Thermocouples are exploited widely throughout the world of science and industry, as they are accurate devices for measuring temperatures, are cheap to produce, easy to use and can be made to withstand a variety of environments including those that are potentially hazardous. They are also accurate temperature measuring devices as they can be made accurately and have a fast response time and have a low heat capacity, so do not affect their surroundings temperatures. It is for the aforesaid reasons that I will be using a thermocouple in this experiment.
Newton’s law of cooling, as it has been previously stated, dictates that the rate at which a body heats up or cools down is proportional to the difference between its temperature and its surrounding’s temperature.
Also, the rate of cooling of a body is related to the area through which cooling may take place and the thickness of the material the vessel is made from.
From the initial formula we arrive at:
Q α AΔt Q=heat energy t=temperature A=area
1 Δx k=constant x=thickness Δ=change in
Heat energy can also be defined in terms of specific heat capacity. The specific heat capacity of a substance is defined as the amount of energy required to raise one gram of a substance through one degree Celsius and can be given the units Jg-1ºC-1 (joules per gram per degree Celsius). The change in heat is equal to the mass of the body multiplied by its specific heat capacity multiplied by the change in temperature experienced by the body, or:
ΔQ=m x c x Δt m= mass c=specific heat capacity
It becomes clear at this stage that these ideas can be rearranged and equated, so:
ΔQ α -kA . (t1-t2)
Δt m x c x (x)
This is a better overall formula as constants such as the thickness of container, temperature difference, surface area and volume are accounted for and it displays the pivotal relationship that states how temperature drop is proportional to these constants.
Cooling is a function that is similar to the exponential graphs for events such as population growth, although the graph for cooling has a negative gradient and never reaches the x axis since it cannot cool to 0ºC without any energy input under standard conditions.
Experiment Preparations
Using a thermocouple to take information from cooling water will result in a set of data where time is related to voltage (mV). To convert these voltages into temperatures a means of conversion is required.
To obtain a means of converting voltage readouts to temperatures, we first had to calibrate the thermocouple.
To do this, boiling water was put in to a test tube that was sitting in a water bath containing 200 ml of water. In to this test tube were also placed a thermometer and one thermocouple junction. The other thermocouple junction was placed in a beaker containing ice and water. The addition of ice to the water in the vessel without the test tube a stable environment for that junction was created, and it stays naturally at 0ºC and so can be used as a reference point for voltage readouts. Across the thermocouple a voltmeter with a suitable millivolt scale was connected to that the size of the potential difference and therefore electromotive force could be measured. By doing this with identical equipment, specific voltages can be determined for fixed temperatures and so provide a means of comparison for the experiment measuring the cooling of the water.
The apparatus was as follows.
(see next page)
The results of this experiment are set out in the table below.
From this set of results a graph can be plotted that will allow us to see if there are any anomalies in the results and therefore if the experiment needs to be corrected and re-done. It will also allow easy conversion from millivolt readings to temperatures in ºC.
(see next page for graph)
The Experiment
To measure the rate of cooling, water was heated to 100˚C and put it straight into a test tube in a water bath, which contained one thermocouple junction. The other junction was in a bath of ice and water to keep it at a fixed 0ºC so it could therefore be used as a reference point for voltage readouts. Across the thermocouple a voltmeter with suitable millivolt scale was connected so the potential difference and therefore electromotive force could be measured. As with the ice regulating the temperature of the fixed thermocouple junction, the water bath should help maintain a static ambient temperature around the subject test tube. This should reduce errors in the experiment through forced convection and heat being transmitted from and to the test tube as the water bath acts as a barrier around the subject test tube, maintaining a static environment.
By fixing the amount of water in the water bath and the amount of water of which the cooling rate will be measured, as well as the material of the containers are constructed from a fair test is ensured and constants are preserved.
The apparatus was as follows.
The results from the experiment are presented in the table below.
From these voltage results we can use the calibration graph to determine temperatures in ºC at given time intervals.
We can now plot a graph of temperature against time.
(See next page)
Newton’s law states that the rate of heat loss is proportional to the temperature difference between the temperature of the body and the ambient temperature (the temperature of the surroundings).
This can be re-stated that the gradient of the graph of temperature against time is proportional to t1-t2 where t1 is the temperature inside the test tube and t2 is the temperature outside the test tube.
So if t2 is 20 and t1 is 60, m (the gradient of the graph) α 40.
By plotting a graph of the gradients of the tangents of the graph showing cooling against temperature difference we can determine whether the rate of heat\loss is proportional to the temperature difference between the temperature of the body and the ambient temperature.
If proportionality between these two factors is displayed then the graph should be straight and pass through the origin.
First a graph of the gradients of all the tangents against the temperature difference these tangents were taken from was plotted.
The data used for the first graph is as follows.
(see next page for graph)
This graph proved inconclusive as the plotted points showed no conclusive correlation. This lack of direct correlation can be traced back to inaccuracies during the taking of measurements. As the rate of cooling was rapid to begin with (the gradient at the beginning was large and negative), the accuracies for the times of each voltage drop were low as there were so many changes in a small space of time.
However, if we take gradients from later on in the experiment the accuracies for the times will be better as the percentage of error will be smaller.
Data for this graph is as follows.
(see next page for graph)
This graph shows a much stronger correlation that is as predicted. Anomalies in this graph can be attributed to the fact that the area on the graph that the gradients were taken from had a generally small overall gradient. This means that to get an accurate tangent drawn to the curve is difficult, and so there is a chance for a large percentage of error. However, the direct correlation of many of the plotted points on the graph means that the graph can be taken as correct along the line of correlation.
We can therefore conclude that the rate at which a body cools is proportional to difference between its temperature and the ambient temperature.
Related to the introduction of this investigation, we can conclude that the best time to add milk to a cup of coffee to get it to drinking temperature as soon as possible is as soon as it is poured, i.e. when time=0, as this is when it is cooling most rapidly and so introducing a cooler liquid will increase this rate of cooling.
Extension work
The same experiment as for the main body of this investigation can be again conducted, but at a certain interval a sample of cold water can be introduced into the warm water that is cooling. This will affect the rate of cooling, but if the spread of heat through the resultant liquid to even temperature is instant then the graph should be the same as for the cooling of the liquid previously in this investigation.
The apparatus used for this experiment will be the same as for the straight cooling experiment, but two 3 ml pipettes will be used to swap, at the same time, 3 ml of water from the ice bath with 3 ml of water from the test tube containing the hot water.
The results for this experiment are as follows.
(see next page)
In theory, the spread of heat energy from the hot water to the newly introduced cold water will be instantaneous, and so there should be a drop at the time of the cold water was introduced.
However, the spread of heat will not be instantaneous, and so the drop will not be vertical but merely steep.
(see next page for graph)
As we can see, the graph for swapping water over has the same gradient as the graph for straight at any given point before adding the water, and is parallel to the straight cooling graph after adding the water, with the same gradient.
We can therefore deduce that Newton’s law of cooling holds true when there are affecting factors other then the four methods of heat transfer discussed at the beginning of this report, Thermal radiation, Evaporation, Convection and Conduction, affecting cooling.
Bibliography
Own Knowledge.
Advancing Physics – AS level physics textbook.