When two systems can exchange energy, they are in thermal contact.

In thermal contact, energy is exchanged between the two systems until thermal equilibrium is reached---heat flows from the hotter to the colder system.

Thermal equilibrium is defined by T₁ = T₂.

Answer: Heat flow in the "wrong" direction is so unlikely that it is impossible!

The flow of heat from the hotter to a colder system corresponds to evolution to the most probable state of the combined system.

The leads to a new concept: Entropy ≡ S
A measure of the probability for a given microscopic state to occur.
Evolving to most probable state   →   increase in entropy!

Example: melting ice cube (mass m₁) in warm water (mass m₂)
initial: ice (low entropy), warm water (high entropy)
final: tepid water (medium entropy)
(m₁ + m₂)s_tepid > m₁s_ice + m₁s_warm

Statement of the Second Law:

The entropy of an isolated system never decreases:

• in equilibrium, the entropy stays the same;
• otherwise the entropy increases until equilibrium is reached.

The second law stems from the extremely large number of particles in a thermodynamic system---10²³.