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 (q), (b) matches (r) or (t), (c) matches (p), (d) matches (s).

Step 1: Understand the Compressibility Factor

The compressibility factor, denoted by \(Z\), is a measure of how much a real gas deviates from ideal gas behavior. For an ideal gas, \(Z = 1\), which implies that the gas behaves exactly according to the ideal gas law (\(PV = nRT\)). If \(Z < 1\), the gas is more compressible than an ideal gas, usually due to attractive forces being dominant. If \(Z > 1\), the gas is less compressible than an ideal gas, often due to repulsive forces.

Step 2: Match Column-I to Column-II

Let's match each item from Column-I to the appropriate item in Column-II based on your understanding from Step 1:- (a) Compressibility factor, \(Z=1\) aligns with (q) \(PV = nRT\) since \(Z = 1\) indicates ideal gas behavior.- (b) Compressibility factor, \(Z>1\) aligns with (r) repulsive forces dominate or (t) gas is less compressible, since \(Z > 1\) means the gas is harder to compress, likely due to repulsive forces.- (c) Compressibility factor, \(Z<1\) aligns with (p) attractive forces dominate because the gas is more compressible due to these forces.- (d) Boyle temperature aligns with (s) attractive force and repulsive forces cancel each other since this is the temperature at which a real gas behaves like an ideal gas over a range of pressures.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes the behavior of an ideal gas. It is represented by the formula \( PV = nRT \), where:

• \(P\) is the pressure of the gas,
• \(V\) is the volume occupied by the gas,
• \(n\) is the number of moles of the gas,
• \(R\) is the universal gas constant, and
• \(T\) is the absolute temperature of the gas.

This equation assumes that the gas particles are point-sized, and there are no interactions between them. However, real gases do not always follow this law, especially at high pressures and low temperatures. In these conditions, the interactions between particles and the volume occupied by gas molecules can cause deviations from ideal behavior.
Therefore, we use the compressibility factor \(Z\) to signify such deviations. When \(Z=1\), the gas behaves ideally, aligning perfectly with the Ideal Gas Law.

Real Gases

Real gases are gases that do not always behave according to the Ideal Gas Law, especially under conditions of high pressure and low temperature. Unlike ideal gases, real gases have finite volume and experience intermolecular forces, which can affect their behavior.
Real gases can have values of the compressibility factor \(Z\) that differ from one. If \(Z < 1\), it suggests that attractive forces between gas molecules dominate, making the gas more compressible than an ideal gas. Conversely, if \(Z > 1\), repulsive forces are stronger, rendering the gas less compressible.
In reality, gases like nitrogen and oxygen can be treated as ideal under normal conditions, but better approximations are needed under extreme conditions to account for intermolecular forces and the volume occupied by the molecules themselves.

Boyle Temperature

Boyle Temperature is a special temperature at which a real gas behaves like an ideal gas. At this specific temperature, over a certain range of pressures, the attractions and repulsions experienced by gas molecules are exactly balanced.
This means that at the Boyle Temperature, the compressibility factor \(Z\) is equal to one, mimicking the behavior described by the Ideal Gas Law.

• Above the Boyle Temperature, the gas behaves more ideally as repulsive forces balance out the attractive forces.
• Below this temperature, attractive forces might dominate, making the gas deviate more from ideal behavior.

Analyzing the Boyle Temperature for different gases helps in understanding how a specific gas would behave under varying conditions of temperature and pressure, and is essential for optimizing industrial processes involving gases.