Question

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

Short answer

Gases can be liquefied when \(T < T_c\); none of the choices explicitly cover this.

Step-by-step solution

Understanding Ideal Gas Conditions

The content tells us that for gases that obey kinetic theory, there are certain critical conditions that must be met for liquefaction to occur. Gases must be below a critical temperature (\(T_c\)) to be liquefied.

Analyzing Option (a)

Option (a) states \(T > T_c\). When a gas is at a temperature above its critical temperature (\(T_c\)), it cannot be liquefied, regardless of the pressure applied.

Analyzing Option (b)

Option (b) suggests \(P > P\) without specifying whether \(P\) refers to the critical pressure or just a general pressure. The statement is incomplete as it does not mention the temperature aspect crucial for liquefaction.

Analyzing Option (c)

Option (c) states \(P > P_c\) and \(T < T_d\). While being above the critical pressure (\(P_c\)) helps, the temperature condition requires clarification with respect to \(T_c\), as \(T < T_c\) is necessary for liquefaction.

Analyzing Option (d)

Option (d) says the gas cannot be liquefied at any values of \(P\) and \(T\). However, this contradicts the kinetic theory and the known fact that gases can be liquefied under the right conditions.

Finding the Correct Condition

Based on the analysis, for an ideal gas to be liquefied, it must be below its critical temperature \(T_c\). Therefore, option (a) is incorrect, and option (c) requires \(T < T_c\). None of the provided options reflect the correct conditions clearly.

Key concepts

Kinetic Theory of Gases

The Kinetic Theory of Gases provides a fundamental explanation of gas behavior. This theory suggests that gases consist of a large number of small particles, mainly atoms or molecules. These particles are in constant random motion, and the energy of this motion is directly related to temperature. The higher the temperature, the more kinetic energy is present, causing particles to move faster.

Understanding this kinetic energy is crucial when considering the conditions under which a gas can be liquefied. For a gas to change into a liquid form, the particles must slow down sufficiently to allow attractive forces to bring them closer together. This typically happens when the gas is cooled, reducing its kinetic energy. The reduction in kinetic energy allows intermolecular forces to overcome the energy keeping particles apart. Consequently, in the context of real gas behavior, kinetic theory guides us in predicting conditions needed for changing gas state.

Critical Temperature

The Critical Temperature, denoted as \(T_c\), is a key parameter when discussing gas liquefaction. It is defined as the maximum temperature at which a substance can exist in liquid form. Above this temperature, no amount of pressure can liquefy the gas.

This phenomenon occurs because the average kinetic energy of particles is too high to allow attractive forces to bind them together into a liquid state. Essentially, \(T_c\) underscores the limit of temperature conditions suitable for liquefaction. In practical terms, if a gas is observed at temperatures below its critical temperature, it may become liquid under the right pressures. This is why options in the original exercise indicate that \(T < T_c\) is a necessary condition for a gas to be liquefied. Critical Temperature thereby acts as a threshold in determining the phases of matter under varied conditions.

Critical Pressure

Critical Pressure, represented as \(P_c\), is another essential variable in the discussion of gas liquefaction. It is the pressure required to liquefy a gas at its critical temperature \(T_c\). When a gas is at or below the critical temperature, applying pressure can help transition it from a gaseous to a liquid state.

The critical pressure, therefore, defines the pressure boundary for a gas to change its phase when at its critical temperature. Above \(P_c\), a gas may become liquid if the temperature is below \(T_c\). Critically, at \(P_c\), small changes in temperature can dramatically influence the state of matter. In essence, critical pressure marks the upper limit of pressure necessary for a gas to sustain its liquid form once it is below its critical temperature, enabling a clearer navigation of conditions required for gas liquefaction.