Problem 81 - Chemistry: The Central Science

Question

The temperature of a \(5.00-\mathrm{L}\) 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; (b) the root- meansquare 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

In summary, when the temperature of the $\mathrm{N}_{2}$ gas is increased from $20^{\circ} \mathrm{C}$ to $250^{\circ} \mathrm{C}$ while holding the volume constant: (a) the average kinetic energy of the molecules will increase; (b) the root-mean-square speed of the molecules will increase; (c) the strength of the impact of an average molecule with the container walls will increase; and (d) the total number of collisions of molecules with the walls per second will also increase.

Step 1: (a) The effect on average kinetic energy of the molecules

According to the Kinetic Theory of Gases, the average kinetic energy of a gas molecule is directly proportional to the temperature in Kelvin (\(K\)). The equation for average kinetic energy is: \[ \text{KE}_{avg} = \frac{3}{2}kT \] where \(k\) is the Boltzmann constant (\(1.38 ×10^{-23} J/K\)) and \(T\) is the temperature in Kelvin. Since the temperature has increased from \(20^{\circ}C\) to \(250^{\circ}C\), we can convert these values to Kelvin: \[T_1 = 20^{\circ}C + 273.15 = 293.15 K\] \[T_2 = 250^{\circ}C + 273.15 = 523.15 K\] Now, comparing \(\text{KE}_{avg}\) of the gas at both temperatures: \[\frac{\text{KE}_{avg2}}{\text{KE}_{avg1}} = \frac{3/2 k T_2}{3/2 k T_1} = \frac{T_2}{T_1} = \frac{523.15 K}{293.15 K}\] Since the ratio is greater than 1, we can conclude that the average kinetic energy of the gas molecules will increase when the temperature is increased.

Step 2: (b) The effect on root-mean-square speed of the molecules

The root-mean-square (\(\text{rms}\)) speed of gas molecules is given by the equation: \[v_{rms} = \sqrt{\frac{3kT}{m}}\] where \(m\) is the mass of a single molecule. For nitrogen gas, \(m = \frac{28.02}{6.022 × 10^{23}} kg\). When the temperature increases, the value inside the square root increases. Therefore, the root-mean-square speed will also increase, as it is directly proportional to the square root of the temperature.

Step 3: (c) The effect on strength of impact of an average molecule with container walls

The strength of the impact of an average molecule with the container walls can be associated with the momentum change during a collision, which is proportional to molecule’s mass and velocity. Since we have already established that the velocity of the gas molecules will increase with the temperature, we can conclude that the strength of the impact of an average molecule with the container walls will also increase.

Step 4: (d) The effect on the total number of collisions of molecules with walls per second

The total number of collisions of molecules with the walls of the container per second is determined by the product of the number of molecules per unit volume, the surface of the walls, and the average component of the velocity of molecules perpendicular to the wall surface. Since the volume of the container and the number of molecules are both constant, we can focus on the effect of temperature on the average component of the velocity of molecules perpendicular to the wall surface. As discussed in part (b), the root-mean-square speed of the molecules will increase with a higher temperature. Consequently, the total number of collisions of molecules with the walls per second will also increase due to the higher average velocity at higher temperature.

Key Concepts

Average Kinetic Energy

The average kinetic energy of gas molecules is fundamentally linked to the temperature of the gas. According to the Kinetic Theory of Gases, this energy is directly proportional to the absolute temperature in Kelvin. The mathematical representation of average kinetic energy is given by:\[\text{KE}_{avg} = \frac{3}{2}kT\]where:

• \( k \) is the Boltzmann constant, \( 1.38 \times 10^{-23} J/K \).
• \( T \) is the temperature in Kelvin.

When temperature rises, the average kinetic energy of gas molecules also rises. For instance, increasing the temperature of a container of nitrogen gas from \(20^{\circ}C \) to \(250^{\circ}C \) shows that the kinetic energy at \(523.15 \, K\) is greater than at \(293.15 \, K\). Hence, more energy is available for molecular motion, making them move faster.

Root-Mean-Square Speed

The root-mean-square speed is a measure of the velocity of particles in a gas. It incorporates the temperature and the mass of the molecules. This speed helps us understand how fast the molecules are moving on average. The formula for root-mean-square (rms) speed is:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]where:

• \( k \) is the Boltzmann constant.
• \( T \) is the temperature in Kelvin.
• \( m \) is the mass of a single molecule.

When temperature increases, such as from \(20^{\circ}C \) to \(250^{\circ}C \), the root-mean-square speed increases as a result of the increased temperature. This happens because the speed is directly related to the square root of the temperature—a higher temperature means higher overall molecular speed.

Molecular Collisions

Molecular collisions in a gas occur when molecules strike each other or the walls of their container. The frequency and force of these collisions can tell us a lot about the behavior of the gas. The strength of an individual collision with the wall is influenced by the molecule's speed and mass, hence related to momentum change during the collision. Since speed increases with temperature, so does the force of these collisions. With the nitrogen gas example, as the temperature rises from \(20^{\circ}C \) to \(250^{\circ}C \), molecules move faster, resulting in more forceful impacts on the container walls.

Temperature Effect on Gases

Temperature profoundly affects gases by changing their molecular motion and energy states. As the temperature of a gas increases, several things happen:

• Average kinetic energy increases, providing molecules with more energy to move.
• Root-mean-square speed increases, leading to faster molecular motion.
• Individual molecular impacts become stronger due to increased speed.
• The frequency of molecular collisions with walls rises as molecules travel more quickly.

All these effects from rising temperatures lead to gas molecules spreading out more and colliding more frequently and powerfully with each other and container walls. This increased activity is a direct outcome of heightened thermal energy.