A bucket full of hot water is kept in a room and it cool from 75^∘ C to 70^∘ C in T1 min, from 7

step 1 As in this question, it says that there is a bucket which is full of hot water, is kept in room

A bucket full of hot water is kept in a room and it cool...

A bucket full of hot water is kept in a room and it cool from $75^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ in $T_{1}$ min, from $70^{\circ} \mathrm{C}$ to $65^{\circ} \mathrm{C}$ in $T_{2}$ $\min$ and from $65^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in $T_{3} \mathrm{~min}$. Then (A) $T_{1}=T_{2}=T_{3}$ (B) $T_{1}<T_{2}<T_{3}$ (C) $T_{1}>T_{2}>T_{3}$ (D) $T_{1}<T_{2}>T_{3}$

A bucket full of hot water is kept in a room and it cool from $75^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ in $T_{1}$ min, from $70^{\circ} \mathrm{C}$ to $65^{\circ} \mathrm{C}$ in $T_{2}$ $\min$ and from $65^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in $T_{3} \mathrm{~min}$. Then
(A) $T_{1}=T_{2}=T_{3}$
(B) $T_{1}<T_{2}<T_{3}$
(C) $T_{1}>T_{2}>T_{3}$
(D) $T_{1}<T_{2}>T_{3}$

Key Concepts

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of a body is proportional to the difference between its temperature and the ambient temperature. This principle forms the basis for modeling the cooling process in many practical situations and leads to a differential equation that describes how temperature changes over time.

Exponential Decay

Exponential decay describes a process in which the rate of decrease of a quantity is proportional to its current value, resulting in a curve that decreases rapidly at first and then more slowly. In cooling problems, this means that as the temperature difference between the object and its environment decreases, the rate of cooling slows down.

Temperature Difference's Role in Cooling

The temperature difference between an object and its surrounding environment is the driving force behind heat transfer. A larger temperature difference results in a faster rate of cooling, while a smaller difference leads to slower cooling. This concept explains why successive equal decrements in the object's temperature may take different amounts of time.

Time Constant

The time constant is a key parameter in the analysis of exponential decay processes. It represents the time required for the system's temperature difference to decrease to 1/e (about 37%) of its initial value. In the context of cooling, the time constant helps quantify how quickly the object approaches thermal equilibrium with its surroundings.

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