Initial commit
This commit is contained in:
commit
2f817d6698
136
analyse-I.tex
Normal file
136
analyse-I.tex
Normal file
@ -0,0 +1,136 @@
|
|||||||
|
\documentclass[a4paper,10pt]{article}
|
||||||
|
%\documentclass[a4paper,10pt]{scrartcl}
|
||||||
|
|
||||||
|
\usepackage{xltxtra}
|
||||||
|
\usepackage{pbox}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{tabularx}
|
||||||
|
\usepackage[top=13pt, bottom=12pt, left=13pt, right=12pt]{geometry}
|
||||||
|
% \setromanfont[Mapping=tex-text]{Linux Libertine O}
|
||||||
|
% \setsansfont[Mapping=tex-text]{DejaVu Sans}
|
||||||
|
% \setmonofont[Mapping=tex-text]{DejaVu Sans Mono}
|
||||||
|
|
||||||
|
\title{}
|
||||||
|
\author{}
|
||||||
|
\date{}
|
||||||
|
|
||||||
|
\let\oldcdot\cdot
|
||||||
|
\let\oldbullet\bullet
|
||||||
|
\let\oldvec\vec
|
||||||
|
\let\olddot\dot
|
||||||
|
\let\oldddot\ddot
|
||||||
|
|
||||||
|
% \renewcommand{\cdot}{ }
|
||||||
|
\renewcommand{\bullet}{\circ}
|
||||||
|
% \renewcommand{\vec}{\mathbf}
|
||||||
|
% \renewcommand{\dot}[1]{\frac{\mathrm{d}#1}{\mathrm{d}t}}
|
||||||
|
% \renewcommand{\ddot}[1]{\frac{\mathrm{d^2}#1}{\mathrm{d}t^2}}
|
||||||
|
\newcommand{\ccdot}{ }
|
||||||
|
|
||||||
|
\setlength{\parindent}{0pt}
|
||||||
|
\setlength{\parskip}{0pt}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
% \maketitle
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X| }
|
||||||
|
\hline
|
||||||
|
|
||||||
|
\textbf{Polynômes de Taylor} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
e^x &= \sum\limits_{k=0}^\infty \frac{x^k}{k!}, &x \in \mathbb{R} \\
|
||||||
|
\sinh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
|
||||||
|
\cosh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
|
||||||
|
\sin(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
|
||||||
|
\cos(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
|
||||||
|
\ln(1+x) &= \sum\limits_{k=0}^\infty (-1)^{k+1} \frac{x^k}{k}, &x \in {]-1,1[} \\
|
||||||
|
\frac{1}{1+x} &= \sum\limits_{k=0}^\infty (-1)^{k} x^k, &x \in {]-1,1[} \\
|
||||||
|
\arctan(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1}, &x \in {]-1,1[} \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
&
|
||||||
|
\textbf{Intégrales} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
&\int \frac{f'(x)}{f(x)}\mathrm{d}x &&= \ln \left|f(x)\right| + C \\
|
||||||
|
&\int \frac{f'(x)}{1+f^2(x)}\mathrm{d}x &&= \arctan \left[f(x)\right] + C \\
|
||||||
|
&\int \left[f(x)\right]^\alpha f'(x) \mathrm{d}x &&= \frac{\left[f(x)\right]^{\alpha+1}}{\alpha + 1} + C, &\forall \alpha \neq -1 \\
|
||||||
|
&\int e^{f(x)} f'(x) \mathrm{d}x &&= e^{f(x)} + C \\
|
||||||
|
&\int \frac{f'(x)}{\sqrt{1-f^2(x)}}\mathrm{d}x &&= \arcsin \left[f(x)\right] + C \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
\\
|
||||||
|
\end{tabularx}
|
||||||
|
\offinterlineskip
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X|X| }
|
||||||
|
\hline
|
||||||
|
|
||||||
|
\textbf{Racine carrée complexe} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
w = u + vi, z = a + bi, z^2 = w \\
|
||||||
|
\begin{cases}
|
||||||
|
a^2 - b^2 &= u \\
|
||||||
|
2ab &= v \\
|
||||||
|
a^2 + b^2 &= \sqrt{u^2 + v^2} \\
|
||||||
|
\end{cases}
