Question

Write the van der Waals equation for a real gas. Explain the corrective terms for pressure and volume.

Short answer

The van der Waals equation: \(\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT\). Pressure and volume corrections account for molecular size and attractions.

Step-by-step solution

Introduction to the van der Waals Equation

The van der Waals equation is used to describe the behavior of real gases by incorporating factors that the ideal gas law does not cover. It accounts for molecular size and the attraction between molecules.

Ideal Gas Law Recap

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This equation assumes that gas particles do not interact with each other and occupy no volume.

Incorporate Volume Correction

Real gas molecules have a finite size, meaning they occupy space within the container. The van der Waals equation adjusts for this by subtracting \( nb \) from the volume \( V \), where \( b \) is the volume occupied by one mole of the gas. This modification is represented as \( (V - nb) \) in the equation.

Incorporate Pressure Correction

In real gases, there are intermolecular attractions that reduce the force exerted against the walls of the container. To correct this, the term \( \frac{an^2}{V^2} \) is added to the pressure \( P \). Here, \( a \) is a constant specific to each gas that represents the strength of intermolecular attraction.

Write the Van der Waals Equation

By incorporating the corrections for volume and pressure, the van der Waals equation is expressed as: \[ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \]. This equation provides a more accurate representation of a real gas compared to the ideal gas law.

Key concepts

Real Gases

Real gases differ from the ideal gases primarily because they occupy volume and have intermolecular forces. In nature, no gas behaves perfectly according to the ideal gas law. While the ideal gas law simplifies calculations by ignoring molecular volume and attractions, real gases deviate from this ideal behavior, especially under high pressure or low temperature conditions.
These conditions cause molecules to come closer together, leading to interactions among them. Real gases require equations that can factor in their molecular size and the forces of attraction between them. The van der Waals equation does this by adding specific correction terms for pressure and volume, offering a more comprehensive model for understanding gas behavior under such conditions.

Ideal Gas Law

The ideal gas law is a fundamental principle in physics and chemistry that describes how gases behave under most conditions. It is represented as \( PV = nRT \), where:

• \( P \) stands for pressure,
• \( V \) is volume,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature measured in Kelvin.

The ideal gas law assumes that gas particles:

• do not attract or repel each other, and
• occupy no volume of their own.

While this model is useful for making quick and easy calculations, it doesn't account for the real-world behaviors seen in gases that don't adhere perfectly to these assumptions. Thus, when precise results are needed, especially in extreme conditions, other models such as the van der Waals equation are utilized.

Pressure Correction

In real gases, intermolecular forces cause a reduction in the pressure exerted on the walls of the container compared to what is predicted by the ideal gas law. These forces slightly pull molecules inward, decreasing the measurable pressure. To account for this, the van der Waals equation introduces a correction term: \( \frac{an^2}{V^2} \).
This correction adds extra pressure back into the equation, reflecting the fact that molecules do attract each other. The constant \( a \) varies for different gases, representing the strength of these intermolecular attractions. This modification provides a better fit for real-gas data by acknowledging attractive forces that lower the pressure in reality.

Volume Correction

The ideal gas law assumes gas particles are point particles with no volume. However, real gas molecules do occupy a certain amount of space. In confined spaces, the available volume for movement is reduced because of the molecules’ finite size. To adjust for this in calculations, the van der Waals equation subtracts a volume correction term, represented as \( nb \), from the total volume \( V \).
This term \( nb \) considers the space occupied by the gas particles themselves, where \( b \) is a constant specific to each type of gas. Subtracting this volume (\( V - nb \)) in the van der Waals equation compensates for the actual volume excluded by the molecules. This makes the equation more accurate for predicting gas behavior by addressing the aspect of molecular size.