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

Question

A gas, which obeys Boyle's law, Charle's law, Avogadro's law etc., or ideal gas equation \(\mathrm{PV}=\mathrm{nRT}\) under all conditions of temperature and pressure, is called ideal gas. No gas is ideal. All gases are real gases. The real gas obeys these gas laws only when the temperature is high or pressure is low. The extent of derivations of a real gas form ideal behaviour is expressed in terms of compressibility factor \(Z\) defined as \(\mathrm{Z}=\frac{\mathrm{PV}}{\mathrm{nRT}}\) Real gases have characteristic temperatures like critical temperature, inversion temperature and Boyle temperature. These temperatures can be calculated using van der Waal constants. The compressibility factor of a gas is less than unity at STP. Therefore (a) Vm \(>22.4\) lit (b) \(\mathrm{Vm}<22.4 \mathrm{lit}\) (c) \(\mathrm{Vm}=22.4 \mathrm{lit}\) (d) \(\mathrm{Vm}=11.2 \mathrm{lit}\)

Step-by-Step Solution

Answer

Option (b) \( \mathrm{Vm}<22.4 \mathrm{lit} \) when \( Z < 1 \).

Step 1: Understanding Compressibility Factor (Z)

The compressibility factor, \( Z \), is a measure of how much the behavior of a real gas deviates from an ideal gas. It is defined by the equation \( Z = \frac{PV}{nRT} \). For an ideal gas at STP, \( Z \) equals 1. If \( Z < 1 \), this indicates that the gas occupies less volume than expected under ideal conditions.

Step 2: Interpret STP Conditions for Real Gases

At standard temperature and pressure (STP), we calculate properties for gases assuming ideal conditions. Here, the molar volume of an ideal gas, \( Vm \), at STP is 22.4 liters. When \( Z < 1 \), it shows that the actual volume \( V \) of the gas is less than it would be for an ideal gas. Thus, \( Vm < 22.4 \) liters when a real gas exhibits \( Z < 1 \).

Step 3: Conclude Based on Given Options

Given that the compressibility factor \( Z \) is less than unity, we need to find the option that correctly describes this scenario. Since \( Vm < 22.4 \) liters when \( Z < 1 \), the correct choice is option (b) \( \mathrm{Vm}<22.4 \mathrm{lit} \).

Key Concepts

Real Gases

Real gases are gases that do not adhere strictly to the ideal gas law under all conditions. Unlike ideal gases, which are hypothetical and follow the ideal gas equation \( PV = nRT \) perfectly, real gases exhibit deviations due to intermolecular forces and the actual volume occupied by gas particles.
These deviations become more pronounced at low temperatures and high pressures. Under these conditions, the assumptions of negligible volume and no intermolecular forces, which are valid for ideal gases, are less accurate. Real gases behave more like ideal gases at high temperatures and low pressures because their particles move faster and the effect of intermolecular attractions and repulsions becomes negligible.
Understanding the behavior of real gases is crucial in practical applications, such as when gases are stored under non-ideal conditions. Using real gas equations like the Van der Waals equation can help to predict their behavior more accurately under these conditions.

Compressibility Factor

The compressibility factor, represented as \( Z \), is a dimensionless number that indicates how much a real gas deviates from ideal gas behavior. It is calculated using the formula \( Z = \frac{PV}{nRT} \).
For an ideal gas, the value of \( Z \) is exactly 1, signifying perfect adherence to the ideal gas law. However, for real gases, \( Z \) can be less than or greater than 1. If \( Z < 1 \), it indicates that the gas occupies less volume than an ideal gas would under the same conditions; if \( Z > 1 \), it means the gas occupies more volume.
This factor is significant because it helps chemists and engineers understand how real gases behave under various pressures and temperatures, enabling more accurate calculations and predictions in real-world scenarios. A compressibility factor helps bridge the gap between theoretical models and actual, observable behavior.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, abbreviated as STP, refers to the conditions of 0°C (273.15 K) and 1 atm pressure. At STP, an ideal gas is said to occupy a molar volume of 22.4 liters per mole.
These conditions are used as a reference point for scientific measurements and calculations, allowing for consistency and comparability in gas behavior studies. Real gases are often measured against STP to identify how their volumes fluctuate in non-ideal settings.
While STP provides a baseline for ideal gases, real gases at STP can behave differently, as seen with the compressibility factor \( Z \) which might be less than 1, indicating deviations in volume and reactivity.

Molar Volume

Molar Volume is a term that refers to the volume occupied by one mole of a gas at a given temperature and pressure. For an ideal gas at Standard Temperature and Pressure (STP), this volume is known to be 22.4 liters.
In real-world applications, the molar volume of a gas can vary due to deviation from ideal behavior, often observed in real gases. The compressibility factor \( Z \) influences the molar volume such that when \( Z < 1 \), the molar volume is less than expected. If \( Z > 1 \), the molar volume is greater.
This concept is vital because it allows for the recalculation of expectations based on real gas behaviors rather than relying solely on ideal gas equations, which can lead to inaccuracies when conditions diverge from ideal.