Cooling: Newton's Law Of Cooling

We know that hot water or milk, when kept at room temperature, cools down gradually. But have you noticed that the rate at which hot milk cools down to a lukewarm state is lesser as compared to that required by a lukewarm milk to cool and attain the room temperature? Have you ever wondered why this happens? In this section, we will learn about Newton’s law of cooling which gives the explanation for this behaviour.

Newton’s Law of Cooling

According to Newton’s law of cooling, the rate of loss of heat from a body is directly proportional to the difference in the temperature of the body and its surroundings.

Mathematically,

Newton’s Law of cooling

Where K is the positive constant that depends on the area and the nature of the surface of the body under consideration.

We can understand this more clearly with the help of the following example.

Consider a body of mass m and specific heat is at a temperature T2 is placed in a surrounding of temperature T1. Let the temperature fall by a small amount ‘dT2′ in time ‘dt’, then the heat loss is given by

dQ = msdT2

The rate of heat loss with respect to temperature is given by

The rate of heat loss

From equation a and b we can write

 Equation

Integrating both the sides we get

ln(T2-T1) = -Kt + C

where K = and C is the constant of integration.

Nature of Graph for Temperature

The Newton’s law of Cooling Formula is used to solve various questions regarding the temperature of a liquid with respect to time. From the equation above, we can see that the nature of graph obtained for temperature with respect to time is exponential in nature, as can be seen in the plot below. Here we notice that the cooling of fluid depends on the difference of its temperature and the surrounding. The plot for the rate of change of temperature is steeper in the beginning and as the temperature decreases with time, it becomes gradual; hence the rate of cooling is higher at the beginning and decreases as the temperature of the body falls.

Newton’s Law of cooling

This plot can be derived experimentally using the setup shown in the figure below. Here we take some water in a calorimeter with the proper stirring mechanism and cover it with a lid consisting of two holes through which a thermometer is dipped in the fluid.

Cooling

The water is heated till it reaches a temperature of 40°C. The heat source is then removed, the system is left for cooling and the stopwatch is then started. The reading of the thermometer and the stopwatch is noted at regular intervals. As the water reaches the room temperature, we stop the experiment. As we plot the temperature on the Y-axis and the time on the X-axis, we get a graph similar to that shown above.

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