Question

At low temperatures and very low pressures, gases behave ideally, but as the pressure is increased the product \(P V\) becomes less than the product \(n R T\). Give a molecular-level explanation of this fact.

Short answer

At high pressures, gas molecules' volumes and intermolecular forces cause deviations from ideal behavior, making \(PV < nRT\).

Step-by-step solution

Understand Ideal Gas Behavior

Ideal gases assume that there are no intermolecular forces between the molecules and that the molecules occupy no volume. Under ideal conditions, the equation \(PV = nRT\) perfectly describes the relationship among pressure \(P\), volume \(V\), number of moles \(n\), gas constant \(R\), and temperature \(T\).

Analyze the Impact of High Pressure

At higher pressures, the volume of the gas is reduced, making the finite size of the gas molecules more significant compared to the total volume. This causes the volume actually occupied by the gas to be less than the ideal assumption, since part of the volume is occupied by the gas molecules themselves.

Consider Intermolecular Forces

When gases are compressed at high pressure, the molecules are brought closer together, enhancing intermolecular interactions such as van der Waals forces. These attractive forces reduce the pressure exerted by the molecules on the walls of the container, thereby causing the product \(PV\) to be less than \(nRT\).

Link to Van der Waals Equation

The van der Waals equation \((P + \frac{an^2}{V^2})(V - nb) = nRT\) accounts for the volume occupied by gas molecules (\(b\)) and the intermolecular forces (\(a\)). As the pressure increases, these deviations become more significant, leading to \(PV < nRT\).

Key concepts

Van der Waals Forces

Van der Waals forces are a type of weak attraction that exists between molecules. In the ideal gas law, these forces are usually neglected. However, as gas molecules are compressed and brought closer together, these forces become more noticeable. At high pressures, gases do not behave ideally, as these attractive forces reduce the kinetic energy of molecules.
This results in a decreased pressure against the container walls than predicted by the ideal gas law. The van der Waals equation modifies the ideal gas equation to include these forces by introducing a term with the constant 'a'. This constant accounts for the strength of attraction between molecules, allowing for a more accurate prediction of gas behavior under non-ideal conditions.

Intermolecular Interactions

Intermolecular interactions refer to the forces that act between neighboring molecules, and they significantly affect the behavior of gases under varying conditions. When molecules are far apart, as in ideal conditions, these forces are negligible. However, under increased pressure, molecules are forced closer together, leading to stronger interactions.
These include not only van der Waals forces but other forms such as dipole-dipole interactions and hydrogen bonding. As these interactions intensify, they inhibit the free movement of molecules, making it harder for them to spread out and collide with container walls. This reduction in molecular activity explains why the product of pressure and volume (PV) is less than what the ideal gas law predicts at higher pressures.

Molecular Volume

Molecular volume is the space occupied by the individual gas molecules, which is effectively ignored in the ideal gas law where gases are considered point masses. But in reality, molecules have size and occupy space.
When the pressure increases, the volume occupied by the gas itself becomes significant. The van der Waals equation introduces the constant 'b', which accounts for the finite size of molecules. This correction acknowledges that not all of the volume of a gas sample is available for other molecules to move into, leading to deviations from ideality and a drop in the observed pressure-volume product.

Compression Effects

Compression effects become significant when gases are strongly pressurized. At this point, the assumptions of ideal gases start breaking down. As gases are compressed, their molecules are forced closer together, revealing the importance of both molecular volume and intermolecular forces.
Under these conditions, the ideal gas law no longer accurately describes the gas behavior as the volume becomes less than expected because the molecules themselves take up space. Additionally, the increased proximity of molecules enhances intermolecular interactions leading to the weakened effectiveness of collisions with container walls. These compression effects are best captured by adjustments made in real gas equations such as the van der Waals equation, enabling a more precise understanding of gas behavior at varying pressures.