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Legendre Transforms on the PDM: Instructor's Guide

Main Ideas

  • Navigating Force-Position space on the PDM to experimentally determine a set of states from which multiple processes can be approximated.
  • Different processes have different independent variables (state variables with $d$s or $\Delta$s in front of them when expressed algebraically).
  • Different types of energies might be more useful to describe certain systems or processes, especially when those energies have the same independent variables as the processes you want to describe.
  • Legendre transformations provide a relationship between different energies (thermodynamic potentials or their PDM analogs.

Students' Task

Estimated Time: 20 minutes

  • Navigate $F_L-x_R$ space on the PDM to obtain the values of $x_L, F_L, x_R$, and $F_R$ for a 'grid' of states, then use these values think about the PDM analogs of different thermodynamic potentials.
  • Obtain an algebraic relationship between the different PDM analogs of thermodynamic potentials.

Prerequisite Knowledge

  • Familiarity taking data on the Partial Derivative Machine (see PDM Variables Activity) FIXME add link
  • The PDM has 4 variables ($F_L$, $F_R$, $x_L$, and $x_R$) and that, at most, 2 of these can be independent.
  • The analog for the first law of thermodynamics/thermodynamic identity on the PDM is $dU=F_Ldx_L+F_Rdx_R,$ where $x_L$ and $x_R$ are the independent variables

Props/Equipment

Activity: Introduction

  • Internal energy $U$ is useful to think about for a number of reasons
    • We know how $U$ changes from the thermodynamic identity $dU=F_Ldx_L+F_Rdx_R,$
    • $dU$ can be experimentally approximated on the PDM by finding $\Delta U = F_L \Delta x_L + F_R \Delta x_R$
    • Rather than measuring an $F \Delta X,$ it is also possible (and sometimes easier) to measure an $x \Delta F$. While these values cannot be used to find $\Delta U,$ there are other types of energies that we can imagine that use different combinations of $F \Delta X$s and $x \Delta F$s.

Activity: Student Conversations

  • When calculating $x \Delta F$ or $F \Delta x$ along a path, if the coefficient (the thing without a $d$ or $\Delta$ in front) changes, then it needs to be approximated as a single value for this calculation. Ask students how they might do this: a reasonable response might be to take an average.
  • The value of $\Delta A, \Delta B,$ and $\Delta C$ depend on various values of $x$ and not just changes in $x$. Thus, it matters where $x=0$ lies. Some groups might choose to keep the values of their measuring takes, others might want to redefine other values as zero.

Activity: Wrap-up

  • Internal energy keeps track of the work (and heat) that goes into and out of a system. These other energies keep track of the energy that goes into or out of the system in other ways (than heat and work). For example, $\Delta A = F_L\Delta x_R-x_R\Delta F_R$ takes into account work on the left side of the PDM but some other form of energy exchange on the right side (for which we have no name). $A$ is a state variable that we can write in terms of other state variables ($A=U-F_Rx_R \implies dA=dU-F_Rdx_R-x_RdF_R$), which is a statement known as a Legendre transformation.
  • It is not always easy to come up with a physical rationalization of these other energies, though it becomes easier if one state variable remains constant throughout a process. For example, if $F_R$ is constant, then $\Delta A = F_L\Delta x_R = \Delta U - F_RdX_R$

Physically interpret $F_Lx_L$ and $F_Rx_R$ as potential energies, though emphasize that such interpretations in thermodynamics are not always possible.

Extensions

This activity is the fourth activity of the Partial Derivative Machine (PDM) Sequence FIXME on measuring partial derivatives and potential energy. This sequence uses the Partial Derivative Machine (PDM).

  • Preceding activities:
    • Quantifying Change: This small group activity introduces students to the PDM by asking them to determine how many measurable quantities exist within the system and how many of these quantities are simultaneously controllable.
    • Isowidth and Isoforce Stretchability: In this small group activity, students are challenged to measure a given partial derivative with the PDM.
    • Easy and Hard Derivatives: This small group activity asks students to write each partial derivative that can be formed from $x_1$, $x_2$, $F_1$, and $F_2$ and then categorize each as “hard” or “easy” to measure on the PDM.
  • Follow-up activity:
    • Potential Energy of an Elastic System: In this integrated laboratory activity, students use the PDM to determine the change in potential energy between two states of a nonlinear system.

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