Question

An ideal gas, obeying kinetic theory of gases, can be liquefied: (a) It cannot be liquefied at any $P$ and $T$ (b) Its pressure is more than $P$ at a temperature less than $T$ (c) Its temperature is more than critical temperature $T$ (d) Its pressure is more than critical pressure $P$

Short answer

The answer is (a). An ideal gas cannot be liquefied at any pressure $P$ and temperature $T$ as ideal gases can't be liquefied according to the kinetic theory of gases.

Step-by-step solution

Understand liquefaction of gases

Liquefaction of a gas happens when it is converted into a liquid state. For this transition to occur, the gas must be below its critical temperature and pressure.

Examine potential scenarios

Examine each situation given in the exercise options: (a) It cannot be liquefied at any $P$ and $T$. (b) Its pressure is more than $P$ at a temperature less than $T$. (c) Its temperature is more than critical temperature $T$. (d) Its pressure is more than critical pressure $P$. It's crucial to remember that ideal gases cannot be liquefied. However, if the gas is below its critical temperature and pressure, it could potentially be liquefied.

Select correct condition

Based on the information given in step 1 and 2, it is clear that the gas can potentially be liquefied if its pressure is more than the critical pressure, and if the temperature is less than the critical temperature. So none of the options gives a condition under which the gas can be liquefied, hence the correct choice is (a)

Key concepts

Kinetic theory of gases

The kinetic theory of gases is a fundamental scientific theory that explains the behavior and properties of gases. It is based on the idea that gases are composed of a large number of tiny particles, usually atoms or molecules, that are in constant, random motion. This motion is what gives gases their unique properties.
- **Molecular Motion**: The gas particles move in straight lines until they collide with either another particle or the walls of a container. These collisions are perfectly elastic, meaning there is no energy loss.
- **Pressure and Temperature**: The pressure exerted by a gas in a container is because of the collisions of its molecules with the walls of the container. The temperature of a gas is related to the average kinetic energy of its molecules. In essence, a higher temperature means the molecules are moving faster.
- **Ideal Gas Assumption**: In the context of the kinetic theory, gases are often considered 'ideal'. This means they follow the ideal gas laws precisely, with assumptions like no intermolecular forces and perfectly elastic collisions. However, real gases deviate from these assumptions under certain conditions, such as high pressure or low temperature, where they might demonstrate tendencies to liquefy.

Critical temperature

Critical temperature, often denoted as $T_c$, is a key concept for understanding the behavior of gases and their ability to transition into a liquid state. It is defined as the maximum temperature at which a gas can be liquefied by pressure alone.
- **Above Critical Temperature**: If the temperature of a gas is above its critical temperature, no amount of pressure, no matter how great, can cause the gas to liquefy. This is because the kinetic energy of the gas particles is too high for intermolecular forces to bring them together into a liquid state.
- **Below Critical Temperature**: When the gas is cooled to its critical temperature or below, the kinetic energy of the particles decreases enough to allow intermolecular forces to take effect, facilitating the transition from a gas to a liquid with sufficient pressure.
This concept is crucial for processes like gas liquefaction and plays an important role when considering real gases, which don't perfectly follow the ideal gas laws.

Critical pressure

Critical pressure, symbolized as $P_c$, is the minimum pressure required to liquefy a gas at its critical temperature. It's a vital aspect of understanding how gases can be converted into liquids and is an intrinsic property of each substance.
- **Role in Liquefaction**: When trying to liquefy a gas, achieving a pressure equal to or greater than the critical pressure is necessary when the temperature of the gas is at or below its critical temperature.
- **Real-World Application**: In practical scenarios, understanding critical pressure allows engineers and scientists to design equipment and processes for efficiently converting gases into liquids. This includes refrigeration, air conditioning, and the production of liquefied natural gases.
- **Interdependence with Critical Temperature**: The relationship between critical temperature and critical pressure illustrates why real gases don't behave identically to ideal gases. Beyond these critical thresholds, the gas cannot simply remain a gas but tends to behave as a liquid.