Problem 69 - A Complete Resource Book in Chemistry for JEE Main

Question

A general form of equation of state for gases is \(\mathrm{PV}=\) \(\mathrm{RT}\left[\mathrm{A}+\mathrm{B} / \mathrm{V}+\mathrm{C} / \mathrm{V}^{2}+\ldots\right]\), where \(\mathrm{V}\) is the molar volume of the gas and \(\mathrm{A}, \mathrm{B}, \mathrm{C} \ldots .\) are constants for the gas. The values of \(\mathrm{A}\) and \(\mathrm{B}\), if the gas obeys van der Waals equation, are respectively (a) \(1, \frac{a}{27 b}\) (b) \(1, \mathrm{~b}-\frac{a}{\mathrm{RT}}\) (c) \(\frac{a}{R T}, b\) (d) \(\frac{2 \mathrm{a}}{\mathrm{RT}}, \mathrm{b}\)

Step-by-Step Solution

Answer

The values of \( A \) and \( B \) are (c) \( \frac{a}{RT}, b \).

Step 1: Understand the van der Waals Equation

The van der Waals equation for gases is given as \( \left( P + \frac{a}{V_m^2} \right)(V_m-b) = RT \), where \( a \) and \( b \) are constants specific to the gas, \( V_m \) is the molar volume, \( P \) is the pressure, and \( RT \) is the ideal gas law term.

Step 2: Expand the van der Waals Equation

Expand the van der Waals equation to match it to the general form provided: \( P(V_m - b) + \frac{a}{V_m^2}(V_m - b) = RT \). Expanding gives \( PV_m - Pb + \frac{a}{V_m} - \frac{ab}{V_m^2} = RT \).

Step 3: Compare with General Equation of State

In the general form, we have \( PV = RT \left[A + \frac{B}{V} + \frac{C}{V^2} + \ldots \right] \). From the expanded van der Waals equation, identify \( A = 1 \), \( B = b \), and \( C = \frac{a}{R} \).

Step 4: Identify Constants A and B

The question asks for the values of \( A \) and \( B \). Comparing the terms, we find \( A = \frac{2a}{RT} \) is incorrect because it matches the coefficient for \( \frac{a}{V_m} \), so the correct answer for \( A \) should be 1; \( B = b \) is correct.

Key Concepts

Equation of State

An equation of state describes the relationship between various state variables of a system. In the context of gases, these variables often include pressure \(P\), volume \(V\), temperature \(T\), and molar volume \(V_m\). One common form is the empirical equation given as \( PV = RT\left[A + \frac{B}{V} + \frac{C}{V^2} + \ldots\right] \). This equation provides a framework to understand how gases behave under different conditions.
The coefficients \(A\), \(B\), and \(C\) are constants specific to the gas being studied and can vary depending on the gas's unique characteristics. These constants adjust the ideal gas behavior to more accurately represent real gases. Such equations help in practical applications, including calculating properties in chemical reactions and industrial processes.

Molar Volume

Molar volume \(V_m\) is the volume occupied by one mole of a substance. For gases, molar volume is particularly significant because it allows the prediction of gas behavior under standard or non-standard conditions.
In van der Waals equation, molar volume is adjusted for actual occupied space and intermolecular forces: \( \left( P + \frac{a}{V_m^2} \right)(V_m - b) = RT \). Here, \(b\) accounts for the finite volume occupied by gas molecules. As pressure or temperature changes, the molar volume will change, demonstrating gas compressibility and expansibility traits. Understanding molar volume is crucial for industries dealing with gas mixtures or reactions.

Gas Constants

Gas constants are fundamental in predicting gas behaviors. In the context of van der Waals and ideal gas equations, these constants include \(R\) (the universal gas constant), \(a\), and \(b\).
- \( \mathbf{R} \): It's a constant that appears in many gas-related equations and has a value of 8.314 J/mol·K. It relates pressure, volume, and temperature of gases.
- \( \mathbf{a} \) and \( \mathbf{b} \): In the van der Waals equation, \(a\) represents the strength of intermolecular forces, while \(b\) corrects for molecular volume. These factors enable the equation to account for deviations from ideal behavior, often necessary for precise scientific calculations. Understanding these constants is key to mastering thermodynamics and predicting gas reactions.

Ideal Gas Law

The ideal gas law is a simplified model to describe the behavior of ideal gases. It is expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the gas constant, and \(T\) is temperature in Kelvin. This law assumes gases are composed of particles with no volume and no intermolecular forces.
While the ideal gas law provides a basic understanding, real gases deviate under high pressure or low temperature. Here, adjustments like those in the van der Waals equation become crucial. Studying the ideal gas law gives a fundamental framework for learning more complex gas behaviors and paves the way for understanding various scientific and engineering phenomena involving gases.