Problem 155 - A Complete Resource Book in Chemistry for JEE Main

Question

The compressibility factor for a real gas at high pressure is - (a) \(1+\mathrm{Pb} / \mathrm{RT}\) (b) \(1+\mathrm{RT} / \mathrm{Pb}\) (c) 1 (d) \(1-\mathrm{Pb} / \mathrm{RT}\)

Step-by-Step Solution

Answer

The correct answer is (a) \( 1+\frac{Pb}{RT} \).

Step 1: Understanding the Compressibility Factor

The compressibility factor (Z) is used to correct the ideal gas equation for real gas behavior. It is defined as \( Z = \frac{PV}{nRT} \) where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature. For ideal gases, \( Z = 1 \). For real gases, especially at high pressures, \( Z \) deviates from 1.

Step 2: Behavior at High Pressure

At high pressures, gases behave less ideally due to intermolecular forces and volume occupied by gas molecules. To account for these deviations, the van der Waals equation or other equations of state modify the ideal gas law parameters. In general, the compressibility factor becomes \( Z = 1 + \frac{Pb}{RT} \), where \( b \) is a constant related to the volume of the gas molecules.

Step 3: Comparing with Given Options

We analyze the given options:(a) \( 1+\frac{Pb}{RT} \) (b) \( 1+\frac{RT}{Pb} \) (c) 1 (d) \( 1-\frac{Pb}{RT} \)Considering the effect of high pressure and our understanding of real gas behavior, option (a) \( 1+\frac{Pb}{RT} \) accurately describes how pressure affects the compressibility factor.

Step 4: Selecting the Correct Option

Based on the known behavior of real gases at high pressure, where the compressibility factor expression involves the ratio \( \frac{Pb}{RT} \), and observing available options, the correct choice is (a) \( 1+\frac{Pb}{RT} \).

Key Concepts

Ideal Gas Equation

The ideal gas equation is a simple but powerful tool to explain the relationship between pressure, volume, and temperature of an ideal gas. It is expressed mathematically as \( PV = nRT \). This relationship assumes that gases consist of a large number of molecules moving randomly with negligible size and no intermolecular forces acting between them. For ideal gases, all collisions between gas particles and with the walls of the container are elastic.

Key aspects to understand include:

• \( P \), the pressure of the gas, typically measured in atmospheres (atm).
• \( V \), the volume containing the gas, often measured in liters (L).
• \( n \), the amount of substance (in moles) contained in the system.
• \( R \), the universal gas constant, which is \( 0.0821 \, \mathrm{L \cdot atm} / \mathrm{K \cdot mol} \).
• \( T \), the absolute temperature measured in Kelvin (K).
  
  

While the ideal gas law provides a robust approximation for many scenarios, real gases deviate from this ideal behavior under certain conditions like high pressure or low temperature, where interactions between molecules become significant.

Real Gas Behavior

Real gas behavior deviates from the ideal gas law due to factors such as intermolecular forces and the volume occupied by gas molecules. In reality, gas molecules are not point masses and they exert attractive and repulsive forces. These deviations are especially noticeable at high pressures and low temperatures.

Some important characteristics of real gases include:

• Intermolecular Attractions: At high pressures, molecules are forced closer together, and the attractive forces become significant, making gases less compressible.
• Finite Molecular Volume: Gas particles occupy space which reduces the free volume available for movement, increasing pressure.
• Compressibility Factor (\( Z \)): It measures deviation from ideal gas behavior, calculated as \( Z = \frac{PV}{nRT} \). For ideal gases \( Z = 1 \), while for real gases, \( Z \) can be greater or less than 1 depending on conditions.

To quantify these deviations, equations like the van der Waals equation are used, altering the ideal gas law to better fit observed data.

Van der Waals Equation

The van der Waals equation is a refined version of the ideal gas law that accounts for the non-ideal behavior of real gases. This equation introduces two parameters, \( a \) and \( b \), to account for the effects of intermolecular attractions and the finite volume of gas molecules, respectively. The formula is given by:\[ \left( P + \frac{n^2a}{V^2} \right)(V - nb) = nRT \]Where:

• \( a \) is a measure of the attraction between particles, specific to each gas.
• \( b \) corresponds to the volume occupied by gas molecules, affecting the available space within a container.

This modified equation helps explain why real gases do not conform precisely to the ideal gas law. At high pressures, the presence of \( b \) becomes more relevant as molecules occupy more volume. Similarly, at low temperatures or high pressures, increased \( a \) values reflect the stronger intermolecular attractions, altering pressure expectations.

Understanding these parameters is crucial for accurately predicting gas behavior in real-world scenarios.