Question

A beaker full of hot water is kept in a room if it cools from $${80}^{\circ }\text{C}$$ to $${75}^{\circ }\text{C}$$ in $$t_1$$ minutes, from $${75}^{\circ }\text{C}$$ to $${70}^{\circ }\text{C}$$ in $$t_2$$ minutes and from $${70}^{\circ }\text{C}$$ to $${65}^{\circ }\text{C}$$ in $$t_3$$ minutes, then
A. $$t_1<t_2<t_3$$
B. $$t_1>t_2>t_3$$
C. $$t_1=t_2=t_3$$
D. $$t_1<t_2=t_3$$

Answer 1

Hint:Use the expression for Newton’s law of cooling. This expression gives the relation between the rate at which heat is exchanged by the body with the surrounding, temperature of the body and temperature of the surrounding. Consider a temperature fixed as temperature of the surrounding and check the difference between the water and the surrounding for three cases and hence, determine the relation between three values of times.

Formula used:
The expression for Newton’s law of cooling is given by
$$Q=hA\left(T-T_{surr}\right)$$ …… (1)
Here, $$Q$$ is the rate at which the heat is transferred to the surrounding, $$h$$ is the heat transfer coefficient, $$A$$ is the heat transfer surface area, $$T$$ is the temperature of the object at time $$t$$ and $$T_{surr}$$ is the temperature of the surrounding.

Complete answer:
We have given that the beaker full of hot water is kept in a room and the hot water is cooling by giving its temperature to the surrounding. The temperature of the hot water is decreasing from $${80}^{\circ }\text{C}$$ to $${75}^{\circ }\text{C}$$ in $$t_1$$ minutes, from $${75}^{\circ }\text{C}$$ to $${70}^{\circ }\text{C}$$ in $$t_2$$ minutes and from $${70}^{\circ }\text{C}$$ to $${65}^{\circ }\text{C}$$ in $$t_3$$ minutes.

We are asked to determine the relation between the times in which the hot water cools.From equation (1), we can conclude that the rate at which the heat is transferred to the surrounding is directly proportional to the temperature difference between the object and the surrounding.Hence, the time required for cooling of the hot water increases as the temperature difference decreases.

If we consider the temperature of the surrounding to a fixed value which is less than the initial temperature of the hot water which is $${80}^{\circ }\text{C}$$ then the temperature difference between the surrounding and the hot water decreases as the hot water cools.Since the temperature difference between the hot water and the surrounding decreases as the hot water cools, the timer required to cool the hot water increases.Therefore, the relation between the times $$t_1$$, $$t_2$$ and $$t_3$$ is $$t_1<t_2<t_3$$.

Hence, the correct option is A.

Note:The students may get confused and take the temperature difference between the two values of the temperatures of the hot water and find that the temperature difference between the hot water is the same for three times. But the students should keep in mind that the time required for cooling is inversely proportional to the temperature difference between the hot water and the surrounding.