Question

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

Short answer

The van der Waals equation for a real gas is \[(P + a(n/V)^2)(V - nb) = nRT\] The corrective term \(a(n/V)^2\) accounts for the intermolecular attractions that effectively reduce the internally experienced pressure. The corrective term \(nb\) accounts for the volume of the gas molecules, reducing the volume accessible to them from the total.

Step-by-step solution

Write the Van der Waals equation

The van der Waals equation for a real gas is given by: \[(P + a(n/V)^2)(V - nb) = nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, T is the temperature, a is a constant that rectifies for the attractive forces between particles, and b is a constant that corrects for the volume of the gas molecules themselves.

Explain the pressure correction term

The term \(a(n/V)^2\) is known as the pressure correction term. It takes into account the attractive forces between particles. As particles get closer together, they exert attractive forces on each other, which effectively reduces the internally experienced pressure. Therefore, this term is being added to the ideal pressure P to compensate for this effect.

Explain the volume correction term

The term nb is called the volume correction term. It compensates for the physical space that gas molecules occupy. In an ideal gas, the volume of the molecules themselves is assumed to be negligible. But in reality, the gas molecules do have a volume, and this term gets subtracted from the total volume to represent the volume that is actually accessible to the gas molecules.

Key concepts

Real Gas Behavior

Real gases deviate from the ideal gas behavior, especially under high pressure or at low temperature conditions. Ideal gas laws assume gas particles have no volume and no intermolecular forces between them. However, in the real world, these assumptions do not hold true. Real gases have particles with finite volume, and interactions between these particles can't be ignored. The van der Waals equation is a mathematical adjustment to the ideal gas law that takes these factors into account. It seeks to more closely mirror the behavior of real gases by incorporating corrective terms for pressure and volume. This results in a more accurate representation of the physical conditions within a given sample of gas.
Understanding the behavior of real gases is crucial in many fields of science and engineering, such as in the design of pressure vessels, the study of atmospheric phenomena, and the manipulation of gases in chemical reactions.

Corrective Terms for Pressure

The corrective terms for pressure in the van der Waals equation address the attractive forces between gas particles. In Johanne van der Waals' model, the term \(a(n/V)^2\) is added to the pressure value 'P' from the ideal gas law. This term corrects for the intermolecular forces by considering the frequency and strength of particle interactions as they become denser or less dense. The constant 'a' is specific to each gas and reflects the magnitude of the attractive forces present amongst the gas molecules.

As the gas molecules come closer together, they attract each other, leading to a reduction in the motion away from each other. Hence, the pressure inside the container is lower than an ideal gas would exert. The addition of this term compensates for the internal pressure being under-reported and ensures that predictions and calculations remain as accurate as possible under a wider range of conditions.

Volume Correction Term

The van der Waals equation modifies the volume term of the ideal gas law to account for the actual space occupied by gas molecules. In the equation, \(V - nb\) represents the volume that is available for gas particles to move freely. The constant 'b' is a measure of the volume occupied by one mole of gas molecules and 'n' is the number of moles, making 'nb' the volume correction for the finite size of the particles.

This correction is necessary because in an ideal gas, particles are considered as points with no volume. However, real gas particles have volume, and as the density of the gas increases, the space that these particles physically occupy cannot be dismissed. Subtracting 'nb' from the volume 'V' gives a more realistic value of the space in which the gas particles can move, hence refining the predictions of the gas law.

Intermolecular Forces

Intermolecular forces are the attractive and repulsive forces that act between molecules. These forces are significant in understanding real gas behavior because they influence how molecules interact with one another. There are several types of intermolecular forces, including London dispersion forces, dipole-dipole interactions, and hydrogen bonds. Each type affects the physical properties of substances, such as boiling point, viscosity, and vapor pressure.

In the context of the van der Waals equation, the intermolecular forces of attraction are particularly important. They cause the gas molecules to clump together, which lowers the pressure compared to what is predicted by the ideal gas law for a given volume and temperature. The corrective term for pressure in the van der Waals equation is a direct response to these intermolecular forces, enabling more accurate behavior modeling of gases under various conditions.