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

The process of a hot object cooling down in a cooler environment is something we experience every day, from a cup of coffee to a car engine. But how exactly does this temperature drop over time? Is it a steady decline, or does it slow down? Let's dive into the physics and mathematics behind this everyday phenomenon.

Imagine you have a piece of metal heated to a high temperature $T=\theta$. You place it in a room that is maintained at a constant, cooler temperature ${\theta }_0$. Because heat always flows from a hotter body to a colder one, the metal will start losing heat to the room.

Intuitively, when the metal is very hot, the temperature difference between it and the room is large, so it loses heat rapidly. As it cools down and its temperature gets closer to the room temperature, the "driving force" for the heat transfer decreases, and the cooling process slows down. This means the temperature-time graph cannot be a straight line; it must be a curve whose slope decreases over time.

Newton's Law of Cooling

To quantify this, we use Newton's Law of Cooling. It states that the rate of heat loss from a body is directly proportional to the temperature difference between the body and its surroundings.

Mathematically, we write: Here, $k$ is a positive constant depending on the area and nature of the surface, and the negative sign indicates that heat is being lost.

We also know from calorimetry that the heat $dQ$ lost by a body of mass $m$ and specific heat capacity $s$ results in a temperature drop $dT$:

Substituting this into our rate equation gives:

To find how temperature $T$ depends on time $t$, we need to solve this differential equation. Let's rearrange the terms to separate the variables:

To simplify, let's define a new constant $K=\frac{k}{ms}$. Now we integrate both sides. At $t=0$, the temperature is $\theta$, and at time $t$, the temperature is $T$:

Evaluating the integrals, we get:

Taking the exponential of both sides to solve for $T$:

This final equation is the mathematical signature of exponential decay. Let's analyze its key features to identify the correct graph: 1. Initial State: At $t=0$, $e^0=1$, so $T={\theta }_0+\theta -{\theta }_0=\theta$. The graph must start at $T=\theta$ on the y-axis. 2. Final State: As $t\to \infty$, $e^{-Kt}\to 0$, so $T\to {\theta }_0$. The graph must asymptotically approach the horizontal line $T={\theta }_0$. 3. Curvature: The rate of cooling $\frac{dT}{dt}$ is negative but its magnitude decreases over time. This means the slope becomes less steep, resulting in a curve that is concave upwards.

Looking at the given options, Graph (c) perfectly matches all these criteria. It starts at $\theta$, decays exponentially, and asymptotically approaches ${\theta }_0$ while remaining concave upwards.

It is important to remember that Newton's Law of Cooling is an approximation. It is highly accurate when the temperature difference $(T-{\theta }_0)$ is relatively small. However, if the metal was glowing white-hot, heat loss via radiation would dominate, governed by the Stefan-Boltzmann Law ($\frac{dQ}{dt}\propto T^4-{\theta }_0^4$). In such extreme cases, the initial cooling would be much more drastic than a simple exponential curve!