Newton’s Law of Cooling

Let us leave a cup of hot water on the table. After some time, the water will start to cool, and finally, its temperature will reach the temperature of the surroundings. We all know that temperature is a measure of the degree of hotness or coldness of a body. Temperature is a sensory perception and can be felt by touch. Taking the example cited here with the hot water, we understand that there is heat transfer from the hot water to the surrounding environment or, in other words, energy transfer in the form of heat has taken place from one system to another due to temperature difference between the two systems.

To understand the main concept of Newton’s law of cooling, some basic concepts must be clear. These are: 

• The First Law of Thermodynamics: It is commonly known as the law of energy conservation. 

Consider an example of water stored at some height above ground level; it has some potential energy. When this water is allowed to fall on a turbine, the energy stored gets converted into electrical energy; So by this, we can observe that energy can be transformed or transferred from one system to another but can never be created or destroyed.

This law can also be explained in terms of internal energy, heat energy, and work done as:

Q = U + W

Where Q = total change in heat energy of the system

U = change in internal energy of the system

W = work done by the system

According to the equation, change in heat energy solely depends on the work done by the system and the change in internal energy of the system.

• The Second Law of Thermodynamics: Second law of thermodynamics states that the entropy of an isolated system always increases. 

It can never decrease, and as our whole universe is an isolated system, its entropy always increases (given that the particular process is not isentropic). According to this law, heat is always transferred from a hot system to a cold one. For example, consider hot soup that you want to cool down. You will add some ice to it. Now the soup is not getting cooled down due to the ice; however, the hotness of the soup is causing the ice to melt and then escape into the environment.

There are three methods by which heat transfer can occur in a system:

• Conduction: occurs when there is direct contact between the systems. For example, when you touch a hot stove, the heat is transferred to your body.
• Convection: occurs when there is a bulk movement of particles of a system. For example, a fan in the summer makes us feel refreshed.
• Radiation: refers to the transfer of heat between two systems without the involvement of any medium.

Newton’s law of cooling formula

Newton’s law of cooling applies to convective heat transfer; it does not imply thermal radiation. It states that the heat exchange rate between a system and its surroundings is directly proportional to the difference in temperature between the system and its surroundings. It can be written as

Rate of cooling ∝ ΔT 

or

– dT/dt = k (T – Ts) 

We put a negative sign here because the temperature of the body is decreasing in this case, where we have considered the temperature of surroundings to be lower than that of the body.

The Newton’s law of cooling formula can be expressed as:

T(t) = Ts + ( To – Ts ) e-kt 

Newton’s law of cooling derivation

dQ/dt ∝ ( T – Ts)

Where T and Ts are the temperatures of the object and surroundings, respectively.

dQ/dt = – k ms [ T – Ts ]

Here again, a negative sign is used because the system loses energy to the surroundings in our example.

This can be directly derived from Stefan’s law,

For a small change in temperature, Newton’s law of cooling is given by 

– dQ/dt = k ms (T2 – T1) ….. (1)

Where k is a constant, m is the mass of the body, and s is its specific heat capacity.

If the temperature falls by a small amount dT2 in time dt, then the heat loss is given by,

dQ = ms dT2

The rate of heat loss is given by 

 dQ/dt = ms (dT2/dt) …… (2)

Comparing equations (1) and (2), we get 

– ms (dT2/dt) = k ms ( T2 – T1)

after rearranging we get,

dT2/(T2 – T1 )= – kdt

Integrating the above equations as

loge ( T2 – T1 ) = – k t + c

or T2 = T1 + C’ e-Kt

where C’ = ec (Constant)

The expression above is used to calculate the cooling time of a body at a particular temperature.

Conclusion

In this Newton’s law of cooling study material, we learnt that heat is a form of energy that gets transferred from one body to another or from a body to the surrounding medium. This transfer occurs due to the temperature differences between the bodies or between the body and the medium. The degree of hotness of a body is known as its temperature. Sir Issac Newton derived the fact that the rate of cooling of any body/object is directly proportional to the excess temperature of the body/object over its surroundings. This law was subsequently known as Newton’s Law of Cooling. This is depicted by the formula

dQ/dT = -k( T2-T1)

Where k is a constant

T1 is the temperature of the medium surrounding the body/object

T2 is the temperature of the body/object.