238 lines
7.5 KiB
TeX
238 lines
7.5 KiB
TeX
\documentclass[fontsize=8pt, paper=a4, pagesize, DIV=calc]{scrartcl}
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\input{../Base.tex}
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\title{Formulaire de Physique II}
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\begin{document}
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\begin{tabu}to \textwidth{ |X|X| }
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\hline
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\textbf{Potentiels} \newline
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$ F_x = -\frac{\partial U}{\partial x} $ \newline
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$ \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x} $
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&
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\textbf{Lagrange} \newline
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$ U = \sum m \cdot g \cdot h + \sum \frac{1}{2} \cdot k \cdot x^2 $ \newline
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$ T = \sum \frac{1}{2} \cdot m \cdot v^2 + \sum \frac{1}{2} \cdot I \cdot \omega^2 $ \newline
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$ L = T -U $ \newline
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$ \frac{\dif}{\dif t} \left( \frac{\partial}{\partial \dot{q_j}} L \right) - \frac{\partial}{\partial q_j} L = 0 $
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\\\hline
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\textbf{Gaz} \newline
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$ P \cdot V = n \cdot R \cdot T = N \cdot k_B \cdot T $ \hfill Parfait \newline
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$ \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
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&
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\textbf{Maxwell-Boltzmann} \newline
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$ P_i = \cte \cdot \e^{-\frac{E_i}{k_B \cdot T}} $ \newline
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$ \sum P_i = 1 $
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\\\hline
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\textbf{Lois thermodynamiques} \newline
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$ \dif U = \delta W + \delta Q $ \hfill 1\textsuperscript{ère} \newline
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$ \dif S = \delta S_{ext} + \delta S_{int} = \frac{\delta Q}{T} + \delta S_{int} $ \hfill 2\textsuperscript{ème}
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&
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\textbf{Énergies} \newline
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$ U = \frac{f}{2} \cdot n \cdot R \cdot T $ \newline
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$ H = U + P \cdot V = \frac{f}{2} \cdot n \cdot R \cdot T + n \cdot R \cdot T $
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\\\hline
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\textbf{Isentropie} \newline
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$ P \cdot V^\gamma = \cte $ \newline
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$ T \cdot V^{\gamma - 1} = \cte $
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&
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\textbf{Énergies II} \newline
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$ U = C_v \cdot \Delta T $ \newline
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$ Q = C_v \cdot \Delta T $ \hfill Isochore \newline
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$ Q = C_p \cdot \Delta T $ \hfill Isobare \newline
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$ W = - \int p_{ext} \cdot \dif V = -W_{ext} $
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\\\hline
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\textbf{Chaleurs} \newline
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$ C_p = C_v \cdot \gamma $ \newline
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$ C_p = C_v + n \cdot R $ \newline
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$ C_v = \frac{\partial U}{\partial T} = \frac{n \cdot R}{\gamma -1} $ \newline
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$ C_p = \frac{\partial H}{\partial T} = \frac{\gamma \cdot n \cdot R}{\gamma -1} $
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&
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\textbf{Rendements} \newline
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$ \eta_{Carnot} = \frac{T_c - T_f}{T_c} $ \newline
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$ \eta = -\frac{W}{Q_c} $ \hfill Moteur \newline
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$ \eta = -\frac{Q_c}{W} $ \hfill Récepteur chauffant \newline
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$ \eta = \frac{Q_f}{W} $ \hfill Récepteur refroidissant
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\\\hline
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\textbf{Cycle} \newline
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$ \circlearrowright $ Cycle moteur \newline
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$ \circlearrowleft $ Cycle récepteur
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&
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\textbf{Cycle II} \newline
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$ \Delta U = 0 = W + Q_c + Q_f $ \newline
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$ \Delta S = 0 = \int \frac{\delta Q_c}{T} + \int \frac{\delta Q_f}{T} + S_{int} $ \newline
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$ W = - \left( Q_c + Q_f \right) $
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\\\hline
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\textbf{Conductibilité} \newline
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$ \lambda = \frac{1}{\rho \cdot 4 \cdot \sqrt{2} \cdot \pi \cdot R^2} $ \newline
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$ \rho = \frac{p}{k_B \cdot T} $ \newline
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$ J_Q = -k \cdot \frac{\partial T}{\partial x} $ \newline
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$ \frac{\partial Q}{\partial T} = A \cdot \alpha \cdot \frac{\partial T}{\partial x} $ \hfill $ \lambda \ll d $ \newline
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$ \frac{\partial Q}{\partial T} = \dif A \cdot \kappa \cdot \Delta T $ \hfill $ \lambda \gg d $
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&
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\textbf{Diffusion} \newline
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$ \frac{\partial \rho \cdot u}{\partial t} + \frac{\partial J_U}{\partial x} = \sigma_U $ \newline
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$ J_U = -\lambda \cdot \frac{\partial T}{\partial x} $ \newline
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$ \frac{\partial \rho \cdot u}{\partial t} - \lambda \cdot \frac{\partial^2 T}{\partial x^2} = \sigma_U $
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\\\hline
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\textbf{Lennard-Jones} \newline
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$ E = 4 \cdot \varepsilon_0 \cdot \left( \left( \frac{r_1}{r} \right)^{12} - \left( \frac{r_1}{r} \right)^6 \right) $ \newline
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$ E = \varepsilon_0 \cdot \left( \left( \frac{r_0}{r} \right)^{12} - 2 \cdot \left( \frac{r_0}{r} \right)^6 \right) $
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&
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\textbf{Lennard-Jones II} \newline
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\includegraphics[width=0.2\textwidth, keepaspectratio=true]{./Potentiel de Lennard-Jones.png}
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\\\hline
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\end{tabu}
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\nointerlineskip
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\begin{tabu}to \textwidth{ |X|X|X| }
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\textbf{Diagramme de phase} \newline
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\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme de phase.png}
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&
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\textbf{Diagramme P-V} \newline
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\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme P-V.png}
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&
