Question

At constant volume, for a fixed number of moles of a gas the pressure of the gas increases with size of temperature due to (1) increase in average molecular speed (2) increase in number of moles (3) increase in molecular attraction (4) decrease in the distance between the molecules

Short answer

Option 1: Increase in average molecular speed.

Step-by-step solution

- Understand the Relationship

Recognize that according to the ideal gas law, which is \[ PV = nRT \], for a constant volume (V) and a fixed number of moles (n), the pressure (P) of the gas is directly proportional to its temperature (T).

- Analyze the Given Options

Consider each of the provided options and how they relate to the temperature increase at constant volume for a fixed number of moles.

- Increase in Average Molecular Speed

Option 1: Increase in average molecular speed. An increase in temperature means an increase in the average kinetic energy and speed of the gas molecules, leading to more frequent and forceful collisions with the walls of the container, resulting in an increase in pressure.

- Increase in Number of Moles

Option 2: Increase in number of moles. This is not relevant here as the number of moles is fixed and not changing with temperature.

- Increase in Molecular Attraction

Option 3: Increase in molecular attraction. Increasing molecular attraction would typically lower the gas pressure, as molecules would stick together more and collide with the container walls less often.

- Decrease in Distance Between Molecules

Option 4: Decrease in the distance between the molecules. For an ideal gas, a temperature increase doesn't affect the distance between molecules at constant volume. The volume occupied by the gas molecules may appear negligible compared to the container volume.

- Conclude the Correct Answer

Option 1 is correct because an increase in temperature results in an increase in the average molecular speed, leading to higher pressure at constant volume.

Key concepts

Average Molecular Speed

When discussing gases, one key concept is the average molecular speed. At a molecular level, gas consists of a large number of particles, all moving randomly in different directions. The temperature of a gas is closely linked to the speed of its molecules. As the temperature increases, so does the average kinetic energy of the molecules.
This can be mathematically expressed as: \( KE = \frac{1}{2}mv^2 \), where \( KE \) is the kinetic energy, \( m \) is the mass of a molecule, and \( v \) is the velocity or speed of the molecule.
At higher temperatures, the average speed of gas molecules increases, resulting in more frequent and forceful collisions with the container walls. This increase in collision frequency and force leads to higher pressure, as described by the Ideal Gas Law.

Pressure-Temperature Relationship

The relationship between pressure and temperature in a gas is a fundamental aspect of the Ideal Gas Law, which is expressed as: \( PV = nRT \). Here, \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. When the volume \( V \) and the number of moles \( n \) are constant, the equation simplifies to show a direct relationship between pressure and temperature: \( P \propto T \).
Simply put, as the temperature \( T \) increases, the pressure \( P \) also increases, as long as the volume and amount of gas remain constant. This is because the molecules move faster at higher temperatures, leading to more energetic collisions with the container walls, which manifests as increased pressure.

Kinetic Energy of Gas Molecules

Kinetic energy plays a crucial role in understanding gas behavior. Kinetic energy is the energy an object has due to its motion. For gas molecules, this kinetic energy depends on both the mass of the molecules and their speed. The relationship can be given by the equation: \( KE = \frac{3}{2} k_B T \), where \( k_B \) is Boltzmann's constant and \( T \) is the absolute temperature.
This equation indicates that the kinetic energy of gas molecules is directly proportional to the temperature. At higher temperatures, molecules have more kinetic energy, which translates to faster movement. Faster movement results in more collisions with the container walls, ultimately increasing the pressure. Understanding this helps explain why temperature has such a significant influence on gas pressure.
These fundamental concepts are crucial for a deeper grasp of gas behaviors and interaction according to the Ideal Gas Law.