Newton's Law Of Cooling

Question

A hot liquid kept in a room cools from $80^\circ	ext{C}$ to $70^\circ	ext{C}$ in 5 minutes while the surroundings stay at $30^\circ	ext{C}$. According to Newton's law of cooling, which temperature interval would take the longest time to traverse for the same liquid in the same room?

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Newton's law of cooling states that the rate at which a body loses heat is proportional to the excess of its temperature over the surroundings, $\frac{dT}{dt} = -k(T - T_s)$. The larger this excess temperature, the faster the body cools; as the body approaches the surroundings the cooling rate slows markedly. For each successive 10-degree drop the average excess over the $30^\circ	ext{C}$ room becomes smaller, so the cooling slows and each step takes longer than the previous one. Among the listed intervals, $50^\circ	ext{C}$ to $40^\circ	ext{C}$ has the smallest average excess temperature, roughly 15 degrees, so it is traversed most slowly and takes the longest time. The interval $80^\circ	ext{C}$ to $70^\circ	ext{C}$ has the largest excess and is actually the quickest, which is why the question's given 5 minutes is the shortest of the four. The intermediate intervals fall in between. This monotonic slowing is the central NCERT consequence of Newton's law of cooling. As a plausibility check, everyday experience confirms that tea cools rapidly at first and then lingers near room temperature, matching the chosen slowest final interval.

Newton's Law Of Cooling

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