PhysicsLAB Lab
Newton's Law of Cooling and the Specific Heat of a Metal Specimen

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In the experiment you will be using a calorimeter of three nested Styrofoam cups, the metal heat specimens, LabPro with a temperature probe, and hot water.
The Physics Behind the Experiment
Newton’s Law of Cooling states that the rate at which convection cools a hot object is proportional to the temperature difference between the hot object and the ambient (room) temperature.
dT/dt  is proportional to (T-Tambient).
This is stated mathematically as  dT/dt = -k(T-Tambient)
Since this cooling rate depends on the instantaneous temperature (and is therefore not a constant value), this relationship is an example of a 1st order differential equation. The constant k in this equation is called the cooling constant. The solution to this differential equation is
In Part I, you will initially graph your data of only the hot water cooling to establish a calibration curve for your apparatus – the blue curve in the graph shown above.
The equation for our sample data equals
In Part II, you will then graph your data for when the metal sample was added to the hot water. The region where the calibration curve for the cooling water is parallel to the curve for each water-metal mixture represents the interval where the cooling constants are the same.
For our example, we used zinc. For the zinc curve, that would start around 185 seconds - see the graph below.
Heat is being exchanged between the hot water and the zinc cylinders in the “more or less” flat region, or plateau, of the graph. Once the water and zinc are at the same temperature, the cooling curve then matched that of the hot water from Part I (our calibration curve).
The matching (parallel sections) of the two curves, will allow us the ability to determine the heat exchange between the zinc cylinders and the hot water without being concerned about the cooling taking place between the apparatus and the environment.
For the hot water in Part I, the original temperature was around 95.5ºC.
In Part II, the zinc was originally at room temperature, 24ºC before it was added to the hot water and eventually warmed to 80.5ºC. The hot water in Part II also had an original temperature of 95.5ºC and reached thermal equilibrium with the zinc cylinders at a final temperature of 80.5ºC.
The vertical line shows us that the hot water from Part I was at 86.5ºC when the two cooling curves first started their parallel decay. T in the diagram would be (86.5-80.5) = 6Cº
Derivation of the equations needed.
Subtracting these two equations yields
Replacing each T with its exact expressions yields Eq #4 shown below.
If the same volume of water (aka, mass of water) is used in both trials, then we can factor out the common mass and specific heat for water.
In the experiment, the initial temperature of the water in both Part I and Part II was 95.5ºC, so the original temperatures subtract to zero, that is, cancel out.
The expression T(f water)-T(f water/zinc) is the T between our cooling curves in our EXCEL data.
Solving for the specific heat of zinc, we get Equation #5
The temperature change for the zinc equals the final temperature of the “plateau” minus room temperature, a positive change.  
Given below is a summary of the data used in this trial:
  •  common mass of water = 205 grams
  •  mass of zinc = 209.8 grams
  •  room temperature of zinc = 24ºC
  •  common initial temperature of water = 95.5ºC
  • T between the two parallel cooling curves = 6Cº
  •  final temperature of water/zinc mixture before starting cooling curve = 80.5ºC
Substituting these values into Eq #5, we obtain the following value for the specific heat of zinc:
The accepted specific heat of zinc is 0.387 J/(g K) giving our experiment a percent error of 12.1%.
Zinc has a molar mass of 65 grams/mole. Using this value, we can calculate zinc’s molar specific heat as
Czinc = (0.434)(65) = 28.21 J/(mole K)
In general, most metals have a molar mass very close to 3R = 3(8.314) = 24.9 J/(mole K), where R is the ideal gas constant.
Your group’s analysis will involve either three copper specimens or three lead specimens. Given below is a graph showing the combined cooling curves for water, lead, zinc, and copper using the same calorimeter. Notice that all of the cooling curves eventually become parallel to each other, but have different amounts of time during which the heat is being exchanged between the metal specimens and the hot water.
Procedure and Data
The following lab is adapted from a lab presented by Dr. Patrick Polley from Beloit College, during the July, 2014 Advanced Placement Physics 2 Workshop held at Florida International University in Miami. It is presented with his permission
Part I.
Mass three nested, empty large Styrofoam cups in grams 

To determine the fill-line for 200 ml of hot water, measure out the correct amount of cool water and mark the water line on the inside of the cup. Then dry out the cup and pour in 200 ml of boiling water and immediately start recording the temperature of the water every 30 seconds for a total of 10 minutes (600 seconds), as it cools in the cups.  This cooling curve will serve as a calibration curve for the cups
Copy the two columns from LabPro into EXCEL (columns A and B) before starting Part II of the experiment. Save your EXCEL file as SpecificHeatCooling_LastnameLastname.xlxs
State the file name for your data.

Mass the cups (your calorimeter) and the hot water when you have finished timing to verify the mass of the hot water actually poured into the cups in grams. 

Part II.
Next, mass three specific heat cylinders (of the same metal) and record your value in grams. 

What type of metal is your group using: copper or lead?

What is the temperature of the metal specimens before they are submerged in the hot water? (room temperature)

In Part II, you want the mass of the hot water to be EXTREMELY close to the same amount of hot water used in Part I.
Once again, add the hot water to the fill line and immediately start measuring the water's temperature. Once the hot water temperature agrees with the original temperature in Part I, carefully lower the three dry cylinders into the cup. As in Part I, record the temperature of the water for a total of 10 minutes (600 seconds) as the water heats the metal cylinders. After a noticeable plunge in temperature, you will see the appearance of a "plateau" where the temperature remains relatively constant. When the plateau ends, thermal equilibrium has been reached and the mixture cools for a total of 600 seconds.

When you have finished timing, mass the cups (your calorimeter) still containing the three metal specimens and the hot water. Perform the necessary subtractions to determine the actual mass of hot water used in Part II. Record your answer in grams. 

For the two experimental trials, what was the average mass of hot water present in grams? 

Before closing the LabPro software, copy the two columns from LabPro into EXCEL (columns D and E). Update your file by saving it one more time to include the new data in columns D and E from Part II.
Graph two cooling curves in EXCEL – the first for Part I when only water was cooling, and the second for Part II when the water and metal specimens were cooling. Place both curves on the same graph. Resave your EXCEL file with the completed graph and printout copies for your group.
Determine the common starting temperature of the hot water for both trials. 

On the graph’s printout, determine the location and value of the T for the two water samples.
What was the final temperature of the plateau on the second cooling curve where the heat exchange between the hot water and the metal specimen ended and the parallel cooling curve started? 

At what time did the plateau end in Part II? 

What was the value for DT between the two cooling curves? 

Calculate your experimental value for your specimens’ specific heat. 

The accepted values for the specific heats of copper and lead are 0.386 J/gK and 0.0128 J/gK respectively. What is the percent error for your group's experimental value? 

What is the molar mass of your metal specimens? 

What is your group's experimental molar specific heat for the metal specimens used? 

Using the equation   as your model, determine the common cooling constant for your curves.
Step 1. Begin by substituting in the values for the original temperature (To) and the final temperature (Tf) for the entire 600 seconds in your calorimeter’s calibration graph created in Part I.
What is your calibration curve's general equation? (before determining the value for k) 

Step 2. To solve for the cooling constant k, substitute in the time and temperature values for the data point on the calibration curve that provided the hot water value of DT (that is, where the parallel cooling curves started).
What are the co-ordinates of this data point? 

What is the value for the cooling constant, k, for your experiment? 

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