236 lines
7.3 KiB
TeX
236 lines
7.3 KiB
TeX
\documentclass[a4paper,10pt]{article}
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%\documentclass[a4paper,10pt]{scrartcl}
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\input{common.tex}
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\begin{document}
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\begin{tabularx}{\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} $ \newline
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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{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial}{\partial \dot{q_j}} L \right) - \frac{\partial}{\partial q_j} L = 0 $ \newline
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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) (V -n \cdot b) = n \cdot R \cdot T $ \hfill Van der Waals \newline
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&
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\textbf{Maxwell-Boltzmann} \newline
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$ P_i = Cst \cdot e^{-\frac{E_i}{k_B \cdot T}} $ \newline
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$ \sum P_i = 1 $ \newline
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\\ \hline
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\textbf{Lois thermodynamiques} \newline
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$ \mathrm{d} U = \delta W + \delta Q $ \hfill 1\textsuperscript{ère} \newline
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$ \mathrm{d} S = \delta S_{ext} + \delta S_{int} = \frac{\delta Q}{T} + \delta S_{int} $ \hfill 2\textsuperscript{ème} \newline
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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 $ \newline
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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 $ \newline
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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 \mathrm{d}V = -W_{ext}$ \newline
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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} $ \newline
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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 \newline
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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 \newline
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&
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\textbf{Cycle} \newline
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$ \Delta U = 0 = W + Q $ \newline
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$ \Delta S = 0 = \int \frac{\delta Q_f}{T} + \int \frac{\delta Q_c}{T} + S_{int}$ \newline
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$ W = - (Q_c + Q_f) $ \newline
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\\ \hline
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\textbf{Condutibilité} \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} = \mathrm{d}A \cdot \kappa \cdot \Delta T $ \hfill $ \lambda \gg d $ \newline
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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 $ \newline
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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) $ \newline
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&
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\textbf{Lennard-Jones II} \newline
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\includegraphics[width=0.2\textwidth,keepaspectratio=true]{./lennard-jones.png} \newline
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\\ \hline
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\end{tabularx}
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\offinterlineskip
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\begin{tabularx}{\textwidth}{ |X|X|X| }
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\textbf{Diagramme de phase} \newline\newline
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\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./diagramme-phase.png} \newline
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&
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\textbf{Diagramme P-V} \newline\newline
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\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./diagramme-PV.png} \newline
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&
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\textbf{Diagramme P-T} \newline\newline
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\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./diagramme-PT.png} \newline
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\\\hline
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\end{tabularx}
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\begin{tabularx}{\textwidth}{ |l|X|X|X|X| }
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\hline
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& 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} \Delta T \\
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&= \frac{p_0}{\gamma - 1} \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} \Delta T \\
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&= \frac{V_0}{\gamma - 1} \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} \Delta T \\
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&= \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \Delta (V^{1-\gamma})
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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 \frac{V_1}{V_0} \\
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&= n \cdot R \cdot T_0 \cdot \ln \frac{p_1}{p_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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Q &= C_p \cdot \Delta T \\
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&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \Delta T \\
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&= \frac{\gamma \cdot p_0}{\gamma - 1} \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} \Delta T \\
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&= \frac{V_0}{\gamma - 1} \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 \frac{V_1}{V_0} \\
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&= -n \cdot R \cdot T_0 \cdot \ln \frac{p_1}{p_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 &= -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} \Delta T \\
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&= \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \Delta (V^{1-\gamma})
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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 \frac{V_1}{V_0} \\
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&= n \cdot R \cdot \ln \frac{p_1}{p_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 S &= C_p \cdot \ln \frac{V_1}{V_0} \\
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&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \frac{V_1}{V_0} \\
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&= C_p \cdot \ln \frac{T_1}{T_0} \\
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&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \frac{T_1}{T_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 S &= C_v \cdot \ln \frac{p_1}{p_0} \\
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&= \frac{n \cdot R}{\gamma - 1} \cdot \ln \frac{p_1}{p_0} \\
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&= C_v \cdot \ln \frac{T_1}{T_0} \\
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&= \frac{n \cdot R}{\gamma - 1} \cdot \ln \frac{T_1}{T_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 S &= 0
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\end{aligned}
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$
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\\\hline
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\end{tabularx}
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\end{document} |