Question

An ideal gas obeying kinetic theory of gases can be liquefied if (1) Its temperature is more than critical temperature, \(T_{e}\). (2) Its pressure is more than critical pressure, \(P_{0}\). (3) Its pressure is more than \(P_{\mathrm{c}}\) at a temperature less than \(T_{\mathrm{v}}\) (4) It cannot be liquefied at any value of \(P\) and \(T\)

Short answer

Option 3 is correct.

Step-by-step solution

Understanding Critical Temperature

Identify the relationship between the critical temperature and the liquefaction of an ideal gas. A gas can be liquefied if it is cooled below a certain temperature known as the critical temperature, \(T_c\).

Understanding Critical Pressure

Identify the relationship between the critical pressure and the liquefaction of an ideal gas. A gas can be liquefied if it is compressed above a certain pressure known as the critical pressure, \(P_c\).

Analyzing the Given Options

Evaluate each option based on the definitions of critical temperature and critical pressure: 1. If the temperature is more than critical temperature \(T_e\), the gas cannot be liquefied.2. If the pressure is more than critical pressure \(P_0\) but the temperature is above critical temperature, the gas cannot be liquefied.3. If the pressure is above \(P_c\) and the temperature is below \(T_v\) (where \(T_v < T_c\)), the gas can be liquefied.4. The statement that a gas cannot be liquefied at any value of \(P\) and \(T\) is incorrect.

Correct Option

From the analysis, option 3 is correct as it aligns with the requirements for liquefaction of an ideal gas: pressure above critical pressure (\(P_c\)) and temperature below critical temperature (\(T_v < T_c\)).

Key concepts

critical temperature

Critical temperature (T_c) is a key concept for understanding when a gas can be liquefied. This is the highest temperature at which a substance can exist as a liquid. Above this temperature, no amount of pressure can cause the gas to liquefy.
For a gas to be liquefied, it must be cooled below its critical temperature.
The critical temperature is unique to each substance.
Some important points to remember about critical temperature include:
- It acts as a boundary - above it, substances remain gaseous regardless of the pressure applied.
- It's a reflection of the strength of intermolecular forces - stronger forces result in a higher critical temperature.
Make sure to understand that exceeding the critical temperature of a gas makes liquefaction impossible, no matter the pressure.

critical pressure

Critical pressure (P_c) is another essential concept. This is the pressure required to liquefy a gas at its critical temperature.
Simply put, it's the minimal pressure needed to change a gas to a liquid at the critical temperature.
A few key points to remember about critical pressure:
- Below the critical temperature, increasing pressure sufficiently will cause liquefaction.
- At the critical temperature, if the pressure is not at least the critical pressure, the gas will not liquefy.
Just as with temperature, each substance has its own critical pressure.
Understanding the relationship between temperature and pressure around these critical points can be tricky but is crucial for predicting when liquefaction can occur.

ideal gas

The concept of an ideal gas is a foundational topic in chemistry and physics. An ideal gas is a theoretical gas that perfectly follows the Ideal Gas Law, which is expressed as:
PV = nRT
Here:
- P is pressure
- V is volume
- n is the number of moles
- R is the gas constant
- T is temperature
Remember, an ideal gas assumes completely elastic collisions and negligible intermolecular forces.
This model simplifies the study of gases and provides a good approximation in many situations.
However, real gases show deviations from this behavior, especially under conditions of high pressure and low temperature.
Understanding the behavior of real gases near their critical points involves recognizing these deviations and using more complex models.
This understanding is essential when dealing with liquefaction, as real gases need consideration of their critical temperature and pressure.