\documentclass[11pt,fleqn]{article}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% template that does not use Revtex4
%%% but allows special fonts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site 
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Page setup
\topmargin -1.5cm      
\oddsidemargin -0.04cm   
\evensidemargin -0.04cm  
\textwidth 16.59cm
\textheight 24cm 
\setlength{\parindent}{0cm} 
\setlength{\parskip}{0cm} 


% Fonts
\usepackage{latexsym}
\usepackage{amsmath} 
\usepackage{amssymb} 
\usepackage{bm}


% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}

% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}

% Mass symbol
\newcommand{\mass}{\mathsf{m}} 
\newcommand{\Mass}{\mathsf{M}} 

% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}} 
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}

% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}

% heading
\newcommand{\exnumber}[1]{\newcommand{\exnum}{#1}}
\newcommand{\heading}[1]{\begin{center} {\Large {\bf Ex\exnum:} #1} \end{center}}
\newcommand{\auname}[1]{\begin{center} {\bf Submitted by:} #1 \end{center}}


\begin{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exnumber{1814}
\heading{Adiabatic versus sudden expansion of an ideal gas}
\auname{Moamen Jbara}


{\bf The problem:}\\

%Baruch's  A14

N atoms of mass $m$ of an ideal classical gas are in a
cylinder with insulating walls, closed at one end by a piston. The
initial volume and temperature are $V_0$ and $T_0$, respectively.
\begin{itemize}
	\item[(a)] If the piston is moving out rapidly the atoms cannot
	perform work, i.e. their energy is constant. Find the condition on
	the velocity of the piston that justifies this result.
	\item[(b)] Find the change in temperature, pressure and entropy if
	the volume increases from $V_0$ to $V_1$ under the conditions
	found in (a).
	\item[(c)] Find the change in temperature, pressure and entropy if
	the volume increases from $V_0$ to $V_1$ with the piston moving
	very slowly, i.e. an adiabatic process.\\
\end{itemize}


\Dn\Dn


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf The solution:}

\Dn
(a) According to the equipartition law, in temperature T the kinetic energy of each particle equals $\frac{T}{2}$, so in order to reach the required situation in the question, the velocity of the piston have to be larger than than the thermal velcoity of the particles : \\


$v_p>v_{Th}=(\frac{T}{m})^{\frac{1}{2}}$  $\Rightarrow$ $v_p>(\frac{T}{m})^{\frac{1}{2}}$\\

%%%%%%%%%%%%%%%
\Dn

(b) It's given in (a) that the energy of the gas is constant.
The energy of an ideal gas is : 

\begin{equation}
E=\frac{3NT}{2}
\end{equation}


Since the energy is constant, we can write :\\
 
(*) $E_0=E_1$ $\Rightarrow$ $T_0=T_1$  \\ 

Therefore :
\begin{equation}
\Delta T=0
\end{equation}
The pressure is : 
\begin{equation}
P=\frac{NT}{V}=nT
\end{equation}
\begin{equation}
\Delta P=P_1-P_0=\frac{NT_1}{V_1}-\frac{NT_0}{V_0}
\end{equation}
We substitute equation (*) in the last equation and we get :
\begin{equation}
\Delta P=\frac{NT_1}{V_1}-\frac{NT_1}{V_0}
\end{equation}
Since $V_1>V_0$ $\Rightarrow$ $P_1<P_0$ $\Rightarrow$ $\Delta P<0$.

\begin{equation}
S=-(\frac{\partial F}{\partial T})_V=N(ln(\frac{1}{n{\lambda_{T_0}}^3})+\frac{5}{2})
\end{equation}
 When : $\lambda_T=(\frac{2\pi}{mT})^{\frac{1}{2}}$\\

 N and T are constants so $\lambda_T$ is also constant and only thing that changes is the volume V.
 
 \begin{equation}
 \Delta S=S_1-S_0=N(\ln(\frac{1}{n_1{\lambda_T}^3})+\frac{5}{2}) - N(\ln(\frac{1}{n_0{\lambda_T}^3})+\frac{5}{2}) = N(\ln(\frac{V_1}{V_0}))
 \end{equation}
 
 Since $V_1>V_0$ $\Rightarrow$ $\ln(\frac{V_1}{V_0})>0$ $\Rightarrow$ $\Delta S>0$.\\
 
 %%%%%%%%%%%%%
 \Dn
 
 (c)  It's given that the process is adiabatic, i.e  $\Delta Q=0$ and $\Delta S=0$.\\ 
 
 $\Delta S=0$ $\Rightarrow$  $S_0=S_1$.\\ 
 
 Because our gas is ideal, we can use equation (7) again (only N is constant) :\\
 
 \begin{equation}
 N(\ln(\frac{1}{n_1{\lambda_{T_1}}^3})+\frac{5}{2}) =  N(\ln(\frac{1}{n_0{\lambda_{T_0}}^3})+\frac{5}{2})\  \ 
  \Rightarrow T_1 = T_0(\frac{V_0}{V_1})^{\frac{2}{3}}
 \end{equation}
\begin{equation}
\Delta T=T_1-T_0= T_0(\frac{V_0}{V_1})^{\frac{2}{3}} - T_0=T_0((\frac{V_0}{V_1})^{\frac{2}{3}}-1)
\end{equation}\\
In the second transition, we used the expression for $T_1$ that we got from equation 8.\\\\
Since $V_0<V_1$ $\Rightarrow$  $\Delta T<0$.\\

\begin{equation}
\Delta P=P_1-P_0=\frac{NT_1}{V_1}-\frac{NT_0}{V_0}=N(\frac{T_1}{V_1}-\frac{T_0}{V_0})=\frac{NT_0}{V_0}((\frac{V_0}{V_1})^{\frac{5}{3}}-1)
\end{equation}\\

In the last transition, we used the expression for $T_1$ that we got from equation 8.\\

Since $V_0<V_1$ $\Rightarrow$  $\Delta P<0$.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document} 
