Problem 126 - A Complete Resource Book in Chemistry for JEE Main

Question

The compressibility factor of a gas is less than unity at STP. Therefore (a) \(\mathrm{Vm}>22.4 \mathrm{lit}\) (b) \(\mathrm{Vm}<22.4 \mathrm{lit}\) (c) \(\mathrm{Vm}=22.4 \mathrm{lit}\) (d) \(\mathrm{Vm}=11.2\) lit

Step-by-Step Solution

Answer

(b) \(V_m < 22.4 \mathrm{liters}\)

Step 1: Understanding Compressibility Factor

The compressibility factor (Z) indicates how much the real gas deviates from ideal gas behavior. It is defined as Z = \(\frac{PV_m}{RT}\), where \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the gas constant, and \(T\) is the temperature. If Z < 1, the gas is more compressible than an ideal gas, usually due to intermolecular attractions.

Step 2: Effect of Z < 1 on Molar Volume

Given that Z < 1, it implies that \(PV_m < RT\). For a gas at STP (Standard Temperature and Pressure), an ideal gas has a molar volume \(V_m = 22.4\, \mathrm{liters}\). When Z < 1 and the gas is more compressible, the molar volume \(V_m\) will be less than that of an ideal gas at the same conditions.

Step 3: Conclusion

Given that the compressibility factor (Z) is less than 1 at STP, it can be concluded that the real gas occupies less volume than an ideal gas. Therefore, \(V_m\) for the real gas is less than \(22.4\, \mathrm{liters}\).

Key Concepts

Real vs Ideal Gas Behavior

Understanding the behavior of real gases as compared to ideal gases is crucial in thermodynamics. Ideal gases follow the Ideal Gas Law, expressed as \(PV = nRT\), perfectly under all conditions. Real gases, however, deviate from this behavior due to interactions between gas molecules and volume occupied by them. These deviations are quantified by the compressibility factor, \(Z\). When \(Z = 1\), the gas behaves ideally. But when \(Z < 1\) or \(Z > 1\), the gas is behaving non-ideally.

• \(Z < 1\): This indicates that the gas molecules experience attractive forces, making the gas more compressible than an ideal gas.
• \(Z > 1\): This suggests repulsive forces dominate, making the gas less compressible than an ideal gas.

At Standard Temperature and Pressure (STP), real gases usually have a compressibility factor less than unity, indicating they occupy less volume than expected by the Ideal Gas Law. This is because of intermolecular attractions that pull the molecules closer together.

Molar Volume

Molar volume is the volume occupied by one mole of a substance, which is significant in understanding gas behavior. For an ideal gas, the molar volume is defined under specific conditions, such as STP (Standard Temperature and Pressure), where it is typically \(22.4\, \text{liters}\) for one mole of gas.

When dealing with real gases, this ideal molar volume may not match due to various forces between molecules. If the compressibility factor \(Z\) is less than one, as in the case mentioned in the exercise, the molar volume \(V_m\) of the gas is less than \(22.4\, \text{liters}\).

This happens because:

• Attractive forces cause the molecules to pull closer together, reducing the volume.
• The real gas's occupancy is lessened compared to an ideal gas under the same STP conditions.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, commonly referred to as STP, is a set of conditions for measurement in chemistry to standardize results. STP is defined as a temperature of \(0^{\circ}\text{C}\) (273.15\, \text{K}) and a pressure of \(1\, \text{atm}\) (101.3\, \text{kPa}). Under these conditions, one mole of an ideal gas occupies \(22.4\, \text{liters}\). This is the basis for comparison with real gas properties.

By using STP conditions, chemists can easily determine and compare the behaviors of various gases. For real gases, knowing the compressibility factor at STP helps in understanding their behavior compared to the idealized scenario. A compressibility factor less than one at STP demonstrates the reduced volume real gases occupy due to molecular attractions.

In summary, STP is critical in providing a baseline for assessing gas behaviors and deviations, which greatly aids in practical applications and predictions of gas properties.