# Newton's Law of Cooling

Instructor: Michael Blosser

Michael has a Masters in Physics and a Masters in International Development. He has over 5 years of teaching experience, teaching Physics, Math, and English classes.

This lesson will introduce the reader to Newton's Law of Cooling, what it is, how it is experimentally verified, and how we can use it to calculate the cooling rate of objects in real-world examples.

## Physics in Action

We have all had that problem of reaching for our hot cup of coffee or tea and realizing that it has already gotten cold. Or we have wondered why certain materials stay cool to the touch under the sun while others are scorching hot. In this lesson, we will learn how and why objects cool and learn how to calculate this cooling with real-world problems and examples.

## Newton's Law of Cooling

Isaac Newton created his revolutionary Law of Cooling in the 17th century. Newton's Law of Cooling is a formula that allows us to determine the temperature of an object during heat loss. Isaac Newton stated that ¨the rate at which a warm body cools is proportional to the difference between the temperature of the warm body and the temperature of its environment.¨ Newton's theory can also be put into an equation, giving us the Law of Cooling equation:

With T (t) being the temperature of an object at a certain time

t being the time in seconds

Ts being the temperature of the surroundings

T0 being the starting temperature of the object

and k being the cooling constant.

Using this equation, we can calculate how fast an object at a certain temperature would cool in a specific environment, and how the rate of cooling of an object is dependent on the difference of temperature between the object and the surroundings but also on the cooling constant of the object.

## Experimental Verification

It is relatively easy to experimentally verify Newton's Law of Cooling. As mentioned above, the Law of Cooling states that ¨the rate at which a warm body cools is proportional to the difference between the temperature of the warm body and the temperature of its environment.¨ We can experimentally verify this law with a spherical calorimeter (a laboratory device that is used to measure the quantity of heat transferred to or from an object) filled with hot water. The calorimeter has mass m and specific heat capacity s and the hot water has mass m1 and specific heat capacity s1. Using the calorimeter we measure the amount of heat energy lost as the temperature of the water and calorimeter falls from temperature T0 to temperature T1 after a set time. As the temperature drops, the temperature is noted for every 30 seconds for a set time. Graphing the change of temperature versus the change in time gives us a cooling curve that we can use to calculate the rate of cooling.

After calculating the rate of the cooling curve, we find that the rate of cooling is proportional to the difference between the temperature of the object and the temperature of the surroundings, verifying Newton's Law of Cooling.

### Example 1

You just bought a boiling cup of coffee of temperature 90 degrees celsius. What would be the temperature of the coffee if you let it cool for 2 minutes? The surrounding temperature is 25 degrees celsius and the cooling constant of the coffee is 0.015 1/s.

#### Answer

We immediately know that we can use Newton's Law of Cooling in order to solve this problem. We are given the initial temperature of the coffee, the temperature of the surroundings, the time of cooling, and the cooling constant. Now all we need to do is convert our temperature to the right units and insert our values into the cooling equation. Our temperature is given in celsius and we need the temperature in degrees Kelvin. The conversion from celsius to Kelvin follows the equation:

Therefore, the initial temperature of the coffee is 363 degrees Kelvin and the temperature of the surroundings is 298 degrees Kelvin. Plugging our values into the equation gives us:

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