Newton’s Law of Cooling: Formula, Derivation and Limitations

# Newton’s Law of Cooling: Formula, Derivation and Limitations

Newton’s Law of Cooling is a fundamental principle in physics that describes the cooling of an object due to the temperature difference between the object and its surroundings. It was formulated by Sir Isaac Newton in the 17th century and is commonly applied to various cooling and heating processes.

The law states that the rate of change of the temperature of an object is directly proportional to the temperature difference between the object and its surroundings.

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Newton’s Law of cooling is given by

dT / dt = k(Tt – Ts)

Where,

T is the time
Tt is temperature of given body at time t
Ts is the surrounding temperature
To is initial temperature of the body
K is the constant

### Newton’s Law of Cooling Formula

The greater the difference in temperature between an object and its surroundings, the faster heat is transferred, causing the object’s temperature to change more quickly.

The formula for Newton’s law of cooling is expressed by,

T(t) = Ts + (To – Ts)e-kt

Where,

t is the time
T (t) is the temperature of the given body at that time
Ts is the surrounding temperature
To is the initial temperature of the body
K is the constant

Newton’s Law of Cooling Derivation

When there’s only a slight temperature difference between an object and its surroundings, the rate at which the object cools down is directly related to both the temperature difference and the amount of its surface area that is exposed.

dQ / dt ∝ (q – qs)

Where, q and qs are temperatures of the object and the surrounding.

From the above expression, dQ / dt = – k (q – qs)

The above expression represents Newton’s law of cooling.

The derivation of Newton’s law of cooling can be done directly from Stefan’s law, given by,

### Limitations of Newton’s Law of Cooling

Newton’s Law of Cooling, while a valuable approximation in certain situations, has its limitations and may not fully capture the complexities of all cooling processes. Some of its limitations include:

• Linear approximation: Newton’s Law assumes a linear relationship between the rate of cooling and the temperature difference. In reality, the cooling process might not always follow a linear pattern, especially if the temperature difference is significant.
• Constant cooling constant: The law assumes a constant cooling constant (k) throughout the cooling process. In many cases, the cooling constant might vary with temperature or other environmental factors, making the law less accurate for extended periods or varying conditions.
• Neglecting internal heat sources: The law assumes that the only heat exchange occurs between the object and its surroundings. However, in some situations, there might be internal heat sources (e.g., chemical reactions or electrical processes) that are not accounted for.
• Limited to conduction and convection: Newton’s Law is primarily applicable to heat transfer through conduction and convection. It may not fully account for radiative heat transfer, which can be significant in certain scenarios.
• Homogeneous medium assumption: The law assumes a homogeneous and isotropic medium for heat transfer. In real-world scenarios, the surrounding medium might not be uniform, which could affect the accuracy of predictions.
• Initial temperature assumption: The law assumes that the initial temperature of the object is significantly higher or lower than the surroundings. If the initial temperature difference is small, the law’s applicability may be limited.

Despite these limitations, Newton’s Law of Cooling remains a valuable and simple tool for estimating cooling behavior in many practical situations. For more complex scenarios, more sophisticated models and equations are used to account for the specific factors influencing the cooling process.

Also Check : Law of Motion

## FAQs on Newton’s law of cooling

### What is Newton's law of cooling by convection short note?

Newton's Law of Cooling by convection states that the rate of cooling of an object is directly proportional to the temperature difference between the object and its surroundings when cooling occurs through the motion of a fluid (e.g., air or water).

### Why is Newton's law of cooling important?

Newton's Law of Cooling is important because it provides a simple and practical way to estimate how an object's temperature changes when it cools down through conduction or convection. It is widely used in various fields, including engineering, meteorology, and environmental sciences.

### What are the limitations of Newton's law of cooling Class 11?

The limitations of Newton's law of cooling include assuming a linear relationship, constant cooling rate, neglecting internal heat sources, and applicability primarily to conduction and convection, among others.

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