In an experiment to verify Newton's law of cooling, a graph is plotted between the temperature difference $(\Delta T)$ of the water and surroundings and time as shown in figure. The initial temperature of water is taken as $80^\circ\text{C}$. The value of $t_2$ as mentioned in the graph will be ______.
Numerical value type. Enter your answer.
Answer: 16
From the graph, $\Delta T$ falls from $60^\circ\text{C}$ to $40^\circ\text{C}$ in the first 6 minutes, and from $40^\circ\text{C}$ to $20^\circ\text{C}$ between $t = 6$ and $t = t_2$.
Using Newton's law of cooling in the average form, rate $= k\times$ (average temperature difference):
$$\frac{60-40}{6} = k\times50\ \Rightarrow\ k = \frac1{15}\ \text{min}^{-1}$$
$$\frac{40-20}{t_2 - 6} = \frac1{15}\times30 = 2\ \Rightarrow\ t_2 - 6 = 10\ \Rightarrow\ t_2 = 16\ \text{min}$$
(The exact exponential form gives $t_2 = 6 + 6\dfrac{\ln2}{\ln1.5}\approx16.3$ min, i.e. $16$ to the nearest integer.)
Solution by Sreeraj P, M.Sc Physics