodium chloride. Therefore, the mole fraction of solute particles, \(X_s\), is twice the mole fraction of sodium chloride: \[X_s = 2 \times X_{NaCl} = 2 \times 0.0494 = 0.0988\]

Calculate the number of moles of sodium chloride

Since the moles of water in the solution is given as 0.5 mol, we can use the mole fraction of solute particles to determine the moles of sodium chloride: \[\text{moles of NaCl} = \frac{X_s}{2} \times \text{moles of }\, H_2O\] \[\text{moles of NaCl} = \frac{0.0988}{2} \times 0.5 = 0.0247\, \text{mol}\]

Convert the number of moles of sodium chloride to grams

We can now convert the number of moles of sodium chloride to grams using its molecular weight (MW = 58.44 g/mol): \[\text{grams of NaCl} = \text{moles of NaCl} \times \text{molecular weight of NaCl}\] \[\text{grams of NaCl} = 0.0247\, \text{mol} \times 58.44\, \frac{\text{g}}{\text{mol}} = 1.44\, \text{g}\] So there are 1.44 grams of sodium chloride in the solution.

Key concepts

Vapor Pressure

Vapor pressure is an essential concept in chemistry, particularly when discussing solutions. It is the pressure exerted by the vapor of a liquid at equilibrium with its liquid phase. In simpler terms, it is how much a liquid evaporates at a given temperature. When a liquid is in a closed container, some molecules escape into the air and create vapor pressure. The vapor pressure depends on the temperature and the nature of the liquid. For water, at 29°C, pure water has a vapor pressure of 4.05 kPa. When we add a solute like sodium chloride to water, the vapor pressure decreases. This is due to the solute particles occupying space at the surface of the liquid, reducing the amount of water that can escape into the vapor phase. This principle is described by Raoult's Law, which relates the reduction in vapor pressure to the mole fraction of the solvent in the solution.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of one component to the total number of moles in the solution. For a solution containing only water and sodium chloride, we consider the mole fraction of water and the mole fraction of sodium chloride. According to Raoult's Law, for a component in a solution, its vapor pressure is directly proportional to its mole fraction. Thus, knowing the vapor pressures, we can calculate the mole fraction of water, as done in the solution. The formula used is:

• \( X_{H_2O} = \frac{P_{H_2O}}{P^*_{H_2O}} \)

Once we find the mole fraction of water, we can easily find that for sodium chloride, which impacts how we calculate its concentration and effects in solution.

Strong Electrolytes

Strong electrolytes are substances that completely dissociate into ions in water. Sodium chloride (NaCl) is an example of a strong electrolyte. When dissolved, it separates into sodium ions (Na\(^+\)) and chloride ions (Cl\(^-\)). This complete dissociation means that in a solution, there are twice as many ions as there were moles of sodium chloride initially added. For instance, if we know the mole fraction of NaCl, we must consider twice the amount when calculating its effect on the solution's properties, such as its vapor pressure. The dissociation into ions is crucial in understanding how solutions alter properties like boiling and freezing points, conductivity, and vapor pressure.

Sodium Chloride Dissociation

The dissociation of sodium chloride in water is a fundamental chemical process. Once dissolved, each formula unit of NaCl separates into one Na\(^+\) ion and one Cl\(^-\) ion. This process is called dissociation and is characteristic of strong electrolytes. The resulting ions are free to move in solution, allowing the solution to conduct electricity. In the context of vapor pressure and Raoult's Law, this dissociation means that when we calculate effects on the solution, we double the mole fraction for sodium chloride because two ions result from one molecule. Understanding this dissociation helps in comprehending the behavior of solutions and their interactions with solvents at a molecular level.