We've learned about energy conservation in a mechanical system, where the energy in microscopic degrees of freedom was considered to be "lost".

For a thermodynamic system, these microscopic degrees of freedom are included in the energy conservation statement.

Energy Conservation (1st Law of Thermodynamics):   dE = dQ + dW

dE ≡ change in the total energy of the system

E = E_mechanical + E_thermal

• E_mechanical = K + U is associated with macroscopic motion
• E_thermal = energy due to microscopic degrees of freedom. For an ideal gas, E_thermal ∝ T!

dQ ≡ heat added to the system

dW ≡ work done on the system

Heat is a form of energy that can be transferred between a system and its surroundings because of a temperature difference between them. Microscopically, heat is associated with molecular motion.