Question

According to kinetic theory of gases the average translational energy \((\mathrm{K} \mathrm{E})\) is (1) \(\frac{K T}{2}\) per molecule (2) KT per molecule (3) RT per moleculc (4) \(\frac{3}{2} \mathrm{KT}\) per molecule

Short answer

Option (4) \( \frac{3}{2} \text{KT} \) per molecule

Step-by-step solution

- Understand the problem

According to kinetic theory of gases, we need to determine the correct expression for the average translational energy per molecule.

- Recall relevant formula

The kinetic theory of gases provides the formula for the average translational kinetic energy per molecule: \[ \text{Average KE} = \frac{3}{2} k_B T \]where \(k_B\) is the Boltzmann constant and \(T\) is the temperature in Kelvin.

- Match the given options with the formula

Compare the given choices with the derived formula: (1) \( \frac{KT}{2} \) per molecule (2) \( KT \) per molecule (3) \( RT \) per molecule (4) \( \frac{3}{2} \text{KT} \) per molecule. Option (4) matches the formula \( \frac{3}{2} k_B T \).

- Conclude the correct option

Based on the comparison, the correct expression for the average translational kinetic energy per molecule is \( \frac{3}{2} k_B T \).

Key concepts

Translational Kinetic Energy

Translational kinetic energy refers to the energy possessed by a molecule due to its motion through space. In the context of the kinetic theory of gases, this energy can be described by the equation: \( \text{Average KE} = \frac{3}{2} k_B T \) Here, \(k_B\) is the Boltzmann constant, and \(T\) is the absolute temperature in Kelvin. Key points to remember:

• Translational kinetic energy is only one component of the total energy of a molecule; it does not account for rotational or vibrational energy.
• The equation \( \frac{3}{2} k_B T \) applies to ideal gases, where interactions between molecules are negligible.

Understanding the translational kinetic energy helps in grasping how gas molecules move and how their energy varies with temperature. This foundational concept is crucial for various applications in thermodynamics and statistical mechanics.

Boltzmann Constant

The Boltzmann constant (\(k_B\)) 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 \(1.38 \times 10^{-23}\) Joules per Kelvin (J/K). Here are some important points about the Boltzmann constant:

• It serves as a bridge between macroscopic and microscopic physics, connecting temperature (a macroscopic property) to the kinetic energy of individual molecules (a microscopic property).
• In the equation \( \frac{3}{2} k_B T \), \(k_B\) helps quantify how much energy each degree of temperature contributes to the kinetic energy of a single molecule.
• The Boltzmann constant is named after Ludwig Boltzmann, an Austrian physicist who made significant contributions to the field of statistical mechanics.

Recognizing the role of the Boltzmann constant can enhance your understanding of how temperature affects molecular behavior in gases.

Temperature

Temperature is a measure of the average kinetic energy of the particles in a substance. In the kinetic theory of gases, it's directly proportional to the average translational kinetic energy of the gas molecules. Here’s what you should know:

• Temperature is measured in units of Kelvin (K) in scientific contexts, ensuring that it starts from absolute zero, the point at which molecular motion ceases.
• Higher temperatures mean higher average kinetic energy among the gas molecules, leading to faster molecular motion.
• Temperature affects various properties of gases, including pressure, volume, and the rate of chemical reactions.

The relationship between temperature and kinetic energy is vital for understanding how gases behave under different conditions, and it forms the basis for many practical applications in science and engineering.