Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

If \(Z\) is a compressibility factor, van der Waals equation at low pressure can be written as: |2014] (a) \(\mathrm{Z}=1-\frac{\mathrm{Pb}}{\mathrm{RT}}\) (b) \(\mathrm{Z}=1+\frac{\mathrm{Pb}}{\mathrm{RT}}\) (c) \(\mathrm{Z}=1+\frac{\mathrm{RT}}{\mathrm{Pb}}\) (d) \(Z=1+\frac{a}{V R T}\)

Step-by-Step Solution

Answer

(b) \( Z = 1 + \frac{Pb}{RT} \) is the correct choice.

Step 1: Understand van der Waals equation at low pressure

The van der Waals equation for real gases is given as \( \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \). At low pressures, the term \( \frac{an^2}{V^2} \) becomes negligible, and the equation simplifies to \( PV = nRT + Pb \) since \( V - nb \approx V \).

Step 2: Define Compressibility factor Z

The compressibility factor \( Z \) is defined as \( Z = \frac{PV}{nRT} \). It indicates how much the behavior of a real gas deviates from an ideal gas.

Step 3: Substitute simplified van der Waals equation into Z

Substitute the simplified form \( PV = nRT + Pb \) into the compressibility factor formula: \( Z = \frac{PV}{nRT} = \frac{nRT + Pb}{nRT} \).

Step 4: Simplify the expression for Z

Divide both terms in the numerator by \( nRT \), which results in: \( Z = \frac{nRT}{nRT} + \frac{Pb}{nRT} = 1 + \frac{Pb}{nRT} \).

Step 5: Choose the correct option

Comparing with the given options, the correct formula for \( Z \) at low pressures matches option (b): \( Z = 1 + \frac{Pb}{RT} \).

Key Concepts

Compressibility Factor

The compressibility factor, often denoted as \( Z \), is a useful concept in understanding the behavior of gases. It tells us how much a real gas deviates from the ideal gas law. For an ideal gas, the compressibility factor is exactly 1, which means that the gas behaves perfectly according to the ideal gas law equation, \( PV = nRT \). However, real gases do not always follow this law due to intermolecular forces and the volume occupied by gas particles.
In the context of van der Waals equation, the compressibility factor can be expressed as:

• \( Z = \frac{PV}{nRT} \)

This factor helps us adjust our calculations and predictions for the behavior of real gases under various conditions. When \( Z \) is greater than 1, the gas is less compressible, often seen when repulsive forces dominate. If \( Z \) is less than 1, the gas is more compressible, suggesting that attractive forces are more significant.

Real Gases

Real gases deviate from ideal gas behavior due to two main factors: intermolecular forces and the actual volume occupied by gas molecules. Ideal gas laws assume no such forces and that particles occupy no volume. However, in reality:

• Intermolecular Attractive Forces: At lower temperatures or higher pressures, these forces become more significant, causing deviations from ideal behavior.
• Finite Molecular Volume: Unlike ideal gas particles, real gas molecules occupy space, affecting how they interact and pack together.

These factors are considered in the van der Waals equation, which adds correction terms for pressure and volume. This equation is expressed as:\[ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \]Where \(a\) represents attraction between particles and \(b\) accounts for the volume occupied by the gas molecules.

Ideal Gas Behavior

Ideal gas behavior is an approximation where gases are assumed to have perfectly elastic collisions and negligible volumes compared to the container. Under such conditions, gases follow the ideal gas law, \( PV = nRT \), without any need for adjustments. This assumption works well under:

• High Temperature: Increased kinetic energy overpowers intermolecular forces.
• Low Pressure: Molecules are spaced far apart, minimizing interactions.

However, it's important to remember that no real gas fits this model perfectly. The ideal gas law is best viewed as a useful approximation that simplifies calculations and helps us understand the limiting behavior of gases under specific conditions.

Low Pressure

At low pressure, gases tend to show behaviors closer to that of an ideal gas because the molecules are spread out over a larger volume, thereby reducing intermolecular interactions. In the case of the van der Waals equation, this means that:

• The correction term for intermolecular attraction, \( \frac{an^2}{V^2} \), becomes negligible.
• The equation simplies to \( PV = nRT + Pb \), emphasizing that at low pressure, the additional forces and volume effects are less crucial.

This simplification helps in calculating the compressibility factor \( Z \) under low pressures, resulting in changes primarily due to the volume occupied by particles. Thus, at low pressure, the real gas behavior can be expressed more conveniently, allowing for easier approximation to the ideal gas law.