|
||||||
|
\end{aligned}$
|
||||||
|
&
|
||||||
|
\textbf{Somme géométrique} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
\sum\limits_{k=0}^n q^k &= \frac{1-q^{n+1}}{1-q} \\
|
||||||
|
\sum\limits_{k=0}^\infty q^k &= \frac{1}{1-q} \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
&
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Exponentielle} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
\cos(\theta) &= \frac{e^{i\theta} + e^{-i\theta}}{2} \\
|
||||||
|
\sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
|
||||||
|
\cosh(\theta) &= \frac{e^{\theta} + e^{-\theta}}{2} \\
|
||||||
|
\sinh(\theta) &= \frac{e^{\theta} - e^{-\theta}}{2} \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
&
|
||||||
|
\textbf{Exponentielle} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
\lim_{n \to \infty} \left(1 + \frac{L}{n}\right)^n &= e^L \\
|
||||||
|
\text{De manière\ générale :} \\
|
||||||
|
\lim_{x \to \infty} f(x) &= +\infty \\
|
||||||
|
\lim_{x \to \infty} f(x)h(x) &= L \\
|
||||||
|
\lim_{x \to \infty} \left[1 + h(x)\right]^{f(x)} &= e^L \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
&
|
||||||
|
\textbf{Trigonométrie} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
\cosh^2(x) - \sinh^2(x) = 1 \\
|
||||||
|
\cos^2(x) + \sin^2(x) = 1 \\
|
||||||
|
\sin(x+y) = \sin x \cos y + \cos x \sin y \\
|
||||||
|
\cos(x+y) = \cos x \cos y + \sin x \sin y \\
|
||||||
|
\sin x + \sin y = 2 \sin(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
|
||||||
|
\sin x - \sin y = 2 \sin(\frac{x-y}{2})\cos(\frac{x+y}{2}) \\
|
||||||
|
\cos x + \cos y = 2 \cos(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
|
||||||
|
\cos x - \cos y = -2 \sin(\frac{x+y}{2})\sin(\frac{x-y}{2}) \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\multicolumn{2}{|X|}{
|
||||||
|
\textbf{Angles particuliers} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
\cos(0) = 1 \quad &\cos(\frac{\pi}{6}) = \frac{1}{2} \quad &\cos(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\cos(\frac{\pi}{3}) = \frac{1}{2}\sqrt{3} \quad &\cos(\frac{\pi}{2}) = 0 \\
|
||||||
|
\sin(0) = 0 \quad &\sin(\frac{\pi}{6}) = \frac{1}{2}\sqrt{3} \quad &\sin(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\sin(\frac{\pi}{3}) = \frac{1}{2} \quad &\sin(\frac{\pi}{2}) = 1 \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
}
|
||||||
|
&
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\multicolumn{2}{|X|}{
|
||||||
|
\textbf{Convergence} \newline
|
||||||
|
$\begin{aligned}
|
||||||
|
&\int_M^\infty x^a e^{-bx} \mathrm{d}x &\text{ converge pour tout } a \in \mathbb{R} \text{ et tout } b > 0 \\
|
||||||
|
&\int_a^\infty \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p > 1 \quad (a > 0) \\
|
||||||
|
&\int_0^b \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p < 1 \\
|
||||||
|
\end{aligned}$ \newline
|
||||||
|
}
|
||||||
|
&
|
||||||
|
\\ \hline
|
||||||
|
\end{tabularx}
|
||||||
|
\end{document}
|
||||||
193
chimie.tex
Normal file
193
chimie.tex
Normal file
@ -0,0 +1,193 @@
|
|||||||
|
\documentclass[a4paper,10pt]{article}
|
||||||
|
%\documentclass[a4paper,10pt]{scrartcl}
|
||||||
|
|
||||||
|
\usepackage{xltxtra}
|
||||||
|
