Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

Match the following Column-I (a) Compressibility factor, \(Z=1\) (b) Compressibility factor, \(Z>1\) (c) Compressibility factor, \(Z<1\) (d) Boyle temperature Column-II (p) Attractive forces dominate (q) \(\mathrm{PV}=\mathrm{nRT}\) (r) Repulsive forces dominate (s) Attractive force and repulsive forces cancel each other (t) Gas is less compressible

Step-by-Step Solution

Answer

(a) matches with (q), (b) with (r), (c) with (p), and (d) with (s).

Step 1: Understand Matching Definitions

In the problem, we are given a list of concepts related to the compressibility factor (Column-I) and a list of corresponding explanations or situations (Column-II). We need to match each element in Column-I to the correct element in Column-II based on the definition of the compressibility factor, denoted as \( Z \).

Step 2: Analyze the Option (a) \( Z=1 \)

The compressibility factor \( Z \) is defined as \( Z = \frac{PV}{nRT} \). When \( Z = 1 \), the gas behaves ideally, thus following the ideal gas equation \( PV = nRT \). Therefore, option (a) should be matched with option (q).

Step 3: Analyze the Option (b) \( Z>1 \)

A compressibility factor \( Z > 1 \) indicates that the gas is less compressible than expected, often due to dominant repulsive forces. Hence, option (b) matches with option (r) as repulsive forces between molecules lead to higher pressure than predicted by an ideal gas.

Step 4: Analyze the Option (c) \( Z<1 \)

The compressibility factor \( Z < 1 \) suggests that the gas is more compressible than expected, due to the dominance of attractive forces. This matches (c) with (p), as these attractive forces cause a reduction in pressure compared to the ideal gas prediction.

Step 5: Analyze the Option (d) Boyle Temperature

At Boyle's temperature, a real gas behaves as an ideal gas because the attractive and repulsive forces between particles cancel each other out. Therefore, option (d) corresponds to option (s).

Key Concepts

Ideal Gas Behavior

Gases that fit the ideal gas behavior laws adhere to a simple equation: \[ PV = nRT \] Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. These gases assume no interactions between molecules and that the molecules occupy no volume. The compressibility factor \( Z \) is essential here, as it helps determine how closely a real gas's behavior compares to the ideal. For ideal gases, \( Z = 1 \), implying perfect alignment with the ideal gas law.

• When \( Z = 1 \), the gas's behavior is identical to the ideal gas assumptions.
• Real gases tend to follow this behavior under low pressure and high temperatures.

In reality, no gas behaves completely ideally because real gases have molecular interactions and occupy space, which can lead to deviations.

Attractive Forces

Attractive forces in gases are the interactions pulling gas molecules towards each other. These forces are often weak but become significant under certain conditions, like low temperatures or high pressures. Attractive forces can lead to the gas being more compressible than predicted by the ideal gas law.

• For gases where \( Z < 1 \), attractive forces dominate.
• This leads to lower pressure at the same volume and temperature because the molecules are closer together and slow the pressure increase.

This behavior implies that the gas particles are more likely to aggregate due to these attractions, deviating from the ideal scenario.

Repulsive Forces

Repulsive forces arise when gas molecules are so close that they start to repel each other, typically occurring at high pressures. These forces make the gas less compressible than an ideal gas. When these repulsive forces dominate:

• \( Z > 1 \), indicating that the molecules are more resistant to being compressed.
• This results in a higher pressure than would be expected from the ideal gas law.

At high pressures, the volume is smaller, causing closer molecular proximity and stronger repulsive forces, leading to the gas exhibiting non-ideal behavior.

Boyle's Temperature

Boyle's temperature is a unique condition for real gases where the gas molecules' attractive and repulsive actions are balanced. At this temperature, the compressibility factor \( Z \) approaches 1, implying ideal gas-like behavior.

• This means that the real gas behaves nearly like an ideal gas, simplifying calculations.
• When attractive and repulsive forces cancel each other, the deviations from ideality are minimized.

Boyle's temperature provides valuable insights into the conditions necessary for a real gas to mimic an ideal gas, crucial for practical applications such as chemical reactions and studies involving gas laws.