Newton’s Law of Cooling – Explanation, Formula, and Applications

Published on: October 15, 2025in Thermodynamics
Newton’s Law of Cooling describes how the temperature of a hot object decreases over time when placed in a cooler surrounding. This principle is widely used in thermodynamics, meteorology, and engineering.
Newton’s Law of Cooling

Newton’s Law of Cooling describes how the temperature of a hot object decreases over time when placed in a cooler surrounding. Hence, this principle is widely used in thermodynamics, meteorology, and engineering.

Statement of Newton’s Law of Cooling

So, Newton’s Law of Cooling states that:

“The rate of cooling of a body is directly proportional to the temperature difference between the body and its surroundings, provided the difference is small.”

Key Points:

  1. Applicable under Natural Convection (no forced cooling like fans).
  2. Furthermore, it works best for small temperature differences (T – Tₛ < ~30°C).
  3. Additionally, it assumes uniform cooling conditions.

Mathematical Formula

The rate of cooling is given by:

    \[\frac{dT}{dt} = -k (T - T_s)\]

where:

  • (\frac{dT}{dt}) = Rate of cooling (temperature drop per unit time).
  • (T) = Temperature of the body at time (t).
  • (T_s) = Temperature of the surroundings.
  • (k) = Cooling constant (depends on surface area, material, etc.).

Integrated Form (For Temperature at Any Time)

If a body cools from (T_0) to (T) in time (t), then:

    \[T = T_s + (T_0 - T_s) e^{-kt}\]

Thus, this shows exponential decay of temperature with time.

Newton's Law of Cooling
Fig. 1: Newton’s Law of Cooling

Derivation Of Newton’s Law of Cooling

Additionally, from the differential form:

    \[\frac{dT}{T - T_s} = -k \, dt\]

Integrating both sides:

    \[\ln (T - T_s) = -kt + C\]

At (t = 0), (T = T_0):

    \[\ln (T_0 - T_s) = C\]

Substituting back:

    \[\ln \left( \frac{T - T_s}{T_0 - T_s} \right) = -kt\]

Taking exponential on both sides:

    \[T - T_s = (T_0 - T_s) e^{-kt}\]

Thus, we get:

    \[T = T_s + (T_0 - T_s) e^{-kt}\]

Applications of Newton’s Law of Cooling

  1. Predicting Cooling Rates – Additionally, it is used in industries to estimate the cooling of hot metals.
  2. Forensic Science – Further, it helps estimate the time of death by measuring body temperature.
  3. Meteorology – Hence, it predicts the cooling of land and water bodies.
  4. Domestic Uses – Furthermore, it explains why coffee cools faster initially.
Applications of Newton's Law
Fig. 2: Applications of Newton’s Law

Limitations

  1. Firstly, Newton’s Law of Cooling is only valid for small temperature differences.
  2. So, it assumes a constant surrounding temperature.
  3. Furthermore, it neglects the effects of radiation at high temperatures.

FAQ’s

Q1: What does Newton’s Law of Cooling state?

Ans: Further, it states that the rate of cooling of a body is proportional to the temperature difference between the body and its surroundings.

Q2: Why is the negative sign used in the formula?

Ans: So, the negative sign indicates that temperature decreases with time (cooling).

Q3: Can this law be applied to all cooling situations?

Ans: No, it applies only to natural convection with small temperature differences.

Q4: How does surface area affect cooling rate?

Ans: Furthermore, a larger surface area increases the cooling rate (higher (k)).

Q5: What is the cooling constant (k) dependent on?

Ans: Also, it depends on surface area, nature of the material, and surrounding conditions.

Conclusion

Furthermore, Newton’s Law of Cooling is a simple yet powerful tool to understand how objects lose heat. Additionally, it helps in analyzing real-world cooling phenomena with reasonable accuracy under normal conditions.

References

  1. Santos, A. (2025). Time-delayed Newton’s law of cooling with a finite-rate thermal quench. Impact on the Mpemba and Kovacs effects. https://doi.org/10.48550/arXiv.2502.13665
  2. Bas, E., Ozarslan, R., & Ercan, A. (2018). A new perspective on Newton’s law of cooling in frame of newly defined fractional conformable derivative.
    https://doi.org/10.48550/arXiv.1811.03696

Additionally, to stay updated with the latest developments in STEM research, visit ENTECH Online. Basically, this is our digital magazine for science, technology, engineering, and mathematics. Furthermore, at ENTECH Online, you’ll find a wealth of information.

Subscribe

Top Posts

Living Cell Secrets

New Microscope Reveals Tiny and Large Structures Inside Living Cells Without Dyes

Small Cars for City Commutes

How to Choose an Affordable Car for City Commutes

finance exchange cloud

The Engine of Modern Finance: Understanding Exchange Clouds

Warning