Problem 61 - A Complete Resource Book in Chemistry for JEE Main

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

Step-by-Step Solution

Answer

(a) frequency of the collisions of the molecules with the walls as well as the momentum imparted by the molecules to the walls.

Step 1: Review given information

The van der Waals equation is provided in a form that appears slightly rearranged or incorrectly formulated. The correct equation typically is: \[\left(P + \frac{a(n/V)^2}{V^2}\right)(V - nb) = nRT\] where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, \(T\) is temperature, and \(a\) and \(b\) are van der Waals constants. This equation considers molecular size and intermolecular forces.

Step 2: Identify the pressure-related component

The exercise asks about factors affecting the pressure exerted by gas molecules on the container walls. Pressure relates to both molecular collisions and the force they exert. In kinetic theory terms, pressure results from direct impacts.

Step 3: Understand kinetic theory related to pressure

The pressure depends on two main aspects: (a) the frequency of molecular collisions with the walls—how often molecules hit the walls, and (b) the momentum they transfer to the walls during collisions. Together, these constitute the pressure according to kinetic molecular theory.

Step 4: Evaluate answer choices

Given the explanations in Steps 2 and 3, choice (a) stating 'frequency of the collisions of the molecules with the walls as well as the momentum imparted by the molecules to the walls' aligns correctly with our understanding of how pressure is exerted according to the kinetic theory.

Key Concepts

Understanding Pressure in Gases

Pressure in gases is essentially the force exerted by gas molecules when they collide with the walls of their container. This concept is crucial in understanding the behavior of gases, especially when dealing with equations like the van der Waals equation. In simple terms:

• The more frequently the gas molecules collide with the walls, the higher the pressure.
• When molecules collide, they transfer momentum to the walls. This momentum transfer is what we perceive as pressure.

This means that both the number of collisions and the force of each collision determine the overall pressure exerted by a gas. In kinetic terms, more energy in the system increases both the number of collisions and the energy imparted during each collision.

Role of Molecular Collisions

Molecular collisions occur when gas molecules bump into each other or into the walls of their container. Here's why they're important:

• The frequency of molecular collisions directly impacts pressure. More frequent collisions mean higher pressure.
• Each collision is an opportunity for a molecule to pass on some of its kinetic energy to the wall. This is why even a small number of molecules in a system can exert significant pressure if they are moving fast enough.

The van der Waals equation modifies the ideal gas law by accounting for the volume occupied by gas molecules and the attraction between them, affecting collision dynamics. Remember, molecular collisions aren't just about the number; how hard they hit also matters, impacting both pressure and system temperature.

Kinetic Molecular Theory: A Deeper Dive

The kinetic molecular theory provides a foundation for understanding the behavior of gases at a molecular level. It's essential for grasping how and why gases exert pressure:

• Molecules are in constant, random motion, colliding with each other and container walls.
• These collisions are elastic, meaning there's no net energy loss. Energy is conserved during the process, leading to a stable system pressure over time.
• The theory assumes that the volume of individual molecules is negligible compared to the total volume. However, real gases deviate from this ideal scenario, which the van der Waals equation accounts for by adjusting for molecular volume and intermolecular forces.

Understanding this theory helps explain why pressure increases with temperature: higher temperatures mean more kinetic energy, leading to more frequent and forceful collisions.