\section{HC-1~~Ideal~Gas~Law~and~Absolute~Zero}

\noindent OBJECTIVES: \smallskip \\ \pind{
 To measure the pressure coefficient of N$_2$; to use a constant
volume gas thermometer to calibrate a digital thermometer; to 
find absolute zero Celsius.}  \\

\noindent APPARATUS: \smallskip \\ \pind{ 
Constant volume gas thermometer having $\sim $~1 atm N$_2$ and
connected to a Hg manometer; steam boiler; ice bucket; ice (opposite 4411 
Sterling);  thermocouple-type thermometer with digital readout 
(accurate to $\pm~2.2^\circ$C);
access to a barometer \& correction charts.}

\begin{figure}[h] \hspace*{.3in}
\includegraphics[width=2.8in]{figs/l103/h02-2.eps} 
\includegraphics[width=2.9in]{figs/l103/h02-3.eps} 
\end{figure} 

\pind{
\noindent Figure~1 shows the constant volume thermometer and the open tube Hg manometer 
for measuring the {\bf gauge} pressure in the bulb.  
Since the Hg manometer measures only the {\bf difference} between 
bulb pressure and atmospheric,
one must also {\bf add the barometer reading} to obtain the {\bf absolute}
pressure in the bulb.  
You will also need the barometer reading to find the 
steam temperature from the b.p.~tables.

\noindent {\bf When the apparatus has been 
leveled}  {\bf (use bubble level and three thumb screws in the base)}, the meter 
scale is vertical; furthermore, the brass index ring has been set to define
a constant volume of N$_2$ gas in the bulb when one adjusts
the {\bf movable} arm of the Hg
manometer to position the Hg meniscus in the {\bf fixed} arm to be just
flush with the upper portion of a brass index ring.  This setting 
of the index ring corresponds to 30.00 cm on the 
vertical meter scale.  Hence the manometer reading is the height of the
{\bf movable} tube meniscus minus 30.00 cm.

\noindent To reduce gas volume not at bulb temperature,
we set the index ring close to the bulb's capillary 
connection but without having the mercury rise into 
the distorted junction.\\

\noindent QUESTION: {\bf Why should the Hg level not be in the capillary?} }  \\

\noindent INTRODUCTION: \smallskip \\ \pind{
The pressure coefficient, $\beta $, of a gas at constant volume is 
$\beta $~=~(1/P)($\partial $P/$\partial $T)$_{V}$.  For a temperature interval 
t~-~t$_0$ 
on the Celsius scale, this becomes 
\begin{equation}
 \beta ~=~\left( \frac{1}{P_0}\right) \left( \frac{P_{t}~-~P_0}{t~-~t_0}
\right). \;  \mbox{ Hence for} \;  t_0=0^\circ C,\; \mbox{we have} 
\nonumber \\
 P_{t}=P_0(1+\beta t)~~  (Celsius) 
\end{equation}
For an ideal gas at constant volume, $\beta $ is independent of temperature;  
on the absolute Kelvin scale 
P$_{T}$/P$_0$~=~T/T$_0$.  Let {T$_0$} correspond to 0$^\circ $C: then
{P$_{T}$/P$_0$~=~P$_{t}$/P$_0$}, and {T/T$_0$ (in Kelvins) = (1 + $\beta $t)}.
Hence {\bf absolute zero, (T=0), corresponds to 1~+~$\beta $t'~=~0 or} 
\begin{equation}
t'\; ^\circ \mbox{C}~=~-~1/\beta \end{equation}
For this experiment, N$_2$ is an
ideal gas to a few parts per thousand.} \\ 

\noindent PRECAUTIONS: 
\begin{enumerate} \epind{
\item[1)] Mercury vapor in high concentration and inhaled 
for an extended time can have serious physiological effects. 
See Appendix 6. {\bf Report any Hg spills.} 
\item[2)] When finished with a reading, lower the movable 
mercury arm? so Hg will not suck into the bulb as the temperature drops.
\item[3)] Stir well any bath surrounding the bulb (and Hg thermometer) 
or error will result from parts of the bath being at different temperatures.
{\bf Gases are poor heat conductors so before a reading
wait till the pressure stabilizes}.
If while taking a reading
you have trouble keeping the bath temperature constant, try 
averaging readings for temperatures rising and then falling.}
\end{enumerate}

