Problem 75 - Chemistry 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 average 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

As the temperature of the $\mathrm{N}_{2}$ gas increases from 20°C to 250°C, (a) the average kinetic energy of the molecules increases, (b) the average speed of the molecules increases, (c) the strength of the impact of an average molecule with the container walls increases, and (d) the total number of collisions of molecules with walls per second increases.

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

According to the molecular theory of gases, the average kinetic energy of the gas molecules is directly proportional to the temperature (measured in Kelvin). So, when the temperature increases, the average kinetic energy of the molecules also increases. To convert temperatures from Celsius to Kelvin, we add 273.15: Initial temperature in Kelvin: \(T_1 = 20 + 273.15 = 293.15 K\) Final temperature in Kelvin: \(T_2 = 250 + 273.15 = 523.15 K\) Since the final temperature is higher than the initial temperature, the average kinetic energy of the molecules increases.

Step 2: (b) The average speed of the molecules

The average speed of the gas molecules is also related to the temperature, as it is related to the square root of the average kinetic energy. If the average kinetic energy increases, as calculated in part (a), then the average speed of the molecules will also increase.

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

The strength of the impact of an average molecule with the container walls depends on the momentum of the molecule, which is the product of its mass and velocity. As the average speed of the molecules increases (from part b), their momentum also increases. Hence, the strength of the impact of an average molecule with the container walls will also increase.

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

The total number of collisions of gas molecules with the container walls per second depends on the pressure of the gas and is also directly proportional to their average speed. Since the average speed of the molecules is increasing (as discussed in part b), the total number of collisions of molecules with walls per second will also increase.

Key Concepts

Gas Laws

The behavior of gases is often predictable and can be described by the gas laws, which are mathematical relationships between the volume, temperature, pressure, and the number of particles in a gas. These relationships are crucial for understanding how changes in one variable can affect the others. Among the most well-known gas laws are Boyle's Law, which states that pressure and volume are inversely proportional at constant temperature; Charles's Law, which indicates that volume and temperature are directly proportional at constant pressure; and the combined gas law, which brings these relationships together and includes pressure.

The gas laws are derived from the kinetic molecular theory, which assumes that gas particles are in continuous random motion and that they collide with each other and the walls of their container without losing energy. The increase in temperature, as described in the textbook exercise, corresponds to an increase in the average kinetic energy per molecule. According to Charles's Law, as the temperature of a gas increases with a constant volume, the pressure must also increase.

Understanding these laws allows us to predict how a gas will behave under different conditions, which is essential in both academic studies and various industrial applications.

Average Kinetic Energy of Gases

The kinetic molecular theory provides insights into the microscopic behavior of gas molecules. A central concept of this theory is the average kinetic energy of gas particles, which is directly proportional to the absolute temperature of the gas, as measured in Kelvin. Essentially, the average kinetic energy determines how much energy the particles have to move and, therefore, how fast they're going when they collide with each other or with the container walls.

The formula for the average kinetic energy of a molecule is given by \( \frac{3}{2}kT \) where \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin. When you increase the temperature, as in the exercise, the kinetic energy increases. This is why an increase from \( 20^\circ C \) to \( 250^\circ C \) results in a higher average kinetic energy. With a greater average kinetic energy, the gas molecules will move faster and collide more often and with greater force, leading to increased pressure if the volume is constant.

Molecular Collisions

Molecular collisions are a foundational concept in the kinetic molecular theory, affecting how we understand gas behaviors and properties. When gas particles collide with each other or the walls of their container, these collisions are perfectly elastic, meaning that there is no net loss of energy from the collisions.

The rate of molecular collisions will increase as a gas's average kinetic energy and thus its temperature increases. This is because the gas particles move faster, leading them to hit the container walls more frequently, as reflected in the exercise problem. These impacts, when totaled over the large number of particles in the gas, account for the gas pressure. The pace at which molecules collide with the walls determines how much pressure they exert. Therefore, when the exercise mentions the increase of temperature, we know that the collisions per second increase, which is also associated with an increase in the pressure exerted by the gas on the walls of its container. It's also worth noting that while the number of collisions increases, the number of molecules remains constant, as the exercise presumes a closed system.