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

Question

Identify the correct statements. (a) Unit of gas constant, \(\mathrm{R}=0.082 \mathrm{kPa} \mathrm{dm}^{3} \mathrm{~K}^{-1}\) mole \(^{-1}\) (b) Inversion temperature, \(T_{i}=\frac{2 a}{R b}\) (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

Statements (a), (b), and (d) are correct.

Step 1: Verify Unit of Gas Constant (a)

The unit of the gas constant R is typically expressed in terms of energy per temperature unit per mole. For instance, it can be \(8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}\) or \((0.0821 \, \text{L} \, \text{atm} \, \text{mol}^{-1} \, \text{K}^{-1})\). The given unit \(\mathrm{kPa} \, \mathrm{dm}^3 \, \mathrm{K}^{-1} \, \text{mol}^{-1}\) matches this description since \([\text{kPa}]\) and \[\text{L}\] are equivalent, therefore this statement is correct.

Step 2: Analyze Inversion Temperature Equation (b)

The inversion temperature \(T_i\) describes a point where the Joule-Thomson coefficient changes sign, typically given by \(T_i = \frac{2a}{Rb}\). This equation correctly relates the parameters \(a\), \(b\), and \(R\) for certain models of gases, and it matches the problem's formula. Thus, this statement is correct.

Step 3: Check Boyle's Temperature Formula (c)

Boyle's temperature is defined as the temperature at which a real gas behaves ideally over a range of pressures. However, its typical formula is given by \(T_B = \frac{a}{Rb}\) rather than \(T_B = \frac{2a}{Rb}\). This suggests an inconsistency with the given formula, thus making the statement incorrect.

Step 4: Validate Critical Temperature Expression (d)

The critical temperature \(T_c\) represents where a substance can exist as two separate phases. Its derived formula often appears as \(T_c = \frac{8a}{27Rb}\), particularly under the Van der Waals equation. This matches the presented format, implying this statement is correct.

Key Concepts

Gas Constant

The gas constant, often symbolized as \( R \), is a fundamental constant in chemistry and physics. It provides a link between the physical properties of gases, such as pressure, volume, and temperature. The most common values of \( R \) include:

• 8.314 \( ext{J} \, ext{mol}^{-1} \, ext{K}^{-1} \)
• 0.0821 \( ext{L} \, ext{atm} \, ext{mol}^{-1} \, ext{K}^{-1} \)
• 8.2057 \( ext{m}^{3} \, ext{Pa} \, ext{mol}^{-1} \, ext{K}^{-1} \)

It is crucial to check the units when working with \( R \) in calculations. The units must align with those of pressure, volume, and temperature in your specific problem. For example, \( 1 \, ext{L} = 1 \, ext{dm}^3 \), and \( 1 \, ext{atm} = 101.325 \, ext{kPa} \). This helps confirm the appropriate usage of the constant in gas laws.

Inversion Temperature

Inversion temperature is a unique temperature where a gas transitions between cooling and heating under adiabatic expansion or compression, respectively. This happens when the Joule-Thomson effect, which describes temperature change during gas expansion, changes its sign. The formula for inversion temperature \( T_i \) is \[T_i = \frac{2a}{Rb}\]where:

• \( a \) and \( b \) are Van der Waals constants that correct for intermolecular forces and volumes, respectively.
• \( R \) is the gas constant.

Understanding inversion temperature is vital in industries, especially where temperature control during gas expansions is critical. For example, in refrigeration systems, it's essential to maintain gas at temperatures lower than its inversion point for efficient cooling performance.

Boyle's Temperature

Boyle's temperature is the specific temperature at which a real gas behaves like an ideal gas over a range of pressures. This simplifies the calculations since ideal gas laws can be used. The correct formula for calculating Boyle's temperature \( T_B \) is:\[T_B = \frac{a}{Rb}\]In this formula:

• \( a \) accounts for the attractive forces between molecules.
• \( b \) is related to the size of the molecules.
• \( R \) is the gas constant.

At this temperature, the compressibility factor \( (Z) \), which defines the deviation of a real gas from ideal behavior, equals one. This means that the gas exhibits no deviation from ideal gas behavior at this particular temperature and pressure conditions.

Critical Temperature

The critical temperature \( T_c \) of a substance is the highest temperature at which it can exist as a liquid and gas simultaneously, achieving a state known as the critical point. Below this temperature, distinct liquid and gas phases exist, while above it, the gas cannot be liquefied regardless of pressure. The formula for critical temperature \( T_c \) is given by:\[T_c = \frac{8a}{27Rb}\]This equation derives from the Van der Waals equation, where:

• \( a \) represents molecular attraction.
• \( b \) measures the finite size of molecules.
• \( R \) is the universal gas constant.

Understanding critical temperature is essential in fields like material science and chemical engineering as it helps in designing and controlling processes involving phase transitions, such as distillation or liquefaction.