Question

What equation relates average kinetic energy to temperature in the high-temperature limit?

Answer 1

Hint: The kinetic energy of an item is the energy it has owing to its motion in physics. It is the amount of effort required to propel a body of a given mass from rest to a certain velocity. The body retains its kinetic energy after gaining it during acceleration unless its speed changes. When the body decelerates from its current speed to a condition of rest, it does the same amount of effort.

Complete answer:
We assume that a molecule is tiny in comparison to the distance between molecules in a gas (contained in a three-dimensional container) and that its interactions with other molecules may be ignored. When molecules collide with the container's wall, we assume elastic collisions.
We saw in the Atom on “Origin of Pressure” that, given our assumptions, for a perfect gas.
 $ \text{P}=\dfrac{\text{Nm}\overline{{{\text{v}}^{2}}}}{3\text{V}} $
where P is the pressure, N is the number of molecules, m is the molecule's mass, v is the molecule's speed, and V is the gas's volume. As a result of the equation, we get:
 $ \text{PV}=\dfrac{1}{3}\text{Nm}\overline{{{\text{v}}^{2}}} $
What can we learn about the ideal gas law at the atomic and molecular level? A connection between temperature and the average translational kinetic energy of molecules in a gas may be calculated. Remember the macroscopic form of the ideal gas law: N is the number of molecules, T is the gas's temperature, and k is the Boltzmann constant.
 $ \text{PV}=\text{NkT} $
Equating the right-hand sides of the macroscopic and microscopic versions of the ideal gas law (Eqs. 1 & 2) results in:
 $ \dfrac{1}{3}\text{m}\overline{{{\text{v}}^{2}}}=\text{kT} $
It's worth noting that a molecule's average kinetic energy (KE) in a gas is: $ \dfrac{1}{2}\text{m}\overline{{{\text{v}}^{2}}} $
As a result, the following is the relationship between average KE and temperature:
 $ \overline{\text{KE}}=\dfrac{1}{2}\text{m}\overline{{{\text{v}}^{2}}}=\dfrac{3}{2}\text{kT} $
Thermal energy refers to a molecule's average translational kinetic energy.
In an ideal gas, a particle's average kinetic energy (KE) is given as:
 $ \overline{\text{KE}}=\dfrac{1}{2}\text{m}\overline{{{\text{v}}^{2}}}=\dfrac{3}{2}\text{kT} $

Note:
A gas is made up of a collection of molecules moving at different speeds in a random pattern. When compared to the volume of the container they occupy, individual gas molecules are incredibly tiny. Until they clash with one other or the container's walls, the molecules travel in straight paths. Except for these impacts, the molecules do not interact with one other or the container's walls. Classical physics rules may be used to examine the motion of gas molecules and compute the gas's bulk characteristics.