Newton’s Law of Cooling
Purpose: When you have completed the experiment, write a statement that describes the purpose of this experiment.
Theory: We all know that an object at higher than ambient temperature will cool as time progresses. This is why your coffee gets cold as time passes, or why forged steel in a foundry will cool to handling temperature relatively quickly. A perceptive question would be to ask how the initial temperature of the system affects the rate of cooling in question. Which cools off faster, the hot steel or the warm (by comparison) coffee? In this lab we will observe the cooling of a “hot” and a “warm” object and compare the rates of cooling to determine which cools at a faster rate.
Newton’s Law of cooling states that “for a body cooling in a draft, the rate of heat loss is proportional to the difference in temperatures between the body and its surroundings.”
Beginning with the specific heat equation,
Q = mcDT
A little algebra and a little calculus later (differentiation with respect to time and integrating over the initial to final temperatures and time = 0 to time t), we end up with the following equation:
where
T(t) = Temperature of the system at time t (in min).
T0 = Initial Temperature of the system being observed
Ta = Ambient temperature
k = a proportionality constant that can be solved for by observing the behavior of the system at some time t1 and observing the temperature at that time (below)
Exciting, huh? Thought you'd like it. Sorry about all of the different T's. Guess what? We are not going to use it. However, at the end of the experiment you will have enough data to confirm something close to the mathematical veracity of the law if you choose.
Materials: 1 Styrofoam cup, 1 Thermometer, (2 cups and thermometers if performing parts a & b simultaneously),1 data table, 1 sheet graph paper
Procedure:
1) Get a reading on the ambient temperature of the classroom from your teacher. Record as Ta below.
Part a)
1) Fill the cup with HOT water (at or near boiling). Immediately record the temperature of the water as T0 in your data table.
2) Record the temperature of the water in the cup every 30 seconds for the first ten minutes of the experiment.
3) Record the temperature of the water every minute for the next fifteen minutes of the experiments.
4) Complete the data table by calculating the rate of change of temperature in each time interval.
Part b)
1) Fill the cup with a similar amount of WARM water (like warm tap water). Immediately record the temperature of the water as T0 in your data table.
2) Record the temperature of the water in the cup every 30 second for the first ten minutes of the experiment.
3) Record the temperature of the water every minute for the next fifteen minutes of the experiments.
4) Complete the data table by calculating the rate of change of temperature in each time interval.
Graph your data as a plot of temperature (y) vs. time (x)
Use the statistical regression tool on your grapher or MS Excel to add a best fit function (use a third or fourth order polynomial regression, we will talk about why later) to your graph. Be sure to include the equation.
Example Data Table: Build your own!
HOT
Ambient Temperature (Ta): ______oC
Time(tn) / Time Value
(Minutes) / Temperature (oC) (Tn) / DT
(Tn+1 - Tn)
To
T1
T2
etc…build your own table!!! Repeat for WARM system.
Questions to consider when writing your conclusion:
1) Which of the systems (Hot or Warm) had the largest rate of cooling during the first ten minutes of the experiment? Give a hypothesis that might explain your results.
2) Which system was (or would be, given enough time) the first to reach the ambient room temperature? Why?
3) Pretend this was a full formal lab report: Write a conclusion that describes your results and includes your answers to questions 1) and 2).