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 compressibility factor at high pressure is given by option (a) \(1 + \frac{Pb}{RT}\).

Step 1: Understand the Compressibility Factor

The compressibility factor, denoted as \(Z\), is a measure of the deviation of a real gas from ideal behavior. It is defined by the equation \(Z = \frac{PV}{nRT}\), where \(P\) is pressure, \(V\) is volume, \(n\) is amount of substance, \(R\) is the ideal gas constant, and \(T\) is temperature.

Step 2: Consider High Pressure Conditions

At high pressure, gases deviate significantly from ideal gas behavior. The molar volume \(V_m\) of a real gas is affected by the pressure, often leading to \(Z\) being greater than 1.

Step 3: Analyze the Given Options

Look at each option to determine which reflects an increase in \(Z\) due to positive deviation at high pressure. (a) suggests an increase as it adds a positive term \(\frac{Pb}{RT}\) to 1. Option (b) results in a decrease, and (d) suggests a decrease, which is contrary to typical high-pressure behavior for real gases.

Step 4: Determine the Correct Expression

For real gases at high pressure, the deviation from ideal gas behavior is typically such that \(Z\) is greater than 1. The correct expression should reflect \(Z > 1\), aligning with option (a) \(1 + \frac{Pb}{RT}\).

Step 5: Validate Option (a)

Since \(P\) and \(b\) are positive values, and division by \(RT\) does not negate this, option (a) increases the compressibility factor as \(P\) increases.

Key Concepts

Real Gases

When we talk about gases, we often refer to them in terms of their behaviors as either ideal or real gases. Ideal gases are a hypothetical concept where the gas particles do not interact with each other except through perfectly elastic collisions. However, real gases, which is what we encounter in the actual world, do interact via forces such as attraction or repulsion. Because of these interactions, real gases do not always follow the fundamental equation of state for ideal gases, given by the formula: \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.This deviation is often quantified using the compressibility factor, \( Z \), which is defined as: \[ Z = \frac{PV}{nRT} \] When \( Z = 1 \), the gas behaves like an ideal gas. However, for real gases, \( Z eq 1 \), showing the difference from ideal behavior due to molecular interactions that occur under specific conditions.

High Pressure

One of the scenarios where real gases show significant deviation from ideal behavior is under high-pressure conditions. At high pressures, the volume of the gas is significantly reduced, bringing the molecules closer together. In such cases:

• The intermolecular forces become much stronger because the molecules are tightly packed together.
• The assumption of negligible molecular size, a key point of the ideal gas law, no longer holds.

Under these conditions, the compressibility factor \( Z \) tends to be greater than 1. This indicates that the gas is less compressible than predicted by the ideal gas law. As the molecules are more crowded, they experience repulsions in addition to attractions, leading to this positive deviation.Therefore, option (a) from the exercise correctly suggests that the gas's behavior results in a compressibility factor of \( 1 + \frac{Pb}{RT} \), which accounts for this increased pressure affecting the molecular interactions, resulting in deviations that make \( Z \) more than 1.

Ideal Gas Behavior

The ideal gas law is an important concept that simplifies our understanding of gases by assuming a world where the individual interactions of molecules are negligible.Ideal gas behavior is predicted accurately in conditions of:

• Low pressure, where the gases have a lot of space to move freely without substantial intermolecular forces affecting them.
• High temperature, which provides the particles with sufficient kinetic energy to overcome any potential interactions.

In these ideal conditions, gases follow the equation \( PV = nRT \) perfectly, and the compressibility factor \( Z \) equals 1. This means the gas's behavior can be predicted using simple mathematical relationships. However, in the real world, especially at high pressures, gases deviate from these ideal predictions, as their molecular interactions can't be ignored.Understanding when and why gases deviate from ideal behavior is crucial, as it helps predict real-world behaviors in various applications, such as in industrial processes and natural phenomena.