In this section, we will discover the quantitative link between a gas’s internal energy and its temperature. We start with the equation of state ( ) and the pressure formula ( ), written in their alternative form.
Combining them, we obtain
Recognize that represents the average translational KE of gas particles in the gas. So
Voila. We now have an understanding of temperature at a molecular level. The thermodynamic temperature is directly proportional to the average (translational) kinetic energy of the gas particles!
Remember that the internal energy ? An ideal gas does not have any potential energy. So for an ideal gas, U can be simplified to be
It turns out that the internal energy of an ideal gas depends only on its temperature. Or in mathematical jargon, we say that U is a function of T only.
 Actually, $ is valid only if the gas is ideal (if not PEtotal won’t be zero) and monatomic (if not KEtotal will include some rotational KE in addition to translational KE). If you’re interested, go read section 11.A.