Q.A cake is removed from an oven at 250°F and left to cool at room temperature which is 70°F. After 30 minutes the temperature of the cake is 150°F. After how much time will it be 100°F?

Newton's Law of Cooling

The Intuition

Imagine you pour a cup of hot coffee and leave it on the table. You know it will cool down. But how does it cool? Does it drop from 90°C to 80°C in the same time it drops from 50°C to 40°C?

Your experience says no. The hotter the coffee is relative to the room, the faster it loses heat. When it's just a few degrees above room temperature, it barely cools at all. That's the core idea: the rate of cooling depends on how much hotter the object is than its surroundings.

The Precise Statement

Newton's Law of Cooling states:

The rate of heat loss from a body is directly proportional to the temperature difference between the body and its surroundings, provided the temperature difference is small and the surroundings are at a constant temperature.

In mathematical form:

where $T$ is the temperature of the body, $T_s$ is the temperature of the surroundings (assumed constant), and $\frac{dQ}{dt}$ is the rate of heat loss.

Since heat loss is related to temperature change ($dQ = m c \, dT$, where $m$ is mass and $c$ is specific heat capacity), we can rewrite it as:

The negative sign is crucial — it tells us the temperature is decreasing when $T > T_s$. The constant $k$ depends on the surface area, nature of the surface, and the specific heat of the object.

What This Equation Tells You

The equation $\frac{dT}{dt} = -k (T - T_s)$ is a differential equation. It says: the instantaneous rate at which temperature changes is proportional to how far the body is from the surrounding temperature.

If you solve it (which you'll learn to do), you get:

where $T_0$ is the initial temperature of the body. This is an exponential decay curve — the temperature difference shrinks by a constant fraction per unit time.

Key Points for Exams

1. Small temperature difference — the law works well only when $T - T_s$ is not too large (say, less than 30–40°C). For very hot objects, radiation dominates and the law breaks down.

2. Constant surroundings — the room temperature $T_s$ must stay fixed. If the surroundings also change temperature, the law doesn't apply directly.

3. The constant $k$ — it's not universal. It depends on the object's surface area, emissivity, and the specific heat of the material. Two objects of the same material but different shapes will have different $k$ values.

4. Cooling vs. heating — the same law works if the body is colder than the surroundings (then $T - T_s$ is negative, and $\frac{dT}{dt}$ becomes positive — the body warms up). …