Problem 149 - A Complete Resource Book in Chemistry for JEE Main

Question

In van der Waals equation of state of the gas law, the constant 'b' is a measure of (a) intermolecular attraction (b) intermolecular repulsions (c) intermolecular collision per unit volume (d) volume occupied by the molecules

Step-by-Step Solution

Answer

The constant 'b' measures the volume occupied by the molecules.

Step 1: Understand the van der Waals Equation

The van der Waals equation for real gases is given by \( \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \), where \( P \) is pressure, \( V_m \) is molar volume, \( R \) is the gas constant, \( T \) is the temperature, \( a \) is the measure of intermolecular attraction, and \( b \) is the correction factor for excluded volume.

Step 2: Interpret the Constant 'b'

In the equation, the constant 'b' accounts for the volume excluded by the gas molecules themselves and is subtracted from the molar volume in the van der Waals equation. It represents the volume occupied by the molecules.

Step 3: Choose the Correct Option

Based on the interpretation of 'b' as the volume occupied by the molecules, the correct answer is option (d).

Key Concepts

Understanding Real Gases

Real gases deviate from the ideal gas behavior due to various factors. In reality, gas molecules do have volume and there are attractive and repulsive forces between them. These factors affect the way gases behave under different conditions, particularly at high pressures and low temperatures where deviations from ideal behavior are more prominent.
Real gas behavior is better described by equations like the van der Waals equation, which consider these deviations. This equation incorporates corrections for molecular volume and intermolecular forces, providing a much more accurate representation of gas behavior than the ideal gas law. Unlike ideal gases, real gases do not perfectly follow the postulates of the Kinetic Molecular Theory, which assumes gases consist of point particles with no interactions between them.

Exploring Molar Volume

Molar volume is a key concept in understanding gases, representing the volume occupied by one mole of a substance. For ideal gases under standard conditions, molar volume is about 22.4 liters per mole. However, for real gases, this is not consistent due to the finite size of gas molecules themselves.
The van der Waals equation modifies the molar volume by subtracting the constant 'b,' which accounts for the volume occupied by the molecules. This adjustment helps in understanding how gases behave more realistically, as it considers the space taken up by the molecules themselves, rather than assuming they occupy no space as in the ideal gas model.

• Molar volume helps determine how many molecules are present in a specified volume of gas.
• In real gases, this volume can vary depending on the pressure and temperature conditions.

Intermolecular Forces in Gases

Intermolecular forces are crucial in understanding the behavior of real gases. These forces include attractions like London dispersion forces, dipole-dipole interactions, and hydrogen bonding, as well as repulsions that occur when molecules are in close proximity.
The van der Waals equation accounts for these forces with the constant 'a'. This constant modifies the pressure term to account for the attractive forces between molecules, which are particularly significant at higher pressures where molecules are closer together.

• Attractive forces can cause real gases to compress more easily than predicted by the ideal gas law.
• These forces help explain why gases condense into liquids at sufficiently low temperatures or high pressures.

Understanding these forces is essential for predicting how gases will behave under various conditions.

Gas Law Corrections in the Van der Waals Equation

The van der Waals equation introduces corrections to the ideal gas law to account for the real behavior of gases. Two key corrections address the two main deviations from ideal scenarios:

• The pressure correction, represented by the term \( \frac{a}{V_m^2} \), adjusts for the attractive forces between gas molecules that reduce the effective pressure.
• The volume correction involves subtracting the constant 'b' from the molar volume \((V_m - b)\), accounting for the finite size of the molecules themselves.

These corrections allow the van der Waals equation to more accurately predict real gas behavior by considering molecular size and intermolecular forces. They are particularly important at conditions where gases are farthest from ideal behavior, such as high pressures and low temperatures.
These adjustments enable chemists and engineers to predict and manipulate the behavior of gases in more practical and realistic scenarios.