\magnification=\magstep1
\def\ven{\vfill\eject\noindent}
%%For print-friendly, comment out the two lines above 
%%and uncomment the two lines below.
%\def\ven{\bigskip\noindent}
%\def\break{\ }
%\voffset=-0.5 true in
\def\dbar{{\mathchar'26\mkern-12mud}}
\vsize = 10.5true in
\hsize=6.5 true in
\nopagenumbers
\centerline{ASSIGNMENT 9}
\smallskip
\centerline{CHEMISTRY 200}
\smallskip
\centerline{Due:  4:30 pm Friday 21 November 2008}
\bigskip
\centerline{\bf The Cycle Assignment}
\medskip\noindent
\item{1.}A statement of the first law is:
$$\Delta U = w_{ad}\ \ {\rm and} \ \ dU = \dbar w_{ad}.$$
The complete differential of U is:
$$dU = \left(\partial U \over \partial T \right)_V dT + \left(\partial U \over
\partial V \right)_T dV$$
where
$$\left( \partial U \over \partial V\right)_T = T\left(\partial p \over
\partial T\right)_V - p.$$

\smallskip\noindent
\item{\ }
Use these to find an expression relating the initial temperature and volume of
a van der Waals gas to the final temperature and volume of the gas undergoing
adiabatic reversible work.  Assume the heat capacity at constant volume of the
gas is $C_v = {5\over 2} nR$.
\medskip\noindent
\item{2.}
Consider a four step cycle which is carried out reversibly.
\smallskip
\itemitem{Step 1:} 
State A to State B.  6.06 mole of a perfect gas, initially at 5.26 atm, is
expanded isothermally at 64.0$^\circ$C to twice its volume.
\smallskip
\itemitem{Step 2:}
State B to State C.  The gas is cooled adiabatically to 7.00$^\circ$C.
\smallskip
\itemitem{Step 3:}
State C to State D.  The gas is compressed isothermally to state D.
\smallskip
\itemitem{Step 4:}
State D to State A.  The gas is heated adiabatically.
\medskip
\itemitem{(a)}  
Calculate p, V, and T at each point A, B, C, D.  (Hint: State D is at the
intersection of the isotherm through C and the adiabat through A.)
\smallskip
\itemitem{(b)}  
Calculate q, w, $\Delta$U, and $\Delta$H for each step of the cycle and for
the entire cycle.
\smallskip
\itemitem{(c)} 
What fraction of the heat transferred in the expansion steps is converted to
work over the cycle?
\medskip\noindent
\item{3.}
Consider a four step cycle which is carried out reversibly.
\smallskip
\itemitem{Step 1:} 
State A to State B.  0.297 mole of a van der Waals gas with $a= 0.1657$ dm$^6$
atm mol$^{-2}$, $b = 58.10$ cm$^3$ mol$^{-1}$, and $C_v ={5\over 2} nR$ ,
initially at a volume of 83.6 L, is expanded isothermally at 90.3$^\circ$C to
twice its initial volume.
\smallskip
\itemitem{Step 2:}
State B to State C.  The gas is expanded adiabatically to five times the
volume it had in state A.
\smallskip
\itemitem{Step 3:}
State C to State D.  The gas is compressed isothermally to state D.
\smallskip
\itemitem{Step 4:}
State D to State A.  The gas is heated adiabatically.
\medskip
\itemitem{(a)}  
Calculate p, V, and T at each point A, B, C, D.  (Hint: State D is at the
intersection of the isotherm through C and the adiabat through A.)
\smallskip
\itemitem{(b)}  
Calculate q, w, $\Delta$U, and $\Delta$H for each step of the cycle and for
the entire cycle.
\smallskip
\itemitem{(c)} 
What fraction of the heat transferred in the expansion steps is converted to
work over the cycle?
\end
\vfill\eject
\noindent{\bf Name:\underbar{\hskip 2.5 true in}\hfill  Student Number:
\underbar{\hskip 1.3 true in}}
\bigskip\bigskip
\vbox
{\tabskip=0pt\offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to 395pt{\strut#&\vrule#\tabskip=1em plus 2em&
\hfil#& \vrule#& \hfil#\hfil& \vrule#&
\hfil#& \vrule#& \hfil#\hfil& \vrule#&
\hfil#& \vrule# \tabskip=0pt\cr\tablerule
%&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
%&&State&&A&&B&&C&&&\cr\tablerule
%&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
%&&p&&&&&&&&&\cr\tablerule
%&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
%&&V&&&&&&&&&\cr\tablerule
%&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
%&&T&&&&&&&&&\cr\tablerule
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&&&&1&&2&&3&&4&\cr\tablerule
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&&A&&&&&&&&&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\tablerule
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&&B&&&&&&&&&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\tablerule
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&&C&&&&&&&&&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr
\tablerule
&\height&\omit&&\omit&&\omit&&\omit&&\omit&\cr

\end




