Question

Which of the following statement(s) is/are incorrect? (a) A gas can be liquefied at a temperature ' \(\mathrm{T}\) ' such that \(\mathrm{T}<\mathrm{T}_{c}\) and \(\mathrm{p}=\mathrm{P}_{\mathrm{C}}-\mathrm{T}_{\mathrm{c}}\) and \(\mathrm{P}_{\mathrm{c}}\) are critical tem- perature and pressure. (b) Rise in the compressibility factor with increasing pressure is due to equal contribution of both a and b (Van der Waal's parameter). (c) The fraction of molecules having speeds in the range of \(\mathrm{u}\) to \(\mathrm{u}+\) du of a gas of molar mass ' \(\mathrm{M}\) ' at temperature ' \(\mathrm{T}\) ' is the same as that of gas of molar mass ' \(2 \mathrm{M}^{\prime}\) at temperature ' \(\mathrm{T} / 2^{\prime}\) (d) The product of pressure and volume of a fixed amount of a gas is independent of temperature.

Short answer

All statements (a), (b), (c), and (d) are incorrect.

Step-by-step solution

Analyze Option (a)

The statement claims that a gas can be liquefied at a temperature \( T \) such that \( T < T_c \), with \( p = P_c - T_c \). In reality, a gas can be liquefied if \( T < T_c \) regardless of pressure. However, the expression \( p = P_c - T_c \) is incorrect because pressure \( p \) should be a positive value and cannot be calculated as a subtraction involving temperature. Thus, the mathematical expression provided does not make sense.

Evaluate Option (b)

The rise in the compressibility factor \( Z \), which indicates deviation from ideal behavior, can indeed be influenced by the van der Waals constants \( a \) and \( b \). However, it is not due to *equal* contributions of both \( a \) and \( b \). The compressibility factor's behavior is much more complex and depends differently on these parameters (\( a \), the attraction term; \( b \), the volume correction). Thus, the statement is incorrect as it oversimplifies the contributions.

Evaluate Option (c)

This option discusses the Maxwell-Boltzmann distribution of molecular speeds. The fraction of molecules with speeds between \( u \) and \( u + du \) depends on temperature and molar mass according to this distribution; changing them affects the speed distribution function. If a gas's molar mass is doubled and temperature is halved, the fraction of molecules having a certain speed range would change. Thus, the statement is incorrect.

Analyze Option (d)

This statement suggests that \( PV \) is independent of temperature for any gas. According to the ideal gas law, \( PV = nRT \), where \( n \) is the number of moles and \( R \) is the gas constant. For a fixed gas quantity, \( PV \) is directly proportional to \( T \), thus dependent on temperature. Therefore, this statement is incorrect.

Key concepts

Critical Temperature and Pressure

Critical temperature and critical pressure are key concepts in understanding when a gas can be converted into a liquid. The critical temperature, denoted as \( T_c \), is the highest temperature at which a gas can be liquefied, no matter how much pressure is applied. Above \( T_c \), the gas cannot become a liquid. Critical pressure, \( P_c \), is the minimum pressure needed to liquefy a gas at its critical temperature. Knowing these two parameters helps us understand the phase behavior of substances.

In the context of the provided exercise, option (a) includes the statement \( p = P_c - T_c \). This suggests an incorrect relation because pressure should not be calculated by subtracting a temperature from a pressure. The key point is that to liquefy a gas, \( T \) must simply be less than \( T_c \), and the pressure should be greater than or equal to \( P_c \) for liquefaction to occur below \( T_c \). Correct understanding of these conditions is essential for processes like industrial gas liquefaction.

Van der Waals Equation

The Van der Waals equation is a modified version of the ideal gas law that accounts for the non-ideal behaviors of real gases. It is expressed as:\[\left(P + \frac{a}{V^2}\right) (V-b) = nRT\]where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature. The constants \( a \) and \( b \) are the Van der Waals parameters. \( a \) accounts for the attractive forces between gas molecules, while \( b \) corrects for the volume occupied by gas molecules themselves.

In the exercise, the statement in option (b) about the rise in the compressibility factor due to "equal contribution" from both \( a \) and \( b \) is incorrect. Each parameter has different effects on gas behavior — \( a \) affects intermolecular attractions, whereas \( b \) adjusts for the volume of the molecules. Therefore, their contributions to non-ideal behavior are not equal or identical, highlighting the complexity of real gases.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in a gas. It is a fundamental principle in statistical mechanics that gives insights into the kinetic energy and behavior of gas molecules at a given temperature. The distribution shows that not all gas molecules have the same speed, but there is a range and distribution of speeds.

For gases with different molecular masses and temperatures, the distribution of speeds changes accordingly. In option (c) of the exercise, it is stated that the fraction of molecules of a certain speed in one gas is the same as that in another gas with twice the molar mass and half the temperature. This is incorrect because doubling the molar mass and halving the temperature will significantly alter the distribution, since both factors directly affect the kinetic energy of molecules, thereby changing their speed distribution. Understanding this helps explain gas behavior at different conditions and is key to mastering concepts related to gas dynamics.