\noindent SUGGESTED~PROCEDURES: 
\begin{enumerate} \epind{
\item {\bf Determine pressure P$_0$ at ice point}:  First lower Hg level so it 
will not suck into the capillary as bulb cools.  Before adding water, 
pack ice around bulb 
so that ice will still be at bottom after adding water; otherwise bottom 
water may warm to 4$^\circ $C.  As per 3) above:
{\bf wait till 
the pressure stabilizes} before taking a reading!  While waiting, 
take the {\bf barometer reading} needed for P$_0$.  See notes below about 
barometer readings and corrections.
{\bf After finding P$_0$, turn on the digital thermometer and insert its probe 
at least 6 cm
into the ice bath.} Stir the bath (why?); if after 5 seconds, 
the reading is not within 1$^\circ $ of 0$^\circ $C, notify the 
instructor.
For easy removal of the ice and to avoid breaking the bulb, add water to float 
the ice to container top.
\item {\bf Measure pressure P$_{t}$ at the steam point}.  Then look up 
for your 
corrected barometer reading (see below) the steam point (in Handbook of Chemistry and
Physics).  Also {\bf calibrate the digital thermometer at the steam 
point.} Is it within the $\pm $~2.2$^\circ $C claimed by the supplier?
\item {\bf Calculate $\beta $} from (1).
\item {\bf Calculate ``absolute zero'' on the Celsius scale} from (2).  Is the 
accepted value for absolute zero within your estimated uncertainty?
\item {\bf Calibrate the digital thermometer at two 
other values between 0$^\circ $ and 100$^\circ $C}.  Use the steam 
jacket as a water bath.  See {\bf PRECAUTION} 3) above.  How much off are the 
digital thermometer readings? }
\end{enumerate}

\noindent BAROMETER READINGS AND CORRECTIONS: 
\begin{enumerate} \epind{
\item {\bf Barometer~readings}:  First set the lower Hg level to just touch the 
ivory index. 
Then read the top of the Hg column; avoid parallax (see 
appendix 4).  If in doubt, read instructions posted by barometer or 
consult instructor.
\item {\bf Barometer~and~manometer~corrections}:  Since barometer and 
manometer are both at room temperature, changes in Hg density with 
temperature will cancel out in calculating the pressures ($\rho $gh)
used in Eq.~1.  Similarly omit capillary corrections, and correcting pressure 
readings to standard ``g''.

\noindent The boiling point of H$_2$0 is, however, a sensitive function 
of {\bf absolute} pressure and the
{\bf handbook tables of b.p.~vs~mm of Hg pressure units refer to 
{\it standard conditions}}; i.e. density of Hg at 0$^\circ $C and standard gravity 
(sea level, latitude 45$^\circ $).  While our gravity corrections 
are negligible, {P\bf to find the b.p.~of water from the 
handbook tables} first 
correct your {\bf barometer reading to 0$^\circ $C}.  
(See correction table
by the barometer or use Handbook of Chemistry and Physics).
{\bf Do {\it not} apply this temperature correction to the barometric pressures used in
Equation 1}. (Why?) }
\end{enumerate}
\pind{
\noindent {\bf QUESTION:}  Most bulbs now contain small droplets of Hg.  How 
large an error results? (The v.p.~of Hg is 0.27 mm at 
100$^\circ $C, but $<~2\times10^{-4}$~mm at 0$^\circ $C.)} \\

\noindent OPTIONAL: \begin{enumerate} \epind{
\item{A.}  What error results from gas in the capillary not being at 
bulb temperature?  Let V$_{B}$ be the volume of the bulb and V$_{C}$
be the 
volume above the meniscus and in the capillary.  As the bulb heats from 
0$^\circ $ to t$^\circ $C, gas flows from V$_{B}$ to V$_{C}$ raising the pressure in 
V$_{C}$ 
by P$_{t}$ - P$_{o}$.  If this gas were forced back into the bulb and warmed from 
room temperature (t$_{R}$) to {\bf t}, its partial pressure in the bulb would be 
$\Delta $P~=~f(P$_{t}$~-~P$_{o}$) where
\begin{equation}
f~=~\frac{V_{C}\left( 273~+~t\right) }{V_{B}\left( 273~+~t_{R}\right) } \mbox{
 and hence} \; \beta =\left( \frac{1 + f}{P_0}\right) \left( \frac{P_{t}
~-~P_0}{t}\right). \nonumber
\end{equation}
Estimate $V_{C}$ and $V_{B}$, find {\bf f} and then recompute 
$\beta $.

\item[B.] Estimate the relative error in $\beta $ which comes from ignoring 
the expansion of the Pyrex bulb.  
(Pyrex has $\alpha = 3.2 \times 10^{-6} / ^{\circ} \mbox{C}$) }
\end{enumerate}

\newpage

