Question

Choose the right statement from the following: (1) The average KE of a molecule of any ideal gas is the same at the same temperature (2) The average translational KE of a molecule of any ideal gas is the same at the same temperature. (3) The average translational KE of a molecule of diatomic gas is \(1.5\) times the average rotational KE of a molecule at a given moderate temperature. (4) The average rotational KE of a molecule of a monoatomic gas increases linearly with increase in temperature.

Short answer

The correct statements are (1) and (2).

Step-by-step solution

- Understanding Kinetic Energy in Ideal Gases

Understand that for an ideal gas, the average kinetic energy (KE) of a molecule is directly proportional to the temperature of the gas. The KE is given by the equation \(\frac{3}{2}k_B T\), where \(k_B\) is the Boltzmann constant and \(T\) is the absolute temperature.

- Examining the Statements

Evaluate each given statement in the context of the ideal gas laws and kinetic theory.

- Analyze Statement (1)

Statement (1) claims the average KE of a molecule of any ideal gas is the same at the same temperature. This is correct according to the kinetic theory of gases.

- Analyze Statement (2)

Statement (2) asserts the average translational KE of a molecule of any ideal gas is the same at the same temperature. This is also true as translational KE is given by \(\frac{3}{2}k_B T\).

- Analyze Statement (3)

Statement (3) describes the average translational KE of a molecule of diatomic gas being 1.5 times the average rotational KE of a molecule at a moderate temperature. For diatomic gases, at moderate temperatures, the average translational KE is indeed proportional to rotational KE, but this proportional factor is \(\frac{3}{2}k_B T\), which is the same for rotational KE.

- Analyze Statement (4)

Statement (4) mentions the average rotational KE of a molecule of a monoatomic gas increasing linearly with temperature. Monoatomic gases do not have rotational KE as they lack rotational degrees of freedom. Therefore, this statement is incorrect.

- Select the Correct Statements

From the analysis, statements (1) and (2) are correct. Statements (3) and (4) are incorrect.

Key concepts

ideal gas law

The ideal gas law is a critical concept in understanding the behavior of gases under various conditions. It is represented by the equation PV = nRT, where P stands for pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. This law helps predict how gases behave under different conditions of pressure, temperature, and volume.
For instance, if you increase the temperature of a gas while keeping the volume constant, the pressure will rise proportionally. This relationship helps us understand how gases interact in various environments and is essential in many scientific and engineering applications.

Boltzmann constant

The Boltzmann constant, denoted as k_B, bridges macroscopic and microscopic physics by relating the average kinetic energy of particles in a gas to the temperature of the gas. Its value is approximately 1.38 x 10^-23 J/K. This constant appears in many fundamental equations, including the equation for the average kinetic energy of gas molecules.
The average translational kinetic energy of a molecule in an ideal gas can be expressed as \(\frac{3}{2}k_BT\), where T is the absolute temperature. The Boltzmann constant thus plays a significant role in thermodynamics and statistical mechanics.

translational kinetic energy

Translational kinetic energy is the energy possessed by a molecule due to its motion through space. For an ideal gas, the average translational kinetic energy per molecule is given by \(\frac{3}{2}k_BT\). This means that the kinetic energy depends directly on the temperature and not on the type of gas.
Higher temperatures mean that the molecules will move more vigorously, increasing their kinetic energy. For all ideal gases at the same temperature, the translational kinetic energy is the same, making it a key aspect of the kinetic theory of gases.

rotational kinetic energy

Rotational kinetic energy is the energy due to the rotation of a molecule about its center of mass. Unlike monoatomic gases, which do not have rotational kinetic energy because they lack rotational degrees of freedom, diatomic and polyatomic gases do have this form of kinetic energy.
For diatomic gases, the rotational kinetic energy at moderate temperatures is given by \(\frac{1}{2}k_BT\) per degree of freedom. Since diatomic molecules have two rotational degrees of freedom, their total average rotational kinetic energy at moderate temperatures is \(\frac{k_BT\). This differentiation helps in understanding the energy distribution within different types of gas molecules.