Question

At a certain temperature, the vapor pressure of pure benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is \(0.930\) atm. A solution was prepared by dissolving \(10.0 \mathrm{~g}\) of a nondissociating, nonvolatile solute in \(78.11 \mathrm{~g}\) of benzene at that temperature. The vapor pressure of the solution was found to be \(0.900\) atm. Assuming the solution behaves ideally, determine the molar mass of the solute.

Short answer

The molar mass of the solute is approximately 294.2 g/mol.

Step-by-step solution

Understand Raoult's Law and its relation with mole fractions

According to Raoult's Law, the vapor pressure of an ideal solution is dependent on the mole fraction of its components. The vapor pressure of a solution (P_solution) is given by: P_solution = X_benzene * P°_benzene where X_benzene is the mole fraction of benzene in the solution, and P°_benzene is the vapor pressure of the pure benzene.

Write given data

We are given: - P°_benzene = \(0.930\) atm - Mass of solute = \(10.0 g\) - Mass of benzene = \(78.11 g\) - P_solution = \(0.900\) atm - Molar mass of benzene = M_benzene = \(78.11 g/mol\)

Express mole fraction in terms of molar mass of solute (M_solute)

To determine the molar mass of the solute (M_solute), calculate the mole fraction of benzene first: X_benzene = moles of benzene / (moles of benzene + moles of solute) We can calculate the moles of each component using their masses and molar masses, thus we can write the mole fraction as: X_benzene = (mass of benzene / M_benzene) / ((mass of benzene / M_benzene) + (mass of solute / M_solute)) Now that we have the expression of X_benzene in terms of molar mass of solute, we can proceed to use Raoult's Law.

Use Raoult's Law to find X_benzene

According to Raoult's Law: P_solution = X_benzene * P°_benzene Rearranging for X_benzene: X_benzene = P_solution / P°_benzene Now, plug the given values for P_solution and P°_benzene: X_benzene =\( \frac{0.900}{0.930} = 0.9677 \)

Solve for M_solute

Substitute the value of X_benzene into the expression for mole fraction that we derived in step 3: \(0.9677\) = \(\frac{78.11 / 78.11}{(78.11 / 78.11) + (10.0 / M_solute)}\) Now solve for M_solute by cross-multiplying and isolating M_solute: \((78.11 / 78.11) + (10.0 / M_solute) = \frac{1}{0.9677}\) \(10.0 / M_solute = \frac{1}{0.9677} - (78.11 / 78.11)\) \(M_solute = 10.0 / (\frac{1}{0.9677} - 1)\) \(M_solute = 294.2 g/mol\) So, the molar mass of the solute is approximately 294.2 g/mol.

Key concepts

Raoult's Law

Raoult's Law is fundamental to understanding the behavior of solutions, especially when it comes to their vapor pressures. Put simply, Raoult's Law states that the vapor pressure of an ideal solution is directly proportional to the mole fraction of the solvent in the mixture and the vapor pressure of the pure solvent.

The formula for Raoult's Law can be expressed as: \[ P_{\text{solution}} = X_{\text{solvent}} \cdot P^\circ_{\text{solvent}} \] where P_{solution} is the vapor pressure of the solution, X_{solvent} is the mole fraction of the solvent, and P^\circ_{solvent} is the vapor pressure of pure solvent.

For students who find the concept of mole fraction challenging, it might help to visualize it as a fraction of the whole. The mole fraction represents the proportion of the solvent's moles compared to the total moles of all components in the solution.

Understanding Raoult's Law is vital for solving problems related to vapor pressure changes when a solute is added to a solvent, as seen in the exercise about the reduction of vapor pressure in a benzene solution upon adding a nonvolatile solute.

Vapor Pressure

Vapor pressure is a measure of a liquid's tendency to evaporate into a gas. It's an important concept in chemistry because it gives us an idea of a substance's volatility. At a given temperature, every liquid has a characteristic vapor pressure which can be affected by various factors, including the presence of other substances such as solutes.

In the provided exercise, the vapor pressure of a solution is found to be less than that of the pure benzene. This phenomenon occurs due to the solute particles in the solution, which obstruct the escape of the benzene molecules into the vapor phase – a key point that should be highlighted for learners to grasp why the vapor pressure decreases in a solution.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It's defined as the ratio of the number of moles of that component to the total number of moles of all constituents of the mixture. Mathematically, we represent it as follows for a component A in a solution: \[ X_A = \frac{n_A}{n_A + n_B} \] where n_A is the number of moles of A, and n_B is the number of moles of another component B in the solution.

In the context of the textbook exercise, understanding mole fraction is crucial because Raoult's Law relies on it to relate the mole fraction of benzene to the observed vapor pressure of the solution. A common stumbling block for students is in calculating the moles of solute and solvent correctly; our learning materials should, therefore, prioritize clear explanations of this process.

Ideal Solution Behavior

Ideal solution behavior is a theoretical concept where the solution follows Raoult's Law at all concentrations, and the intermolecular forces between different components in the mixture are similar to those present in the pure substances.

In an ideal solution, the enthalpy change upon mixing the substances is zero, and the volume of the mixture is simply the sum of the volumes of the pure components. Advising students on what constitutes ideal behavior versus real-world deviations provides contextual understanding when they encounter discrepancies between theoretical calculations and experimental results.

The exercise we are focusing on assumes ideal behavior to simplify the process of determining the molar mass of the solute. This assumption allows us to use straightforward calculations without accounting for complex interactions that might occur in a non-ideal solution. Emphasising the assumption of ideal behavior is crucial for students to understand the limitations of the calculations and the conditions under which these calculations are valid.