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 (TT_{ambient}).
This is stated mathematically as dT/dt = k(TT_{ambient})
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 watermetal 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 (86.580.5) = 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
C_{zinc} = (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. 