\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}

\input{../Base.tex}

\title{Formulaire de Physique II}

\begin{document}

\begin{tabu}to \textwidth{ |X|X| }
\hline

\textbf{Potentiels} \newline
  $ F_x = -\frac{\partial U}{\partial x} $ \newline
  $ \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x} $
&
\textbf{Lagrange} \newline
  $ U = \sum m \cdot g \cdot h + \sum \frac{1}{2} \cdot k \cdot x^2  $ \newline
  $ T = \sum  \frac{1}{2} \cdot m \cdot v^2 + \sum \frac{1}{2} \cdot I \cdot \omega^2 $ \newline
  $ L = T -U $ \newline
  $ \frac{\dif}{\dif t} \left( \frac{\partial}{\partial \dot{q_j}} L \right) - \frac{\partial}{\partial q_j} L = 0 $
\\\hline

\textbf{Gaz} \newline
  $ P \cdot V = n \cdot R \cdot T = N \cdot k_B \cdot T $ \hfill Parfait \newline
  $ \left( p + \frac{n^2 \cdot a}{V^2} \right) \left( V -n \cdot b \right) = n \cdot R \cdot T $ \hfill Van der Waals
&
\textbf{Maxwell-Boltzmann} \newline
  $ P_i = \cte \cdot \e^{-\frac{E_i}{k_B \cdot T}} $ \newline
  $ \sum P_i = 1 $
\\\hline

\textbf{Lois thermodynamiques} \newline
  $ \dif U = \delta W + \delta Q $ \hfill 1\textsuperscript{ère} \newline
  $ \dif S = \delta S_{ext} + \delta S_{int} = \frac{\delta Q}{T} + \delta S_{int} $ \hfill 2\textsuperscript{ème}
&
\textbf{Énergies} \newline
  $ U = \frac{f}{2} \cdot n \cdot R \cdot T $ \newline
  $ H = U + P \cdot V = \frac{f}{2} \cdot n \cdot R \cdot T + n \cdot R \cdot T $
\\\hline

\textbf{Isentropie} \newline
  $ P \cdot V^\gamma = \cte $ \newline
  $ T \cdot V^{\gamma - 1} = \cte $
&
\textbf{Énergies II} \newline
  $ U = C_v \cdot \Delta T $ \newline
  $ Q = C_v \cdot \Delta T $ \hfill Isochore \newline
  $ Q = C_p \cdot \Delta T $ \hfill Isobare \newline
  $ W = - \int p_{ext} \cdot \dif V = -W_{ext} $
\\\hline

\textbf{Chaleurs} \newline
  $ C_p = C_v \cdot \gamma $ \newline
  $ C_p = C_v + n \cdot R $ \newline
  $ C_v = \frac{\partial U}{\partial T} = \frac{n \cdot R}{\gamma -1} $ \newline
  $ C_p = \frac{\partial H}{\partial T} = \frac{\gamma \cdot n \cdot R}{\gamma -1} $
&
\textbf{Rendements} \newline
  $ \eta_{Carnot} = \frac{T_c - T_f}{T_c} $ \newline
  $ \eta = -\frac{W}{Q_c} $ \hfill Moteur \newline
  $ \eta = -\frac{Q_c}{W} $ \hfill Récepteur chauffant \newline
  $ \eta = \frac{Q_f}{W} $ \hfill Récepteur refroidissant
\\\hline

\textbf{Cycle} \newline
  $ \circlearrowright $ Cycle moteur \newline
  $ \circlearrowleft $ Cycle récepteur
&
\textbf{Cycle II} \newline
  $ \Delta U = 0 = W + Q_c + Q_f $ \newline
  $ \Delta S = 0 = \int \frac{\delta Q_c}{T} + \int \frac{\delta Q_f}{T} + S_{int} $ \newline
  $ W = - \left( Q_c + Q_f \right) $
\\\hline

\textbf{Conductibilité} \newline
  $ \lambda = \frac{1}{\rho \cdot 4 \cdot \sqrt{2} \cdot \pi \cdot R^2} $ \newline
  $ \rho = \frac{p}{k_B \cdot T} $ \newline
  $ J_Q = -k \cdot \frac{\partial T}{\partial x} $ \newline
  $ \frac{\partial Q}{\partial T} = A \cdot \alpha \cdot \frac{\partial T}{\partial x} $ \hfill $ \lambda \ll d $ \newline
  $ \frac{\partial Q}{\partial T} = \dif A \cdot \kappa \cdot \Delta T $ \hfill $ \lambda \gg d $
&
\textbf{Diffusion} \newline
  $ \frac{\partial \rho \cdot u}{\partial t} + \frac{\partial J_U}{\partial x} = \sigma_U $ \newline
  $ J_U = -\lambda \cdot \frac{\partial T}{\partial x} $ \newline
  $ \frac{\partial \rho \cdot u}{\partial t} - \lambda \cdot \frac{\partial^2 T}{\partial x^2} = \sigma_U $
\\\hline

