Limitations and functions: Four examples of integrating thermodynamics

A summary of the four examples along with the corresponding theories and prevalent inappropriate theories are listed in Table 1.

Table 1: Major theories (

) and prevalent inappropriate reasoning (×) of the real-life examples

While most students can easily invoke “thermal expansion” to explain the “floating” of the lava, they encounter serious difficulty in explaining the “sinking”¹.

The “lava” has slightly higher density at room temperature than the liquid, and a greater coefficient of thermal expansion. Thus, when heated by the lamp, the lava floats due to buoyant force.

In order to “sink” the lava, the containers need to taper towards the top, causing the temperature to drop².To derive the cooling rate (DT/t) depending on the radius (r) of the container, the heat transfer rate adopted is

, where

, where Q is heat exhausted, T: temperature, t: time, m: mass, s: specific heat, and h is the height of the container. Thus, the narrower section has a faster cooling rate, causing the floating lava to sink. Therefore, this example illustrates that in addition to the concepts of buoyant force and thermal expansion, thermal conduction is required in order to explain the sink of the lava.

Why do hot-air balloons (Figure 2) float upwards? Which force lifts the system? While many students successfully respond: “buoyant force”, many of them inappropriately attribute the buoyant force to “thermal expansion”. They illustrate thermal expansion as magnifying the oscillation of the chemical bonds for gas, implying the inadequate static model of ideal gas, rather than the kinetic model.

Instead of thermal expansion, the effective principle is the ideal-gas law (PV=nRT or PM=ρRT), where P is pressure, T: temperature, r: density, M: Molar molecular mass, and V: volume. Since a hot-air balloon is an open system, Pin=Pout, from PM=ρRT

, then buoyant force F_B = Δρ_(out－in)·V·g can be applied. Thus, a hot-air balloon not only connects the ideal-gas law with Archimedes’s buoyant force, but also confronts the restriction of “thermal expansion” in the gas system. In addition, the students’ pitfall can be utilized to distinguish the static model for liquids and solids, and the kinetic model for gas, as well as to highlight the neglect of chemical bonds (nil potential energy) of ideal gas. Thus, the total internal energy of the gas system contains kinetic energy of molecules only.

Figure 3 shows the color variation of the atomic explosion, which changes from yellowish at the bottom to white (clouds) at the top. While explaining the phenomenon, many students successfully respond that the white clouds are results of dramatic cooling. However, they tend to explain that based on PV=nRT, the explosion leads to a rise in temperature (V↗=> T↗). This justification is ineffective because they neglect the third variable (P), which is significantly reduced. The students’ logic flaw found by the author is consistent with that found by Rozier and Viennot’s (1991) study.

After challenging the ineffectiveness of the ideal-gas law, the 1st Law (Q+W=ΔU) is initiated, where Q is heat absorbed by the gas system, W: work done on the gas, and ΔU: change of the internal energy of the system. The explosion is fast enough to be treated as an adiabatic process; due to an adiabatic explosion (Q=0, W:-), the internal energy reduces (ΔU:-). By equating the internal energy (of the 1st law) to the kinetic energy (of kinetic theory), ΔU=ΔE_k= for6

for air, the reduction of the internal energy leads to a temperature drop, which becomes cold enough to condense the water vapor (the white cloud) at the top.

Since the notion of “internal energy difference (ΔU) due to work (W)” in Q+W=ΔU is less intuitive, the next example may reinforce the concept further.

To explain this common meteorological phenomenon, the 1^st Law (Q+W=ΔU) is more effective than the ideal-gas law (PV=nRT). When wind ascends from a mountain, the atmospheric pressure decreases, resulting in air expansion, which is fast enough to be treated as adiabatic (Q=0). Since work is done by the expanding air (W:－), the air loses internal energy (ΔU:－). Then, combining the 1st Law and the kinetic theory, the temperature drops (ΔU=ΔE_k

ΔT), usually becoming cold enough to condense the damp air into rain

When the air blows from windward to leeward, dry wind descends and results in warm air, called Foehn (or Chinook). Foehn is adiabatic compression (Q=0, W:+ => ΔU:+ => ΔT:+). Therefore, Foehn is warm owing to the “work” done via air compression. Since Foehn is dryer than ascending wind, it has a greater temperature gradient compared with that of the ascending wind.

Examples 3 and 4 may help students to appreciate the function of the First Law in terms of distinguishing among heat (Q), work (W), and hotness (T). Warm Foehn is due to “work” (W) rather than “heat” (Q). In addition, hotness (ΔT>0) does not equate to “being heated” (Q>0), which students often have difficulty distinguishing (Erickson, 1979).

¹The narratives quoted in this paper are based on the author’s teaching in introductory physics classes for engineering.

²Leif explained the narrowed top of the container causing the lava to sink verbally without sufficient argumentation.

Theories Examples	Thermal conduction	Buoyant force	Thermal expansion	Ideal-gas Law (PM=ρRT)	The 1st law (Q+W=ΔU)	Kinetic theory E_k∝T
1. Lava lamp
2. Hot-air balloon			×
3. Atomic bomb explosion				×
4. Cold mountain-tops & warm foehn (chinook)				×