Problem 75 - Chemistry: The Central Science in SI Units, Global Edition

Question

Determine whether each of the following changes will increase, decrease, or not affect the rate with which gas molecules collide with the walls of their container: (a) increasing the volume of the container, \((\mathbf{b})\) increasing the temperature, (c) increasing the molar mass of the gas.

Step-by-Step Solution

Answer

(a) Increasing the volume of the container will \(\textbf{decrease}\) the rate of collisions, as it allows the gas molecules more space to move around and causes a decrease in pressure. (b) Increasing the temperature will \(\textbf{increase}\) the rate of collisions, as it increases the kinetic energy of the gas molecules, leading them to move faster and collide with the walls more frequently. (c) Increasing the molar mass of the gas will \(\textbf{decrease}\) the rate of collisions, as it leads to a decrease in the number of moles and subsequently, a decrease in pressure.

Step 1: Effects of increasing the volume of the container

When the volume of the container increases, the gas molecules will have more space to move around. According to the ideal gas law, when the volume increases while the temperature and the number of moles remain constant, the pressure decreases. Since the pressure is the result of collisions of gas molecules with the walls of the container, a decrease in pressure means that the rate of collisions will also decrease. Therefore, increasing the volume of the container will decrease the rate of collisions.

Step 2: Effects of increasing the temperature

When the temperature of the gas increases, the kinetic energy of the gas molecules increases as well. This increase in kinetic energy means that the gas molecules will move faster, thus colliding with the walls of the container more frequently. Using the ideal gas law, we see that an increase in temperature while keeping the volume and the number of moles constant leads to an increase in pressure. Therefore, increasing the temperature will increase the rate of collisions.

Step 3: Effects of increasing the molar mass of the gas

Increasing the molar mass means that the gas has heavier molecules. When we examine the ideal gas law equation, we see that the molar mass (M) is not directly present in the equation. However, we could relate molar mass to the number of moles (n) if we consider the mass (m) of gas constant in the container. If the mass remains constant and the molar mass increases, the number of moles (n) will decrease (n = m/M) since n and M are inversely related. Consequently, from the ideal gas law equation, we can see that decreasing the number of moles while keeping the volume and temperature constant will decrease the pressure. As a result, the rate of collisions will also decrease. Therefore, increasing the molar mass of the gas will decrease the rate of collisions. In conclusion: (a) Increasing the volume of the container decreases the rate of collisions. (b) Increasing the temperature increases the rate of collisions. (c) Increasing the molar mass of the gas decreases the rate of collisions.

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory offers a fascinating view of how gas molecules behave and interact with their surroundings. This theory posits that gas particles are in constant, random motion, colliding with the walls of their container and with each other.
These collisions are perfectly elastic, meaning that no energy is lost.
One key aspect of this theory is its prediction that increasing the temperature of a gas will cause the molecules to move faster.

• Faster moving molecules collide more frequently, explaining why gas pressure increases with temperature.
• In contrast, if the volume of the container increases, the same number of molecules has more space to move around, resulting in fewer collisions.

The Kinetic Molecular Theory provides a framework for understanding how changes in temperature, volume, and molecular mass affect gas behavior, helping us predict outcomes in various scenarios.

Ideal Gas Law

The Ideal Gas Law is a cornerstone of understanding gas behavior. It's expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
This equation summarizes the relationships between these variables and describes an ideal gas where interactions between molecules are negligible.
From this law, we can see that pressure is directly proportional to temperature and the number of moles, but inversely proportional to volume.

• If we increase the temperature while keeping the volume constant, the pressure increases due to more frequent and energetic collisions.
• Conversely, if we increase the volume and keep the temperature constant, the pressure decreases as molecules collide with the walls less often.

The Ideal Gas Law provides a simple way to model real gases, though it's important to note that real gases deviate from this law under high pressure or low temperature.

Collision Rate

In the context of gases, collision rate refers to how often gas molecules hit the walls of their container. This rate is influenced by several factors:

• Temperature: Higher temperatures increase the speed at which molecules move, leading to more frequent collisions.
• Volume: Increasing the volume of a container spreads out the molecules, reducing collision frequency.
• Molecular Mass: Heavier molecules, with higher molar mass, typically move slower at a given temperature, leading to fewer collisions.

Understanding collision rates is crucial since it directly affects the gas pressure and is integral to applying both the Kinetic Molecular Theory and the Ideal Gas Law.
Temperature and volume changes are intuitive influences, but the weight of the molecules also plays a subtle yet important role in determining the overall behavior of the gas in a defined space.

Gas Pressure

Gas pressure is the force exerted by gas molecules as they collide with the walls of their container. Each collision exerts a small amount of force, and the sum of these forces translates to the pressure we observe. This pressure depends on multiple factors.
According to the Ideal Gas Law:

• Increasing temperature will increase the gas pressure as molecules move faster and collide more forcefully and frequently.
• Decreasing the volume while keeping temperature constant will also increase pressure due to a higher collision rate.

Moreover, gas pressure is inversely related to the molar mass when the mass of the gas is constant.
This detail highlights that while lighter molecules might move faster, leading to higher pressure at a given temperature, heavier molecules might decrease pressure due to lower collision rates. Grasping the fundamentals of gas pressure aids in comprehending how gases behave under varying conditions.