Heat & Temperature

The change of thermal energy, or heat, Q in an object when its temperature changes from T₁ to T₂ is

Q=ρVc(T₂-T₁)

where ρ is the density, V is the volume, and c is the specific heat.

Imagine an object is placed into a completely insulated box filled with water (see the image below). The initial temperatures of the object and the water may be different. The water will either warm up or cool down through the diffusion of thermal energy. The direction of heat diffusion depends on the difference of the initial temperatures. Two physical laws dictate that the following things must happen:

1. The First Law of Thermodynamics (i.e., the Law of Conservation of Energy): Regardless of which way the heat diffuses, the total energy of the system must stay the same as no thermal energy is lost or gained in the completely insulated system.
2. The Second Law of Thermodynamics: Regardless of which way the heat diffuses, the temperature inside the system must finally become the same everywhere in the completely insulated system.

The thermal energy that flows into or out of the water in the box is:

Q_wat=ρ_watV_watc_wat(T_final-T_wat)

The thermal energy that out of or into the object is:

Q_obj=ρ_objV_objc_obj(T_final-T_obj)

The Law of Conservation of Energy requires that Q_wat+Q_obj=0. This allows us to determine that the final temperature must be:

T_final= (ρ_watV_watc_watT_wat+ρ_objV_objc_objT_obj)/ (ρ_watV_watc_wat+ρ_objV_objc_obj)

Computer modeling provides opportunities of virtual experimentation, which could make the understanding of the above concepts more concrete. The model below presents a dynamic view of how heat transfer actually happens between the test piece and the water, as opposed to just providing a value of final temperature.

In the model, there are two thermometers for monitoring the temperatures inside and outside the test piece. While the model is running, you can turn a graph on to view the temperatures recorded by the thermometers. The following things can be explored with this model (remember to reset the simulation using the "Reset" button before you change the model):

• Change the initial temperatures: Use the radio buttons below the model to change the initial temperatures of the test piece and the water and re-run the simulation to watch the changes of the temperatures the thermometers record.
• Change the size of the test piece: Click on the test piece, resize it, and re-run the simulation to see how the final temperature changes.
• Change the specific heat: Use the "Specific heat" drop-down menu to change the specific heat of the test piece and re-run the simulation to see how the final temperature changes.
• More realistic simulations: A 2D model simulates a cross section of a 3D system. In reality, the cross section also loses or gains energy from the perpendicular direction (we call it the z-direction). To make our simulation more realistic, you can choose to allow heat loss in the z-direction. When this is on, the entire system will gradually lose energy, simulating what happens in reality.

1. Does a higher initial temperature of the test piece result in a higher final temperature recorded by the thermometer on the left (T2)?
2. Does a larger, initially colder test piece result in a lower final temperature recorded by the thermometer on the left (T2)?
3. Does a test piece with a higher specific heat and a higher initial temperature result in a higher final temperature recorded by the thermometer on the left (T2)?
4. In how many different ways can you get both the thermometers to read about 25°C at the end?