Question

The compressibility factor of a gas is less than unity at STP. Therefore: (a) \(\mathrm{V}_{\mathrm{m}}>22.4 \mathrm{~L}\) (b) \(\mathrm{V}_{\mathrm{m}}<22.4 \mathrm{~L}\) (c) \(\mathrm{V}_{\mathrm{m}}=22.4 \mathrm{~L}\) (d) \(\mathrm{V}_{\mathrm{m}}=44.8 \mathrm{~L}\)

Short answer

(b) \( V_m < 22.4 \, L \)

Step-by-step solution

Understanding Compressibility Factor

The compressibility factor, denoted as \( Z \), is a measure of how much a real gas deviates from ideal gas behavior. It is defined by the equation \( Z = \frac{PV_m}{RT} \), where \( P \) is pressure, \( V_m \) is molar volume, \( R \) is the gas constant, and \( T \) is temperature. For an ideal gas, \( Z \) is equal to 1.

Interpreting Z < 1

When the compressibility factor \( Z \) is less than 1, it indicates that the gas has stronger intermolecular forces than expected in an ideal gas. This implies that the gas is more compressible, occupying a smaller volume than predicted by ideal gas laws.

Molar Volume of Ideal Gas at STP

At standard temperature and pressure (STP), the molar volume of an ideal gas is 22.4 L. This is calculated using \( V_m = \frac{RT}{P} \) where \( R \) is the ideal gas constant, \( T \) is 273.15 K, and \( P \) is 1 atm.

Determining the Effect on Molar Volume

Since \( Z < 1 \) suggests the real gas is more compressible and occupies less space compared to the ideal scenario, the molar volume \( V_m \) of the real gas must be less than that of an ideal gas at STP, which is 22.4 L.

Conclusion

Given that \( Z < 1 \) implies \( V_m < 22.4 \, L \), the correct answer for the molar volume of a gas at STP with a compressibility factor less than 1 is option (b) \( V_m < 22.4 \, L \).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry, relating the pressure, volume, temperature, and amount of a gas. It is represented by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

This law assumes that gases consist of small particles moving randomly in straight lines, with no volume and no interaction forces between them, which is a simplified model of a real gas. However, real gases have intermolecular forces, and those can affect how they behave under different conditions. Ideal Gas Law helps us understand theoretical conditions but often requires adjustments for real gas scenarios.

Molar Volume

Molar volume is the volume occupied by one mole of a substance, commonly referenced in the context of gases. At standard temperature and pressure (STP)—where temperature is 273.15 K and pressure is 1 atm—the molar volume of an ideal gas is known to be 22.4 liters. This standardized measurement simplifies calculations. When introducing the concept of compressibility, you must understand that if a gas deviates from this ideal behavior (e.g., when experiencing stronger intermolecular forces), it could have a smaller molar volume than 22.4 liters due to increased compression. This is an example of how real gas behavior varies from the ideal predictions using references like the compressibility factor, \( Z \).

Intermolecular Forces

Intermolecular forces are the forces that occur between molecules. Unlike the forces within molecules (covalent or ionic bonds), these forces are responsible for the physical properties of substances, such as boiling and melting points, solubility, and gas compressibility. For gases, these forces can significantly alter volume and pressure conditions predicted by the Ideal Gas Law. The stronger the intermolecular forces, the more a gas can deviate from ideal behavior. When a gas's compressibility factor \( Z \) is less than one, it indicates that the gas molecules experience relatively strong attractions, causing them to come closer than expected under ideal conditions. This results in a reduced volume or increased compressibility, explaining why real gases can deviate from the molar volume of an ideal gas. Understanding these interactions is crucial for predicting how gases behave in different environments and under various conditions.