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\textbf{Diagramme P-T} \newline
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\includegraphics[width=0.3\textwidth, keepaspectratio=true]{./Diagramme P-T.png}
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\\\hline
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\end{tabu}
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\begin{tabu}to \textwidth{ |X|X|X|X|X| }
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\hline
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\textit{Résultats uniquement pour le cas réversible} & Isotherme & Isobare & Isochore & Adiabatique
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\\\hline
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Constantes &
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$ \begin{aligned} P \cdot V = \cte \end{aligned} $ &
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$ \begin{aligned} \frac{V}{T} = \cte \end{aligned} $ &
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$ \begin{aligned} \frac{P}{T} = \cte \end{aligned} $ &
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$ \begin{aligned} P \cdot V^\gamma = \cte \\ T \cdot V^{\gamma - 1} = \cte \end{aligned} $
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\\\hline
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Énergie interne &
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$
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\begin{aligned}
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\Delta U & = 0
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\end{aligned}
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$ &
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$
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\begin{aligned}
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\Delta U & = C_v \cdot \Delta T \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
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& = \frac{p_0}{\gamma - 1} \cdot \Delta V \\
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& = C_v \cdot \frac{T_0}{V_0} \cdot \Delta V
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\end{aligned}
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$ &
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$
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\begin{aligned}
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\Delta U & = C_v \cdot \Delta T \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
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& = \frac{V_0}{\gamma - 1} \cdot \Delta p \\
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& = C_v \cdot \frac{T_0}{p_0} \cdot \Delta p
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\end{aligned}
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$ &
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$
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\begin{aligned}
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\Delta U & = C_v \cdot \Delta T \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
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& = \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \cdot \Delta \left( V^{1-\gamma} \right)
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\end{aligned}
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$
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\\\hline
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Chaleur &
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$
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\begin{aligned}
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Q & = n \cdot R \cdot T_0 \cdot \ln \left( \frac{V_1}{V_0} \right) \\
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& = n \cdot R \cdot T_0 \cdot \ln \left( \frac{p_1}{p_0} \right) \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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Q & = C_p \cdot \Delta T \\
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& = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \Delta T \\
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& = \frac{\gamma \cdot p_0}{\gamma - 1} \cdot \Delta V \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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Q & = C_v \cdot \Delta T \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
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& = \frac{V_0}{\gamma - 1} \cdot \Delta p \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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Q & = 0
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\end{aligned}
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$
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\\\hline
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Travail &
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$
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\begin{aligned}
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W & = -n \cdot R \cdot T_0 \cdot \ln \left( \frac{V_1}{V_0} \right) \\
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& = -n \cdot R \cdot T_0 \cdot \ln \left( \frac{p_1}{p_0} \right) \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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W & = -p_0 \cdot \Delta V \\
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& = -n \cdot R \cdot \Delta T \\
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& = -p_0 \cdot \frac{V_0}{T_0} \cdot \Delta V \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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W & = 0 \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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W & = C_v \cdot \Delta T \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \Delta T \\
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& = \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \cdot \Delta \left( V^{1-\gamma} \right)
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\end{aligned}
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$
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\\\hline
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Entropie &
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$
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\begin{aligned}
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\Delta S & = n \cdot R \cdot \ln \left( \frac{V_1}{V_0} \right) \\
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& = n \cdot R \cdot \ln \left( \frac{p_1}{p_0} \right) \\
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\end{aligned}
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$ &
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$
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\begin{aligned}
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\Delta S & = C_p \cdot \ln \left( \frac{V_1}{V_0} \right) \\
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& = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{V_1}{V_0} \right) \\
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& = C_p \cdot \ln \left( \frac{T_1}{T_0} \right) \\
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& = \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{T_1}{T_0} \right)
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\end{aligned}
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$ &
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$
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\begin{aligned}
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\Delta S & = C_v \cdot \ln \left( \frac{p_1}{p_0} \right) \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{p_1}{p_0} \right) \\
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& = C_v \cdot \ln \left( \frac{T_1}{T_0} \right) \\
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& = \frac{n \cdot R}{\gamma - 1} \cdot \ln \left( \frac{T_1}{T_0} \right)
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\end{aligned}
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$ &
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$
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\begin{aligned}
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\Delta S & = 0
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\end{aligned}
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$
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\\\hline
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\end{tabu}
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\end{document}
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