Introduction to Newton's Law of Cooling - Class 11 Physics Notes

Newton’s Law of Cooling is a principle that governs the rate at which an exposed body changes its temperature through radiation. The law is named after Sir Isaac Newton, who first formulated it. It plays a crucial role in understanding heat transfer between a body and its surroundings and has wide applications in areas such as thermodynamics, engineering, and everyday life scenarios like cooling of hot objects.

In this chapter, we will discuss the law in detail, its derivation, mathematical expression, and real-world applications.



What is Newton's Law of Cooling?

Newton's Law of Cooling states that the rate of loss of heat from a body is directly proportional to the temperature difference between the body and its surroundings, provided the temperature difference is small and the mode of heat transfer is primarily through radiation.

Statement of Newton's Law of Cooling

The law can be mathematically stated as:

dT​/dt ∝ (T−Ts​)

where:

- dT​/dt is the rate of change of temperature of the object.

-  T is the temperature of the object at time t.

- Ts is the surrounding temperature (assumed constant).

- (T - Ts) is the temperature difference between the body and its surroundings.

The negative sign indicates that the object is losing heat and hence cooling down, reducing its temperature.

Derivation of Newton's Law of Cooling

Let's derive Newton’s Law of Cooling step by step:

1. Assumption: A hot body is kept in a cooler surroundings, and heat is lost to the surroundings mainly by radiation.

2. Heat Loss Equation: The amount of heat lost by a body in a small time interval ( dt ) can be given by:

dQ = −mcdT

  where:

  - m is the mass of the body.

  - c is the specific heat capacity of the material.

  - dT is the small decrease in temperature in time dt.

  The negative sign represents the loss of heat.

3. According to Newton's Law of Cooling, the rate of heat loss is proportional to the temperature difference between the body and its surroundings:

dQ​/dt ∝ (T−Ts​)

  or,

dQ​/dt = -k (T - Ts)

where k is a constant that depends on the surface area and nature of the surface of the body.

4. Using the relation dQ = mcdT, we substitute into the equation above:

mc dT/dt = -k (T - Ts)

5. Rearranging the equation, we get:

dT/dt = -k/mc(T - Ts)

This is the differential form of Newton's Law of Cooling.

Solution to the Differential Equation

To find the temperature of the body as a function of time, we need to solve this differential equation. The solution is obtained by integrating both sides:

∫ dT/ T−Ts​ = - k/mc ∫dt

On integrating, we get:

ln|T - Ts| = -k/mc.t + C

where C is the constant of integration. Taking the exponential of both sides:

|T - Ts| = e^ −k/mc​.t+C

This simplifies to:

T−Ts= Ae^-k/mc.t

where A = e^C is a constant. At t = 0, the initial temperature of the body is  T0, so:

T0 - Ts = A

Thus, the temperature of the body at any time t is:

T = Ts + (T0 - Ts) e^-k/mc.t

ALSO READ: Ideal-Gas Equation and Absolute Temperature

Factors Affecting Cooling Rate

1. Surface Area of the Object: The larger the surface area, the greater the rate of heat loss due to increased exposure to the surroundings.

2. Nature of the Surface: Dark, rough surfaces radiate more heat compared to shiny, smooth surfaces.

3. Temperature Difference: A larger temperature difference between the object and the surroundings increases the rate of cooling.

4. Specific Heat Capacity: A higher specific heat capacity means the body will require more heat to change its temperature, hence cooling more slowly.

Applications of Newton’s Law of Cooling

1.   Cooling of Hot Beverages: The law explains how a hot coffee or tea cools down over time as the temperature difference with the surroundings decreases.

2.   Forensic Science: The law is used to estimate the time of death by measuring the body temperature and using it to calculate how long the body has been cooling.

3.   Industrial Applications: In manufacturing processes, the law is applied to control the cooling rate of materials like metals to achieve desired mechanical properties.

4.   Climate Control: Understanding heat transfer in buildings helps in designing efficient heating, ventilation, and air conditioning systems.

Limitations of Newton’s Law of Cooling:

Small Temperature Differences: The law is valid only for relatively small temperature differences between the body and the surroundings.

Radiation Dominant: It assumes radiation is the primary mode of heat transfer, but other modes like convection and conduction may also play a role in practical situations.

Constant Surrounding Temperature: The law assumes that the temperature of the surroundings remains constant, which is not always the case in real scenarios.

Conclusion:

Newton’s Law of Cooling provides a fundamental understanding of how bodies lose heat to their environment. It finds applications across multiple disciplines, from everyday phenomena to advanced industrial processes. However, its assumptions limit its applicability, making it more accurate for systems where the temperature difference is small and radiation is the dominant mode of heat transfer.

By understanding this law, students can develop a stronger grasp of thermodynamic concepts and appreciate the interplay between heat, temperature, and environmental conditions.