Unit 2: Exploring the Nature of Thermal Phenomena
VII. Developing a Mathematical Representation of Thermal Phenomena Based on Theoretical Considerations
The experimental approach discussed above focused only upon the relative masses of the hot and cold water and their relative changes in temperature. No mention was made of the energy flowing from the hot to the cold water. An earlier section in this unit (II.B, Table II.1, page 11) had developed the following central idea:
A temperature difference implies a flow of energy from hotter objects to colder objects.
In this section, we consider how to express theoretical ways of thinking about that transfer of energy from the hot water into the cold water. This involves use of the Law of Conservation of Energy, that the total energy in a closed system does not change, that energy lost by the hot water is gained by the cold water and its surroundings.
A. Considering what happens when energy flows from hot water into cold
Question 2.11 What theoretical considerations can provide insights into what is happening when energy flows from the hot water into the cold water?
Energy is measured in different units in different contexts. (see https://www.aps.org/policy/reports/popa-reports/energy/units.cfm . In this course, we use the energy unit of a calorie.
A calorie is the amount of energy needed to change the temperature of one gram of water by one degree Celsius at standard atmosphere pressure and 20°C. This amount of energy is very small; the calories typically discussed in the context of food are kilocalories, 1000 of the calories discussed here. A kilocalorie is the energy needed to change the temperature of one kilogram of water by one degree Celsius at standard atmospheric pressure and 20°C.
There are many different materials one could use in exploring thermal phenomena. The amount of energy needed to raise the temperature of one gram of a material by one degree Celsius is called the material’s specific heat.
Water is the standard. One calorie of energy is needed to raise a mass of one gram of water by one degree Celsius at standard atmospheric pressure and 20°C. The symbol for specific heat is c. In this course, water’s specific heat is written cw = 1.0 cal/(g°C).
Note that the units for mass and temperature both are in the denominator of this expression for specific heat. This is a mathematical way of stating that the change in energy for water is one calorie to change each gram by one Celsius degree. The dimensions of specific heat, c, are:
Dimensions of specific heat: [latex]\frac{\text{(energy)}}{\text{(mass)(temperature)}}[/latex]
The units are: [latex]\frac{\text{(calories)}}{\text{(grams)(degrees Celsius)}}[/latex]
To express mathematically how much energy is gained or lost when something is warmed or cooled:
- What might be the effect of how much “stuff” is involved?
- What might be the effect of the kind of “stuff” one has?
- What might be the effect of how much the temperature changes?
- How c