Why does a hot cup of coffee cool quickly at first and then more slowly (Newton's law of cooling)?
Newton's law of cooling says the rate of cooling is proportional to the temperature difference from the surroundings: dT/dt = −k(T − Tₛ).
Newton's law of cooling says the rate of cooling is proportional to the temperature difference from the surroundings: dT/dt = −k(T − Tₛ). When the coffee is much hotter than the room, that difference is large, so it loses heat fast; as T approaches room temperature the difference shrinks, so cooling slows to a crawl. Solving this differential equation gives an exponential approach to room temperature — rapid at first, then leveling off.
Key point
dT/dt = −k(T − Tₛ): cooling rate ∝ temperature gap, giving fast-then-slow exponential cooling.
Common mistake
thinking the coffee cools at a constant rate.
Memory tip
big temperature gap → fast cooling; small gap → slow — it's proportional.
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