Problem 96 - Chemistry: The Central Science

Question

A solution contains \(0.115 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) and an unknown number of moles of sodium chloride. The vapor pressure of the solution at \(30^{\circ} \mathrm{C}\) is \(25.7\) torr. The vapor pressure of pure water at this temperature is \(31.8\) torr. Calculate the number of moles of sodium chloride in the solution. (Hint: remember that sodium chloride is a strong electrolyte.)

Step-by-Step Solution

Answer

The number of moles of sodium chloride in the solution is approximately 0.027. This result was calculated by first using Raoult's law to find the mole fraction of water in the solution. The mole fraction of sodium chloride was then found by subtracting the mole fraction of water from 1. Finally, this mole fraction was used to find the number of moles of sodium chloride in the solution.

Step 1: Write down Raoult's law

Raoult's law states that the vapor pressure of a solution is the product of the mole fraction of the solvent and its vapor pressure in the pure state: \(P_{solution} = x_{H_2O} \times P^{\circ}_{H_2O}\) Where: - \(P_{solution}\) is the vapor pressure of the solution, - \(x_{H_2O}\) is the mole fraction of water in the solution, - \(P^{\circ}_{H_2O}\) is the vapor pressure of pure water.

Step 2: Calculate the mole fraction of water

Rearrange Raoult's law to solve for the mole fraction of water: \(x_{H_2O} = \frac{P_{solution}}{P^{\circ}_{H_2O}}\) Substitute the given values: - \(P_{solution} = 25.7\) torr - \(P^{\circ}_{H_2O} = 31.8\) torr \(x_{H_2O} = \frac{25.7}{31.8} \approx 0.808\)

Step 3: Calculate the mole fraction of sodium chloride

The sum of the mole fractions in a solution is always equal to 1. Thus, we can write: \(x_{NaCl} = 1 - x_{H_2O}\) Substitute the calculated value of \(x_{H_2O}\): \(x_{NaCl} = 1 - 0.808 \approx 0.192\)

Step 4: Determine the number of moles of sodium chloride using mole fractions

We can relate the mole fraction of sodium chloride to the moles of water, which we know, and the moles of sodium chloride, which we want to find: \(x_{NaCl} = \frac{n_{NaCl}}{n_{H_2O} + n_{NaCl}}\) Rearrange the equation to solve for the number of moles of sodium chloride: \(n_{NaCl} = \frac{x_{NaCl} \times n_{H_2O}}{1 - x_{NaCl}}\) Substitute the calculated value of \(x_{NaCl}\) and the given value of \(n_{H_2O} = 0.115\) mol: \(n_{NaCl} = \frac{0.192 \times 0.115}{1 - 0.192} \approx 0.027\) Hence, there are approximately 0.027 moles of sodium chloride in the solution.

Key Concepts

Vapor Pressure

Vapor pressure is a measure of the tendency of a substance's molecules to evaporate into the gas phase. It represents the pressure exerted by the vapor of a liquid when it is in equilibrium with its liquid phase. In simpler terms, vapor pressure indicates how easily a liquid evaporates. A higher vapor pressure means the liquid evaporates more quickly, which is characteristic of volatile substances.

In the context of solutions, Raoult's Law helps us understand how adding a solute, such as sodium chloride, affects the vapor pressure of a solvent. The presence of non-volatile solutes generally lowers the vapor pressure of the solvent because it reduces the number of solvent molecules at the surface that can escape into the vapor phase. This is why the vapor pressure of the solution is often less than that of the pure solvent. Understanding vapor pressure changes is critical in fields ranging from meteorology to the manufacturing of perfumes and pharmaceuticals.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present. For example, in a solution containing water and sodium chloride, the mole fraction of water is given by:\[ x_{H_2O} = \frac{n_{H_2O}}{n_{H_2O} + n_{NaCl}} \]where \(n_{H_2O}\) is the number of moles of water, and \(n_{NaCl}\) is the number of moles of sodium chloride.

Mole fraction provides a simple and direct way to relate the proportion of a solute to a solvent without involving unit conversions typically required with other concentration measures, like molarity. It's also a key player in Raoult's Law, which uses mole fractions to calculate the vapor pressure of a solution, giving insights into the behavior of solute-solvent interactions.

Electrolyte Solutions

Electrolyte solutions are varieties of solutions that conduct electricity due to the presence of free ions in the solvent. These ions are produced when a solute, like sodium chloride, dissociates in a solvent such as water. Sodium chloride (NaCl) is a strong electrolyte, which means it completely dissociates into its ions, \(Na^+\) and \(Cl^-\), when dissolved in water.

The dissociation of sodium chloride affects the calculation of mole fractions and vapor pressures, as the dissociation increases the number of particles in the solution. This alteration must be accounted for in calculations involving colligative properties, such as vapor pressure lowering. Understanding how strong electrolyte solutions behaved is crucial in a range of applications, including biochemistry and electrochemistry, where the flow of ions is a routine part of the phenomena being studied.