\usepackage{pbox}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{tabularx}
|
||||||
|
\usepackage{siunitx}
|
||||||
|
\usepackage[top=13pt, bottom=12pt, left=13pt, right=12pt]{geometry}
|
||||||
|
% \setromanfont[Mapping=tex-text]{Linux Libertine O}
|
||||||
|
% \setsansfont[Mapping=tex-text]{DejaVu Sans}
|
||||||
|
% \setmonofont[Mapping=tex-text]{DejaVu Sans Mono}
|
||||||
|
|
||||||
|
\title{}
|
||||||
|
\author{}
|
||||||
|
\date{}
|
||||||
|
|
||||||
|
\setlength{\parindent}{0pt}
|
||||||
|
\setlength{\parskip}{0pt}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
% \maketitle
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X| }
|
||||||
|
\hline
|
||||||
|
|
||||||
|
\textbf{Bohr / Hydrogène} \newline
|
||||||
|
$ E_{photon} = h \cdot \nu $ \newline
|
||||||
|
$ E_{n} = \frac{-B}{n^2} $ \newline
|
||||||
|
$ \Delta E = E_f - E_i = B \cdot \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right) $ \newline
|
||||||
|
$ \lambda = \frac{h}{m \cdot v} = \frac{c}{\nu} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Thermodynamique} \newline
|
||||||
|
$ \Delta_r H^0 = \sum n_P \cdot \Delta_f H^0_P - \sum n_R \cdot \Delta_f H^0_R $ \newline
|
||||||
|
$ \Delta_r S^0 = \sum n_P \cdot S^0_P - \sum n_R \cdot S^0_R $ \hfill Exoth. si $ \Delta_r H^0 < 0 $ \newline
|
||||||
|
$ \Delta_r G^0 = \sum n_P \cdot \Delta_f G^0_P - \sum n_R \cdot \Delta_f G^0_R $ \newline
|
||||||
|
$ \Delta_r G^0 = \Delta_r H^0 - T \cdot \Delta_r S^0 $ \hfill Spont. si $ \Delta_r G^0 < 0 $ \newline
|
||||||
|
$ \Delta S_{univers} = \Delta_r S^0 - \frac{\Delta_r H^0}{T} $ \newline
|
||||||
|
$ \Delta_r H^0 = \Delta_r U^0 + P \cdot \Delta V = \Delta_r U^0 + R \cdot T \cdot \Delta n $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Équilibres} \newline
|
||||||
|
$ K_c = \frac{\prod [P]^{n_P}}{\prod [R]^{n_R}} $ \newline
|
||||||
|
$ K_p = \frac{\prod P_P^{n_P}}{\prod P_R^{n_R}} $ \newline
|
||||||
|
$ K_c = K_p \cdot ( R \cdot T)^{-\Delta n} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Activités} \newline
|
||||||
|
$ a_i = \frac{P_i}{P_0} $ \hfill Gaz \newline
|
||||||
|
$ a_i = \frac{c_i}{c_0} $ \hfill Solutés \newline
|
||||||
|
$ a_i = 1 $ \hfill Liquides et solides \newline
|
||||||
|
$ K = \frac{\prod a_P^{n_P}}{\prod a_R^{n_R}} $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Équilibres II} \newline
|
||||||
|
$ \Delta_r G = \Delta_r G^0 + R \cdot T \cdot \ln(Q) $ \newline
|
||||||
|
$ \Delta_r G^0 = -R \cdot T \cdot \ln(K) $ \newline
|
||||||
|
$ \ln\left(\frac{K_{T_2}}{K_{T_1}}\right) = \frac{\Delta_r H^0}{R} \cdot \frac{T_2 - T_1}{T_2 \cdot T_1} $ \newline
|
||||||
|
$ \Delta n = \sum n_P - \sum n_R $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Acide-Base} \newline
|
||||||
|
$ K_a = \frac{[A^-][H_3O^+]}{[HA]} $ \newline
|
||||||
|
$ K_b = \frac{[HA][OH^-]}{[A^-]} $ \newline
|
||||||
|
$ pX = -\log([X]) $ \newline
|
||||||
|
$ pK_e = pK_a + pK_b = pH + pOH = 14 $ \hfill Eau \newline
|
||||||
|
$ \alpha = \sqrt{\frac{K_a}{M}} $ \hfill $ \alpha \leqslant 0.05 $ \newline
|
||||||
|
$ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) $ \hfill Solution tampon \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Électrochimie} \newline
