%\section{{\Large Heat:} \smallskip \\ HC-1~~The Ideal Gas Law and Absolute Zero}
\section{HC-1~~The Ideal Gas Law and Absolute Zero}

\input{toolbar/h1_tb0}

\noindent OBJECTIVES:
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}To observe the behavior of an ``ideal'' gas (air) and to determine absolute zero (in Centigrade).
\end{enumerate}

\noindent PRECAUTIONS:
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}Liquid $N_2$  (go to the large Dewar in room 4329 Chamberlin) is fascinating to work with.  However, please keep in mind the following simple safety precautions.
\begin{enumerate}
\item \latex{\vspace*{-.05in}}{\bf{Never stopper a flask of liquid $N_2$ with an unperforated stopper. This can result in a dangerous explosion with risk of major injury. Students found attempting this will be referred to the Dean of Students for disciplinary action.}}
\item \latex{\vspace*{-.05in}}Have a perforated stopper on the Dewar throughout the experiment to prevent condensation of moisture from the air on the inside of the flask.
\item \latex{\vspace*{-.05in}}{\bf{Avoid contact of liquid $N_2$ with your skin.}} The insulating vapor may disappear and severe frost-bite may result.\latex{\vspace*{-.03in}}
\end{enumerate}
\end{enumerate}

\noindent
FUNDAMENTAL CONCEPTS: 
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}The behavior of an ideal gas under varying 
conditions of pressure and temperature is described by the ``Ideal Gas Law'':
%\marginpar{\flushleft THE IDEAL \\ GAS LAW }
$$ PV = nRT \eqno(1)$$ 
\noindent
Where  
$P$ is the pressure (in Pascals, $Pa$), 
$V$ is the volume (in $m^3$), $T$ is the temperature in Kelvins (K= C$ + 273$),
$n$ is the number of moles present (1 mole $\equiv 6.023 \times 10^{23} $),
and $R$ is the gas constant ($8.31~J/mol \cdot K$). \\
This equation is a combination of two laws which were discovered previously:

\begin{enumerate}
\item \latex{\vspace*{-.05in}}{\em{Boyle}}'s Law relates volume and pressure of a fixed quantity of gas at a constant temperature: $PV = constant$
\item \latex{\vspace*{-.05in}}{\em{Charles'}} Law relates volume and temperature of a fixed quantity of gas at a constant pressure $V/T = constant$\latex{\vspace*{-.03in}}
\end{enumerate}
Gay-Lussac finally combined the two laws into the ideal gas law shown in 
Eqn.~1.  \\ 
In EXPT.~I  Boyle's law will be verified by varying $V$ and $P$ (assuming fixed $T$). \\ 
In EXPT.~II You will vary $T$ and $V$ to determine the value of absolute zero.  \smallskip
\latex{\vspace*{-.05in}}
\end{enumerate}
 


\noindent APPARATUS: 
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}
{\it Basic equipment:} EXPT.~I - Lab stand; 60 cm$^3$ plastic syringe attached
to a plastic ``quick-connect'' coupling. \\
EXPT.~II - PASCO Steam generator; water jacket container; steel Dewar flask; small stainless steel can with a volume of $98$ cm$^3$ with a ``quick release'' connector; FLUKE digital thermometer and temperature probe 

\item[] \latex{\vspace*{-.04in}}{\it Computer equipment:}   Personal computer set to the HC1 lab manual web-page; PASCO interface module;  PASCO pressure sensor (mounted on lab stand).\latex{\vspace*{-.04in}}
\end{enumerate}


\noindent EXPERIMENT I:  BOYLE'S LAW 
\begin{enumerate} 
\item \latex{\vspace*{-.05in}}CLICK on the telescope icon below (web version only) to initiate the PASCO$^{\mbox{\copyright}}$ interface software. The computer, monitor and PASCO interface  must already be on and pressure sensor plugged into the {\bf A} DIN connector position.  (If not see your instructor.) 

\input{toolbar/h1_tb1}

\item \latex{\vspace*{-.05in}}The computer screen should look something like the figure \html{immediately below}  \latex{at top on the next page} .
\begin{figure}[h!] 
\centering 
\includegraphics[width=4.2in]{figs/h1_pasco.eps}\\
\vspace*{-.1in}\caption{Image of PASCO DataStudio display.}
\vspace*{-.1in}\end{figure}
In this window there is a graph configured to display pressure $vs$ volume and a table for these three values, a panel meter for the instantaneous pressure reading.  Since there is no automated volume measurement you will have to enter these data points manually.  After initiating the \includegraphics[height=0.18in]{figs/start_button.eps} button it will then change to the  \includegraphics[height=0.18in]{figs/keep_button.eps}.  CLICKing on this button will generate a pop-up window for data entry. To terminate the run you must CLICK on the red square immediately to the right
of the \includegraphics[height=0.18in]{figs/keep_button.eps}  icon.\\
NOTE:  The volume is recorded in  {\em milliliters} (1~mL = 1 cm$^3$) and the pressure in {\em kilopascals}  ($kPa$).  In this case the ideal gas law ($PV=nRT$, $T$ in Kelvin) uses $R = 8.31 \times 10^3$(kPa$\cdot$cm$^3$)/(mol$\cdot$K).