\textbf{Lennard-Jones} \newline
  $ E = 4 \cdot \varepsilon_0 \cdot \left( \left( \frac{r_1}{r} \right)^{12} - \left( \frac{r_1}{r} \right)^6 \right) $ \newline
  $ E = \varepsilon_0 \cdot \left( \left( \frac{r_0}{r} \right)^{12} - 2 \cdot \left( \frac{r_0}{r} \right)^6 \right) $
&
\textbf{Lennard-Jones II} \newline
  \includegraphics[width=0.2\textwidth, keepaspectratio=true]{./Potentiel de Lennard-Jones.png}
\\\hline

\end{tabu}

\nointerlineskip

\begin{tabu}to \textwidth{ |X|X|X| }
\textbf{Diagramme de phase} \newline
  \includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme de phase.png}
&
\textbf{Diagramme P-V} \newline
  \includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme P-V.png}
&
\textbf{Diagramme P-T} \newline
  \includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme P-T.png}
\\\hline
\end{tabu}

\begin{tabu}to \textwidth{ |X|X|X|X|X| }
\hline
\textit{Résultats uniquement pour le cas réversible} & Isotherme & Isobare & Isochore & Adiabatique
\\\hline
Constantes &
$ \begin{aligned} P \cdot V = \cte \end{aligned} $ &
$ \begin{aligned} \frac{V}{T} = \cte \end{aligned} $ &
$ \begin{aligned} \frac{P}{T} = \cte \end{aligned} $ &
$ \begin{aligned} P \cdot V^\gamma = \cte \\ T \cdot V^{\gamma - 1} = \cte \end{aligned} $
\\\hline
Énergie interne &
$
\begin{aligned}
  \Delta U & = 0
\end{aligned}
$ &
$
\begin{aligned}
  \Delta U & = C_v \cdot \Delta T \\
           & = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
           & = \frac{p_0}{\gamma - 1} \cdot \Delta V \\
           & = C_v \cdot \frac{T_0}{V_0} \cdot \Delta V
\end{aligned}
$ &
$
\begin{aligned}
  \Delta U & = C_v \cdot \Delta T \\
           & = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
           & = \frac{V_0}{\gamma - 1} \cdot \Delta p \\
           & = C_v \cdot \frac{T_0}{p_0} \cdot \Delta p
\end{aligned}
$ &
$
\begin{aligned}
  \Delta U & = C_v \cdot \Delta T \\
           & = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
           & = \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \cdot \Delta \left( V^{1-\gamma} \right)
\end{aligned}
$
\\\hline
Chaleur &
$
\begin{aligned}
  Q & = n \cdot R \cdot T_0 \cdot \ln \left( \frac{V_1}{V_0} \right) \\
    & = n \cdot R \cdot T_0 \cdot \ln \left( \frac{p_1}{p_0} \right) \\
\end{aligned}
$ &
$
\begin{aligned}
  Q & = C_p \cdot \Delta T \\
    & = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \Delta T \\
    & = \frac{\gamma \cdot p_0}{\gamma - 1} \cdot \Delta V \\
\end{aligned}
$ &
$
\begin{aligned}
  Q & = C_v \cdot \Delta T \\
    & = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
    & = \frac{V_0}{\gamma - 1} \cdot \Delta p \\
\end{aligned}
$ &
$
\begin{aligned}
  Q & = 0
\end{aligned}
$
\\\hline
Travail &
$
\begin{aligned}
  W & = -n \cdot R \cdot T_0 \cdot \ln \left( \frac{V_1}{V_0} \right) \\
    & = -n \cdot R \cdot T_0 \cdot \ln \left( \frac{p_1}{p_0} \right) \\
\end{aligned}
$ &
$
\begin{aligned}
  W & = -p_0 \cdot \Delta V \\
    & = -n \cdot R \cdot \Delta T \\
    & = -p_0 \cdot \frac{V_0}{T_0} \cdot \Delta V \\
\end{aligned}
$ &
$
\begin{aligned}
  W & = 0 \\
\end{aligned}
$ &
$
\begin{aligned}
  W & = C_v \cdot \Delta T \\
        & = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
    & = \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \cdot \Delta \left( V^{1-\gamma} \right)
\end{aligned}
$
\\\hline
Entropie &
$
\begin{aligned}
  \Delta S & = n \cdot R \cdot \ln \left( \frac{V_1}{V_0} \right) \\
           & = n \cdot R \cdot \ln \left( \frac{p_1}{p_0} \right) \\
\end{aligned}
$ &
$
\begin{aligned}
  \Delta S & = C_p \cdot \ln \left( \frac{V_1}{V_0} \right) \\
           & = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{V_1}{V_0} \right) \\
           & = C_p \cdot \ln \left( \frac{T_1}{T_0} \right) \\
           & = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{T_1}{T_0} \right)
\end{aligned}
$ &
$
\begin{aligned}
  \Delta S & = C_v \cdot \ln \left( \frac{p_1}{p_0} \right) \\
           & = \frac{n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{p_1}{p_0} \right) \\
           & = C_v \cdot \ln \left( \frac{T_1}{T_0} \right) \\
           & = \frac{n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{T_1}{T_0} \right)
\end{aligned}
$ &
$
\begin{aligned}
  \Delta S & = 0
\end{aligned}
$
\\\hline
\end{tabu}

\end{document}