Question

How is the kinetic energy of a gas related to temperature? How is the root mean square velocity of a gas related to its molar mass?

Short answer

The average kinetic energy of a gas is directly proportional to its absolute temperature. The root mean square velocity of a gas is inversely proportional to the square root of its molar mass.

Step-by-step solution

Understanding Kinetic Energy and Temperature Relation

The kinetic energy of a gas can be related to its temperature through the formula for the average kinetic energy of a particle in a gas: \( KE_{avg} = \frac{3}{2}k_{B}T \), where \( KE_{avg} \) is the average kinetic energy per molecule, \( k_{B} \) is the Boltzmann constant, and \( T \) is the absolute temperature in Kelvin. Higher temperature means higher average kinetic energy per molecule.

Relating Root Mean Square Velocity to Molar Mass

The root mean square (rms) velocity of a gas \( (v_{rms}) \) can be related to its molar mass \( (M) \) through the formula \( v_{rms} = \sqrt{\frac{3k_{B}T}{M}} \), where \( k_{B} \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( M \) is the molar mass. Here, it's important to note that the unit of the molar mass M must be kg/mol for the units to be consistent.

Key concepts

Relationship Between Temperature and Kinetic Energy

The kinetic energy of gas molecules is fundamentally linked to the temperature of the gas. This is described by a principle of thermodynamics which states that the average kinetic energy (\(KE_{avg}\)) of gas molecules is directly proportional to the absolute temperature (\(T\) in Kelvin) of the gas. The formula that represents this relationship is given as:
\[ KE_{avg} = \frac{3}{2}k_{B}T \]
The Boltzmann constant (\(k_{B}\)) in this equation provides a bridge between the macroscopic world of temperature measurements and the microscopic behavior of individual gas molecules. As the temperature increases, the molecules move faster indicating an increase in their kinetic energy. Conversely, as the temperature decreases, the molecules slow down, showing a reduction in kinetic energy. Understanding this relationship is crucial for grasping the behavior of gases under various temperature conditions.

Root Mean Square Velocity

The root mean square (rms) velocity is a statistical measure used to determine the average speed of gas molecules. The rms velocity (\(v_{rms}\)) takes into account the distribution of individual molecule speeds. The relationship between rms velocity and temperature is vital in kinetic theory, and it’s given by the formula:
\[ v_{rms} = \sqrt{\frac{3k_{B}T}{M}} \]
In this expression, \(k_{B}\) represents the Boltzmann constant, \(T\) is the absolute temperature, and \(M\) is the molar mass of the gas. Remember, the molar mass must be in kilograms per mole (kg/mol) to ensure consistency in units. Higher temperatures will result in higher rms velocities, signifying faster-moving particles typically associated with higher energy states. Students should grasp that a greater rms velocity signifies not just faster average particle speeds but also greater kinetic energy of the gas.

Boltzmann Constant

Named after Ludwig Boltzmann, a key figure in the development of statistical mechanics, the Boltzmann constant (\(k_{B}\)) is a proportionality factor that links the average kinetic energy of particles in a gas with the temperature of the gas. The value of the Boltzmann constant is
\[ k_{B} = 1.38 \times 10^{-23} \text{ J/K} \]
where J stands for joules and K for Kelvin. The constant is essential to understanding the energy scales at molecular levels and plays a crucial role in not just kinetic theory but also in defining the entropy of a system. It's ubiquitous in equations that describe the behavior of gases, such as the ideal gas law and the aforementioned formulas for kinetic energy and rms velocity.

Molar Mass of Gases

Molar mass (\(M\)) expresses the mass of one mole of a given substance and is quantified in units of grams per mole (g/mol) or kilograms per mole (kg/mol). It is a critical factor when examining gas behaviors, as it is intricately connected to the rms velocity of gas molecules. A lighter gas, with a lower molar mass, will have a higher rms velocity at the same temperature compared to a heavier gas. This can be seen from the formula mentioned earlier, which illustrates that the rms velocity of a gas is inversely proportional to the square root of its molar mass. In practical terms, this means that at room temperature, lighter gases like hydrogen will move much faster than heavier gases such as oxygen.