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

(d) volume occupied by the molecules

Step 1: Understanding the van der Waals Equation

The van der Waals equation is used to correct the ideal gas law for the behavior of real gases. It is given by the formula:\[(P + \frac{a}{V_m^2})(V_m - b) = RT\]where \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the gas constant, and \(T\) is the temperature. The constants \(a\) and \(b\) are specific to each gas.

Step 2: Identifying the Role of Constant 'b'

In the van der Waals equation, the constant \(b\) refers to the volume correction. It accounts for the volume occupied by the gas molecules themselves, which means it corrects the molar volume \(V_m\) by subtracting the volume occupied by the gas molecules.

Step 3: Relating 'b' to Physical Properties

The value of \(b\) specifically relates to the physical volume occupied by the molecules, meaning it is the volume excluded by a mole of particles. It does not account for attractions or collisions, but rather the finite space taken up by the molecules due to their physical size.

Key Concepts

Volume Correction in Van der Waals Equation

The concept of volume correction in the van der Waals equation is crucial for understanding the behavior of real gases. In the ideal gas law, it is assumed that gas molecules do not take up any space. However, in reality, molecules occupy a certain volume. Instead of treating molecules as point particles, the van der Waals equation introduces a correction factor known as 'b'.

This constant 'b' represents the volume occupied by the gas molecules themselves. By subtracting 'b' from the molar volume (\(V_m\)), we correct for the volume that the molecules exclude due to their finite size.

In simpler terms, 'b' is the effective space each mole of particles occupies, effectively decreasing the overall available volume for molecular movement.

• The higher the 'b' value, the larger the molecules, since more space is taken up by the molecules themselves.
• The value of 'b' differs depending on the type of gas, reflecting the size of the molecules in that gas.

Understanding this aspect is essential for predicting how gases will behave under different conditions, particularly when gases are highly compressed or under high pressure.

Importance of Real Gases

Real gases exhibit behaviors that differ from the assumptions of the ideal gas law, which makes understanding real gases important for various scientific and industrial applications. The ideal gas law assumes no attractive or repulsive forces between gas molecules and disregards their volume. However, in real gases, these factors come into play, particularly at high pressures or low temperatures.

The van der Waals equation accommodates these real gas behaviors by introducing two constants, 'a' and 'b'.

• The constant 'a' corrects for intermolecular attractions, which are neglected in the ideal gas law.
• The constant 'b' accounts for the physical volume occupied by the molecules.

This equation helps accurately predict real gas behavior under various conditions by adjusting for molecular volume and attractions. Thus, understanding real gases involves recognizing how these deviations manifest and how they impact gas behavior in practical terms.

Molar Volume and its Significance

Molar volume is a fundamental concept in chemistry, representing the volume occupied by one mole of a substance. In the context of gases, it is particularly significant because it allows for the determination of how gases behave under different conditions.

In the ideal gas framework, molar volume is described by the formula:\[V_m = \frac{RT}{P}\]where \(V_m\) is the molar volume, \(R\) is the gas constant, \(T\) is the temperature, and \(P\) is the pressure.

The van der Waals equation modifies this to incorporate volume correction and intermolecular forces by adjusting the ideal assumptions.

Considering molar volume in light of real gas behavior helps to ensure accurate calculations and predictions, particularly when precise conditions must be met for industrial or scientific purposes. By adjusting for real-world factors, the concept of molar volume becomes more applicable across diverse scenarios.