Problem 79 - Chemistry The Central Science In Si Units

Question

The temperature of a 5.00-L container of \(\mathrm{N}_{2}\) gas is increased from \(20^{\circ} \mathrm{C}\) to \(250^{\circ} \mathrm{C}\). If the volume is held constant, predict qualitatively how this change affects the following: (a) the average kinetic energy of the molecules; \((\mathbf{b})\) the rootmean- square speed of the molecules; (c) the strength of the impact of an average molecule with the container walls; d) the total number of collisions of molecules with walls per second.

Step-by-Step Solution

Answer

(a) Increases; (b) Increases; (c) Increases; (d) Increases.

Step 1: Understand the Relationship between Temperature and Kinetic Energy

The average kinetic energy of gas molecules is directly proportional to the absolute temperature (in Kelvin). As the temperature increases, the average kinetic energy of the molecules increases. The equation for average kinetic energy is \( KE_{avg} = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

Step 2: Convert Temperature to Kelvin

The initial temperature is \( 20^{\circ} \text{C} \) and the final temperature is \( 250^{\circ} \text{C} \). Convert these temperatures to Kelvin by adding 273.15. Hence, \( T_{1} = 293.15 \text{ K} \) and \( T_{2} = 523.15 \text{ K} \).

Step 3: Analyze Average Kinetic Energy Change

Since the temperature increases, the average kinetic energy of the molecules will also increase. This is because the average kinetic energy is proportional to temperature in Kelvin.

Step 4: Calculate Root-Mean-Square Speed Relationship

The root-mean-square speed \( v_{rms} \) is calculated using \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( m \) is the mass of a molecule. As the temperature increases, \( v_{rms} \) increases, which means the molecules move faster on average.

Step 5: Impact with Container Walls Analysis

The strength of the impact (force) of an average molecule with container walls is related to both how fast the molecule moves and how frequently it collides with the walls. As speed increases, each impact is stronger due to the relationship \( F \propto v \).

Step 6: Collision Frequency at Increased Temperature

Due to increased speed, molecules hit the walls more often per second. With the volume held constant, the total number of collisions with the walls per second increases because the molecules are moving faster and covering more distance in the same amount of time.

Key Concepts

Gas Laws

Gas laws describe the behavior of gases and establish relationships between pressure, volume, and temperature. One of the fundamental ideas is that gas particles move independently and their speed increases with temperature.

• According to Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume. More collisions at higher temperatures suggest increased pressure if the volume does not expand.
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature. An increase in temperature, keeping volume the same, leads to increased energy per particle and impacts pressure.
• Gay-Lussac’s Law: At constant volume, the pressure of a gas is directly proportional to its temperature. Hence, higher temperatures lead to increased gas pressure, assuming volume remains unchanged.

These relationships imply that with a rise in temperature while holding the volume constant, the pressure must increase, leading to more energetic molecular interactions with the container walls.

Temperature and Kinetic Energy

Temperature and kinetic energy share a direct relationship: as the temperature of a gas rises, so does the average kinetic energy of its molecules. The equation \[ KE_{avg} = \frac{3}{2}kT \] demonstrates this relationship, where:

• \( KE_{avg} \) is average kinetic energy.
• \( k \) is the Boltzmann constant (approximately \( 1.38 \times 10^{-23} \text{ J/K} \)).
• \( T \) is temperature measured in Kelvin (K).

By converting Celsius to Kelvin using \( T(K) = T(°C) + 273.15 \), it becomes evident that a rise from 20°C to 250°C results in notable kinetic energy increase. This is because the temperature increment significantly raises the amount of energy that each gas particle holds, leading to faster and more forceful particle movements.

Root Mean Square Speed

The root mean square (rms) speed provides insight into the average speed of gas molecules under certain conditions. Defined by the formula \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where:

• \( v_{rms} \) is the root mean square speed.
• \( k \) stands for the Boltzmann constant.
• \( T \) represents temperature in Kelvin.
• \( m \) is the mass of a single molecule.

An increase in temperature, at a constant mass, elevates the rms speed. In our scenario, elevating the temperature from 293.15 K to 523.15 K results in faster molecular travel within the container. Larger velocities imply that molecules strike container walls more energetically and swiftly, impacting collision frequency and impact force due to enhanced velocity momentum relation.

Molecular Collisions

Molecular collisions in a gas encompass two major aspects: the number of times they collide with container walls per second and the force of these collisions. Both these dynamics are influenced by temperature:

• Increased temperature results in higher kinetic energy, causing molecules to move faster and thus increase the frequency of collisions per second.
• Faster molecules exert stronger forces upon impact with container walls, as impact force depends on molecular speed. This is manifested through the equation \[ F \propto v \] where greater speed \( v \) leads to greater force \( F \).

At heightened speeds and impact forces, the container experiences intensified interactions. Understanding these behaviors is crucial as they encapsulate the dynamic nature of gas particles under varying thermal conditions.