Average Kinetic Energy of Gases: Videos & Practice Problems

The average kinetic energy (KE) of an ideal gas is described by the equation K = 3/2⁢k⁢T, where K is the average kinetic energy, k is the Boltzmann constant, and T is the temperature in Kelvins. This relationship shows that the average energy per particle depends solely on temperature, not the type of gas. For example, at 300 K, the average kinetic energy of oxygen molecules is approximately 6.21⁢⁢10−21 joules.

Introduction to Kinetic Theory of Gasses

Understanding the average kinetic energy of an ideal gas is essential in the study of thermodynamics and the kinetic molecular theory. The average kinetic energy per particle is directly related to the temperature of the gas, which is a crucial concept in this field. The equation that describes this relationship is:

\( K = \frac{3}{2} k_B T \)

In this equation, \( K \) represents the average kinetic energy, \( k_B \) is the Boltzmann constant (approximately \( 1.38 \times 10^{-23} \, \text{J/K} \)), and \( T \) is the absolute temperature measured in Kelvins. It is important to convert temperatures from Celsius to Kelvins to avoid negative values, which would yield nonsensical results for energy calculations. The conversion is done using the formula:

\( T(K) = T(°C) + 273 \)

For example, to calculate the average kinetic energy of oxygen molecules at 27 degrees Celsius, first convert the temperature to Kelvins:

\( T = 27 + 273 = 300 \, K \)

Substituting this value into the kinetic energy equation gives:

\( K = \frac{3}{2} (1.38 \times 10^{-23}) (300) \)

Calculating this results in:

\( K \approx 6.21 \times 10^{-21} \, \text{J} \)

This value represents the average energy per particle in the gas at that temperature. A key conceptual takeaway is that the average kinetic energy depends solely on the temperature of the gas and not on its type. Therefore, whether the gas is oxygen or nitrogen at the same temperature, the average kinetic energy per particle remains the same.

In a sample of gas, you pick a particle at random. The mass of the particle is 1.67 × 10^-27 kg and you measure its speed to be 1600 m/s. If that particle's kinetic energy is equal to the average kinetic energy of the gas particles, what is the temperature of the sample of gas?

• A. 232 K
• B. 0.065 K
• C. 103.3 K
• D. 206.5 K

Answer: (C)

Here’s what students ask on this topic:

What is the equation for the average kinetic energy of an ideal gas?

The equation for the average kinetic energy (KE) of an ideal gas is given by:

$K=\frac{3}{2}kT$

where $K$ is the average kinetic energy, $k$ is the Boltzmann constant, and $T$ is the temperature in Kelvins. This equation shows that the average kinetic energy per particle depends solely on the temperature of the gas.

How do you calculate the average kinetic energy of oxygen molecules at 27 degrees Celsius?

To calculate the average kinetic energy of oxygen molecules at 27 degrees Celsius, follow these steps:

1. Convert the temperature to Kelvins: $27+273=300 K$

2. Use the equation for average kinetic energy:

$K=\frac{3}{2}kT$

3. Plug in the values: $K=\frac{3}{2} 1.38 \times {10}^{-23} J K \times 300 K$

4. Calculate the result: $K=6.21 \times {10}^{-21} J$

So, the average kinetic energy of oxygen molecules at 27 degrees Celsius is approximately $6.21 \times {10}^{-21} J$.

Does the average kinetic energy of a gas depend on the type of gas?

No, the average kinetic energy of a gas does not depend on the type of gas. According to the equation $K=\frac{3}{2}kT$, the average kinetic energy depends solely on the temperature (T) and the Boltzmann constant (k). The type of gas, including its mass or molecular composition, does not appear in the equation. Therefore, at a given temperature, all ideal gases will have the same average kinetic energy per particle.

Why must temperature be in Kelvins when calculating average kinetic energy?

Temperature must be in Kelvins when calculating average kinetic energy because the Kelvin scale is an absolute temperature scale. The equation $K=\frac{3}{2}kT$ requires an absolute temperature to ensure that the kinetic energy is always positive. Using Celsius or Fahrenheit could result in negative temperatures, leading to nonsensical negative kinetic energies. The Kelvin scale starts at absolute zero, where molecular motion ceases, making it the appropriate scale for thermodynamic calculations.

What is the Boltzmann constant and its value?

The Boltzmann constant (k) is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the temperature of the gas. Its value is approximately:

$k=1.38 \times {10}^{-23} J K-1$

This constant is crucial in statistical mechanics and thermodynamics, as it bridges the macroscopic and microscopic worlds by linking temperature to energy at the particle level.