Question

At constant volume, for a fixed number of moles of a gas, the pressure of the gas increases with increase in temperature due to: (a) Increase in the average molecular speed (b) Increase rate of collision amongst molecules (c) Increase in molecular attraction (d) Decrease in mean free path

Short answer

(a) Increase in the average molecular speed and (b) Increase rate of collision amongst molecules are responsible for the increase in pressure with temperature at constant volume.

Step-by-step solution

Understand the Relationship Between Pressure and Temperature

Recall the kinetic molecular theory, which states that at constant volume, the pressure exerted by a gas depends on the temperature and the number of gas molecules' collisions against the walls of the container. With an increase in temperature, the kinetic energy of the molecules increases.

Analyze the Potential Effects Listed

With higher kinetic energy, the molecules move faster (a) which leads to more frequent and forceful collisions (b) against the walls. Molecular attraction (c) generally decreases with increased kinetic energy, and this option doesn't lead to increased pressure. The mean free path (d) relates to the average distance a molecule travels before a collision, but doesn't directly explain the increase in pressure with increased temperature at constant volume.

Identify the Correct Causes

Items (a) and (b) are directly related to the increase in gas pressure when the temperature rises at constant volume. Increased average molecular speed leads to an increased rate of collisions, which in turn increases the pressure. Hence, (a) and (b) are the causes of increased pressure.

Key concepts

Pressure-Temperature Relationship

The relationship between the pressure and temperature of a gas is a fundamental aspect of the Kinetic Molecular Theory. This theory explains the behavior of gases in terms of the motion of individual molecules. When dealing with gases, one key principle to bear in mind is that heating a gas will increase its pressure if the volume is held constant. This is because heat adds energy to the system which translates into the kinetic energy of gas molecules.

As the kinetic energy increases, gas molecules move more rapidly and collide with the walls of their container more forcefully and frequently. This is dictated by the direct proportionality between temperature (measured in Kelvin) and pressure. These collisions are the cause behind the pressure a gas exerts. Thus, an increase in temperature will inevitably lead to an increase in pressure, which is described quantitatively by Gay-Lussac's law, one of the gas laws that relates pressure and temperature.

Kinetic Energy of Gases

The kinetic energy of gas molecules is intimately linked to the temperature of the gas. To understand this, we can recall the basic kinetic energy formula for an individual particle: \( KE = \frac{1}{2}mv^2 \), where \(m\) is the mass and \(v\) is the velocity of the particle. For gases, we consider the average kinetic energy of all molecules. When the temperature of a gas increases, the average kinetic energy of its molecules also increases, which means they move at higher speeds.

This increase in speed enhances the momentum of the molecules upon collision with the surfaces of the container, resulting in greater force and hence greater pressure exerted by the gas. The direct relationship between kinetic energy and temperature is foundational to explaining gas behavior under varying conditions of pressure and volume, tying back to the principles of thermodynamics and statistical mechanics.

Gas Law Concepts

Gas laws are empirical relationships that describe how different physical properties of a gaseous system interrelate with one another. They are essentially simplified models of gas behavior that, while not accounting for molecular size or intermolecular forces, provide a useful basis for understanding and predicting how gases will respond to changes in conditions.

• Boyle's law states that, at constant temperature, the volume and pressure of a gas are inversely proportional.
• Charles' law asserts that volume and temperature are directly proportional when pressure is held constant.
• Gay-Lussac's law links pressure and temperature as directly proportional at constant volume.
• The Combined Gas Law combines Boyle's, Charles', and Gay-Lussac's laws into one equation, showing the relationship between pressure, volume, and temperature.
• The Ideal Gas Law, represented as \(PV = nRT\), brings together these relationships and introduces the amount of gas (n) and the ideal gas constant (R) to cover conditions under which gases do not behave ideally.

Understanding and applying these laws allow us to predict how a gas will behave under different thermodynamic scenarios, and they are essential to the fields of chemistry and physics.