Problem 101 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

(b) \( V_m < 22.4 \, \text{L} \)

Step 1: Understanding Compressibility Factor

The compressibility factor (Z) is defined as the ratio of the molar volume of a real gas to the molar volume of an ideal gas at the same temperature and pressure. Mathematically, it is given as \( Z = \frac{V_m}{V_{m,ideal}} \). A gas with \( Z < 1 \) means the real gas's volume is less than expected for an ideal gas.

Step 2: Identifying STP Conditions

At standard temperature and pressure (STP), the volume of 1 mole of an ideal gas is 22.4 L (\( V_{m,ideal} = 22.4 \, \text{L} \)). This is the baseline for comparing the real gas's behavior with that of an ideal gas.

Step 3: Calculating Real Molar Volume

Given \( Z < 1 \), \( V_m = Z \times V_{m,ideal} \) implies that \( V_m < V_{m,ideal} \). With \( V_{m,ideal} = 22.4 \, \text{L} \), it follows that \( V_m < 22.4 \, \text{L} \).

Step 4: Choosing the Correct Option

Since we have determined \( V_m < 22.4 \, \text{L} \), option (b) is correct: \( V_m < 22.4 \, \text{L} \).

Key Concepts

Real Gas Behavior

Real gases deviate from ideal behavior due to interactions between their particles. Unlike ideal gases, where there are no intermolecular forces and the particles occupy no volume, real gases have particles that do interact. This deviation is especially noticeable under high pressure and low-temperature conditions.

When discussing the compressibility factor, denoted by Z, we're examining how closely a real gas's behavior matches that of an ideal gas. The equation used is \( Z = \frac{V_m}{V_{m,ideal}} \). If \( Z = 1 \), the gas behaves ideally. However, if \( Z < 1 \), the real gas's volume is less than what would be expected for an ideal gas. This indicates attractive forces among particles, leading them to be closer together. Conversely, \( Z > 1 \) would suggest repulsive forces are dominant.

Understanding these behaviors is vital in applications where precise gas measurements are required, such as in industrial gas processes and atmospheric science.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used in chemistry to describe the behavior of gases. It's expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin. This law works best under conditions of low pressure and high temperature, where gases are more likely to behave ideally.

The Ideal Gas Law assumes that gas particles do not attract or repel each other, and that they occupy negligible volume. These assumptions are seldom met fully in real-world gases, leading to discrepancies between predicted and observed behaviors. Despite this, the Ideal Gas Law is invaluable for making first approximations in chemical calculations and conceptualizing gas behavior under varying conditions.

Volume at STP

Standard Temperature and Pressure, abbreviated as STP, is a set of conditions typically used to quantify gas volumes. At STP, the temperature is 273.15 K (0°C or 32°F) and the pressure is 1 atm. One mole of an ideal gas occupies 22.4 L at these conditions. This is referred to as the molar volume of an ideal gas at STP.

When assessing real gas behavior at STP, it's essential to compare the real molar volume \( V_m \) to this standard molar volume. Deviations from 22.4 L are evaluated using the compressibility factor \( Z \), revealing whether the gas expands more or contracts compared to an ideal gas.

Understanding the concept of volume at STP allows chemists to easily convert between different units and conditions when measuring gas quantities, enhancing the accuracy of laboratory calculations and providing a baseline for understanding gas behavior under standardized conditions.