\item \latex{\vspace*{-.1in}}If the syringe is attached to the pressure sensor, disconnect it at the twist-lock quick connect coupling.  
Set the syringe plunger to the 60 cm$^3$ mark and then  reconnect it to the pressure sensor.  Make sure that the equipment looks as sketched in Fig.~2.
\begin{figure}[h!] 
\centering 
\includegraphics[width=3.2in]{figs/hc1-01.eps}\\
\vspace*{-.1in}\caption{Sketch of pressure sensor and syringe/plunger assembly.}
\vspace*{-.1in}\end{figure}

\item \latex{\vspace*{-.05in}}Initiate the data acquisition  by CLICKing on the \includegraphics[height=0.18in]{figs/start_button.eps} icon. (Data entry is enabled each time you CLICK the \includegraphics[height=0.18in]{figs/keep_button.eps} icon.)\\  Slowly push the plunger in to the 50 cm$^3$ position (about 10 seconds) and back to the original setting while watching the panel pressure meter and record in your lab-book the readout precision of the pressure sensor.

\item \latex{\vspace*{-.05in}} Now begin the actual data logging by again CLICKing the mouse on the KEEP icon.  Data logging should be confirmed by observing a new data point on the plots and new values in appropriate row of the table.  
%The PASCO software will try to {\em anticipate} your next volume setting in the Keyboard Sampling window. You should change this value to the appropriate value before logging your next data point.

\item \latex{\vspace*{-.05in}}Reduce the volume slowly in discrete 5 cm$^3$ increments while logging the data at each step (see the previous item) until you reach 20 cm$^3$.  Then as a check, slowly increase the volume in 5 or 10 cm$^3$ steps (remember to log the data) until you reach 60 cm$^3$.  

\item \latex{\vspace*{-.05in}}Stop the data acquisition and transfer your
data to a table in your lab write-up and comment on the reproducibility of your measurements.  You should make two plots, one with respect to volume and one with respect to {\em inverse} volume.  Do your plots have the correct functional behavior?  Your can refrain from printing out the graphs until your have performed a linear regression using either the PASCO graphical analysis capabilities or Excel  (or by whatever additional graphical analysis packages are available).

\item \latex{\vspace*{-.05in}}Which of the two graphs do you expect to be a
straight line and why?  For this ``curve'' what is the expected $y$ intercept? Now fit this data to a line (using linear regression) and record the slope
and intercepts with the appropriate units.  Print out the respective plots and include them in your lab write-up.

\item \latex{\vspace*{-.05in}}OPTIONAL:  Repeat the procedure of item 4 except move from 60 to 20 cm$^3$ as rapidly as possible and record the pressure data at the start and stop points.  Are the pressure readings consistent with those of the ``slow'' moving experiment?  If not can you suggest a reason?

\end{enumerate}
%\newpage
\noindent QUESTIONS:
\begin{enumerate} 
\item[I.] \latex{\vspace*{-.05in}}Does the line go through the origin as expected?  By how much should you change the pressure readings so that the line
goes through the origin?
\item[II.] What are the possible sources of error in this experiment?
For each source, try to decide what effect it might have on the experimental
results.

\item[III.] Read the barometer that is located  in your laboratory
room, ask your instructor for help if you are having trouble. Convert the
pressure from cm of Hg to $kPa$ and compare this value with the one
obtained at 60 cm$^3$ in step 5. Is the difference between the two readings important?  Does it affect your conclusions?

\item[IV.] OPTIONAL: Assuming that there are 22.4 liters per mole of an ideal gas, how well does your observed slope agree with your expected slope? 

