Problem 12 - A Complete Resource Book in Chemistry for JEE Main

Question

The compressibility factor of an ideal gas is (a) 1 (b) 2 (c) 4 (d) 0

Step-by-Step Solution

Answer

(a) 1

Step 1: Understand Compressibility Factor

The compressibility factor, often represented as \( Z \), is a measure of how much a real gas deviates from ideal gas behavior. For an ideal gas, \( Z = 1 \).

Step 2: Analyze the Choices

Review the provided options: (a) 1, (b) 2, (c) 4, and (d) 0. Recall that for an ideal gas, the compressibility factor \( Z \) must be exactly 1.

Step 3: Select the Correct Answer

Since an ideal gas has a compressibility factor of 1, and option (a) matches this, select option (a).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure, volume, temperature, and amount of an ideal gas. It is written as:\[ PV = nRT \]where:

• P is the pressure of the gas
• V is the volume of the gas
• n is the number of moles of the gas
• R is the universal gas constant
• T is the temperature of the gas in Kelvin

This law assumes that gases are made up of tiny particles that are in constant random motion. These particles collide perfectly elastically with each other and the walls of the container, without experiencing intermolecular attractions. The law is an approximation that works well under many conditions, but fails when gases are at high pressure or low temperatures.

Real Gases

Real gases differ from ideal gases because the particles that compose them do interact with each other. These interactions can be attractive or repulsive, affecting properties like pressure and volume. Real gases can be described by equations of state such as the Van der Waals equation:\[ \left( P + \frac{an^2}{V^2} \right)(V - nb) = nRT \]where:

• a and b are constants specific to each gas, representing attractive forces and volume of particles respectively.

At high pressure, particles are forced close together, and intermolecular attractions become significant. At low temperatures, particles do not have enough kinetic energy to overcome these attractions. These conditions cause real gases to deviate from the ideal gas behavior.

Deviation from Ideal Behavior

Deviation from ideal behavior occurs when the assumptions of the ideal gas law are not valid due to intermolecular forces and the finite size of gas particles. The compressibility factor, denoted as \( Z \), helps measure this deviation and is defined as:\[ Z = \frac{PV}{nRT} \]For an ideal gas, \( Z \) is always 1, indicating that the predictions of the ideal gas law match exactly with actual behavior. In real gases, however, \( Z \) can differ from 1.

• If \( Z > 1 \), repulsive forces dominate, and the gas appears less compressible than expected.
• If \( Z < 1 \), attractive forces dominate, making the gas more compressible.

Understanding \( Z \) allows you to grasp the extend of deviation, guiding the use of more accurate models for real gases in various conditions.