Problem 83 - A Complete Resource Book in Chemistry for JEE Main

Question

An ideal gas obeying kinetic gas equation can be liquefied if (a) it cannot be liquefi ed at any value of \(\mathrm{P}\) and \(\mathrm{T}\) (b) its temperature is more than Boyle's temperature (c) its temperature is more than critical temperature (d) its pressure is more than critical pressure

Step-by-Step Solution

Answer

(c) Its temperature is more than critical temperature.

Step 1: Understanding the Gas Law

An ideal gas is a hypothetical gas whose molecules occupy negligible space and have no interactions. It perfectly obeys the laws expressed in Boyle's law, Charles's law, and Avogadro's law.

Step 2: Kinetic Gas Equation

The kinetic gas equation describes the properties of gases in terms of molecular motion and is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

Step 3: Conditions for Liquefaction

A real gas can be liquefied when its temperature is below its critical temperature and when sufficient pressure is applied. The critical temperature is the highest temperature at which a gas can be liquefied by pressure alone.

Step 4: Eliminating Incorrect Options

Option (a) is incorrect because some real gases can be liquefied at appropriate conditions. Option (b) is incorrect because Boyle's temperature relates to the deviation from ideal behavior, not liquefaction. Option (d) is incorrect as pressure alone does not determine liquefaction without considering temperature.

Step 5: Correct Option

An ideal gas cannot be liquefied if its temperature is above the critical temperature, as no amount of pressure can liquefy it under these conditions. Therefore, option (c) is the correct choice.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure, volume, and temperature of an ideal gas. This law is expressed in the formula \( PV = nRT \), where \( P \) stands for pressure, \( V \) is volume, \( n \) is the number of moles of the gas, \( R \) is the universal gas constant, and \( T \) represents temperature in Kelvin.
An ideal gas is a theoretical construct in which gas molecules do not interact and occupy negligible space. Therefore, it follows this law perfectly. While real gases deviate from these assumptions under high pressure and low temperatures, the Ideal Gas Law provides a good approximation under normal conditions.
One interesting aspect of the Ideal Gas Law is its integration of several other gas laws:

• Boyle's Law: which states that pressure and volume are inversely related at constant temperature.
• Charles's Law: which establishes that volume and temperature are directly related when pressure is constant.
• Avogadro's Law: which shows that volume and number of moles of gas are proportional at a fixed temperature and pressure.

Kinetic Molecular Theory

Kinetic Molecular Theory is a simple model that explains the behavior of gases based on the motion of their molecules. According to this theory, gases consist of tiny particles in constant, random motion. These particles collide with each other and the walls of any container they are in, leading to the exertion of pressure.
The theory assumes that the actual volume of gas molecules is negligible compared to the volume they occupy. It also suggests that there are no forces of attraction or repulsion between the particles. As a result, the movement and collisions of these particles obey Newton's laws of motion.
Here are some key takeaways from the kinetic molecular theory:

• Gas pressure is a result of collisions between gas particles and the walls of the container.
• The average kinetic energy of gas molecules is directly proportional to the absolute temperature.
• At higher temperatures, particles move faster, which increases the number of collisions and thus the pressure and volume.

Understanding this theory provides foundational knowledge crucial for studying changes in states of matter, such as liquefaction.

Critical Temperature

Critical Temperature is a vital concept in understanding the liquefaction of gases. It is defined as the highest temperature at which a gas can be turned into a liquid, solely through the application of pressure.
Above the critical temperature, it is impossible for a gas to become a liquid, regardless of how much pressure is applied. This is because the kinetic energy of the particles at such temperatures is too high to allow them to bond into a liquid phase. For practical purposes, knowing a gas's critical temperature is important in industrial processes where the liquefaction of gases is required, such as in cooling systems and the transport of gases.
The concept emphasizes the limitation of pressure’s role in phase changes and highlights temperature's influence. In the context of the original exercise, choosing option (c) is justified because it is a principle rooted in the behavior of real gases.