Problem 120 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statement(s) is/are incorrect? (a) A gas can be liquefied at a temperature 'T' such that \(\mathrm{T}<\mathrm{T}_{\mathrm{C}}\) 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 \(u\) to \(\mathrm{u}+\mathrm{du}\) of a gas of molar mass ' \(\mathrm{M}\) ' at temperature ' \(\mathrm{T}\) ' is the same as that of gas of molar mass ' \(2 \mathrm{M}\) ' at temperature ' \(\mathrm{T} / 2\) ' (d) The product of pressure and volume of a fixed amount of a gas is independent of temperature.

Step-by-Step Solution

Answer

Statements (b), (c), and (d) are incorrect.

Step 1: Analyze Statement (a)

Statement (a) is discussing the conditions under which a gas can be liquified. The statement says a gas can be liquified at a temperature \( T \) such that \( T < T_C \) and pressure \( p = P_C \). This statement is true because the conditions describe being below the critical temperature which allows for liquefaction when critical pressure is reached.

Step 2: Analyze Statement (b)

Statement (b) claims that the rise in the compressibility factor with increasing pressure is due to equal contribution from both 'a' and 'b' of the Van der Waals equation. This is incorrect, as the 'a' parameter relates to intermolecular forces, while 'b' relates to molecular volume, and their contribution is not necessarily equal.

Step 3: Analyze Statement (c)

Statement (c) compares the fraction of molecules within a speed range for two gases of different molar masses and temperatures. Using the Maxwell-Boltzmann distribution, the fraction of molecules with speeds \( u \) to \( u + du \) depends on \( \sqrt{T/M} \). For molar masses \( M \) at temperature \( T \) and \( 2M \) at temperature \( T/2 \), they are not equivalent, so this statement is incorrect.

Step 4: Analyze Statement (d)

Statement (d) implies that the product of pressure and volume of a gas is independent of temperature. According to the ideal gas law \( PV = nRT \), pressure and volume are dependent on temperature, so this statement is incorrect.

Key Concepts

Critical Temperature

The critical temperature is the highest temperature at which a substance can be liquefied, regardless of the pressure applied. If the temperature of a gas is below its critical temperature, it can be transformed into a liquid by increasing the pressure to the critical pressure. For example, carbon dioxide has a critical temperature of 31.0°C. Below this temperature, applying the critical pressure will turn it into liquid.
The importance of critical temperature lies in various applications, including refrigeration and chemical engineering, where manipulating temperature and pressure is essential for controlling the state of gases.

Van der Waals Equation

The Van der Waals equation is an adjusted version of the ideal gas law, accounting for the volume occupied by gas molecules and the intermolecular forces between them. It is expressed as:\[ \left(P + \frac{a}{V^2}\right)(V-b) = nRT \]Where:

• \( P \) and \( V \) are the pressure and volume of the gas,
• \( n \) is the number of moles,
• \( R \) is the gas constant,
• \( T \) is the temperature,
• \( a \) reflects the magnitude of intermolecular attractions.
• \( b \) accounts for the volume occupied by the gas molecules.

The "a" parameter is crucial when addressing why compressors work differently with various gases, influencing how their volume changes under pressure. Meanwhile, the "b" parameter adjusts for the physical size of the molecules. This equation helps us understand real gas behaviors better than the ideal gas law.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in a gas. This concept is vital in understanding molecular kinetics. The distribution indicates that, at a given temperature, different molecules have different speeds, which leads to diverse kinetic energies within the gas.
The distribution function is expressed as:\[ f(u) = \left(\frac{m}{2\pi kT}\right)^{3/2} 4\pi u^2 \exp\left(-\frac{mu^2}{2kT}\right) \]Where:

• \( f(u) \) is the probability density function for a molecule to have speed \( u \),
• \( m \) is the mass of a molecule,
• \( k \) is the Boltzmann constant,
• \( T \) is the absolute temperature.

In practice, this means gases at higher temperatures have more molecules moving at higher speeds, illustrating how temperature increases can enhance reaction rates in chemical processes.

Ideal Gas Law

The ideal gas law, one of the core principles of chemistry, describes how gases behave under various conditions of pressure, volume, and temperature. It is represented by the equation:\[ PV = nRT \]Where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume,
• \( n \) denotes the number of moles,
• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin.

This law assumes a perfect gas where intermolecular forces and volumes of molecules are negligible. It is very useful for basic calculations involving gas properties such as those found in everyday scenarios like inflating tires or balloons, where pressure, volume, and temperature are often interrelated.