Newton’s Law of Cooling

An Experimental Investigation

by Elizabeth Gieseking

When an object is at a different temperature than its surroundings, it will gradually cool down or heat up until the temperatures are equal.  Everyone has experienced this.  You boil water to make tea and then wait several minutes until it is at a temperature at which you can drink it.  You place a cold turkey in the hot oven for Thanksgiving dinner and after several hours it has reached the desired temperature.  Newton’s Law of Cooling relates the rate of change in the temperature to the difference in temperature between an object and its surroundings.

where  the temperature of the object of interest as a function of time

the time

the ambient temperature

the initial temperature

a proportionality constant specific to the object of interest

This differential equation can be integrated to produce the following equation.

Experimental Investigation

For this exploration, Newton’s Law of Cooling was tested experimentally by measuring the temperature in three beakers of water as they cooled from boiling.  The purpose of this investigation was twofold.  First I wanted to determine how well Newton’s law of cooling fit real data.  Second, I wanted to investigate the effect of changing the volume of water being cooled.

Three beakers of water were used for this experiment.  The first held 100 ml of water, the second 300 ml, and the third 800 ml.  All three beakers originally held water at 100°C.  Each beaker had its own thermometer and the thermometers were kept in the beakers between measurements so there would be no temperature lag.  The temperature of the water in each beaker was measured every minute, always in the same order.  The ambient temperature for this investigation was 23°C.  The experimental setup is shown below.

The temperature was measured every minute for 35 minutes and then every 5 minutes for the remainder of one hour.  The following data was obtained.

 Time (min) 100 ml Temperature °C 300 ml Temperature °C 800 ml Temperature °C 0 100 100 100 1 95 95 96 2 82 91 95 3 79 87 92 4 74 84 90 5 70 81 88 6 67 78 85 7 65 76 83 8 61 73 80 9 59 71 78 10 57 70 76 11 56 68 75 12 54 66 74 13 52 64 73 14 51 63 71 15 50 61 70 16 49 60 68 17 48 58 66 18 47 58 66 19 45 56 65 20 45 55 63 21 44 55 62 22 43 54 61 23 42 53 60 24 42 52 60 25 41 51 59 26 41 50 58 27 40 49 56 28 39 48 56 29 38 48 55 30 38 47 54 31 38 46 53 32 38 46 52 33 37 45 52 34 36 45 51 35 36 45 50 40 34 42 47 45 33 40 45 50 31 38 43 55 30 37 41 60 29 36 40

From this data, it can be observed that the water in the smaller beakers cooled more quickly than the water in the larger beakers.  Below is a graph of the data.

The next step is to determine the value of k for each of the beakers of water.  We will look back at the integrated equation and then solve for k.

We will use Excel to calculate k at different times for each beaker and then find the average k value for each beaker.

For the 100 ml sample of water, the calculated k value was -0.0676.  This resulted in a root mean square error of 4.80°.

For the 300 ml sample, the calculated k value was -0.0447 and the root mean square error was 3.71°.

For the 800 ml sample, the calculated value of k was -0.0327 and the root mean square error was 2.24°.  We note that the value of k for the 800 ml sample was about half of that for the 100 ml sample.

In all of these cases, the experimental temperature fell more quickly at the beginning of the experiment than that predicted by the theoretical model and more slowly than predicted toward the end.  The larger water sample followed the Newton’s Law of Cooling model more closely than the smaller samples did.  There are several explanations for this from a thermodynamics standpoint.  Newton’s Law of Cooling accounts primarily for conductive heat exchange and assumes that the only heat lost by the system to the surroundings is that due to the temperature difference.  At temperatures near boiling, the rate of evaporation is high.  The heat lost through the phase change is greater than the heat lost through convective heat exchange with the environment.  Additionally, since the beakers were placed on a granite countertop, the heat lost through conduction with the countertop at the beginning of the experiment is significant and is higher than later on when the countertop has warmed up.  If the countertop is now warmer than the surrounding air, the temperature gradient is not what it was assumed to be from the initial temperature measurement.  Despite these complications, we conclude that Newton’s Law of Cooling provides a reasonable approximation of the change in temperature for an object cooling in a constant ambient temperature.