Problem 3 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

At a given temperature \(\mathrm{T}\), gases \(\mathrm{Ne}, \mathrm{Ar}, \mathrm{Xe}\) and \(\mathrm{Kr}\) are found to deviate from ideal gas behaviour. Their equation of state is given as $$ \mathrm{P}=\frac{\mathrm{RT}}{\mathrm{V}-\mathrm{b}} \text { at } \mathrm{T} $$ Here, \(\mathrm{b}\) is the van der Waals constant. Which gas will exhibit steepest increase in the plot of \(Z\) (compression factor) vs \(\mathrm{P}\) ? [Main April 9, 2019 (II)] (a) Xe (b) \(\mathrm{Kr}\) (c) \(\mathrm{Ne}\) (d) Ar

Step-by-Step Solution

Answer

(a) Xe.

Step 1: Understanding the Compression Factor

The compression factor, denoted as \( Z \), is defined as \( Z = \frac{PV}{RT} \). For an ideal gas, \( Z = 1 \). For real gases described by the given equation of state \( P = \frac{RT}{V - b} \), we need to understand how \( Z \) behaves in response to the gas's behavior compared to an ideal gas.

Step 2: Substitute the Equation of State into Z

Using the equation \( \mathrm{P} = \frac{\mathrm{RT}}{\mathrm{V}-\mathrm{b}} \), we substitute \( P \) back to find \( Z \). Rearranging gives \( PV = RT + Pb \), leading to \( Z = \frac{PV}{RT} = 1 + \frac{Pb}{RT} \). This shows \( Z > 1 \), as the pressure increases, the factor \( \frac{Pb}{RT} \) becomes significant.

Step 3: Analyzing the Impact of the Constant b

The term \( b \) in the van der Waals equation is different for each gas and represents the volume excluded by the gas molecules themselves. A higher \( b \) means the gas molecules take up more space, which affects the steepness of \( Z \) vs \( P \) plot. As pressure increases, gases with a larger \( b \) will have a steeper increase in \( Z \).

Step 4: Identifying the Correct Gas

Amongst the options, \( \mathrm{Xe} \) typically has the largest atomic size and thus the largest van der Waals constant \( b \). This means that the deviation from ideality, reflected in the plot of \( Z \) vs \( P \), would be steepest for \( \mathrm{Xe} \) as \( P \) increases due to its large \( b \) value relative to the other gases.

Key Concepts

Compression Factor

The compression factor, often represented as \( Z \), is a measure that indicates how much a real gas deviates from ideal gas behavior. It is defined by the equation:

• \( Z = \frac{PV}{RT} \)

For an ideal gas, \( Z \) equals 1, meaning there are no deviations from ideality. If \( Z \) is greater than 1, the gas is less compressible than an ideal gas. Conversely, if \( Z \) is less than 1, the gas is more compressible. This factor becomes important in assessing real gases, especially at high pressures or low temperatures. By comparing \( Z \) to 1, we can better understand the extent of a gas's deviation from ideal behavior.
In the context of van der Waals gases, the compression factor explains how different gases will react under changes in pressure and volume. Understanding the value of \( Z \) helps in the adjustment of gas processes and calculations for industrial applications.

Ideal Gas Behavior

Ideal gases are theoretical gases that perfectly follow the ideal gas law, which is expressed as:

• \( PV = nRT \)

Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) the gas constant, and \( T \) temperature. Ideal gases are not subject to any intermolecular forces and assume that the volume of gas particles is negligible compared to the space between them.
While the concept of an ideal gas is convenient for calculations and theory, real gases deviate from this behavior under certain conditions. Specifically, at high pressures and low temperatures, real gases begin to behave in ways that do not align with the ideal gas law, due to the size of molecules and intermolecular forces becoming significant. Thus, understanding ideal gas behavior serves as a foundation for when we study real gases with more complex models like the van der Waals equation.

van der Waals Constant

The van der Waals equation adjusts the ideal gas law to better describe real gases. It is expressed as:

• \( \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \)

In this equation, the constants \( a \) and \( b \) are the van der Waals constants. The constant \( a \) accounts for the attractive forces between molecules, while \( b \) corrects for the volume occupied by the molecules themselves.
The importance of \( b \) lies in its role in deviating from ideal gas behavior. Larger values of \( b \) mean that the molecules occupy a greater volume, reducing the space available for movement. This impacts the pressure and, thus, the compressibility of the gas.
In exercises that ask about the impact of these constants, understanding that a larger \( b \) typically leads to greater deviations from ideality helps in predicting gas behavior under changes in conditions such as pressure.

Real Gases

Real gases refer to gases that do not adhere strictly to the ideal gas law, especially under conditions like high pressure and low temperature. Unlike ideal gases, real gases have intermolecular attractions and the molecules themselves occupy significant space.
To understand how real gases deviate from ideal gases, scientists use more complex equations like the van der Waals equation, which takes into account molecular size and intermolecular forces. When studying real gases, it is crucial to consider how these factors affect the observed pressure, volume, and temperature, which can significantly differ from the predictions made by the ideal gas law.
In practical terms, understanding real gases involves recognizing when to apply different models to non-ideal conditions, which is essential in fields ranging from chemistry to engineering. Being able to predict the behavior of real gases leads to more accurate scientific and industrial processes.