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.)

Short answer

The number of moles of sodium chloride (NaCl) in the solution is approximately 0.0273 moles.

Step-by-step solution

Write down the given values.

The given values are: - Moles of water (H2O) = 0.115 mol - Vapor pressure of the solution (Psol) = 25.7 torr - Vapor pressure of pure water (Pwater) = 31.8 torr

Write down Raoult's Law equation for the solution.

Raoult's Law states that the partial vapor pressure of each component in an ideal solution is equal to the product of the mole fraction of the component and its vapor pressure in the pure state. Mathematically, it can be written as: \(P_{A} = X_{A} * P_{A}^{\circ}\) For water in the solution, we can write: \(P_{H2O} = X_{H2O} * P_{H2O}^{\circ}\) Where, \(P_{H2O}\) is the vapor pressure of water in the solution, \(X_{H2O}\) is the mole fraction of water in the solution, and \(P_{H2O}^{\circ} = 31.8\, torr\) is the vapor pressure of pure water.

Calculate the mole fraction of water in the solution.

As sodium chloride (NaCl) is a strong electrolyte, it dissociates completely into ions (Na+ and Cl-) in the solution. Let's represent the unknown number of moles of NaCl as 'x'. Then, the total moles present in the solution will be (0.115 + x). The mole fraction of water in the solution can then be calculated as: \(X_{H2O} = \frac{moles \, of \, H2O}{total \, moles \, in \, solution} = \frac{0.115}{0.115+x}\)

Substitute the given vapor pressure of the solution in Raoult's Law equation.

We have the vapor pressure of the solution (Psol) = 25.7 torr. According to Raoult's Law for water, we can write: \(P_{H2O} = X_{H2O} * P_{H2O}^{\circ}\) Substituting the values, we have: \(25.7 = \frac{0.115}{0.115+x} * 31.8\)

Solve for x (number of moles of NaCl).

We can now solve the above equation for the unknown number of moles of NaCl (x): \(25.7 = \frac{0.115}{0.115+x} * 31.8\) Cross-multiplying, we get: \(25.7(0.115+x) = 0.115 * 31.8\) Now, simplify and solve for x: \(25.7(0.115+x) = 3.657\) Expanding and simplifying, we get: \(2.9555 + 25.7x = 3.657\) Subtracting 2.9555 from both sides, we have: \(25.7x = 0.7015\) Now, dividing by 25.7: \(x = \frac{0.7015}{25.7} = 0.0273 \, moles\) Therefore, the number of moles of sodium chloride (NaCl) in the solution is approximately 0.0273 moles.

Key concepts

Vapor Pressure

Vapor pressure is an important concept in understanding solutions and Raoult's Law. When we talk about vapor pressure, we are referring to the pressure created by the vapor that hovers over a liquid in a closed container, at a given temperature. This pressure occurs because some molecules at the liquid's surface gain enough energy to enter the gas phase. The vapor pressure increases as temperature rises because more molecules have sufficient energy to escape.

In the context of solutions, vapor pressure can be altered by adding a solute. According to Raoult's Law, the presence of a non-volatile solute (one that doesn’t easily vaporize) will lower the vapor pressure of the solvent. This is because the solute molecules occupy space on the surface, leaving fewer solvent molecules available to escape into the vapor phase. This effect is prominent in our problem, where sodium chloride acts as the solute, reducing the vapor pressure from 31.8 torr to 25.7 torr.

Mole Fraction

The mole fraction is a way to express concentration, specifically within a mixture. It is a dimensionless number that represents the ratio of the number of moles of one component to the total number of moles in the solution.

Imagine you have a container with 100 apples and 50 oranges. The mole fraction of apples would be the number of apples divided by the total number of fruits:

• Mole fraction of apples = 100 / (100 + 50) = 2/3

In solutions, knowing the mole fraction helps us determine how the presence of each component affects the overall properties of the solution.

In Raoult's Law, the mole fraction plays a critical role. It dictates how much each component contributes to the total vapor pressure of the solution. In the exercise, we calculate the mole fraction of water to determine its reduced vapor pressure in the presence of sodium chloride. You can think of it as dividing the contribution of each molecule by their available amount.

Electrolyte Dissociation

Electrolyte dissociation involves a process where an electrolyte, like sodium chloride, dissolves in water and breaks down into ions. This process greatly affects the properties of the solution, including its ability to conduct electricity.

Sodium chloride (NaCl) is considered a strong electrolyte. When NaCl dissolves in water, it completely dissociates into sodium (Na⁺) and chloride (Cl^-) ions. This complete dissociation is crucial because it effectively increases the number of particles in the solution. More particles mean that the colligative properties, like vapor pressure, are significantly altered.

In our problem, NaCl dissociates to contribute a higher number of moles in the solution, which in turn influences the calculation of the mole fraction. This demonstrates how the presence of electrolytes can affect physical aspects such as vapor pressure by adjusting how solutions interact at a molecular level.