\end{enumerate}

\noindent EXPERIMENT II:  IDEAL GAS LAW 
\begin{enumerate} 
\item[] \latex{\vspace*{-.05in}}In this part the Ideal Gas Law is verified by observing the pressure of a fixed volume of gas at four different temperatures: Room temperature, 100~$^\circ$C, 0~$^\circ$C and -197~$^\circ$C.  Since $PV=nRT$ and $T=T_{\mbox{\tiny Celsius}}+T_0$  we can rewrite this as

$$ P(T_c) = \frac{nR}{V} T_c  + \frac{nR}{V} T_0 $$ and  thereby determine the value of absolute zero ($T_0$) from the intercept. \\ 
\latex{\vspace*{-.05in}}\end{enumerate}

\noindent SUGGESTED PROCEDURE: 
\begin{enumerate} 
\item \latex{\vspace*{-.05in}}CLICK on the telescope icon below (web version) to initiate the PASCO$^{\mbox{\copyright}}$ interface software. The computer, monitor and PASCO interface  must already be on and pressure sensor plugged into the {\bf A} DIN connector position.  (If not see your instructor.) 

\input{toolbar/h1_tb2}

\item Attach the small stainless steel can (volume of $98$ cm$^3$) as shown below to the PASCO pressure sensor using the plastic ``quick release'' 
connector.   Raise the assembly up if necessary.   Turn on the Fluke thermometer.  Unless the stainless steel can has been recently used it should now be at room temperature.

\begin{figure}[h!] \centering 
\includegraphics[width=2.8in]{figs/hc1-02.eps} 
\vspace*{-.1in}\caption{Sketch of pressure sensor and stainless-steel can assembly.}
\end{figure} 
\vspace*{-.1in}

\item  \latex{\vspace*{-.05in}}Once again initiate the experiment by CLICKing on the \includegraphics[height=0.18in]{figs/start_button.eps} icon.  There should be a single plot (pressure vs. keyboard entry or, equivalently, temperature).  Record the Fluke meter reading by CLICKing on the  \includegraphics[height=0.18in]{figs/keep_button.eps} icon and then
entering the temperature in the keyboard entry window.

\item \latex{\vspace*{-.05in}}Fill the PASCO steam generator with water and turn it on. The level of the water should be $\sim 3 \ cm $ below the top, or about 1/2 -- 3/4 full (Fig. 3).  Turn the knob to 9 until the water boils and then turn it down somewhat.


\begin{figure}[h] \centering
\includegraphics[width=2.8in]{figs/l103/h02-2.eps}
\vspace*{-.1in}\caption{PASCO steam generator.}
\end{figure}

\vspace*{-.1in}

\item \latex{\vspace*{-.05in}}As soon as the water in the steamer is boiling, raise the pressure sensor stand and then lower it  so as to immerse  the stainless  steel can in the boiling water.

\item \latex{\vspace*{-.05in}}Wait until the water is boiling again and then put the Fluke thermometer in the boiling water.  Resume the data acquisition by clicking on the REC button.  Watch the pressure readout and after it is stable (with time) record the Fluke temperature in the data entry window.  Turn the PASCO steam generator off and remove the small stainless steel can from the bath.  Pause the data acquisition.

\item  \latex{\vspace*{-.05in}}Fill the water jacket container  with water and ice. The level of the water should be $\sim 5 \ cm $ below the top. Repeat the last two steps for obtaining the 0~$^\circ$C pressure reading. Empty and dry the stainless steel dewar.

\item \latex{\vspace*{-.05in}}Take the stainless steel dewar and  
ask the instructor to fill your dewar with liquid Nitrogen and repeat step 3 for the container at  liquid Nitrogen temperature.

\item \latex{\vspace*{-.05in}}Stop the data acquisition and transfer your
data to a table in your write-up.  Does your plot have the correct
functional behavior?  Your can refrain from printing out the data until
your have performed a linear regression using either the PASCO graphical
analysis capabilities or any other analysis package.

\item \latex{\vspace*{-.05in}}For your curve what is the expected $y$ intercept?   Now fit this data to a line (using linear regression) and record the slope and intercepts with the appropriate units.   Print out the plot and include it in your lab write-up.
\latex{\vspace*{-.08in}}
 \end{enumerate} 

\noindent QUESTIONS
\begin{enumerate} 
\item[I.] \latex{\vspace*{-.05in}}What is the percentage difference between the value you found  and the accepted value for absolute zero?
\item[II.] \latex{\vspace*{-.05in}}Assuming that ice water is exactly at 0~$^\circ$C and the boiling water is at 100~$^\circ$C, estimate the systematic error introduced into your absolute zero measurement?  Does this improve your results?
\item[III.] \latex{\vspace*{-.05in}}OPTIONAL: Qualitivity what error results from the gas in the small tube not being always at the can temperature?
\item[IV.] \latex{\vspace*{-.05in}}OPTIONAL: Does thermal expansion of the can affect your results?  In what way?

\end{enumerate} 