|
||||||
|
$ n = \frac{I \cdot t}{z \cdot F} $ \newline
|
||||||
|
$ \eta = \frac{\Delta_r G^0}{\Delta_r H^0} $ \newline
|
||||||
|
$ \Delta E^0 = E^0_+ - E^0_- $ \hfill Spont. si $ \Delta E^0 > 0 $ \newline
|
||||||
|
$ \Delta_r G^0 = -z \cdot F \cdot \Delta E^0 $ \newline
|
||||||
|
$ \ln(K) = -\frac{\Delta_r G^0}{R \cdot T} = \frac{z \cdot F \cdot \Delta E^0}{R \cdot T} $ \newline
|
||||||
|
$ E_{Ox/Red} = E^0_{Ox/Red} + 2.3 \cdot \frac{R \cdot T}{z \cdot F} \cdot \log\left(\frac{[Ox]^{n_{Ox}}}{[Red]^{n_{Red}}}\right) $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Cinétique} \newline
|
||||||
|
$ v = -\frac{1}{n_R} \cdot \frac{\mathrm{d}[R]}{\mathrm{d}t} = \frac{1}{n_P} \cdot \frac{\mathrm{d}[P]}{\mathrm{d}t} $ \newline
|
||||||
|
$ \tau_{1/2} = \frac{\ln(2)}{k} $ \hfill Ordre 1 \newline
|
||||||
|
$ \tau_{1/2} = \frac{1}{k \cdot [A]_0} $ \hfill Ordre 2 \newline
|
||||||
|
$ k = A_f \cdot e^{-\frac{E_a}{R \cdot T}} $ \newline
|
||||||
|
$ \ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \cdot \left( \frac{1}{T_1} - \frac{1}{T_2} \right) $ \newline
|
||||||
|
$ \Delta_r H^0 = E_a^\rightarrow - E_a^\leftarrow $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Constantes} \newline
|
||||||
|
$ N_A = \SI{6.02e23}{mol^{-1}} $ \newline
|
||||||
|
$ h = \SI{6.63e-34}{J.s} $ \newline
|
||||||
|
$ B = \SI{2.179e-18}{J} $ \newline
|
||||||
|
$ F = \SI{96487}{C.mol^{-1}} $ \newline
|
||||||
|
$ R = \SI{0.0821}{L.atm.K^{-1}.mol^{-1}} $ \newline
|
||||||
|
$ R = \SI{0.0831}{L.bar.K^{-1}.mol^{-1}} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Conditions} \newline
|
||||||
|
Conditions normales : \SI{101.3}{kPa} et \SI{0}{°C} \newline
|
||||||
|
Conditions standard : \SI{1}{bar} et \SI{25}{°C} \newline \newline
|
||||||
|
$ R = \SI{8.314}{L.kPa.K^{-1}.mol^{-1}} $ \newline
|
||||||
|
$ R = \SI{8.314}{J.K^{-1}.mol^{-1}} $ \newline
|
||||||
|
$ R = \SI{8.314}{m^3.Pa.K^{-1}.mol^{-1}} $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\offinterlineskip
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X|X| }
|
||||||
|
\textbf{Loi de vitesse} & \textbf{Loi intégrée} & \textbf{Forme linéaire} \\
|
||||||
|
$ -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k $ \hfill Ordre 0 & $ [A]_t = [A]_0 - k \cdot t $ & $ [A]_t = [A]_0 - k \cdot t $ \\
|
||||||
|
$ -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k \cdot [A] $ \hfill Ordre 1 & $ [A]_t = [A]_0 \cdot e^{-k \cdot } $ & $ \ln([A]_t) = \ln([A]_0) - k \cdot t $ \\
|
||||||
|
$ -\frac{\mathrm{d}[A]}{\mathrm{d}t} = k \cdot [A]^2 $ \hfill Ordre 2 & $ [A]_t = \frac{[A]_0}{1 + k \cdot t \cdot [A]_0} $ & $ \frac{1}{[A]_t} = \frac{1}{[A]_0} + k \cdot t $ \newline \\
|
||||||
|
\hline
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
|
||||||
|
\offinterlineskip
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X| }
|
||||||
|
\textbf{Construction d'une molécule} \newline
|
||||||
|
\begin{itemize}
|
||||||
|
\item Dénombrer les électrons de valence de tous les atomes de la molécule ou de l’ion.
|
||||||
|
\item Dessiner le squelette de la molécule en relient les atomes les un aux autres par une pair d’électrons; l’atome le moins électronégatif occupe la place centrale.
