Problem 2 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider the van der Waals constants, a and \(\mathrm{b}\), for the following gases, Gas \(\quad \mathrm{Ar} \quad \mathrm{Ne} \quad \mathrm{Kr} \quad \mathrm{Xe}\) \(\begin{array}{lllll}\mathrm{a} /\left(\mathrm{atm} \mathrm{dm}^{6} \mathrm{~mol}^{-2}\right) & 1.3 & 0.2 & 5.1 & 4.1\end{array}\) \(\begin{array}{lllll}\mathrm{b} /\left(10^{-2} \mathrm{dm}^{3} \mathrm{~mol}^{-1}\right) & 3.2 & 1.7 & 1.0 & 5.0\end{array}\) Which gas is expected to have the highest critical temperature? [Main April 9, 2019 (I)] (a) \(\mathrm{Kr}\) (b) \(\mathrm{Ne}\) (c) Xe (d) \(\mathrm{Ar}\)

Step-by-Step Solution

Answer

The gas Kr is expected to have the highest critical temperature.

Step 1: Understanding the Relationship

The critical temperature of a gas is related to its van der Waals constants, specifically the constant 'a'. The critical temperature is given by the formula:\[ T_c = \frac{8a}{27bR} \]where \( R \) is the universal gas constant. The critical temperature is directly proportional to the constant 'a' and inversely proportional to 'b'.

Step 2: Comparing Constant 'a' Values

List the van der Waals constant 'a' values for the gases:- Ar: \( a = 1.3 \)- Ne: \( a = 0.2 \)- Kr: \( a = 5.1 \)- Xe: \( a = 4.1 \)Since the critical temperature is mainly influenced by 'a' when 'a' is larger and 'b' are comparable (as they are in different units), 'Kr' with the highest 'a' value will have the highest contribution from 'a'.

Step 3: Comparing Constant 'b' Values

List the van der Waals constant 'b' values for the gases:- Ar: \( b = 3.2 \times 10^{-2} \)- Ne: \( b = 1.7 \times 10^{-2} \)- Kr: \( b = 1.0 \times 10^{-2} \)- Xe: \( b = 5.0 \times 10^{-2} \)The influence of 'b' on \( T_c \) is smaller due to its larger atomic interpretation and unit. Comparing the 'b' values, Ne and Xe would suggest a lower critical value, especially as they are different gas atoms.

Step 4: Determine the Gas with Highest Critical Temperature

Combining the effects of both 'a' (higher is better) and 'b' (lower or mid-values is better), Kr, having both the highest 'a' and a relatively reasonable 'b', is the expected gas with the highest critical temperature among the four options.

Key Concepts

Critical Temperature

Critical temperature plays a crucial role in understanding the behavior of gases, especially as they transition into liquids. This is the temperature above which a gas cannot be liquefied, regardless of the applied pressure. The van der Waals equation provides insight into this concept through the constants 'a' and 'b'.
The critical temperature \( T_c \) is calculated using the formula:

• \( T_c = \frac{8a}{27bR} \)

where \( R \) is the universal gas constant.
The constant 'a' relates to the attractive forces between gas molecules; higher 'a' implies stronger attractions, which generally results in a higher critical temperature. Therefore, comparing 'a' values across different gases is one method to predict which gas has the highest critical temperature. Factor 'b', however, is inversely proportional to the critical temperature, accounting for the finite size of molecules. By assessing these constants, it's clear Kr (Krypton) with its highest 'a' value, logically has the highest critical temperature among Xe, Ar, and Ne.

Gas Constants

Van der Waals constants serve as adjustments to the ideal gas law, accounting for real gas behavior. Understanding these constants helps us predict how gases will behave under different conditions. The two main constants are:

• Constant 'a': Represents the magnitude of attractive forces between molecules. Higher 'a' values suggest stronger intermolecular forces and typically lead to higher critical temperatures.
• Constant 'b': Represents the volume occupied by gas molecules, accounting for their size. This is crucial for understanding the space gases take up and their compressibility.

When evaluating gases using these constants, it's important to remember how they influence the gas's physical properties. The interplay between 'a' and 'b' also aids in the accurate describing of phase transitions and conditions under which gases can be condensed into liquids.
Each gas has unique 'a' and 'b' values, accounting for their specific physical characteristics and interactions.

Phase Transitions

Phase transitions refer to the changes between different states of matter such as gas, liquid, and solid. These transitions occur due to variations in temperature and pressure and are a key concept in thermodynamics.

• Gas to Liquid (Condensation): Happens when the gas is cooled or compressed, allowing intermolecular attractions to bring molecules closer into a liquid form. The critical temperature is the point where further compression cannot induce this transition.

Moreover, analyzing phase transitions through the lens of critical temperature and gas constants provides insights into the conditions needed for these transitions.

• Van der Waals Equation: Helps model and predict these transitions by providing a more practical representation over the ideal gas law, particularly near the condensation point where ideal gas assumptions no longer hold.

This detailed understanding of phase transitions is fundamental in fields like materials science and chemical engineering, where controlled manipulation of matter states is often necessary.