Problem 42 - A Complete Resource Book in Chemistry for JEE Main

Question

The term that accounts for intermolecular force in van der Waals equation for non ideal gas is (a) \(\mathrm{RT}\) (b) \(\mathrm{V}-\mathrm{b}\) (c) \(\left(\mathrm{P}+\alpha / \mathrm{V}^{2}\right)\) (d) \((\mathrm{RT})^{-1}\)

Step-by-Step Solution

Answer

Option (c) \(\left(\mathrm{P} + \alpha / \mathrm{V}^{2}\right)\) accounts for intermolecular forces.

Step 1: Understanding the van der Waals Equation

The van der Waals equation for a non-ideal gas is given by: \[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \] where \( P \) is the pressure, \( V_m \) is the molar volume, \( T \) is the temperature, \( R \) is the universal gas constant, \( a \) is a measure of the attraction between particles, and \( b \) accounts for the volume occupied by gas particles.

Step 2: Identify the Term for Intermolecular Forces

In the van der Waals equation, the intermolecular forces are accounted for by the term \( \frac{a}{V_m^2} \). This adjusts the pressure \( P \) to account for the attraction between molecules, thus modifying it to \( P + \frac{a}{V_m^2} \).

Step 3: Match with Given Options

Compare the term \( P + \frac{a}{V_m^2} \) to the multiple-choice options provided. Option (c) \( \left( \mathrm{P} + \alpha / \mathrm{V}^{2} \right) \) closely resembles the pressure correction in the van der Waals equation.

Key Concepts

Non-Ideal Gas

When we talk about gas behavior, we often start with the ideal gas model, which assumes that gases follow certain simplistic rules. However, the real world is a bit more complex. **Non-ideal gases** are a more accurate representation. Their behavior diverges from the ideal gas law, especially under high pressure and low temperature.
Understanding non-ideal gas behaviors involves considering factors like:

• **Intermolecular attractions:** Unlike ideal gases, real gases have molecules that attract each other, affecting pressure and volume.
• **Finite molecular volume:** Real gas molecules occupy space, meaning the volume available to move isn't as much as one might think.

These deviations led to the development of the van der Waals equation, which accounts for these real-world behaviors.

Intermolecular Forces

In the context of gases, **intermolecular forces** are the attractions or repulsions between molecules. While negligible in ideal gas calculations, these forces play a significant role in non-ideal gas behaviors.
The van der Waals equation incorporates these forces using the term \( \frac{a}{V_m^2} \):

• **Attractive forces:** Shorten the distance between molecules, effectively lowering the pressure exerted by the gas.
• **Repulsive forces:** Prevent molecules from sticking too closely together, safeguarding against compression.

Including these forces provides a truer picture of gas dynamics, correcting the simplistic assumptions made by the ideal gas law.

Molar Volume

Understanding the concept of **molar volume** is essential in applying the van der Waals equation. It refers to the volume occupied by one mole of a gas. While the ideal gas law assumes that gases spread out evenly, in reality, gases take up space because of the volume of their particles.
Molar volume is affected by:

• **Temperature:** Higher temperatures increase molecular motion, often expanding volume.
• **Pressure:** Increasing pressure compresses gas, decreasing volume.

The correction factor 'b' in the van der Waals equation represents this finite volume that gas particles take up, reminding us that not all available space is free for movement.

Universal Gas Constant

The **universal gas constant (R)** is a constant that appears in various equations describing the states of gases, including the ideal gas law and van der Waals equation.
Its value is universally consistent:

• **8.314 J/mol·K:** This value is used when considering the energy relations in gases.
• **0.0821 L·atm/mol·K:** Typically used within the framework of standard equations with pressure in atmospheres and volume in liters.

In the van der Waals equation, \( RT \) represents the energy available in the system under specific temperature conditions. This term stays consistent, making it a reliable measure despite changes in other properties, like pressure and volume.