Question

The van der Waals equation of state is $$ \mathrm{P}+\frac{(\mathrm{V}-\mathrm{nb})}{\mathrm{V}^{2}}=\mathrm{nRT} $$ The pressure exerted by individual gas molecules on the walls of the container depends upon the (a) frequency of the collisions of the molecules with the walls as well as the momentum imparted by the molecules to the walls (b) frequency of molecular collision (c) mean free path of the molecules (d) momentum and critical pressure of the gas molecules

Short answer

The correct answer is (a), as pressure depends on collision frequency and momentum transfer.

Step-by-step solution

Identifying Key Equation

The given equation is the van der Waals equation, which we can rearrange to highlight its components: \( P = \frac{nRT}{V - nb} - \frac{a}{V^2} \). This equation attempts to correct the ideal gas law by accounting for intermolecular forces and finite molecular size.

Analyzing Forces in the Equation

The term \( \frac{a}{V^2} \) is a correction for intermolecular forces, while \( \frac{nRT}{V - nb} \) is related to ideal gas behavior adjusted for the volume of gas particles (\( b \) factor). However, for assessing individual molecule impact, the terms indicate corrections for pressure due to molecular size and attraction.

Connecting Pressure with Collisions

Pressure in gases is fundamentally linked to the behavior of gas molecules colliding with the container walls. It's determined by how often molecules collide with walls (frequency) and the change in momentum imparted by these collisions (impulse).

Identifying Correct Dependencies

Option (a) highlights both frequency and momentum, which are directly related to how gases exert pressure according to kinetic theory. Other options mention components like mean free path or additional parameters, not directly tied to pressure exerted on the walls as described.

Key concepts

Understanding Intermolecular Forces

Intermolecular forces are the interactions that occur between molecules. These forces have significant roles in determining the physical properties of gases.
In the van der Waals equation, intermolecular forces are accounted for by the term \( \frac{a}{V^2} \), which corrects for these attractions. This term considers that real gas particles are not entirely free and are subject to attractive forces that pull them closer together.
Intermolecular forces can be of various types:

• London dispersion forces: These are weak forces that occur due to the temporary shifts in electron density in atoms or molecules.
• Dipole-dipole interactions: These occur between molecules that have permanent dipoles, such as hydrogen chloride.
• Hydrogen bonding: A stronger type of dipole-dipole interaction occurring in molecules where hydrogen is covalently bonded to highly electronegative atoms like nitrogen, oxygen, or fluorine.

Considering these forces helps explain why real gases deviate from ideal behavior, especially at high pressures and low temperatures.

Kinetic Theory of Gases

The kinetic theory of gases offers a model that describes a gas's macroscopic properties in terms of its molecular movement.
This theory assumes that gas molecules are in constant random motion and the collisions between them and the container walls result in pressure.
The key assumptions include:

• Molecules move with a range of speeds: The speed distribution can be described statistically.
• Collisions are elastic: There's no net loss in kinetic energy during collisions.
• Molecules are point masses: They have negligible volume compared to the container's volume.
• No intermolecular forces: In an ideal gas, molecules do not attract or repel each other.

By considering the frequency and momentum change during collisions, the kinetic theory connects molecular movements with measurable parameters like pressure. The theory offers a foundation for deriving equations like the ideal gas law.

Ideal Gas Law Corrections: Adjusting for Reality

The ideal gas law, expressed as \( PV = nRT \), is a simple equation that relates the pressure, volume, and temperature of a gas.
However, this model assumes no volume occupied by gas molecules and no intermolecular forces, which is rarely the case in real gases.
The van der Waals equation introduces two critical corrections to better match real-life observations:

• Volume correction \( nb \): The term \( nb \) accounts for the finite size of gas molecules, reducing the free volume available for movement.
• Pressure correction \( \frac{a}{V^2} \): This term attributes pressure adjustments due to intermolecular attractions, lessening the pressure exerted on the walls by the gas.

These corrections help in understanding real gas behaviors, particularly under conditions of high pressure and low temperature where deviations from ideality are more pronounced.