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

Option (a): Frequency of collisions and momentum determine pressure.

Step-by-step solution

Identify Key Terms

First, let's identify the key terms and concepts involved in the given options. The pressure in a real gas as described by the Van der Waals equation accounts for interactions between molecules and volume occupied by molecules.

Analyze Frequency and Momentum

In option (a), it mentions 'frequency of collisions' and 'momentum imparted by the molecules.' Focus on understanding that the force exerted by gas molecules on container walls arises from collisions. The 'frequency of collisions' refers to how often molecules hit the walls, and 'momentum' refers to the product of mass and velocity of molecules, which changes when they hit the walls and thus affects exerted pressure.

Consider Other Options

Evaluate options (b), (c), and (d). Option (b) only mentions 'frequency of molecular collision,' which doesn't account for momentum. Option (c) involves 'mean free path' (average distance traveled by a molecule between collisions) which affects collision frequency but not pressure directly. Option (d) mentions 'momentum and critical pressure,' which doesn't directly relate to the pressure exertion mechanism described.

Select the Best Option

Combine insights from analyzing the options. The pressure exerted by gas molecules directly depends on both the 'frequency of the collisions' with walls and the 'momentum imparted.' Hence, option (a) is consistent with the physics of the gas molecules' interactions, considering how pressure is derived in this context.

Key concepts

Pressure in Real Gases

Real gases do not behave ideally under all conditions because intermolecular forces and the volume occupied by gas molecules become significant. To account for these non-ideal behaviors, the van der Waals equation is used:

\[\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT\]

In this equation, \(a\) accounts for the attraction between molecules, and \(b\) corrects for the volume occupied by gas molecules. The terms capture the deviations from ideal gas law assumptions. In real gases, pressure is affected by:

• Intermolecular attractions which diminish the force a molecule exerts on container walls before colliding.
• The effective volume that gas molecules can occupy in a container.

The van der Waals equation helps modify the ideal gas law to better represent the behavior of real gases, especially under high pressure and low temperature conditions where deviations are most pronounced.

Molecular Collisions

Molecular collisions occur when gas molecules frequently collide with each other and with the walls of their container. These collisions are elastic, meaning there is no loss of kinetic energy during the process. The frequency of collisions impacts:

• The pressure exerted by the gas on the walls.
• The overall behavior of the gas as molecules transfer energy upon collision.

It's important to understand that increased collision frequency typically leads to increased pressure if the speed and number of gas molecules remain constant. Collision frequency is influenced by:

• Gas temperature: Higher temperatures increase molecular speed and collision rates.
• Gas density: More molecules in a given volume increase the likelihood of collisions.

Under real conditions, molecular interactions and \(a\) and \(b\) factors in van der Waals equation further modulate collision outcomes.

Momentum Imparted by Molecules

When gas molecules collide with the walls of their container, they impart momentum. Momentum is a product of mass and velocity; hence, changes in either affect the force exerted by the gas.

Gas pressure, in simple terms, is the total force applied by molecules hitting the walls of the container divided by the area of the wall. Each collision transfers momentum from the molecule to the wall:

• The change in momentum for a molecule is roughly twice its initial momentum (considering the elastic nature of collisions).
• Higher velocities or higher molecular mass results in greater momentum transfer, thus increasing pressure.

In real gases, these momentum transfers are adjusted for internal interactions and volume adjustments, which is why the van der Waals corrections are vital for accurately predicting gas behaviors.

Mean Free Path

The mean free path is the average distance a gas molecule travels between collisions. This concept is crucial to understand because:

• It dictates how frequently molecules collide in a given volume.
• Longer mean free paths imply fewer collisions and lower collision frequency.

Factors that affect the mean free path include:

• Gas density: Lower density increases the mean free path as molecules are farther apart.
• Temperature: Higher temperatures generally increase molecular speed, affecting collision rates and spacing.

While the mean free path doesn't directly dictate pressure, it indirectly affects pressure by influencing collision frequency, which in turn, impacts momentum transfer and cumulative pressure exerted by the gas.

Gas Laws Analysis

Gas laws help predict how gases respond to changes in temperature, volume, and pressure. The ideal gas law, \(PV = nRT\), assumes no interactions between molecules and negligible volume for gas particles. However, in reality:

• Deviations occur due to intermolecular forces, especially at high pressures and low temperatures.
• These deviations are accounted for by the van der Waals equation, offering a more accurate prediction of gas behavior under non-ideal conditions.

Analyzing gas behaviors through various gas laws:

• Allows prediction of how gases compress, expand, or exert pressure under varying physical conditions.
• Forms the foundation for understanding more complex systems in fields like chemistry and physics.

Understanding the transition from ideal to real gas behavior is critical for fields that deeply rely on accurate gas predictions, such as engineering and environmental technology.