Question

If we continuously compress a real gas in a cylinder with a movable piston while keeping the container at a constant temperature in an ice bath: (a) What will eventually happen to the gas and why? (b) How well will the ideal gas law apply to the real gas shown when it is close to doing what you said it will do in part (a)?

Short answer

(a) The real gas will eventually condense into a liquid due to the continuous compression and dominance of intermolecular forces. (b) The ideal gas law will not apply well to the real gas when it is close to condensing into a liquid, because the assumptions of negligible volume and intermolecular forces are no longer valid under these conditions. Instead, a more realistic model like the Van der Waals equation should be used.

Step-by-step solution

(a) What will eventually happen to the gas and why?

As we continuously compress the real gas inside the cylinder with a movable piston, the pressure of the gas will increase, and the volume of the gas will decrease. The constant temperature inside an ice bath implies that the gas's internal energy will remain constant, as will the temperature of the gas. Eventually, due to the continuous compression, the gas molecules will be in very close proximity to each other, and the intermolecular forces will start to dominate. When intermolecular forces become dominant, the gas will eventually condense into a liquid. This occurs because the increased pressure forces the gas particles to come closer together, and when they are close enough, the attractive forces between the particles will cause the gas to change its phase and become a liquid. So, the answer to part (a) is that the real gas will eventually condense into a liquid due to the continuous compression and dominance of intermolecular forces.

(b) How well will the ideal gas law apply to the real gas shown when it is close to doing what you said it will do in part (a)?

The ideal gas law, given by PV = nRT, works well under the assumption that gas particles have negligible volume and negligible intermolecular forces between them. However, when the real gas is close to condensing into a liquid, as described in part (a), the volume of the gas particles becomes significant compared to the overall volume of the gas. Moreover, the intermolecular forces also become significant due to the close proximity of the gas particles. Under these conditions, the ideal gas law no longer holds true, and we need to consider a more realistic model like Van der Waals equation of state to incorporate these deviations from the ideal gas law. The Van der Waals equation is given by \[ \left( P + \frac{an^2}{V^2}\right) (V-nb) = nRT,\] where a and b are Van der Waals constants specific to the gas under consideration. So, the answer to part (b) is that the ideal gas law will not apply well to the real gas when it is close to condensing into a liquid, because the assumptions of negligible volume and intermolecular forces are no longer valid under these conditions.

Key concepts

Ideal Gas Law Limitations

The ideal gas law, expressed as \(PV = nRT\), is a mathematical model that describes the behavior of an ideal gas. This equation assumes that gas molecules have no volume and do not interact with each other, meaning they possess no intermolecular forces. These assumptions allow for the derivation of this simple relationship between pressure \(P\), volume \(V\), number of moles \(n\), gas constant \(R\), and temperature \(T\). However, in reality, all gases deviate from these ideal assumptions. One of the primary limitations of the ideal gas law is its inability to accurately describe the behavior of gases under high pressure or low temperature conditions. In such cases, the volume of gas molecules becomes non-negligible, and interactions between them cannot be ignored.

• The volume occupied by the gas particles themselves becomes a significant part of the total volume.
• Intermolecular forces start to affect the behavior of gases, leading them to deviate from ideality.

Under conditions that deviate from ideal, other models like the Van der Waals equation are used to provide better approximations.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between neighboring particles, such as atoms, molecules, or ions. In the context of gases, these forces become significant when particles are very close to each other. As compression of a real gas progresses, these forces, which are usually weak under normal conditions, grow stronger. This can lead to the gas transitioning into a different phase, such as from a gas to a liquid, because the attractive forces pull the particles closer together.

• Intermolecular forces can be van der Waals forces, hydrogen bonds, or dipole-dipole interactions.
• They influence various properties of gases, such as boiling points and condensation.

Understanding intermolecular forces is crucial when studying the behavior of real gases and their phase transitions during compression processes.

Compression and Condensation

When a gas is compressed, the particles are forced closer together, increasing both pressure and the frequency of particle collisions. If this compression occurs at a constant temperature, as in the scenario described with a gas trapped in an ice bath, the internal energy remains constant, but physical proximity between particles increases significantly. The critical point in this scenario is that as the gas is compressed further, intermolecular forces start to dominate. As a result, these forces can cause the gas to transition into a liquid.

• Compression reduces the volume available to gas molecules increasing the interactions between them.
• Condensation is the process where a gas changes to a liquid phase due to these interactions.

This phase change is crucial to understand why gases do not adhere to ideal gas laws when heavily compressed.

Van der Waals Equation

The Van der Waals equation is used to account for deviations from ideal gas behavior by incorporating the effects of intermolecular forces and the finite size of particles. This equation is expressed as:\[\left( P + \frac{an^2}{V^2}\right) (V-nb) = nRT\]In this equation, constants \(a\) and \(b\) are unique to each gas:

• The term \(\frac{an^2}{V^2}\) corrects for intermolecular attractions, decreasing the effective pressure since particles attract each other.
• The term \(nb\) accounts for the finite volume occupied by gas molecules, reducing the effective volume available.

The Van der Waals equation thus provides a more accurate model for real gases than the ideal gas law, especially under conditions where compression and intermolecular forces become significant. It effectively bridges the gap by acknowledging that particles are not point masses and that they do indeed interact with one another.