Problem 53 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

An evacuated glass vessel weighs \(50.0 \mathrm{~g}\) when empty, \(148.0 \mathrm{~g}\) when filled with a liquid of density \(0.98 \mathrm{~g} \mathrm{~mL}^{-1}\) and \(50.5 \mathrm{~g}\) when filled with an ideal gas at \(760 \mathrm{~mm} \mathrm{Hg}\) at \(300 \mathrm{~K}\). Determine the molar mass of the gas.

Step-by-Step Solution

Answer

The molar mass of the gas is approximately 122.8 g/mol.

Step 1: Calculate the volume of the glass vessel

First, we find the volume of the glass vessel by using the mass of the liquid. The vessel weighs 148.0 g when filled, meaning the mass of the liquid is 148.0 g - 50.0 g = 98.0 g. With the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), we can rearrange this to find the volume: \( \text{Volume} = \frac{98.0 \text{ g}}{0.98 \text{ g/mL}} = 100.0 \text{ mL} \).

Step 2: Find the mass of the gas

Next, determine the mass of the gas. The mass of the empty vessel is 50.0 g, and it weighs 50.5 g when filled with the gas. Therefore, the mass of the gas is \( 50.5 \text{ g} - 50.0 \text{ g} = 0.5 \text{ g} \).

Step 3: Calculate the number of moles of gas

The number of moles of gas can be found using the Ideal Gas Law: \( PV = nRT \). Rearrange to solve for \( n \): \( n = \frac{PV}{RT} \). Substitute in the known values: pressure \( P = 760 \text{ mmHg} = 1 \text{ atm} \), volume \( V = 100.0 \text{ mL} = 0.1 \text{ L} \), \( R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} \), \( T = 300 \text{ K} \). Calculate the moles: \( n = \frac{1 \times 0.1}{0.0821 \times 300} = 0.00407 \text{ moles} \).

Step 4: Determine the molar mass of the gas

Finally, calculate the molar mass of the gas using the formula \( \text{Molar Mass} = \frac{\text{Mass}}{\text{Moles}} \). Here, mass of the gas is 0.5 g, and the number of moles is 0.00407 moles. Therefore, \( \text{Molar Mass} = \frac{0.5 \text{ g}}{0.00407 \text{ mol}} \approx 122.8 \text{ g/mol} \).

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is an equation of state for an ideal gas, helping us to understand the relationship between pressure, volume, temperature, and the number of moles of a gas. The equation is given by:

• \( PV = nRT \)
• where \( P \) is the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

This equation assumes that the gas particles:

• Have negligible volume compared to the container
• Do not exert any forces on each other except during elastic collisions

In practical terms, it allows us to connect different properties of a gas under certain conditions. By rearranging the equation, we can solve for any one variable if the other three are known, which is what enables us to determine the moles of gas in a given problem. This is useful in situations such as the one in our exercise, where we needed to calculate the number of moles to find the molar mass of a gas.

Density Calculation Explained

Density is a measure of how much mass is contained within a specific volume. The formula for density (\( \rho \)) is:

• \( \rho = \frac{\text{Mass}}{\text{Volume}} \)

This formula can be rearranged to find other variables:

• \( \text{Mass} = \rho \times \text{Volume} \)
• \( \text{Volume} = \frac{\text{Mass}}{\rho} \)

Density helps us understand how "packed" the substance is. In the exercise, we used the known density of a liquid to determine the volume of the glass vessel. By dividing the mass of the liquid by its density, we were able to find how much space the liquid occupied inside the vessel, which is equivalent to the volume of the vessel itself. This was crucial for later calculations involving the gas.

Volume Determination in Gas Problems

The exact measurement of volume is fundamental in gas law problems, and there are various methods to determine it depending on the data available. For liquids, volume can be directly measured by using tools like graduated cylinders or by using density as shown in our exercise. Here, the volume was indirectly measured using the liquid's mass and density. For gases, volume often has to be considered at Standard Temperature and Pressure (STP), unless specified otherwise. In our exercise, the volume was calculated based on the physical characteristics of the glass vessel using the liquid's data first, which was later applied to the gas problem.

• At STP, 1 mole of an ideal gas occupies \(22.4 \text{ L}\)
• Conversion to STP might be needed if other conditions are given

Remember, knowing how to determine and convert volume is essential to manage the Ideal Gas Law effectively in homework exercises.

Mole Calculation and Its Importance

Calculating the number of moles (\(n\)) in a gas is a central task when applying the Ideal Gas Law. In our exercise, moles were calculated using the rearranged formula from the Ideal Gas Law:

• \( n = \frac{PV}{RT} \)

Here, pressure (\(P\)) is given in atmospheres, volume (\(V\)) in liters, and temperature (\(T\)) in Kelvin. **\(R\)** is the universal gas constant, which depends on the units of pressure and volume, and in our example, it was \(0.0821 \text{ L atm}\text{ K}^{-1} \text{ mol}^{-1} \). Calculating moles helped us advance towards finding the molar mass, an essential step in understanding the properties of the gas.Understanding mole calculation is critical because:

• It helps in comparing quantities of substances in a "chemical counting" sense
• Moles serve as a bridge between microscopic (atoms, molecules) and macroscopic (grams, liters) worlds

By mastering this concept, students can tackle a wide array of problems involving gases in their studies.