Problem 123 - A Complete Resource Book in Chemistry for JEE Main

Question

Identify the correct statements. (a) Unit of gas constant, \(\mathrm{R}=0.082 \mathrm{kPa} \mathrm{dm}^{3} \mathrm{~K}^{-1} \mathrm{~mole}^{-1}\) (b) Inversion temperature, \(\mathrm{T}_{\mathrm{i}}=\frac{2 \mathrm{a}}{\mathrm{Rb}}\) (c) Boyle's temperature, \(\mathrm{T}_{\mathrm{B}}=\frac{2 \mathrm{a}}{\mathrm{Rb}}\) (d) Critical temperature, \(\mathrm{T}_{\mathrm{C}}=\frac{8 \mathrm{a}}{27 \mathrm{Rb}}\)

Step-by-Step Solution

Answer

Correct statements are (b) and (d).

Step 1: Analyze Statement (a)

The gas constant, denoted as \( R \), has a well-known SI unit of \( 8.314 \text{ J K}^{-1} \text{ mol}^{-1} \). However, the provided unit in statement (a) is not consistent with this. The unit \( 0.082 \mathrm{kPa} \mathrm{dm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \) is incorrect, as it seems to confuse different measurement systems.

Step 2: Analyze Statement (b)

Inversion temperature, \( T_i \), is typically defined in terms of the van der Waals constants \( a \) and \( b \). The mathematical expression for inversion temperature is \( T_i = \ \frac{2a}{Rb} \), which matches statement (b). Therefore, this statement is correct.

Step 3: Analyze Statement (c)

Boyle's temperature \( T_B \) is defined as \( T_B = \ \frac{a}{Rb} \). The given expression \( T_B = \ \frac{2a}{Rb} \) does not match the standard definition, so this statement is incorrect.

Step 4: Analyze Statement (d)

The critical temperature \( T_C \) for a van der Waals gas is given by the formula \( T_C = \ \frac{8a}{27Rb} \). The expression stated in (d) matches this definition, so statement (d) is correct.

Key Concepts

Gas Constant

The gas constant, commonly denoted by the symbol \( R \), is a fundamental constant that forms a bridge between the energy scale and the temperature scale in various gas equations, including the van der Waals equation and the ideal gas law.
The universally accepted SI unit for the gas constant \( R \) is \( 8.314 \text{ J K}^{-1} \text{ mol}^{-1} \). This unit aligns energy per temperature increment per mole. It allows us to calculate crucial properties such as the amount of work done by or on a gas as it expands or compresses.
In various contexts, sometimes alternative units are presented, like \( 0.0821 \, \text{L atm K}^{-1} \text{ mol}^{-1} \), which are more convenient for specific calculations in chemistry due to their ability to simplify expressions involving volumes in liters and pressures in atmospheres. However, always ensure that the unit used matches the rest of the problem's measurements to avoid inconsistencies and errors.

Inversion Temperature

The inversion temperature, denoted by \( T_i \), is an important concept when discussing the cooling effects in gas expansion, particularly related to the Joule-Thomson effect. It plays a crucial role in refrigeration technologies where gases are expanded.
For a gas described by the van der Waals equation, the inversion temperature can be represented mathematically by the formula:\[T_i = \frac{2a}{Rb}\]Here, \( a \) and \( b \) are van der Waals constants specific to each gas, accounting for molecular attraction and volume, respectively. \( R \) is the gas constant.Some important points to remember:

• Below the inversion temperature, a gas cools upon expansion.
• Above the inversion temperature, the opposite occurs; the gas heats upon expansion.

Understanding the inversion temperature helps in predicting the thermal behavior of gases during processes like throttling.

Boyle's Temperature

Boyle's Temperature, represented as \( T_B \), is unique to the study of gases and their behaviors at specific temperatures.
It is named after Robert Boyle, who was instrumental in formulating the relationship between pressure and volume for gases. Boyle's Temperature is an indication of the point where a real gas behaves like an ideal gas over a wide range of pressures.The correct mathematical model for Boyle's Temperature in the van der Waals context is:\[T_B = \frac{a}{Rb}\]Where \( a \) and \( b \) are constants that account for molecular interactions and volume, and \( R \) is the gas constant.

• At Boyle’s Temperature, the second coefficient in the virial equation of state becomes zero.
• Hence, gases approximate ideal behavior at this temperature for a certain pressure range.

This is critical when designing experiments or industrial applications where near-ideal gas behavior is desired.

Critical Temperature

Critical temperature, noted as \( T_C \), is a defining characteristic of gases, particularly when discussing phase transitions.
It specifically represents the temperature above which a gas cannot be liquefied, regardless of the pressure applied.In the framework of the van der Waals equation, the critical temperature is defined by the expression:\[T_C = \frac{8a}{27Rb}\]Here:

• \( a \) measures the attraction forces between molecules.
• \( b \) accounts for the finite size of molecules.
• \( R \) is the gas constant.

Understanding the critical temperature is crucial for both theoretical physics and practical applications such as supercritical fluid extraction and in the design of pressure vessels, to predict the behavior of substances near their phase boundaries.