PV = 1/3Nm𝑣̄²

Definition

The equation PV = 1/3Nm$\bar{v}^2$ is a fundamental relationship in the kinetic theory of gases, which provides an atomic and molecular explanation for the concepts of pressure and temperature. This equation establishes a connection between the macroscopic properties of a gas, such as pressure (P) and volume (V), and the microscopic properties of the gas molecules, including the number of molecules (N), the average molecular speed ($\bar{v}$), and the mass (m) of the individual molecules.

5 Must Know Facts For Your Next Test

1. The equation PV = 1/3Nm$\bar{v}^2$ is derived from the kinetic theory of gases, which assumes that gas molecules are in constant random motion and collide with each other and the container walls.
2. The term 1/3Nm$\bar{v}^2$ represents the average kinetic energy of the gas molecules, which is directly proportional to the absolute temperature of the gas.
3. The pressure (P) of a gas is related to the average force exerted by the gas molecules on the container walls, which is proportional to the average kinetic energy of the molecules.
4. The volume (V) of the gas is related to the space occupied by the gas molecules, which is inversely proportional to the pressure (P) of the gas.
5. The equation PV = 1/3Nm$\bar{v}^2$ demonstrates the direct relationship between the macroscopic properties of a gas (pressure and volume) and the microscopic properties of the gas molecules (number, mass, and average speed).

Review Questions

• Explain how the equation PV = 1/3Nm$\bar{v}^2$ connects the macroscopic properties of a gas to its microscopic properties.
  • The equation PV = 1/3Nm$\bar{v}^2$ establishes a direct relationship between the macroscopic properties of a gas, such as pressure (P) and volume (V), and the microscopic properties of the gas molecules, including the number of molecules (N), the average molecular speed ($\bar{v}$), and the mass (m) of the individual molecules. This equation demonstrates that the pressure and volume of a gas are directly proportional to the average kinetic energy of the gas molecules, which is represented by the term 1/3Nm$\bar{v}^2$. This connection between the macroscopic and microscopic properties of a gas is a fundamental principle of the kinetic theory of gases.
• Describe how changes in temperature affect the terms in the equation PV = 1/3Nm$\bar{v}^2$.
  • According to the kinetic theory of gases, an increase in temperature corresponds to an increase in the average kinetic energy of the gas molecules, as represented by the term 1/3Nm$\bar{v}^2$. As the average molecular speed ($\bar{v}$) increases with higher temperatures, the term 1/3Nm$\bar{v}^2$ will also increase. This increase in average kinetic energy will result in a corresponding increase in the pressure (P) of the gas, as the gas molecules will exert a greater force on the container walls. Conversely, a decrease in temperature will lead to a decrease in the average molecular speed and the average kinetic energy, resulting in a decrease in the pressure of the gas.
• Analyze how the equation PV = 1/3Nm$\bar{v}^2$ can be used to predict the behavior of a gas under different conditions.
  • The equation PV = 1/3Nm$\bar{v}^2$ can be used to predict how a gas will behave under different conditions by analyzing the relationships between the variables. For example, if the volume (V) of a gas is increased while the number of molecules (N) and their average speed ($\bar{v}$) remain constant, the pressure (P) of the gas will decrease, as indicated by the inverse relationship between pressure and volume. Similarly, if the temperature of the gas is increased, leading to a higher average molecular speed ($\bar{v}$), the pressure (P) will increase, as the average kinetic energy of the gas molecules (1/3Nm$\bar{v}^2$) has risen. By understanding these relationships, the equation PV = 1/3Nm$\bar{v}^2$ can be used to predict and explain the behavior of gases under various conditions, which is crucial for applications in fields such as thermodynamics and engineering.

Related terms

Kinetic Theory of Gases: The kinetic theory of gases is a model that explains the macroscopic properties of gases, such as pressure, temperature, and volume, in terms of the microscopic behavior of the individual gas molecules.

Pressure: Pressure is the force exerted per unit area by the gas molecules as they collide with the walls of the container. It is a measure of the average force exerted by the gas molecules on the surrounding surfaces.

Temperature: Temperature is a measure of the average kinetic energy of the gas molecules, which is directly related to their average speed ($\bar{v}$). Higher temperatures correspond to greater average kinetic energy and faster molecular motion.