Representations of partial derivatives in thermodynamics

John R. Thompson (University of Maine), David Roundy (Oregon State University), Donald B. Mountcastle (University of Maine)

Submitted Abstract

One of the mathematical objects that students become familiar with in thermodynamics, often for the first time, is the partial derivative of a multivariable function. The symbolic representation of a partial derivative and related quantities present difficulties for students in both mathematical and physical contexts, most notably what it means to keep one or more variables fixed while taking the derivative with respect to a different variable. Material properties are themselves written as partial derivatives of various state functions (e.g., compressibility is a partial derivative of volume with respect to pressure). Research in courses at the University of Maine and Oregon State University yields findings related to the many ways that partial derivatives can be represented and interpreted in thermodynamics. Research has informed curricular development that elicits many of the difficulties using different representations (e.g., geometric) and different contexts (e.g., connecting partial derivatives to specific experiments).

  • Definitions, file cards, dimensions,
  • Sometimes true, true by definition, true from mathematics, physical laws
  • Loverude problem: which students use first law reasoning?
  • Meaning of partial derivatives→name the experiment, what things are held constant.
  • Chain rule diagrams (add these to the bridge book)
    • partial derivative version
    • exact differential version
    • one variable doesn't change version–see our two physical examples from the interlude
  • Lagrange multipliers–geometric interpretation and figure
    • how is this an example of holding things constant–check out interlude notes from Tevian
  • What is a function of what?
    • 6 state variables (in pairs), 3 are independent.
    • which variables are independent changes throughout the problem.
    • no canonical independent variables (in contrast to E&M).
  • different kinds of d's (dbar, delta, curly d, straight d)
    • dU = dbar Q + dbar W
    • delta U = Q + W
    • sign of work in the first law
  • $exp(-\frac{\delta E}{kT})$
    • high and low termperature limits for diatomic molecule: vibrations, rotation, translation
    • connection with QM
    • new interpretation for students: what do you mean by limits?
    • in low temp. limit, $\delta E$ matters, you automatically get a series expansion.
    • in high temperature limit, size of $\delta E$ is small, so replace sum with integral–this is a new idea for students.
    • pull out all dimensionful information first–this is an important strategy–should we teach this in first few paradigms?

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