In this session, students will investigate the equilibrium temperature produced when water and metal of different temperatures come into contact.
Previously, we have mixed masses of water at different temperatures and discovered a mathematical expression for predicting the equilibrium temperature of the mixture. In this session, we want to determine if mixing water and metal of the different temperatures follows the same rule. If it does, we can extend our hypothesis to cover more materials, If it does not, we must modify our hypothesis in some way. (Hopefully, the students will discover the concept of "Specific Heat".)
Begin class by dropping metal samples into a tub of ice cold water. (The instructor would be wise to have several liters of ice water and boiling water prepared before class.) Pull out the samples, dry them, and give them to the students. (One sample per group.) If there is time, also give them a metal sample at about 40°C. This is fairly hot be not so hot as to be dangerous. Be sure to dry it before giving it to the students. Ask them to conjecture about bringing identical metal samples into temperature equilibrium. Have them talk about the experimental procedure, measuring the temperature, keeping the metal in contact, external temperature influences, etc.
Now ask them to hypothesize about bringing water and metal into contact so as to produce thermal equilibrium in the "mixture". At this point, they should be free to conduct experiments and see if the equilibrium "rule" they produced last session applies to the new mixture. The students should initially note that the agreement is not very good. Encourage them to hypothesize about differences in ways different materials may respond to thermal conditions. Then encourage them to experiment using their new hypothesis.
For the instructor's sake, the answer is...well, I don't know about the ci in the denominator but I think it is correct because it makes the units right.
Discuss errors. Use the equations from the last lesson to begin the discussion and then extend it to include the results of this lesson as appropriate.
Provide the students with a short homework set where the questions introduce the term "heat capacity". The homework set should lead them through examples where using the concept of heat capacity allows them to correctly calculate the equilibrium temperature in a system similar to the one they experienced in the classroom. Possible questions include:
What if the metal samples used in class today do not have the same thermal "oom-pa" of water. That is, a mass of metal at some temperature T raises the equilibrium temperature of water by 10°C but an identical mass of water at the same temperature T might raise the equilibrium temperature of water by 14°C. Physicists usually call the thermal "oom-pa" of a material its "specific heat". Using your best conjecture, write an expression for finding the equilibrium temperature of mixed materials with different initial temperature, different initial masses and different specific heats.
Hint: One way to think of specific heat is that it changes the thermal "weight" of the material by a constant factor. Physicists often use the symbol c to denote the specific heat of a material. They sometimes use a subscript to denote specific heat of a material. So for water, the symbol might be cw and for aluminum it might be ca.
Suppose you had to mix 100 g of water at 25°C with 40 g of aluminum at 95°C. What what be their equilibrium temperature. Oh, by the way, the specific heat of water is cw=4.2 and the specific heat of aluminum is ca=0.9.
There should be more questions than just these two. Also, everything depends on answering the first question correctly. Maybe the students should be given a Mulligan if they get the first part wrong.