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# Energy and Integrals

## Prerequisites

Students should be able to:

- Manipulate differentials equations.
- Relate the net work done on a system to its change in internal energy.
- Integrate numerically using a graph or a table of data.

## In-class Content

- Potential Energy of an Elastic System (SGA - 50 min)
- Note from Paul - we didn't quite do all of this activity in 2017

- Lots of Ways to Change a State (Guided Activity: 10 min)

- New Surfaces activity - Quantifying Change in Thermal Systems (SGA - 10 min)

## Alternate In-class Content

## Homework for Energy and Entropy

- (ContoursAll)
*What goes here?*Consider the diagram of $p$ vs $V$ for different constant values of $T$ (blue) and of $S$ (green). The system under consideration is a gas in a piston–but not necessarily an ideal gas!

\begin{figure}[ht] \begin{center} \includegraphics[height=.85\textwidth]{\TOP Figures/PVST}\\ \end{center} \end{figure}

Suppose your system is in an initial state (A) defined by $T$ = 400 K and $S$ = 615 entropy units. What are the values of $p$ and $V$ for the system? Explain why these values are well defined here.

Calculate the work required to expand the gas from the initial state (A) to a new state with $V = 0.5 m^3$ at constant entropy. Then, calculate the work required to expand the gas from this state to a final state (B) with $V = 0.9 m^3$ at constant temperature. What is the total work to go from A to B along this path?

If you were instead to expand the gas from the initial state (A) to $V = 0.55 m^3$ at constant temperature, and then from this state to the same final state (B) $V = 0.9 m^3$ at constant entropy, is the total work

*greater than, less than,*or*equal to*the total work you calculated previously? Give an answer in words and then support it with a calculation.Shown below is a contour graph of the total energy of this system. Use this graph to determine the change in total energy of the system between the initial state (A) and the final state (B) you considered in each of the previous problems. Be as precise as you can.

\begin{figure}[ht] \begin{center} \includegraphics[height=.85\textwidth]{\TOP Figures/Energy}\\ \end{center} \end{figure}

You considered two quantities in this problem (work and energy). Which of these quantities depends on the path taken between two states? Explain why your answer makes sense.