Problem 23 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The compression factor (compressibility factor) for one mole of a van der Waals gas at \(0^{\circ} \mathrm{C}\) and 100 atmospheric pressure is found to be \(0.5\). Assuming that the volume of a gas molecule is negligible, calculate the van der Waals constant \(a\).

Step-by-Step Solution

Answer

The van der Waals constant \(a\) is approximately \(5.62 \, \text{L}^2 \text{atm} \, \text{mol}^{-2}\).

Step 1: Identify Given Values

We are given the compressibility factor \(Z = 0.5\), temperature \(T = 0^{\circ}C = 273 \text{ K}\), pressure \(P = 100\, \text{atm}\), and the assumption that the volume \(b\) is negligible.

Step 2: Use van der Waals Equation with Compressibility Factor

The van der Waals equation is \[ \left(P + \frac{a}{V^2}\right)(V-b) = RT \]. Using the compressibility factor formula \( Z = \frac{PV}{RT} \) and \( Z = 0.5\), we deduce that \( PV = 0.5RT \).

Step 3: Rearrange to Find V

Using \( Z = 0.5 \), we rearrange the ideal gas law to solve for volume: \( V = \frac{RT}{2P} \).

Step 4: Substitute V in van der Waals Equation

Substitute \( V = \frac{RT}{2P} \) back into the van der Waals equation: \[ P + \frac{a}{\left(\frac{RT}{2P}\right)^2} = \frac{RT}{\left(\frac{RT}{2P}\right)} \].

Step 5: Solve for a

Simplify and solve the equation for \( a \): \[ P + \frac{a(2P)^2}{R^2T^2} = 2P \]. Aggregate terms to find \[ a = 2P \left( \frac{R^2T^2}{(2P)^2} \right) \] and solve for \( a \).

Step 6: Use Known Constants

Use \( R = 0.0821 \, \text{L atm K}^{-1} \, \text{mol}^{-1} \) to calculate \( a \): \[ a = 2 \times 100 \text{ atm} \times \left( \frac{(0.0821 \, \text{L atm K}^{-1} \, \text{mol}^{-1})^2 (273 \text{ K})^2}{(2 \times 100 \text{ atm})^2} \right) \].

Step 7: Calculate Result for a

Calculate \( a \) using the simplified expression: \( a = 5.62 \, \text{L}^2 \text{atm} \, \text{mol}^{-2} \).

Key Concepts

Compressibility Factor

The compressibility factor, often denoted as \( Z \), is a vital part of understanding real gases. It helps us measure how much a real gas deviates from ideal gas behavior. This factor is defined using the formula \( Z = \frac{PV}{RT} \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature of the gas.

If \( Z = 1 \), the gas behaves like an ideal gas. When \( Z \)< 1, the interactions between molecules are more pronounced than expected. In the original exercise, the compressibility factor was found to be \( 0.5 \), indicating significant interactions between molecules, deviating from ideal behavior. Understanding \( Z \) aids in illustrating how real gases deviate from the simplistic model of an ideal gas, especially under high pressure or low temperature conditions. The van der Waals equation incorporates \( Z \) to address these deviations by adding correction terms for pressure and volume, which we'll explore further.

Volume of Gas Molecule

When we talk about the volume of a gas molecule in the context of the van der Waals equation, we're looking at the term \( b \). This constant is often described as the 'volume excluded per mole due to the finite size of the molecules.' It's key to recognize that real gas molecules take up space, unlike ideal gases, where molecules are assumed to have no volume.In ideal conditions, gas molecules are considered points with negligible volume. However, this isn't true for real gases. By incorporating \( b \), the van der Waals equation modifies the available volume for a molecule by subtracting \( b \) from the actual volume \( V \). This accounts for the actual space the molecules occupy and their inherent volume.In some cases, including our original task, the assumption is made to neglect \( b \). This simplifies calculations but is only valid in certain conditions, typically when the pressure is not very high or the molecules are small.

Ideal Gas Law

The ideal gas law is a simple equation that describes the behavior of an ideal gas. Expressed as \( PV = nRT \), it relates pressure \( P \), volume \( V \), and temperature \( T \) of the gas, where:

• \( n \) represents the number of moles.
• \( R \) is the ideal gas constant.

This law asserts that gases consist of negligible volume particles in constant, random motion with no attractions or repulsions between them. The equation assumes a linear relationship between \( PV \) and \( T \), only holding accurate at high temperatures and low pressures.However, real gases often deviate from the ideal gas law under certain conditions such as high pressures or low temperatures, leading to the necessity of more complex models like the van der Waals equation. The Ideal Gas Law is crucial to setting a baseline so students and scientists understand standard gas behavior, which can then be adjusted for real-world applications by models like van der Waals.