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 \(_{7}\) 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) \(1, \mathrm{~b}-\frac{\mathrm{a}}{\mathrm{RT}}\) (c) \(\frac{\mathrm{a}}{\mathrm{RT}}, \mathrm{b}\) (d) \(\frac{2 \mathrm{a}}{\mathrm{RT}}, \mathrm{b}\)

Step-by-Step Solution

Answer

Option (c): \( \frac{a}{RT}, b \) is correct.

Step 1: Understand the Problem

The general equation of state provided is written as \( \mathrm{PV} = \mathrm{RT} \left[ \mathrm{A} + \frac{\mathrm{B}}{\mathrm{V}} + \frac{\mathrm{C}}{\mathrm{V}^2} + \ldots \right] \). We need to relate this to the van der Waals equation to find constants \(\mathrm{A}\) and \(\mathrm{B}\).

Step 2: Recall the Van der Waals Equation

The van der Waals equation for real gases is expressed as \( (P + \frac{a}{V_m^2})(V_m - b) = RT \), where \(P\) is the pressure, \(V_m\) is the molar volume, \(T\) is the temperature, \(a\) and \(b\) are constants specific to the gas.

Step 3: Rearrange the Van der Waals Equation

Rearrange the van der Waals equation to a form similar to the general equation of state. Start by expanding and simplifying:\[ P = \left( \frac{RT}{V_m - b} \right) - \frac{a}{V_m^2} \]

Step 4: Identify Corresponding Terms

To match the equations, compare the simplified van der Waals equation with the general form. The term \(\frac{RT}{V_m - b}\) creates a similar form with \(\mathrm{A}=1\), by an approximation when \( V_m >> b \), and \(\frac{a}{V_m^2}\) relates to \(\mathrm{B} = b\) considering small volume corrections in the state equation.

Step 5: Finalize the Values of A and B

Based on the rearranged van der Waals equation matched to the general form, we identify \(\mathrm{A} = 1\) and \(\mathrm{B} = b\).

Key Concepts

Van der Waals Equation

Gas laws are vital in understanding how gases behave under different conditions. Among these, the Van der Waals Equation gives a more accurate representation for real gas behavior compared to the ideal gas law. This equation is expressed as \[ (P + \frac{a}{V_m^2})(V_m - b) = RT \] where:

• \(P\) is the pressure exerted by the gas.
• \(V_m\) is the molar volume of the gas.
• \(T\) is the temperature in Kelvin.
• \(a\) and \(b\) are constants specific to each gas.

These constants help account for two main real-world factors: the attractive forces between molecules and the volume occupied by the gas molecules themselves.
The term \(\frac{a}{V_m^2}\) corrects for these attractive forces, considering that they reduce the effective pressure of the gas.
The constant \(b\), on the other hand, corrects for the physical volume of gas particles. By subtracting \(b\) from \(V_m\), it acknowledges that gas particles take up space that the ideal gas law does not consider. This refined understanding can better predict behavior under high pressure and low temperature conditions, where deviations from ideal behavior are more pronounced.

Real Gases

In the realm of thermodynamics, real gases differ from ideal gases, primarily due to intermolecular forces and finite molecular sizes. These factors lead to deviations from the ideal gas law, described by the equation \(PV=nRT\) where no individual gas properties are considered.
When pressures are high or temperatures are low, these assumptions of ideality no longer hold, and gases behave as real gases. Real-world conditions involve:

• Intermolecular attractions: Molecules in a gas attract each other weakly which affects the overall gas pressure.
• Finite molecular volume: Molecules occupy space, which effectively reduces the volume available to the gas.

Hence, real gases exhibit varying behavior differing from ideal predictions. As a solution, equations like the Van der Waals Equation integrate these considerations into their calculations, offering a more practical and usable formulation for real conditions.
This means real gases require corrections for molecular interaction and size, making their study crucial for accurate scientific and industrial applications.

Molar Volume

Molar volume is a critical concept in understanding the properties of gases both at ideal and real-world conditions. It is defined as the volume occupied by one mole of a gas at a given temperature and pressure, typically denoted as \(V_m\). Under standard conditions of temperature and pressure (STP), the molar volume of an ideal gas is approximately 22.4 liters per mole.
This figure can change when considering real gases since the presence of molecular forces and finite volume affect how much space a gas's molecules take up.

• For an ideal gas, the molar volume is straightforward as it neglects molecular interactions and volume.
• For real gases, however, \(V_m\) must consider the adjustments made by factors \(a\) and \(b\) in the Van der Waals equation, as these factors influence how close together the molecules can get.

Understanding molar volume helps in calculating other important gas properties such as density, and in predicting how gases will respond when subjected to changes in temperature or pressure. It also underscores important deviations that occur when switching from ideal to real gas behavior.