Step 1 of 4: The Clausius statement — the law's axiom
Intuition
Hot coffee left in a room always cools; a cup never 'sucks' heat from the air to warm itself — even though that would perfectly conserve energy. The first law permits it; the second forbids it.
Analysis
This is an empirical axiom that classical mechanics cannot derive from below: no one has ever observed heat spontaneously flowing from a cold body to a hot one. An equivalent formulation (Kelvin–Planck): no heat engine can convert all absorbed heat into work. The two statements can be proven equivalent: violating either lets you build a device violating the other.