Problem 4 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

At very high pressures, the compressibility factor of one mole of a gas is given by: [Main Online April 9, 2016] (a) \(1+\frac{P b}{R T}\) (b) \(\frac{P b}{R T}\) (c) \(1-\frac{P b}{R T}\) (d) \(1-\frac{b}{(V R T)}\)

Step-by-Step Solution

Answer

The answer is (a) \(1+\frac{P b}{R T}\).

Step 1: Understand the Compressibility Factor

The compressibility factor, denoted as \( Z \), is defined as \( Z = \frac{P V}{n R T} \). It is a measure of how much the behavior of a real gas deviates from an ideal gas.

Step 2: Determine Compressibility at High Pressure

At very high pressures, the real gas behavior becomes significant and deviations from ideal behavior occur. Typically, the compressibility factor \( Z \) becomes greater than 1 as the volume decreases significantly under high pressure.

Step 3: Identify the Correct Expression

For high pressure, the concept of repulsive forces increases due to particle crowding, leading an ideal gas to deviate such that the compressibility factor \( Z \) is greater than 1. The formula \( Z = 1 + \frac{P b}{R T} \) accounts for these conditions by adding a positive correction factor.

Key Concepts

Real Gas Behavior

In the world of gases, not all behave perfectly under all conditions. Real gases are gases that do not always stick to the rules of the Ideal Gas Law: \[ PV = nRT \]. In reality, gases have interactions between particles. These can be attractions or repulsions. And while these interactions are small under normal conditions, they become significant when the gas is under high pressure or at low temperatures.
Real gases have properties such as:

• Intermolecular forces: These include both attraction and repulsion between gas molecules.
• Finite volume: Gas particles actually take up space. In contrast, the Ideal Gas Law assumes particles have no volume.

As we study gases under different conditions, it becomes clear that their behavior varies widely from the ideal predictions. Understanding these variations helps in designing experiments and equipment that use gases as part of their processes.

Deviation from Ideal Gas

Most students learn about the Ideal Gas Law first, which is a great approximation under many conditions. But when you're dealing with real gases, things can go off track. This deviation, known as non-ideal behavior, is expressed using the compressibility factor \( Z \). This factor tells us how much a real gas deviates from an ideal gas under given temperatures and pressures.
A compressibility factor \( Z \) is calculated as:
\[ Z = \frac{PV}{nRT} \]
When \( Z = 1 \), the gas shows ideal behavior, indicating perfect compliance with the Ideal Gas Law. However, when \( Z \) deviates from 1:

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

Understanding these deviations is crucial, especially for students planning to delve into fields involving thermodynamics, chemical engineering, or any field where gas behavior under pressure is a consideration.

High Pressure Effects

When gases are subjected to high pressures, their behavior changes significantly compared to normal atmosphere pressure conditions. At high pressures, gas molecules are forced closer together which has a few implications, especially regarding their interactions and space constraints.
One of the key factors that changes under high pressure is the compressibility factor \( Z \). Each increase in pressure typically increases \( Z \), which can be expressed mathematically as:
\[ Z = 1 + \frac{Pb}{RT} \]
The above equation suggests that at high pressures, \( Z \) adjusts by adding a positive correction factor \( \frac{Pb}{RT} \) to account for the increased repulsion between particles. Why does this happen?

• Increased repulsive interactions: Molecules are packed so tightly that they start to push each other apart.
• Reduced effective volume: The assumption that molecules have no volume doesn’t hold. Real particles' space matters more.

Such conditions are common in industrial processes, especially in fields such as petroleum and chemical engineering. Understanding these effects helps in designing systems that operate efficiently despite non-ideal conditions.