\subsection{Courbes de \tkzname{Van der Waals}} \bigskip Soient $v$ le volume d'une masse fluide et $p$ sa pression. $b$ et $k$ sont deux nombres réels strictement positifs. On souhaite étudier une formule exprimant la dépendance de ces variables proposée par Van~der~Waals. \[ p(v)= \frac{-3}{v^2} + \dfrac{3k}{v-b} \] définie sur l'intervalle $I=\big]b~;~+\infty\big]$ \subsubsection{Tableau de variations} \begin{center} \begin{tkzexample}[] \begin{tikzpicture} \tkzTab% { $v$ /1,% $g'(v)$ /1,% $g(v)$ /3% }% { $b$ ,% $3b$ ,% $+\infty$% }% {0,$+$,$0$,$-$,t} {-/ $0$ /,% +/$\dfrac{8}{27b}$ /,% -/ $0$ /}% \end{tikzpicture} \end{tkzexample} \end{center} \newpage \subsubsection{ Première courbe avec \emph{b}=1} Quelques courbes pour $r\leq\ v \leq\ 6$ \medskip \begin{center} \begin{tkzexample}[] \begin{tikzpicture}[xscale=2,yscale=2.5] \tkzInit[xmin=0,xmax=6,ymax=0.5,ystep=0.1] \tkzDrawX[label=$v$] \tkzDrawY[label=$g(v)$] \tkzGrid(0,0)(6,0.5) \tkzFct[color = red,domain =1:6]{(2*(x-1)*(x-1))/(x*x*x)} \tkzDrawTangentLine[color=blue,draw](3) \tkzDefPointByFct(1) \tkzText[draw, fill = brown!30](4,0.1){$g(v)=2\dfrac{(v-1)^2}{v^3}$} \end{tikzpicture} \end{tkzexample} \end{center} \newpage \subsubsection{ Deuxième courbe \emph{b}=1/3 } \medskip \begin{center} \begin{tkzexample}[] \begin{tikzpicture}[scale=1.2] \tkzInit[xmin=0,xmax=2,xstep=0.2,ymax=1,ystep=0.1] \tkzAxeXY \tkzGrid(0,0)(2,1) \tkzFct[color = red,domain =1/3:2]{(2*(\x-1./3)*(\x-1./3))/(\x*\x*\x)} \tkzDrawTangentLine[draw,color=blue,kr=.5,kl=.5](1) \tkzDefPointByFct(1) \tkzText[draw,fill = brown!30](1.2,0.3)% {$g(v)=2\dfrac{\left(v-\dfrac{1}{3}\right)^2}{v^3}$} \end{tikzpicture} \end{tkzexample} \end{center} \newpage \subsubsection{ Troisième courbe \emph{b}=32/27 } \medskip \begin{center} \begin{tkzexample}[] \begin{tikzpicture}[scale=1.2] \tkzInit[xmin=0,xmax=10,ymax=.35,ystep=0.05]; \tkzAxeXY \tkzGrid(0,0)(10,.35) \tkzFct[color = red, domain =1.185:10]{(2*(\x-32./27)*(\x-32./27))/(\x*\x*\x)} \tkzDrawTangentLine[draw,color=blue,kr=2,kl=2](3.555) \tkzText[draw,fill = brown!30](5,0.3)% {$g(v)=2\dfrac{\left(v-\dfrac{32}{27}\right)^2}{v^3}$} \end{tikzpicture} \end{tkzexample} \end{center} \newpage \subsection{Valeurs critiques} \subsubsection{Courbes de \tkzname{Van der Walls} } %<–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––> \begin{tkzexample}[] \begin{tikzpicture}[scale=4] \tkzInit[xmax=3,ymax=2]; \tkzAxeXY \tkzGrid(0,0)(3,2) \tkzFct[color = red,domain =1/3:3]{0.125*(3*\x-1)+0.375*(3*\x-1)/(\x*\x)} \tkzDefPointByFct[draw](2) \tkzDefPointByFct[draw](3) \tkzDrawTangentLine[draw,color=blue](1) \tkzFct[color = green,domain =1/3:3]{0.125*(3*x-1)} \tkzSetUpPoint[size=8,fill=orange] \tkzDefPointByFct[draw](3) \tkzDefPointByFct[draw](1/3) \tkzDefPoint(1,1){f} \tkzDrawPoint(f) \tkzText[draw,fill = white,text=red](1,1.5)% {$f(x)=\dfrac{1}{8}(3x-1)+\dfrac{3}{8}\left(\dfrac{3x-1}{x^2}\right)$} \tkzText[draw,fill = white,text=green](2,0.4){$g(x) = \dfrac{3x-1}{8}$} \end{tikzpicture} \end{tkzexample} \newpage \subsubsection{Courbes de \tkzname{Van der Walls} (suite)} \begin{tkzexample}[] \begin{tikzpicture}[xscale=4,yscale=1.5] \tkzInit[xmin=0,xmax=3,ymax=3,ymin=-4] \tkzGrid(0,-4)(3,3) \tkzAxeXY \tkzClip \tkzVLine[color=red,style=dashed]{1/3} \tkzFct[color=red,domain = 0.35:3]{-3/(x*x) +4/(3*x-1)} \tkzFct[color=blue,domain = 0.35:3]{-3/(x*x) +27/(4*(3*x-1))} \tkzFct[color=orange,domain = 0.35:3]{-3/(x*x) +8/(3*x-1)} \tkzFct[color=green,domain = 0.35:3]{-3/(x*x) +7/(3*x-1)} \tkzText[draw,fill = white,text=Maroon](2,-2)% {$f(x)=-\dfrac{3}{x^2}+\dfrac{8\alpha}{3x-1}$ \hspace{.5cm}% avec $\alpha \in% \left\{\dfrac{1}{2}~;~\dfrac{27}{32}~;~\dfrac{7}{8}~;~1\right\}$} \end{tikzpicture} \end{tkzexample} \endinput