|
||||||
|
\item Compléter les octets des atomes liés à l’atome central.
|
||||||
|
\item Placer les électrons restants sur l’atome centrale.
|
||||||
|
\item Si les nombres d’électrons disponibles est insuffisant, introduire des liaisons multiples et attribuer les charges de l’ion.
|
||||||
|
\end{itemize}
|
||||||
|
&
|
||||||
|
\textbf{Équilibrage d'une réaction} \newline
|
||||||
|
\begin{itemize}
|
||||||
|
\item Repérer les éléments dont le degré d’oxydation (DO) change au cours de la réaction.
|
||||||
|
\item Le nombre d’électrons cédés par le réducteur doit être égal au nombre d’électrons acquis par l’oxydant. Ceci permet de trouver quatre coefficients.
|
||||||
|
\item S’il figure dans l’équation d’autres substances dont le DO n’est pas modifié, le coefficient de ces substances est déterminé par un bilan de masse.
|
||||||
|
\item Si des réactifs et/ou des produits sont des ions, il faut vérifier le calcul par un bilan de charges.
|
||||||
|
\end{itemize}
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Formes} \newline
|
||||||
|
\begin{itemize}
|
||||||
|
\item Linéaire (sp).
|
||||||
|
\item Coudée (sp²).
|
||||||
|
\item Trigonale plane (sp²).
|
||||||
|
\item Pyramidale à base triangulaire (sp³).
|
||||||
|
\item Tétraèdrique (sp³).
|
||||||
|
\end{itemize}
|
||||||
|
&
|
||||||
|
\textbf{Nombres quantiques} \newline
|
||||||
|
\begin{itemize}
|
||||||
|
\item Principal : $ n \geqslant 1 $ \hfill Couche
|
||||||
|
\item Secondaire : $ 0 \leqslant l \leqslant n-1 $ \hfill Forme
|
||||||
|
\item Magnétique : $ -l \leqslant m_l \leqslant l $ \hfill Orientation
|
||||||
|
\item Spin : $ m_s = \pm 1/2 $ \hfill Sens de rotation sur lui-même
|
||||||
|
\end{itemize}
|
||||||
|
\\ \hline
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X| }
|
||||||
|
\hline
|
||||||
|
|
||||||
|
\textbf{Rayon atomique} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./rayon.png} \newline
|
||||||
|
&
|
||||||
|
\textbf{Électronégativité} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./electronegativite.png} \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Pouvoir oxydant} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./oxydant.png} \newline
|
||||||
|
&
|
||||||
|
\textbf{Énergie de ionisation} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./ionisation.png} \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Caractère métallique} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./metallique.png} \newline
|
||||||
|
&
|
||||||
|
\textbf{Résumé} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./relations.png} \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Géométrie} \newline\newline
|
||||||
|
\includegraphics[width=0.45\textwidth,keepaspectratio=true]{./geometrie.png} \newline
|
||||||
|
&
|
||||||
|
\textbf{Titrage} \newline\newline
|
||||||
|
{
|
||||||
|
\begin{tabularx}{\textwidth}{cc}
|
||||||
|
\includegraphics[width=0.22\textwidth,keepaspectratio=true]{./titrage-fort.png} \newline &
|
||||||
|
\includegraphics[width=0.22\textwidth,keepaspectratio=true]{./titrage-faible.png} \newline
|
||||||
|
\end{tabularx}
|
||||||
|
}
|
||||||
|
\\ \hline
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\end{document}
|
||||||
BIN
diagramme-PT.png
Normal file
BIN
diagramme-PT.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 37 KiB |
BIN
