Consider a house filled with air of a monatomic ideal gas, shared with the
outside atmosphere, so that PV = NkT can model the house air. The gas
energy is E = (3/2)NkT.
If I turn on a furnace I increase the temperature T of the house air.
Since P and V do not change values, N must decrease in value. Air
molecules have left the house with ALL of the added thermal energy. The
house air energy has not changed. The energy per molecule has increased,
but N has decreased. We have heated the outside atmosphere! (Of course we
have also heated the furniture.)