Problem 5 - A Complete Resource Book in Chemistry for JEE Main

Question

Ideal gas obeying kinetic theory of gases can be liquefied if (a) \(\mathrm{T}>\mathrm{T}\) (b) \(\mathrm{P}>\mathrm{P}_{\mathrm{c}}\) (c) \(\mathrm{P}>\mathrm{P}_{\mathrm{c}}\) and \(\mathrm{T}<\mathrm{T}_{\mathrm{c}}\) (d) it cannot be liquefied at any value of \(\mathrm{P}\) and \(\mathrm{T}\).

Step-by-Step Solution

Answer

Option (c) is correct: \(P > P_c\) and \(T < T_c\).

Step 1: Understanding key concepts

To solve this problem, we need to use the key concepts of critical temperature (\(T_c\)) and critical pressure (\(P_c\)) in thermodynamics. An ideal gas can only be liquefied if the temperature is below its critical temperature and pressure is above its critical pressure.

Step 2: Analyzing each option

Let's analyze each option based on the concept of critical temperature and pressure: (a) \(T > T_c\): if the temperature is above the critical temperature, the gas cannot be liquefied. (b) \(P > P_c\): pressure alone above the critical pressure is not sufficient, as temperature must also be below \(T_c\). (c) \(P > P_c\) and \(T < T_c\): this is correct, as both conditions are necessary for liquefaction. (d) The statement claiming it cannot be liquefied under any conditions is incorrect because it ignores the critical temperature and pressure concepts.

Step 3: Selecting the correct option

Based on our analysis of each option, we conclude that option (c) is correct. An ideal gas can be liquefied if the pressure is greater than the critical pressure and the temperature is less than the critical temperature.

Key Concepts

Critical Temperature

Critical temperature is the utmost temperature at which a gas can be turned into a liquid. At this temperature, the kinetic energy of a gas's molecules is very high. Beyond this, the thermal energy causing the molecules to move apart is stronger than the forces that pull them together. Thus, gases can only be liquefied below the critical temperature because at lower temperatures, the attractive forces between molecules dominate. It is a unique characteristic of every gas and helps in determining how easily a gas can transition into a liquid state.

Critical Pressure

Critical pressure refers to the minimum pressure that must be applied to a gas to transform it into a liquid at its critical temperature. Think of it as the push needed to compress a gas enough for its molecules to interact closely and form a liquid. It ensures that even at the highest permissible temperatures for liquefaction, the gas is dense enough for the intermolecular attractions to take effect. If pressure alone is elevated without paying heed to temperature, liquefaction will still not be possible if the gas is above its critical temperature.

Liquefaction of Gases

Liquefaction is the process where a gas transitions into a liquid. For this to occur, both the pressure and temperature of the gas must adhere to specific conditions. The ideal condition for liquefaction occurs when the gas is cooled below its critical temperature and subjected to a pressure greater than its critical pressure. In this state, the kinetic energy of molecules is reduced, allowing intermolecular attractions to pull molecules together to form a liquid. Understanding these conditions aids in diverse applications, including industrial gas liquefaction and refrigeration systems.

Ideal Gas

An ideal gas is a theoretical model that simplifies the behavior of gases. It assumes that gas molecules are point particles that do not interact with each other except for perfect elastic collisions. This model disregards any intermolecular forces, which are non-ideal conditions observed in real gases. However, in real-world applications, corrections are made through concepts such as critical temperature and pressure to account for deviations from ideal behavior, especially when tackling gas liquefaction processes.

Thermodynamics

Thermodynamics is the study of energy, heat, and work within physical systems. It provides insights into how different forms of energy transition and transform, playing a crucial role in understanding gas behaviors. Most relevant to this context, it encompasses laws that govern the conditions for liquefaction of gases, including essential parameters like temperature, pressure, and volume. A grasp of thermodynamic principles allows us to predict when and how a gas might liquefy when the appropriate critical pressure and temperature conditions are met.