diagramme-PV.png
Normal file
BIN
diagramme-PV.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 337 KiB |
BIN
diagramme-phase.png
Normal file
BIN
diagramme-phase.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 238 KiB |
BIN
electronegativite.png
Normal file
BIN
electronegativite.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 240 KiB |
BIN
geometrie.png
Normal file
BIN
geometrie.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 72 KiB |
BIN
ionisation.png
Normal file
BIN
ionisation.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 169 KiB |
BIN
lennard-jones.png
Normal file
BIN
lennard-jones.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 62 KiB |
BIN
metallique.png
Normal file
BIN
metallique.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 109 KiB |
BIN
oxydant.png
Normal file
BIN
oxydant.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 91 KiB |
212
physique-I.tex
Normal file
212
physique-I.tex
Normal file
@ -0,0 +1,212 @@
|
|||||||
|
\documentclass[a4paper,10pt]{article}
|
||||||
|
%\documentclass[a4paper,10pt]{scrartcl}
|
||||||
|
|
||||||
|
\usepackage{xltxtra}
|
||||||
|
\usepackage{pbox}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{tabularx}
|
||||||
|
\usepackage[top=13pt, bottom=12pt, left=13pt, right=12pt]{geometry}
|
||||||
|
% \setromanfont[Mapping=tex-text]{Linux Libertine O}
|
||||||
|
% \setsansfont[Mapping=tex-text]{DejaVu Sans}
|
||||||
|
% \setmonofont[Mapping=tex-text]{DejaVu Sans Mono}
|
||||||
|
|
||||||
|
\title{}
|
||||||
|
\author{}
|
||||||
|
\date{}
|
||||||
|
|
||||||
|
\let\oldcdot\cdot
|
||||||
|
\let\oldbullet\bullet
|
||||||
|
\let\oldvec\vec
|
||||||
|
\let\olddot\dot
|
||||||
|
\let\oldddot\ddot
|
||||||
|
|
||||||
|
% \renewcommand{\cdot}{ }
|
||||||
|
\renewcommand{\bullet}{\circ}
|
||||||
|
% \renewcommand{\vec}{\mathbf}
|
||||||
|
% \renewcommand{\dot}[1]{\frac{\mathrm{d}#1}{\mathrm{d}t}}
|
||||||
|
% \renewcommand{\ddot}[1]{\frac{\mathrm{d^2}#1}{\mathrm{d}t^2}}
|
||||||
|
% \renewcommand{\frac}[2]{#1 / #2}
|
||||||
|
\newcommand{\cdotbis}{ }
|
||||||
|
|
||||||
|
\setlength{\parindent}{0pt}
|
||||||
|
\setlength{\parskip}{0pt}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
% \maketitle
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X|X| }
|
||||||
|
\hline
|
||||||
|
|
||||||
|
\textbf{Produits vectoriels} \newline
|
||||||
|
$ \vec{e}_x \times \vec{e}_y = -\vec{e}_y \times \vec{e}_x = \vec{e}_z $ \newline
|
||||||
|
$ \vec{e}_y \times \vec{e}_z = -\vec{e}_z \times \vec{e}_y = \vec{e}_x $ \newline
|
||||||
|
$ \vec{e}_z \times \vec{e}_x = -\vec{e}_x \times \vec{e}_z = \vec{e}_y $ \newline
|
||||||
|
$ \vec{e}_x \times \vec{e}_x = \vec{e}_y \times \vec{e}_y = \vec{e}_z \times \vec{e}_z = \vec{0} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{MRUA} \newline
|
||||||
|
$ r = \frac{1}{2} \cdot a_0 \cdot t^2 + v_0 \cdot t + r_0 $ \newline
|
||||||
|
$ v = a_0 \cdot t + v_0 $ \newline
|
||||||
|
$ a = a_0 $ \newline
|
||||||
|
&
|
||||||
|
\textbf{MCU} \newline
|
||||||
|
$ a = \frac{v^2}{r} = \omega^2 \cdot r$ \newline
|
||||||
|
$ \vec{v} = \vec{\omega} \times \vec{r} $ \newline
|
||||||
|
$ \vec{a} = \vec{\alpha} \times \vec{r} $ \newline
|
||||||
|
$ \omega \cdot T = 2 \cdot \pi $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Moments / Centre de masse} \newline
|
||||||
|
