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

The correct answer is (a): frequency of the collisions and momentum imparted.

Step 1: Understanding the Concepts

The van der Waals equation is an improvement over the ideal gas law, describing non-ideal behavior by including the volume of gas particles and the attraction between them. For the given options, we need to understand what affects the pressure exerted by gas molecules on the container walls.

Step 2: Identifying Key Factors for Pressure

Pressure is defined as the force per unit area exerted by gas molecules when they collide with the walls of a container. Thus, to analyze how pressure is generated, we should consider both the frequency of collisions and the momentum transferred during these collisions.

Step 3: Analyzing the Options

Option (a) states that pressure depends on the frequency of collisions and the momentum imparted during collisions, which aligns with our understanding of how pressure is exerted. Option (b) only considers frequency, while options (c) and (d) do not appropriately combine the factors affecting pressure.

Step 4: Final Decision

Based on the definitions and analysis, option (a) encompasses both primary factors affecting the exertion of pressure: collision frequency and momentum transfer. Therefore, option (a) is the correct answer.

Key Concepts

Understanding Non-Ideal Gas Behavior

In real-world scenarios, gases do not always behave as perfectly as they do in the idealized version. This deviation from the ideal behavior is what we call non-ideal gas behavior. Unlike ideal gases, where interactions between gas molecules are negligible and the volume occupied by the molecules themselves is ignored, non-ideal gases acknowledge these factors. The van der Waals equation is a key formula that accounts for non-ideal gas behavior. It does this by introducing parameters that correct for the actual volume occupied by gas molecules and the attraction or repulsion between them. By considering these factors, the van der Waals equation provides a more accurate representation of a gas's behavior, especially under high pressure and low temperature conditions.
When gases are dense, such as at high pressures, the volume of molecules themselves becomes significant compared to the space between them. Furthermore, the slight attractive forces between them can affect how they move and collide. These factors lead to behavior that diverges from ideality, necessitating the adjustments made by the van der Waals equation.

Pressure Exertion by Gas Molecules

The pressure exerted by gas molecules within a container is a fundamental concept when dealing with gases. It involves understanding how molecules collide with the walls of their container. Pressure is created by the force these collisions exert over a specified area.
Every time a gas molecule hits the wall of a container, it imparts a certain amount of momentum to the wall. This transferred momentum translates into a force. The greater the number of such impacts in a given time frame, the higher the pressure exerted on the walls. Hence, both the frequency of molecular collisions and the amount of momentum transferred during each collision are critical in influencing the pressure exerted by gas molecules.

Collision Frequency in Gases

Collision frequency refers to how often gas molecules collide with the walls of their container. This concept is pivotal in determining pressure. The more frequently molecules hit the walls, the higher the pressure. In a dense gas sample or one with high kinetic energy, collisions are more frequent. This is why increasing the temperature of a gas results in stronger molecular motion, leading to more frequent and energetic collisions. As such, managing conditions like temperature and volume can directly influence collision frequency. Understanding this helps in comprehending how pressure changes under different conditions without needing to change the actual amount of gas present.

Momentum Transfer in Gases

Momentum transfer during molecular collisions is another critical aspect that affects the exertion of pressure. When a gas molecule hits the wall of its container, it bounces back, transferring momentum to the wall. Momentum is defined as the product of an object's mass and velocity, and in this context, it describes how much motion gas molecules impart during collisions. The greater the mass and speed of the gas particles, the more momentum is transferred.
This means that heavier molecules or those moving faster (typically at higher temperatures) will impart more force during a collision, contributing to greater pressure. Momentum transfer is thus an essential piece in the puzzle of understanding gas behaviors, particularly when reconciling real gas behavior with idealized models.

The Ideal Gas Law Simplified

The ideal gas law is a fundamental equation in chemistry that provides insight into the behavior of gases under ideal conditions. It is given by the equation: \[ PV = nRT \] where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. This equation assumes no interaction between molecules and that the volume of the molecules is negligible. In essence, it provides a simple relationship between the four variables, allowing predictions about one variable if others are known.
However, under real-world scenarios, these assumptions break down, particularly at high pressures and low temperatures. That's where extensions like the van der Waals equation come into play, offering corrections for these non-ideal interactions by modifying the ideal gas law to account for molecular volume and intermolecular forces. Despite its simplicity, the ideal gas law serves as the cornerstone for understanding gas behavior, especially when proceeding to more complex equations like van der Waals.