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

(a) \(1+\frac{Pb}{RT}\) is the correct choice for the compressibility factor at high pressure.

Step 1: Understand the Compressibility Factor

The compressibility factor, denoted as \(Z\), is used to measure the deviation of a real gas from ideal gas behavior. For an ideal gas, \(Z = 1\). However, for real gases, particularly under high-pressure conditions, \(Z\) tends to deviate from 1, accounting for interactions between gas particles.

Step 2: Identify the Correct Relationship for High Pressure

At high pressures, gases typically deviate from ideal behavior. The formula for the compressibility factor under such conditions is modeled by the relationship \(Z = 1 + \frac{Pb}{RT}\) to account for molecular volumes and interactions. This expression approximates that the pressure-volume product becomes significant compared to the ideal gas law due to the real size of molecules.

Step 3: Compare Given Options with Known Relationship

Let's compare the answer options with our knowledge:- (a) \(1+\frac{Pb}{RT}\) matches directly with the known relationship for high pressure.- (b) \(1+\frac{RT}{Pb}\) does not make sense physically as it suggests incorrect dimensionality and context.- (c) 1 applies only to ideal gases, not real gases at high pressure.- (d) \(1-\frac{Pb}{RT}\) incorrectly suggests opposite correction under high pressure.

Step 4: Choose the Correct Option

Given our analysis and knowledge of the compressibility factor under high-pressure conditions, option (a) \(1+\frac{Pb}{RT}\) is the correct choice as it fits the behavior of real gases deviating from ideal behavior.

Key Concepts

Real Gases

Real gases differ from ideal gases due to interactions and volume of molecules. In ideal gas calculations, we assume no interactions and negligible molecular volume. However, real gases show deviations from this theory. This is where the compressibility factor \(Z\) comes into play. It provides a quantitative measure to indicate how much a real gas deviates from ideal behavior.

• The compressibility factor \(Z = \frac{PV}{nRT}\).
• For ideal gases, \(Z = 1\).
• For real gases, \(Z\) can be more or less than 1.

When \(Z > 1\), it means weaker attractive forces between molecules or that the volume is significantly influenced by molecular size. When \(Z < 1\), it indicates strong attractions between molecules, causing the gas to occupy lesser volume.

Ideal Gas Behavior

Ideal gas behavior is a theoretical concept that simplifies the equations to relate pressure, volume, temperature, and number of moles. It assumes:

• No interactions between molecules.
• The volume of molecules is negligible compared to the volume the gas occupies.

Under these assumptions, the ideal gas law is expressed as:\[ PV = nRT \]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. While practical for many conditions, this model falls short under high pressures or low temperatures, where deviations become significant. For instance, gases tend to condense into liquids under such extremes, violating the ideal assumptions.

High Pressure Gas Behavior

When gases are subjected to high pressures, they deviate significantly from ideal gas behavior, resulting in different compressibility factors. The real size of molecules and intermolecular forces become more pronounced, affecting how gases behave.

• Under high pressure, the volume no longer remains negligible compared to the container.
• The gas molecules occupy more space due to repulsive forces.

To describe these deviations, we use the compressibility factor equation for high pressure:\[Z = 1 + \frac{Pb}{RT}\]This formula indicates that as pressure \(P\) increases, and due to the molecular volume \(b\), the deviation \(Z\) from ideal behavior is more pronounced. Thus, the pressure-volume \((PV)\) product shows significant deviation from conventional ideal predictions.