Question

In van der Waals equation for a non-ideal gas, the term that accounts for intermolecular force is (a) \((V-b)\) (b) \(\left(P+\frac{a}{V_{-}^{2}}\right)\) (c) \(R T\) (d) \(P V\)

Short answer

The term in the van der Waals equation that accounts for intermolecular force is (b) \(\left(P + \frac{a}{V_{-}^{2}}\right)\).

Step-by-step solution

Understanding van der Waals Equation

The van der Waals equation is an equation of state that extends the ideal gas law to account for the finite size of molecules and the interactions between them. It is generally written as: \(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\). The terms that account for molecular size and intermolecular forces need to be identified.

Identifying the Intermolecular Force Term

The term \(\frac{a}{V^2}\) in the van der Waals equation, present in the pressure correction part of the equation, accounts for the intermolecular attractive forces. The constant \(a\) is a measure of the magnitude of these attractive forces.

Selecting the Correct Answer

Given the choices, the term that includes the intermolecular force correction is \(\frac{a}{V^2}\), which is part of the pressure correction term \(\left(P + \frac{a}{V^2}\right)\). Therefore, the correct answer is: (b) \(\left(P + \frac{a}{V_{-}^{2}}\right)\).

Key concepts

Intermolecular Forces

Intermolecular forces are the forces that act between molecules to hold them together. These forces can be attractive or repulsive and are much weaker than the intramolecular forces that hold atoms together within a molecule.

In the context of gases, intermolecular forces become significant when the gas molecules are pushed closer together, which can happen under high pressures or low temperatures. An important category of intermolecular forces in gases is the Van der Waals forces, named after the Dutch scientist Johannes Diderik van der Waals. These include attractions and repulsions between atoms and molecules that are not bound in the same molecule.

Understanding the role of intermolecular forces can help us explain behaviors of gases that deviate from ideal conditions, such as condensation into liquids when cooled or applying high pressure. In the van der Waals equation, the term \(\frac{a}{V^2}\) represents the correction for these attractive forces, enabling the equation to more accurately predict the behavior of real gases.

Non-Ideal Gas Behavior

The behavior of non-ideal or real gases differs from the predictions of the ideal gas law under certain conditions. The ideal gas law, given by PV = nRT, assumes that there are no intermolecular forces between the gas particles and the particles do not occupy any space.

However, in the real world, gases do exhibit intermolecular forces, and the molecules occupy space. When the pressure of a gas is high or the temperature is low, the gas particles are close together and these assumptions are no longer valid. The van der Waals equation modifies the ideal gas law to account for the volume occupied by gas molecules with the term \(V - nb\) and for the attractive forces between particles with the term \(\frac{a}{V^2}\).

The deviation from ideal behavior is especially important to consider when dealing with gases at high pressures or low temperatures where the ideal gas law would not provide accurate or reliable results. By including corrective factors for particle volume and intermolecular attractions, the van der Waals equation enables us to make more precise calculations for the behavior of real gases.

Equation of State

An equation of state is a mathematical model that describes the properties of matter under a given set of physical conditions. It provides a relationship between variables such as pressure (P), volume (V), temperature (T), and the amount of substance (n) for a particular state of matter, such as a gas.

The simplest equation of state for gases is the ideal gas law, which assumes that the molecules in the gas do not interact and do not take up space. However, this is an approximation and only accurate for gases under conditions of low pressure and high temperature. The van der Waals equation is a more sophisticated equation of state that corrects for non-ideal gas behavior by introducing two additional parameters: \(a\) and \(b\).

Parameter \(a\) is a measure of the attractive forces between particles, while \(b\) is a measure of the effective volume occupied by the gas particles. Together, they provide a more accurate description of the behavior of real gases than the ideal gas law. The van der Waals equation thus supports a deeper understanding of the thermodynamic properties of substances under various conditions and is essential for chemical engineering and physical sciences.