Problem 40 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The vapour pressure of pure benzene at a certain temperature is 640 \(\mathrm{mm}\) Hg. A non-volatile non-electrolyte solid weighing \(2.175 \mathrm{~g}\) is added to \(39.0 \mathrm{~g}\) of benzene. The vapour pressure of the solution is 600 \(\mathrm{mm} \mathrm{Hg}\). What is the molecular weight of the solid substance?

Step-by-Step Solution

Answer

The molecular weight of the solid is approximately 64.93 g/mol.

Step 1: Write Down Known Values

Identify and write down all known values given in the problem.- Vapour pressure of pure benzene, \( P^0 = 640 \, \text{mm Hg} \).- Vapour pressure of benzene when solid is added, \( P = 600 \, \text{mm Hg} \).- Mass of the non-volatile solid, \( m_\text{solid} = 2.175 \, \text{g} \).- Mass of benzene, \( m_\text{benzene} = 39.0 \, \text{g} \).We aim to find the molecular weight (molar mass) of the solid substance.

Step 2: Use Raoult's Law

Use Raoult's Law to find the mole fraction of benzene in the solution. Raoult's Law is given by:\[ P = P^0 \times X_\text{benzene} \]Where \( X_\text{benzene} \) is the mole fraction of benzene in the solution. Rearrange to solve for \( X_\text{benzene} \):\[ X_\text{benzene} = \frac{P}{P^0} = \frac{600}{640} = 0.9375 \]

Step 3: Calculate Moles of Benzene

Find the moles of benzene using its molar mass. The molar mass of benzene is approximately 78 g/mol.\[ n_\text{benzene} = \frac{m_\text{benzene}}{M_\text{benzene}} = \frac{39.0}{78} = 0.5 \, \text{moles} \]

Step 4: Use Mole Fraction to Find Total Moles

Use the mole fraction of benzene to find the total moles of the solution:\[ X_\text{benzene} = \frac{n_\text{benzene}}{n_\text{benzene} + n_\text{solid}} = 0.9375 \]Let \( n_\text{solid} \) be the moles of solid. Rearrange and solve for \( n_\text{solid} \):\[ 0.9375 = \frac{0.5}{0.5 + n_\text{solid}} \]\[ 0.5 + n_\text{solid} = \frac{0.5}{0.9375} \]\[ n_\text{solid} = \frac{0.5}{0.9375} - 0.5 = 0.0335 \, \text{moles} \]

Step 5: Calculate Molecular Weight of the Solid

Finally, calculate the molecular weight (molar mass) of the solid using its mass and moles:\[ M_\text{solid} = \frac{m_\text{solid}}{n_\text{solid}} = \frac{2.175 \, \text{g}}{0.0335 \, \text{moles}} \approx 64.93 \, \text{g/mol} \]

Key Concepts

Vapour Pressure

Vapour pressure is a key concept in understanding how different substances behave when mixed together in a solution. It is essentially the pressure exerted by the vapor of a liquid in equilibrium with its liquid phase at a given temperature. In the context of the exercise, we are given the vapor pressures of benzene before and after a solid is added. This is crucial because the vapor pressure of a liquid is influenced by the presence of a non-volatile solute, which is the solid added to benzene in our example.

When a non-volatile substance is dissolved in a volatile solvent like benzene, the vapor pressure of the solution is lower than that of the pure solvent. This happens because the solute molecules occupy space at the surface, reducing the number of solvent molecules that can escape into the vapor phase. The reduction in vapor pressure can be calculated using Raoult’s Law, which provides a way to determine how much the vapor pressure decreases based on the concentration of the solute.

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. It is a dimensionless quantity, meaning it has no units, and it values between 0 and 1. This concept is integral to Raoult's Law, as it helps determine the influence of a solute on the solvent's vapor pressure.

In our calculation, the mole fraction of benzene was found after rearranging Raoult's Law. This value was calculated as the ratio of the vapor pressure of the solution to the vapor pressure of pure benzene: \[ X_{benzene} = \frac{600}{640} = 0.9375 \] This means that 93.75% of the moles in the solution are benzene. By understanding the mole fraction, we can work backward to find the total moles in the solution, which ultimately reveals the number of moles of the unknown solid.

Molar Mass Calculation

Molar mass, or molecular weight, is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). Calculating molar mass is key to identifying unknown substances, as it connects the mass of a substance with the number of particles or molecules it contains.

In the exercise, we were tasked to find the molar mass of a non-volatile solid added to benzene. We accomplished this by first determining the moles of the benzene and the solid in the solution using the mole fraction. Once we knew the number of moles of the solid, we used the formula:\[ M_{solid} = \frac{m_{solid}}{n_{solid}} \]This final step directly gives us the molar mass of the solid. Plugging in the values from the original solution, we calculated:\[ M_{solid} = \frac{2.175}{0.0335} \approx 64.93 \, \text{g/mol} \]Understanding this process can help students grasp how colligative properties and compound identification work hand in hand.