Temperature

Intuitively, hotter air suggests faster movement of air molecules. In this atom, we will derive an equation relating the temperature of a gas (a macroscopic quantity) to the average kinetic energy of individual molecules (a microscopic quantity). This is a basic and extremely important relationship in the kinetic theory of gases.

Microscopic View

We assume that a molecule is small compared with the separation of molecules in the gas (confined in a three dimensional container), and that its interaction with other molecules can be ignored. Also, we assume elastic collisions when molecules hit the wall of the container, as illustrated in .

Elastic Collisions When Molecules Hit the Wall of the Container

Gas in a box exerts an outward pressure on its walls. A molecule colliding with a rigid wall has the direction of its velocity and momentum in the x-direction reversed. This direction is perpendicular to the wall. The components of its velocity momentum in the y- and z-directions are not changed, which means there is no force parallel to the wall.

We have seen in the Atom on "Origin of Pressure" that, for an ideal gas under our assumptions:

$P = \frac{Nm \overline{v^2}}{3V}$,

where P is the pressure, N is the number of molecules, m is the mass of the molecule, v is the speed of molecules, and V is the volume of the gas. From the equation, we get:

$PV = \frac{1}{3} Nm \overline{v^2}$ (Eq. 1).

What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas. Recall the macroscopic expression of the ideal gas law:

$PV = NkT$ (Eq. 2),

where N is the number of molecules, T is the temperature of the gas, and k is the Boltzmann constant.

Equating the right hand sides of the macroscopic and microscopic versions of the ideal gas law (Eq. 1 & 2) gives:

$\frac{1}{3}m\overline{v^2}=kT$.

Thermal Energy

Note that the average kinetic energy (KE) of a molecule in the gas is:

$\frac{1}{2}m\overline{v^2}$.

Therefore, we derive the relation between average KE and temperature as follows:

$\overline{KE} = \frac{1}{2} m\overline{v^2} = \frac{3}{2} kT$, (Eq. 3).

The average translational kinetic energy of a molecule is called thermal energy.

RMS Speed

Eq. 3 is a molecular interpretation of temperature. It has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy. It is sometimes useful to know the average speed of molecules in a gas in terms of temperature:

$\sqrt{\overline{v^2}} = v_{rms} = \sqrt{\frac{3kT}{m}}$,

where v_rms stands for root-mean-square (rms) speed.