A liquid takes 5 minutes to cool from 80^∘ C to 50^∘ C. How much time will it take to cool from

step 1 Hello students in this question we have a liquid which take time T1 5 minute from temperature T1

A liquid takes 5 minutes to cool from 80^∘ C to 50^∘ C. How...

A liquid takes 5 minutes to cool from $80^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$. How much time will it take to cool from $60^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$ ? The temperature of surroundings is $20^{\circ} \mathrm{C}$.

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Key Concepts

Differential Equations

The cooling process is modeled using a first-order linear differential equation. This mathematical approach is essential for describing how the temperature changes over time and for solving for variables such as the cooling constant or the time required for a given temperature drop.

Time Constant

The time constant is a measure of the rate at which the temperature difference decreases by a factor of 1/e (approximately 36.8%). It is derived from the cooling constant in the exponential decay model and provides a useful metric for comparing cooling processes across different scenarios.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate at which an object cools (or warms) is proportional to the difference between its temperature and the ambient temperature. This principle leads to the formulation of a differential equation that models the cooling process, which is foundational in understanding heat transfer in a variety of scientific and engineering applications.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of Newton's Law of Cooling, the temperature difference between the object and its surroundings decays exponentially over time, resulting in a predictable cooling curve governed by a decay constant.

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