Animated Solution for Physics - Properties of Solids and Liquids: If a piece of metal is heated to temperature $\theta$ and then allowed to cool in a room which is at temperature ${\theta }_0$. The graph between the temperature $T$ of the metal and time $t$ will be closed to

Imagine you've just poured yourself a steaming cup of tea, or perhaps you've taken a red-hot piece of metal out of a furnace and placed it on a table. What happens next is something we all intuitively know: it cools down. But how exactly does it cool down? Does it cool at a constant rate, or does the rate change over time?

To answer this, we turn to a principle formulated by Sir Isaac Newton, aptly named Newton's Law of Cooling.

Newton observed that the rate at which an object loses heat is directly proportional to the temperature difference between the object and its surroundings. Mathematically, we can express the rate of change of temperature as a differential equation:

Here, $T$ is the temperature of the object at any time $t$, ${\theta }_0$ is the constant temperature of the surroundings (the room), and $k$ is a positive constant that depends on the physical properties of the object, like its surface area and material. The negative sign is crucial—it indicates that the temperature is decreasing over time.

To find out how the temperature $T$ behaves as a function of time $t$, we need to solve this differential equation. We start by separating the variables, bringing all the $T$ terms to one side and the $t$ terms to the other:

Next, we integrate both sides:

To find the constant of integration $C$, we use our initial conditions. We know that at the very beginning, when $t=0$, the temperature of the metal is its initial hot temperature, $T=\theta$. Substituting these into our equation gives:

Now, we substitute $C$ back into our integrated equation:

Using the properties of logarithms, we can rearrange this to:

Finally, taking the exponential of both sides yields the master equation for the temperature as a function of time:

This final equation is a classic exponential decay function. Let's break down what it tells us visually:

1. At $t=0$: The term $e^0=1$, so $T={\theta }_0+(\theta -{\theta }_0)=\theta$. The graph starts exactly at the initial high temperature. 2. The Shape: Because of the $e^{-kt}$ term, the temperature drops rapidly at first (when the temperature difference is large) and then cools more slowly as it gets closer to the room temperature. 3. As $t\to \infty$: The term $e^{-kt}$ approaches $0$. Therefore, $T$ approaches ${\theta }_0$.

This means the graph will be a curve that starts at $\theta$ and asymptotically approaches the horizontal line $T={\theta }_0$. It will never cross this line, nor will it abruptly become a straight horizontal line. It smoothly glides towards the room temperature, making the exponential decay curve the only physically and mathematically correct representation.