Problem 111 - A Complete Resource Book in Chemistry for JEE Main

Question

Match the following: \(\begin{array}{ll}\text { List I } & \text { List II }\end{array}\) 1\. Critical temperature (i) a \(/ \mathrm{R}_{\mathrm{b}}\) 2\. Boyle's temperature (ii) \(2 \mathrm{a} / \mathrm{R}_{\mathrm{b}}\) 3\. Inversion temperature (iii) \(\mathrm{T} / \mathrm{T}_{\mathrm{c}}\) 4\. Reduced temperature (iv) \(8 \mathrm{a} / 27 \mathrm{R}_{\mathrm{b}}\) The correct matching is \(\begin{array}{lllll}\mathrm{I} & 2 & 3 & 4\end{array}\) (a) (ii) (iv) (i) (iii) (b) (iv) (i) (ii) (iii) (c) (iii) (ii) (i) (iv) (d) (iv) (iii) (ii) (i)

Step-by-Step Solution

Answer

The correct match is option (d): 1-(iv), 2-(i), 3-(ii), 4-(iii).

Step 1: Understanding the Terms

We need to pair terms from List I with expressions in List II based on their meanings. The goal is to match each term with the correct formula or expression that defines or is associated with it.

Step 1: Matching Critical Temperature

The critical temperature is defined in terms of the Van der Waals constants as \(T_c = \frac{8a}{27Rb}\). The corresponding expression from List II is (iv), so 1 matches with (iv).

Step 2: Matching Boyle's Temperature

Boyle's temperature is given by the expression \(T_b = \frac{a}{Rb}\). In List II, (ii) matches this form if the expression is adjusted to the typical representation for Boyle's temperature, implying that the problem might contain a typo or error. Still, based on their close relation and our context knowledge (ii) should match with 2.

Step 3: Matching Inversion Temperature

Inversion temperature is calculated as \(T_i = \frac{2a}{Rb}\). Therefore, 3 matches best with (ii) as per conventional definitions used in gas laws.

Step 4: Matching Reduced Temperature

Reduced temperature is the ratio of the actual temperature \(T\) to the critical temperature \(Tc\), described as \( \frac{T}{T_c} \). Thus, 4 matches with (iii).

Step 6: Drawing Conclusions

Based on the traditional definitions and expressions that match the provided list, the correct pairings are derived as: 1 - (iv), 2 - (i), 3 - (ii), 4 - (iii).

Key Concepts

Critical Temperature

Critical temperature is a fundamental concept in thermodynamics describing the highest temperature at which a substance can exist in a liquid phase regardless of pressure. Beyond this temperature, the substance can only be in a gaseous phase. This characteristic is crucial when studying the behavior of gases and liquids, especially in processes like liquefaction.
The critical temperature is linked with the Van der Waals equation of state. It's determined using the constants of the equation, expressed as:

• \[ T_c = \frac{8a}{27Rb} \]

where:

• \(a\) is the measure of attraction between particles,
• \(Rb\) is the ideal gas constant multiplied by the volume adjustment constant \(b\).

This formula helps understand how intermolecular forces affect the state of a substance.

Boyle's Temperature

Boyle's temperature is the temperature at which a real gas behaves like an ideal gas for an extended range of pressures. At this specific temperature, the real gas's compressibility factor \(Z\) approaches 1, signifying ideal gas behavior.
The formula to find Boyle's temperature using the Van der Waals constants is:

• \[ T_b = \frac{a}{Rb} \]

This expression links pressure and volume to the characteristics of the gas itself, showing its tendency to deviate less from ideal behavior at specific conditions.
Understanding Boyle's temperature is crucial for applications involving gas law calculations, ensuring accurate predictions of gas behavior under varying conditions.

Inversion Temperature

Inversion temperature is significant in the context of gas cooling and heating processes, particularly during throttling - a type of adiabatic expansion where no heat is exchanged with the environment. It refers to the point at which a gas neither heats up nor cools down when expanded.
If the initial temperature of the gas is above its inversion temperature, the gas would heat up upon expansion. Below this temperature, it would cool. Thus, it defines the boundary for cooling by expansion, crucial for refrigeration cycles.
The formula for inversion temperature using Van der Waals constants is:

• \[ T_i = \frac{2a}{Rb} \]

Recognizing the inversion temperature is essential for designing systems that rely on Joule-Thomson effects, relevant in various industrial applications like the liquefaction of gases.

Reduced Temperature

Reduced temperature describes a nondimensional parameter that compares the temperature of a real gas to its critical temperature. It simplifies the study of gases by normalizing temperature scales for different substances.
The reduced temperature is defined mathematically as:

• \[ \frac{T}{T_c} \]

where:

• \(T\) is the actual temperature of the gas,
• \(T_c\) is the critical temperature.

By employing the reduced temperature, one can compare the behaviors of different gases under varying conditions, facilitating a broader understanding of thermodynamic principles without being bogged down by units.
This concept is especially useful in fields exploring phase diagrams and critical phenomena where transformation properties need unifying across varying substances.