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\centerline{\textbf{Rubber Band Pre-Lab}}
\bigskip


In this lab, we will measure the tension of a rubber band stretched to
a fixed length, as a function of temperature.  By measuring this at
several similar lengths, we will be able to determine the change in
internal energy $U$ and entropy $S$ for a rubber band that is
stretched at fixed temperature.

\begin{multicols}{2}
\paragraph{Materials:}
\begin{itemize}
\item One rubber band
\item Tube
\item Stopper with hook
\item Vernier force guage
\item Vernier temperature guage
\item Several clamps
\item Boiling water
\item Ice
\item Pan
\end{itemize}
\end{multicols}

\newcommand\inexactd{d\hspace{-0.9ex}^-\!}
\newcommand\hs{\hspace{6pt}}

\Large \textbf{Background}
\normalsize

The inexact differential of work for a system such as a rubber band
that is stretched in just one direction is
$$
  \inexactd W = -\tau dL.
$$
This follows naturally from the definition of work as force dotted
with distance---provided one takes into account the sign convention
for tension, which is opposite that of pressure.  Thus the
thermodynamic identity is:
$$
  dU = TdS + \tau dL
$$

We could use the thermodynamic identity directly, but since we are
working at constant $T$, it is more helpful to consider the Helmholtz
free energy
$$
  F \equiv U - TS
$$
The total differential of $F$ is
$$
  dF = \tau dL - SdT
$$
from which we can extract a Maxwell relation:
$$
  \frac{\partial^2F}{\partial L\partial T} =
  \frac{\partial^2F}{\partial T\partial L} 
$$
$$
  \left(\frac{\partial}{\partial L}
  \left(\frac{\partial F}{\partial T}\right)_{L}\right)_{T} =
  \left(\frac{\partial}{\partial T}
  \left(\frac{\partial F}{\partial L}\right)_T\right)_{L} 
$$
$$
  -\left(\frac{\partial S}{\partial L}\right)_{T} =
  \left(\frac{\partial \tau}{\partial T}\right)_{L}
$$
By measuring the variation of \emph{tension} with \emph{temperature}
at fixed \emph{length}, we can find out how \emph{entropy} will change
when the \emph{length} is changed at fixed \emph{temperature}!  At the
same time, a measurement of the tension will tell us how the free
energy will vary under the same isothermal stretch:
$$
  \tau = \left(\frac{\partial F}{\partial L}\right)_{T}
$$
Thus, provided we know $\tau$ and $\partial\tau/\partial T_{L}$ for our
rubber band, as a function of the length, we can integrate to find
$\Delta S$, $\Delta F$ and $\Delta U$ for an isothermal stretch.
%\hfill\textcolor{white}{.} % very hokey tex trick!
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  \includegraphics[width=2.5in]{eerubberbandhandfig1.png}
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\Large \textbf{The setup}

\normalsize
You will stretch your rubber band between a force meter and a hook in
the stopper in the bottom of a pipe.  If you attach the rubber band to
the force meter by means of a chain of paper clips, then you can
ensure that when the pipe is filled with water, the rubber band is
completely immersed.

You will need to insert a thermometer probe into the top of the pipe
in order to measure the temperature of your water---and thus the
temperature of your rubber band.

\bigskip

\Large \textbf{Collect data}

\normalsize
During the experiment you will pour water of various different
temperatures into the pipe in order to heat up your rubber band, and
record the tension as a function of length.  When changing to a
different temperature, you will need to empty your pipe and refill it.
Each time, you should measure the temperature at the top and bottom to
ensure that the water is well mixed.

\Large \textbf{Conclusions and Questions}

\normalsize
Please answer the following questions.  As always, show your work.  In
each case, discuss whether your result agrees with your predictions.
If it disagrees, please attempt to explain what was wrong with your
reasoning.

\textbf{Tension vs. temperature} \hs Plot the tension versus temperature
  for each of your lengths on the same plot.

\textbf{Tension vs. length} \hs \label{tvsl}Plot the tension versus
  length for a few temperatures. 

\textbf{$\left(\frac{\partial S}{\partial L}\right)_{T}$
  vs. length} \hs Plot $\left(\frac{\partial S}{\partial L}\right)_{T}$
  versus length for the same set of temperatures you chose for
  Question~\ref{tvsl}.

\begin{samepage}
\textbf{Isothermal stretch} \hs Pick a temperature and a range of lengths
  for which you have clean data.\label{isothermal-stretch}
  \begin{enumerate}
  \item What is the change in free energy for an isothermal stretch at
    this temperature from the smallest length to the largest?
  \item What is the change in entropy?
  \item What is the change in the internal energy?
  \item What is $Q$?
  \item What is $W$?
  \item What is $\left|Q/W\right|$?
  \end{enumerate}

\end{samepage}

\textbf{Adiabatic stretch} \hs What additional experiments would you
  need to perform in order to answer Question~\ref{isothermal-stretch}
  for a stretch that is adiabatic rather than isothermal? 
  
   
\vfill
\leftline{\textit{by David Roundy}}
\leftline{\copyright DATE David Roundy}
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