Problem 68 - Chemistry The Molecular Science

Question

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

Step-by-Step Solution

Answer

The product \(PV\) exceeds \(nRT\) at high pressures due to finite molecular volume and intermolecular forces.

Step 1: Define Ideal Gas Behavior

An ideal gas follows 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. Under ideal conditions, gas molecules are considered to have no volume and exhibit no intermolecular forces.

Step 2: Understand Real Gas Deviations

In real gases, intermolecular forces and the finite volume of gas molecules play a role. These deviations become significant at high pressures and low temperatures, leading to observed differences from ideal behavior.

Step 3: Explain the Effect of Finite Molecular Volume

As pressure increases, the volume occupied by gas molecules themselves becomes non-negligible. Gas molecules thus take up more space than predicted by the ideal gas law, increasing \(PV\) compared to \(nRT\).

Step 4: Explain Intermolecular Forces

At higher pressures, molecules are closer together, making intermolecular attractions and repulsions more prominent. These interactions impact the volume and pressure exerted by the gas, further deviating \(PV\) from \(nRT\).

Step 5: Combine Explanations for Deviation

Both the finite molecular volume and intermolecular forces contribute to \(PV > nRT\) at high pressures. The ideal gas law does not account for these factors, which affect the relationship between pressure, volume, and temperature in real gases.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of gas. It is represented as \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume occupied by the gas.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) is the temperature of the gas in Kelvin.

For a gas to behave ideally, it assumes certain conditions:
- Gas molecules have no volume.
- No attractive or repulsive forces exist between the molecules.
Under these assumptions, gases tend to follow the Ideal Gas Law closely at high temperatures and low pressures.
This is because increased temperature and reduced pressure allow molecules to move freely without significant interaction.

Intermolecular Forces

In reality, all gas molecules exert forces on each other. These forces are termed intermolecular forces. They can be either attractive or repulsive.

• Attractive forces: These pull molecules closer together. Examples include van der Waals forces and hydrogen bonding.
• Repulsive forces: These push molecules apart. They occur when molecules are very close to each other, as electrons begin to repel each other.

At high pressures, gas molecules are packed closer.
This proximity amplifies the influence of intermolecular forces, altering the gas’s behavior from that predicted by the Ideal Gas Law.
This is because these forces cause deviations in both pressure and volume, impacting the overall gas behavior.

Finite Molecular Volume

When considering real gases, the volume of individual molecules cannot be ignored. Unlike ideal gases, real gas molecules occupy space.
As the pressure increases or the volume decreases, this finite molecular volume affects how gases behave.

• At high pressure, molecules are forced closer together, and their inherent volume takes up more space than an ideal gas would predict.
• Instead of the gas particles flying freely, they now collide with the container walls more frequently, thereby occupying measurable space.

This causes the product \(PV\) to be greater than \(nRT\) under these conditions.
By understanding finite molecular volume, it is clearer why real gases deviate from ideal behavior, especially under varying conditions of pressure and volume.