Internal energy has ingrediants of potential energy and kinetic energy.
Potential energy arises due to interaction of atoms in the molecules and also one atom with another atom
Kinetic energy arises due to translatory,rotational and vibrational motion molecules. Mainly, translatory motion of molecules occurs due to thermal energy applied on the system. So, different molecules of gas gets different energies, so that there is random motion of molecules. For that, according to classical mechanics, we can get only average translatory kinetic energy of gas molecules which is given by

There is no change in the internal energy due to exchange of temperature with the system, but to there is change in the internal energy due to exchange of heat, because when heat is the energy quantity.
One of the important point to remember is that when external energy is applied on the system, the internal energy increases due to translational kinetic energy of molecules of the gas.
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