Question

An ideal gas obeying kinetic gas equation can be liquefied if: (a) It cannot be liquefied 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

Short answer

(c) Its temperature is more than critical temperature.

Step-by-step solution

Understanding Ideal Gas Behavior

An ideal gas is considered in theory to have no forces of attraction or repulsion between particles. It follows the ideal gas law but does not account for real conditions affecting gas liquefaction, like intermolecular forces.

Critical Temperature Explained

The critical temperature of a gas is the highest temperature at which it can be converted into a liquid by applying pressure. Above this temperature, even high pressures cannot liquefy the gas.

Analyzing Given Options

Option (a) suggests that the gas can't be liquefied at any pressure or temperature which is incorrect as gases can generally liquefy below critical temperature. Option (b) mentions Boyle's temperature which is unrelated to liquefaction. Option (c) states that if the temperature is over the critical temperature, the gas cannot be liquefied, which is true. Option (d) misdirects with pressure alone, as temperature also plays a key role.

Conclusion and Answer

The correct condition for an ideal gas to not be liquefied is when its temperature is higher than the critical temperature, as no pressure can liquefy it beyond this point.

Key concepts

Critical Temperature

The critical temperature of a gas is a fundamental concept in understanding gas behavior and liquefaction processes. It is defined as the maximum temperature at which a gas can be liquefied, regardless of the pressure applied. If the temperature of the gas exceeds the critical temperature, no amount of pressure will succeed in turning the gas into a liquid. This is because, above this threshold, the kinetic energy of gas molecules is too high for attractive forces to pull them together into a liquid state. Additionally, the critical temperature varies for different substances based on their molecular structures. For example, carbon dioxide has a critical temperature of about 31.1°C, meaning it can only be liquefied below this temperature. Understanding this concept is crucial for industrial applications that involve gas storage or transportation, where maintaining gases below their critical temperatures is often essential.

Boyle's Temperature

Boyle's Temperature, also known as the Boyle Temperature, is another important thermodynamic concept. It is the temperature at which a real gas behaves like an ideal gas across a range of pressures. At this temperature, the deviations from ideal gas behavior, as predicted by the ideal gas law, are minimized.
- Unlike the critical temperature, Boyle's temperature is not directly related to the ability of a gas to liquefy. - It is named after Robert Boyle, who is known for Boyle's Law, which states that pressure is inversely proportional to volume at constant temperature for gases. Boyle's temperature is a crucial point where a gas's compressibility factor is close to one, indicating that the gas obeys the ideal gas law more accurately at this temperature. Remember, while Boyle's temperature provides insights into temperature and pressure effects on gas behavior, it does not influence the gas's liquefaction process.

Ideal Gas Behavior

The concept of ideal gas behavior forms the backbone of classical thermodynamics in gases. An ideal gas is assumed to have perfectly elastic collisions and no intermolecular forces. This means that the particles are only influenced by collisions and follow the ideal gas law: \[ PV = nRT \] Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) for the universal gas constant, and \( T \) for temperature. Though an ideal gas is a theoretical model, it serves as a helpful approximation for understanding real gas behavior under many conditions. For example, at low pressures and high temperatures, real gases tend to exhibit behavior similar to ideal gases. However, at conditions such as high pressures or low temperatures, real gases deviate from the ideal model, primarily due to the presence of intermolecular attractive forces. Recognizing ideal gas behavior helps in analyzing how real gases might deviate under various conditions, which is vital for processes like liquefaction.

Kinetic Gas Equation

The kinetic gas equation is a quantitative expression that combines kinetic theory with the behaviors of gases. It provides a molecular-level understanding of pressure and temperature as functions of the movement and speed of gas particles. The equation is expressed as: \[ PV = \frac{1}{3} mN\bar{v}^2 \] In this equation, \( P \) is the pressure of the gas, \( V \) is the volume, \( m \) is the mass of a single molecule, \( N \) is the number of molecules, and \( \bar{v}^2 \) is the mean square speed of the molecules.
The kinetic gas equation connects macroscopic gas properties, like pressure and volume, with microscopic properties like molecular speed and mass. It underscores how temperature affects molecular speeds: higher temperatures result in faster molecular speeds, contributing to greater pressures. This insight is crucial when studying gas behavior, especially under extreme conditions where real gases deviate from the ideal gas laws. Understanding this framework helps in more accurately predicting how and when a gas might be capable of undergoing liquefaction, which is pivotal in understanding real-world gas processes.