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# Variables and Representations

## Prerequisites

Students should be able to:

• Record data from measurements.
• Plot data by hand.

## In-class Content

### Activity: PDM Variables

Activity Highlights

1. This small group activity is designed to help students become familiar with the Partial Derivative Machine (PDM), a mechanical system which allows students to measure partial derivatives and determine various physical quantities, including potential energy.
2. Students use the PDM to determine which properties of the physical system they can control and which ones they can measure.
3. The whole class wrap up discussion focuses on the fact that not all of the properties of the physical system are independently controllable, and leads students to think about what is being held fixed when they measure a derivative.

### Lecture: Contour Graphs of PDM Variables (25 minutes)

Fix $x_R$ and make a graph of $x_L$ vs. $F_L$. (Don't let students be too precise.)

Repeat for 3 different fixed values of $x_R$ and graph them all on the same set of axes. (Make sure students have explored enough of parameter space to see the nonlinearity.)

What variables is $x_L$ a function of? (How do you indicate what you are holding constant on the graph?)

This is a contour graph! You can write it using functional notation as $x_R(x_L,F_L)$.

Did we have to hold $x_R$ fixed? What might we hold fixed instead?

Try it again with fixed $F_R$! (What variables is $x_L$ a function of now?)

Wrap-up: this is a good opportunity to discuss the different kinds of representations we use in thermodynamics, such as experiments, tables, graphs, and symbols. This can be open-ended and student generated, but a good conclusion is that in each case we want to represent certain information like which variables depend on which other variables and which things are being held constant.

### Activity: Thermodynamic States I

Activity Highlights

1. This small group activity is designed to teach students that the set of numbers that describes a state is unique and can be located on different representations of a thermodynamic system.
2. Students work in small groups to reason about how many states can and must be specified to indicate a state.
3. The whole class wrap-up discussion emphasizes how many pieces of information are needed to communicate a state.

### Homework

1. (mbIntensiveExtensive) Categorize extensive and intensive variables for a 1-D system

In class, we divided the PDM variables $U$, $x_L$, $F_L$, $x_R$, and $F_R$, and the thermodynamic variables $T$, $U$, $V$, $p$, into categories of intensive and extensive. Consider a one-dimensional object such as a stretched rubber band. Categorize as intensive or extensive its properties of length, tension, mass and internal energy.

2. (mbTranslateContours) Translate from one contour diagram to another.

Consider the diagram of $T$ vs $V$ for several different constant values of $p$.

\begin{figure}[ht] \begin{center} \includegraphics[height=.85\textwidth,angle=270]{\TOP Figures/TV}\\ \end{center} \end{figure}

1. Translate this diagram to a $p$ vs $V$ w/ constant $T$ graph, including the point $A$. Please print out the provided frame and complete your graph by hand.

\begin{figure}[ht] \begin{center} \includegraphics[height=.85\textwidth,angle=270]{\TOP Figures/PV}\\ \end{center} \end{figure}

2. Are the lines that you drew straight or curved? What feature of the $T$ vs. $V$ graph would have to change to change this result?

3. Sketch the line of constant temperature that passes through the point $A$.

4. What are all the thermodynamic values associated with the point A? More generally, what does this (or any other) point represent?

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