Question

A hot liquid is kept in a big room. The rate of fall of temperature of the liquid is plotted against time. The plot will be very nearly,

• A. a straight line
• B. a circular arc
• C. a parabola
• D. an ellipse

Answer 1

The correct option is A a straight line

When a hot liquid is kept in a big room, the liquid will begin to lose heat. Since the room is big, we can assume that the energy dissipated by the liquid will not change the temperature of the surroundings.
According to Newton's law of cooling, the rate of fall of temperature (i.e. rate of cooling) of the hot liquid is directly proportional to the temperature difference between the liquid and the surroundings of the room. If $T$ is the temperature of the hot liquid and $T_s$ is the temperature of the surroundings of the room, then according to Newton's cooling law,

$\frac{dT}{dt}\propto (T-T_s)$

$\frac{dT}{dt}=-k(T-T_s)$ (here, k is a constant)

The negative sign indicated that as $t$ increases, $\frac{dT}{dt}$ decreases.

So, the graph between $\frac{dT}{dt}$ and time will be a straight line, as shown below.


Hence, $\left(A\right)$ is the correct answer.

$$\begin{array}{c}\text{Why this question?}\\ \text{Caution: Newton's law of cooling is only applicable to small temperature differences. If the room temperature would have been small, the temperature difference between room and liquid would have been large and hence, Newton's law would not be applicable. In this case, we should use, stefan's law.}\end{array}$$