Question

A body cools down from 65°C to 60°C in minutes. It will cool down from 60°C to 55°C in

(a) 5 minutes
(b) less than 5 minutes
(c) more than 5 minutes
(d) less than or more than 5 minutes depending on whether its mass is more than or less than 1 kg.

Answer 1

(c) more than 5 minutes

Let the temperature of the surrounding be T$°\mathrm{C}$.
Average temperature of the liquid in first case = 62.5$°\mathrm{C}$
Average temperature difference from the surroundings = (62.5 $-$ T)$°\mathrm{C}$

From newton law of cooling,

$$\begin{gathered} 1^{°}\mathrm{C}\min {}^{-1}=-bA(62.5-T{)}^{°}\mathrm{C} \\ \Rightarrow -bA\mathit{}=\frac{1}{62.5-T}{\min }^{-1}...(\mathrm{i}) \end{gathered}$$

For the second case,
Average temperature = 57.5$°\mathrm{C}$
Temperature difference from the surroundings = (57.5 $-$ T)$°\mathrm{C}$

From Newton's law of cooling and equation (i),
$$\begin{gathered} \frac{5^{°}\mathrm{C}}{t}=-bA(57.5-T{)}^{°}\mathrm{C} \\ \Rightarrow \frac{5^{°}\mathrm{C}}{t}=\frac{1}{62.5-T}(57.5-T{)}^{°}\mathrm{C} \\ \Rightarrow t\mathit{}=\frac{5(62.5-T)}{(57.5-T)} \\ \Rightarrow t>5\mathrm{minutes} \end{gathered}$$