Problem 18 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The vapour pressure of pure benzene at a certain temperature is \(0.85\) bar. A \(\begin{array}{ll}\text { non-volatile, } & \text { non-electrolyte } & \text { solid }\end{array}\) weighing \(0.5 \mathrm{~g}\) is added to \(39.0 \mathrm{~g}\) of benzene. The vapour pressure of the solution then is \(0.845\) bar. Molar mass of the solid substance is (a) \(169 \mathrm{~g} / \mathrm{mol}\) (b) \(170 \mathrm{~g} / \mathrm{mol}\) (c) \(85 \mathrm{~g} / \mathrm{mol}\) (d) \(39 \mathrm{~g} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The molar mass of the solid substance is (b) \(170 \mathrm{~g} / \mathrm{mol}\).

Step 1 - Understand the Concept of Raoult's Law

Raoult's Law states that for a solution with non-volatile solute, the vapor pressure of the solvent decreases proportionally to the presence of solute particles. It's formulated as \( P = X_{\rm solvent} \cdot P^0_{\rm solvent} \) where \( P \) is the vapor pressure of the solution, \( X_{\rm solvent} \) is the mole fraction of the solvent, and \( P^0_{\rm solvent} \) is the vapor pressure of the pure solvent.

Step 2 - Calculate the Mole Fraction of Benzene in the Solution

The mole fraction of benzene, \( X_{\rm benzene} \) can be calculated using the change in vapor pressure. The new vapor pressure after adding the solute is \( P = 0.845 \) bar and the vapor pressure of pure benzene is \( P^0 = 0.85 \) bar. According to Raoult's Law, \( X_{\rm benzene} = \frac{P}{P^0} = \frac{0.845}{0.85} \)

Step 3 - Find the Mole Fraction of the Solute

Using the relationship that the sum of mole fractions in a solution equals 1, the mole fraction of the solute is \( X_{\rm solute} = 1 - X_{\rm benzene} \)

Step 4 - Calculate the Moles of Benzene

First, calculate the moles of benzene using its mass and the known molar mass of benzene, which is \( 78.11 \, \mathrm{g/mol} \) (referencing a known data source). \( n_{\rm benzene} = \frac{\text{Mass of benzene}}{\text{Molar mass of benzene}} = \frac{39.0 \, \mathrm{g}}{78.11 \, \mathrm{g/mol}} \)

Step 5 - Calculate the Moles of Solute Using Its Mole Fraction

From the mole fraction of the solute and the moles of benzene, calculate the moles of solute: \( n_{\rm solute} = X_{\rm solute} \cdot n_{\rm benzene} \)

Step 6 - Determine the Molar Mass of the Solute

Use the mass of the solute and calculate the molar mass: \( M_{\rm solute} = \frac{\text{Mass of solute}}{n_{\rm solute}} \) Using the mass of the solute given as \(0.5 \, \mathrm{g} \) and the moles of solute calculated from step 5.

Step 7 - Match the Calculated Molar Mass with the Given Options

After calculating the molar mass of the solid substance from the previous step, determine which of the given options (a-d) matches the calculated value.

Key Concepts

Vapor Pressure

When learning about vapor pressure, it's essential to understand that it's the pressure exerted by a vapor in equilibrium with its liquid phase at a given temperature. In a closed system, when a liquid evaporates and molecules transition into gas, they exert pressure on their surroundings. Vapor pressure is determined solely by temperature and the physical properties of the liquid.

For example, the vapor pressure of pure benzene at a certain temperature was given as 0.85 bar in the exercise. When a non-volatile substance is added, vapor pressure changes because the number of molecules of the original liquid (benzene) at the surface decreases, leading to fewer molecules escaping into the vapor phase. This concept is directly tied to Raoult's Law, which allows us to quantify this change in pressure based on the composition of the solution.

Solution Chemistry

Solution chemistry involves the study of homogeneous mixtures composed of two or more substances. In the exercise, we're working with a binary solution made of benzene and a non-volatile solute. The important aspect of such solutions is that they exhibit a uniform distribution of particles, and the solute does not change into vapors at the given temperature; therefore it doesn't contribute to the solution's vapor pressure.

The changes that occur when a solute dissolves in a solvent involve interactions at the molecular level, which can affect properties such as vapor pressure, boiling point, and freezing point. When a non-volatile solute is dissolved in a liquid, as in this exercise, it disrupts the solvent's ability to evaporate, which manifests as a drop in vapor pressure.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It's defined as the number of moles of a particular component divided by the total number of moles of all components in the mixture. Represented by the symbol \(X\), mole fraction is a dimensionless quantity and helps in employing Raoult's Law.

In our scenario, the exercise guides us through finding the mole fraction of benzene, the solvent, and then using it to determine the mole fraction of the solute. This calculation is crucial as it relates directly to how much the presence of a solite affects the vapor pressure of the solvent. Remember, the sum of mole fractions in a solution always equals one because it's essentially a ratio of parts out of a whole.

Molar Mass Calculation

Molar mass calculation is a foundational topic in chemistry that often confuses students. It refers to the weight in grams of one mole of a substance. The unit for molar mass is grams per mole (\(\mathrm{g/mol}\)). Molar mass allows chemists to convert between the mass of a substance and the amount in moles, which is key for stoichiometric calculations.

In this textbook scenario, we calculate the molar mass of a non-volatile solute by using the mass provided and the mole fraction calculated from the given solution. The steps outlined take us from understanding the impact of the solute on the solution's vapor pressure to finding out how much solute is present in moles, and finally to determining the molar mass which is the ratio of the mass of the solute to the number of moles of the solute. This is a critical calculation for identifying unknown substances based on measurable changes in solution properties.