Newton’s Law of Cooling

When hot water in a vessel is kept on a table, it begins to cool gradually. To study how a given body cools on exchanging heat with its surroundings, the following experiment is performed.

Setup:

• A calorimeter is filled up to two-thirds of its capacity with boiling water and is covered.
• A thermometer is fixed through a hole in the lid. Its position is adjusted so that the bulb of the thermometer is fully immersed in the water.
• The calorimeter vessel is kept in a constant temperature enclosure or in open air, since room temperature will not change much during the experiment.​

Procedure

1. Note the temperature on the thermometer at one-minute intervals.
2. Continue until the temperature of the water decreases by about 25°C.

Observation

• Initially, the rate of cooling is higher.
• The rate of cooling decreases as the temperature of the water falls — the water cools more slowly as it approaches room temperature.​

A graph of temperature T (along the y-axis) is plotted against time t (along the x-axis). This graph is called the Cooling Curve.

Fig 7.13 (a) — Cooling Curve: T vs t

• The curve shows that cooling is rapid initially and slows down over time.
• A tangent is drawn to the curve at suitable points.
• The slope of each tangent gives the rate of fall of temperature (dT/dt) at that temperature.​

Slope at point A = \[\lim_{\Delta t\to0}\frac{\Delta T}{\Delta t}=\frac{dT}{dt}\]

Fig. 7.13 (b) — Rate of Cooling vs Temperature Difference: dT/dt vs (T − T₀)

• Taking (0, 0) as the origin, a graph of dT/dt is plotted against the corresponding temperature difference (T−T₀).
• The curve is a straight line, confirming direct proportionality between the rate of cooling and the temperature difference.

Statement: The rate of loss of heat \[\frac {dT}{dt}\] of the body is directly proportional to the difference of temperature (T − T₀) of the body and the surroundings, provided the difference in temperatures is small.

Mathematical Form:

Graphical Representation:

• Graph of rate of cooling \[\left(\frac{dT}{dt}\right)\] vs (T − T₀) → straight line through origin.

• Graph of Temperature T vs time t → exponential decay curve (temperature drops steeply at first, then gradually).

Proportionality Form

\[\frac{dT}{dt}\propto(T-T_0)\]

Introducing the constant of proportionality C:

\[\frac{dT}{dt}=C\left(T-T_0\right)\]

T = Temperature of the body at time t
T₀ = Temperature of the surroundings (constant)
C = Constant of proportionality
\[\frac {dT}{dt}\] = Rate of fall of temperature (rate of cooling)

Problem: A metal sphere cools at the rate of 1.6°C/min when its temperature is 70°C. At what rate will it cool when its temperature is 40°C? The temperature of the surroundings is 30°C.

Given

Solution

Step 1 — Apply Newton's Law at temperature T₁ to find C:

\[\left(\frac{dT}{dt}\right)_1=C\left(T_1-T_0\right)\]

1.6 = C(70 − 30)

1.6 = 40 C ⇒ C = \[\frac {1.6}{40}\] = 0.04 min⁻¹ 

Step 2 — Apply Newton's Law at temperature T₂:

\[\left(\frac{dT}{dt}\right)_2=C\left(T_2-T_0\right)=0.04\times(40-30)=0.04\times10\]

\[{\left(\frac{dT}{dt}\right)_2=0.4°\mathrm{C/min}}\]

Result: The rate of cooling drops by a factor of four when the difference in temperature of the metal sphere and its surroundings drops by a factor of four – directly confirming Newton's Law of Cooling.

• A hot body loses heat to its surroundings in the form of heat radiation.​
• The rate of cooling is directly proportional to the temperature difference between the body and its surroundings.
• The cooling curve (T vs t) shows rapid initial cooling that gradually slows down.
• Plotting \[\frac {dT}{dt}\] vs (T−T₀) gives a straight line through the origin, confirming Newton's law.​
• Mathematically: dT/dt = C(T − T₀), where C is the constant of proportionality.​
• The rate of cooling is proportional to — not independent of — the temperature difference. A 4× drop in temperature difference produces a 4× drop in cooling rate.