## Newton Law of Cooling- Class 11 | Chapter – 11 | Physics Short Notes Series PDF for NEET & JEE

Newton Law of Cooling: Newton law of cooling is a fundamental principle in physics that describes how the rate of cooling of an object is related to the temperature difference between the object and its surroundings. The law states that the rate of heat loss of an object is proportional to the temperature difference between the object and its surroundings.

More formally, Newton’s law of cooling can be expressed mathematically as:

dQ/dt = -hA(T – Ts)

Where:

• dQ/dt is the rate of heat loss of the object over time
• h is the heat transfer coefficient, which depends on the properties of the object and the surrounding medium
• A is the surface area of the object
• T is the temperature of the object
• Ts is the temperature of the surroundings

The negative sign in the equation indicates that heat is transferred from the object to its surroundings, and the proportionality constant h represents the rate at which this heat transfer occurs. The larger the value of h, the more rapidly the object will cool.

Newton law of cooling applies to a wide range of physical systems, from simple objects like cups of coffee to more complex systems like refrigeration units and industrial cooling systems. It is an important tool for understanding the behavior of thermal systems and for designing efficient cooling strategies.

## Formula of Newton Law of Cooling

Newton law of cooling relates the rate of cooling of an object to the temperature difference between the object and its surroundings. It is given by the following formula:

dT/dt = -k(T – Ts)

where:

• dT/dt is the rate of change of temperature of the object over time
• k is the cooling constant, which depends on the properties of the object and the surrounding medium
• T is the temperature of the object
• Ts is the temperature of the surroundings

The negative sign in the formula indicates that the rate of temperature change is negative, meaning the temperature of the object is decreasing with time. The constant k represents the rate at which this cooling occurs, and is determined by the properties of the object and its surroundings. The larger the value of k, the more rapidly the object will cool.

It is important to note that Newton law of cooling is an empirical law that applies to objects that are cooling in still air or a stagnant medium. For objects that are cooling in moving air or other flowing media, the cooling rate may be different and can be affected by a variety of factors, such as the speed and direction of the flow, the properties of the fluid, and the geometry of the object.

## Derivation of Newton Law of Cooling

Newton’s law of cooling is an empirical law that relates the rate of cooling of an object to the temperature difference between the object and its surroundings. The law can be derived using basic principles of thermodynamics and heat transfer. Consider an object of mass m and specific heat capacity c, initially at temperature T0, that is placed in a medium at temperature Ts. The object will lose heat to the surroundings, causing its temperature to decrease with time.

The rate of heat transfer from the object to the surroundings is given by:

dQ/dt = -hA(T – Ts)

where dQ/dt is the rate of heat transfer, h is the heat transfer coefficient, A is the surface area of the object, T is the temperature of the object, and Ts is the temperature of the surroundings.

Using the definition of specific heat capacity, we can express the change in temperature of the object as:

dT/dt = – (1 / (mc)) dQ/dt

Substituting the expression for dQ/dt from above, we get:

dT/dt = – (1 / (mc)) (-hA(T – Ts))

Simplifying the expression, we get:

dT/dt = k(Ts – T)

where k = hA / (mc) is the cooling constant.

The resulting differential equation is a first-order linear ordinary differential equation, which can be solved using standard techniques. The solution is:

T(t) = Ts + (T0 – Ts) e^(-kt)

This equation describes the temperature of the object as a function of time, and it shows that the temperature of the object will exponentially approach the temperature of the surroundings as time goes on.

This is the basic derivation of Newton’s law of cooling. However, it should be noted that the law is an empirical law that does not take into account many of the complex factors that can affect the rate of cooling of an object in a real-world setting. Nonetheless, the law is useful for understanding the basic principles of heat transfer and for making simple predictions about the behavior of cooling systems.

## Limitation of Newton Law of Cooling

Newton law of cooling is an empirical law that provides a simple model for predicting the cooling rate of an object in still air or a stagnant medium. However, the law has some limitations that make it less accurate for predicting the behavior of cooling systems in certain situations. Some of the main limitations of Newton’s law of cooling include:

• Assumption of constant cooling constant: Newton law of cooling assumes that the cooling constant k is constant throughout the cooling process. In reality, the cooling constant can change as the temperature of the object changes, or as the conditions of the surrounding medium change. For example, if the temperature difference between the object and the surrounding medium is very large, the heat transfer coefficient may decrease due to changes in the fluid properties, such as viscosity or thermal conductivity.
• Neglects heat conduction within the object: Newton law of cooling assumes that the heat transfer occurs only at the surface of the object, and neglects the effects of heat conduction within the object. In reality, heat can be conducted through the object, which can affect the cooling rate and cause temperature gradients within the object.
• Neglects radiative heat transfer: Newton law of cooling assumes that the only heat transfer mechanism is convection, which is valid only for objects that are large compared to the wavelength of radiation. For smaller objects or at high temperatures, radiative heat transfer can become important, and must be taken into account.
• Neglects fluid flow effects: Newton’s law of cooling assumes that the fluid surrounding the object is stationary, which is not always the case in practice. In situations where the fluid is moving or flowing, the cooling rate can be affected by factors such as turbulence, flow direction, and flow velocity.

Overall, while Newton’s law of cooling is a useful tool for making simple predictions about the behavior of cooling systems, it is important to recognize its limitations and to use more advanced models when greater accuracy is required.

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