$ \vec{L}_O = \vec{r} \times \vec{p} = m \cdot \vec{r} \times \vec{v} $ \newline
|
||||||
|
$ \vec{M}_O = \vec{r} \times \vec{F} = \frac{\mathrm{d}\vec{L}_O}{\mathrm{d}t} $ \newline
|
||||||
|
$ \vec{r}_{cm} = \frac{1}{M} \int_{M} \vec{r} \cdot \mathrm{d}m = \frac{1}{M} \int_{V} \vec{r} \cdot \rho(\vec{r}) \cdot \mathrm{d}V $ \newline
|
||||||
|
$ I_{cm,\Delta} = \int_{M} r_\bot^2 \cdot \mathrm{d}m $ \newline
|
||||||
|
$ \vec{L}_{cm,\Delta} = I_{cm,\Delta} \cdot \vec{\omega} $ \newline
|
||||||
|
$ \vec{M}_{cm,\Delta} = I_{cm,\Delta} \cdot \vec{\alpha} $ \newline
|
||||||
|
$ I = I_{cm} + M \cdot r^2 $ \newline
|
||||||
|
$ \vec{r}_{cm} = \frac{1}{M} \sum m_i \cdot \vec{r}_i $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Forces} \newline
|
||||||
|
$ \vec{p} = m \cdot \vec{v} $ \newline
|
||||||
|
$ \vec{F} = m \cdot \vec{a} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} $ \newline
|
||||||
|
$ \vec{F}_f = \mu \cdot \vec{N} $ \newline
|
||||||
|
$ \vec{F}_f = -K \cdot \eta \cdot \vec{v} $ \newline
|
||||||
|
$ W = \int \vec{F} \bullet \mathrm{d}\vec{r} $ \newline
|
||||||
|
$ P_{inst} = \frac{\mathrm{d}W}{\mathrm{d}t} = \vec{F} \bullet \vec{v} $ \newline
|
||||||
|
$ P_{moy} = \frac{W}{\Delta t} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Énergie} \newline
|
||||||
|
$ W = \Delta E $ \newline
|
||||||
|
$ E_{mec} = E_{cin} + E_{pot} $ \newline
|
||||||
|
$ E_{mec,sat} = - \frac{G \cdot M \cdot m}{2 \cdot r} $ \newline
|
||||||
|
$ E_{cin} = \frac{1}{2} \cdot m \cdot v^2 $ \newline
|
||||||
|
$ E_{cin} = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot (A^2 - x^2) $ \newline
|
||||||
|
$ E_{cin} = \frac{1}{2} \cdot I_{cm,\Delta} \cdot \omega^2 $ \newline
|
||||||
|
$ E_{pot} = m \cdot g \cdot h $ \newline
|
||||||
|
$ E_{pot} = \frac{1}{2} \cdot k \cdot x^2 = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot x^2 $ \newline
|
||||||
|
$ E_{pot} = - \frac{G \cdot M \cdot m}{r} $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Référentiel non-galiléen} \newline
|
||||||
|
$ m \cdot \vec{a}' = \sum \vec{F}_{ext} - m \cdot \vec{a}_e - m \cdot \vec{a}_{Cor} $ \newline
|
||||||
|
$ - m \cdot \vec{a}_e = - m \cdot \vec{\omega} \times (\vec{\omega} \times \vec{r})$ \newline
|
||||||
|
$ - m \cdot \vec{a}_{Cor} = - 2 \cdot m \cdot \vec{\omega} \times \vec{v}' $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Balistique} \newline
|
||||||
|
$ h_{max} = \frac{(v_0 \cdot \sin(\alpha))^2}{2 \cdot g} $ \newline
|
||||||
|
$ p = \frac{v_0^2 \cdot \sin(2 \cdot \alpha)}{g} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Intégrales volumiques} \newline
|
||||||
|
$ V = \iiint\limits_{cube} \mathrm{d}V = \iiint \mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z $ \newline
|
||||||
|
$ V = \iiint\limits_{cylindre} \mathrm{d}V = \iiint \rho \cdot \mathrm{d}\rho \cdot \mathrm{d}\varphi \cdot \mathrm{d}z $ \newline
|
||||||
|
$ V = \iiint\limits_{boule} \mathrm{d}V = \iiint r^2 \cdot \sin(\theta) \cdot \mathrm{d}r \cdot \mathrm{d}\theta \cdot \mathrm{d}\varphi $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Kepler} \newline
|
||||||
|
$ \frac{a^3}{T^2} = \frac{G \cdot M}{4 \cdot \pi^2} $ \hfill 1\textsuperscript{ère} loi \newline
|
||||||
|
$ \frac{\mathrm{d}\vec{A}}{\mathrm{d}t} = \frac{1}{2} \cdot \vec{r} \times \vec{v} = \frac{\vec{L}_O}{2 \cdot m} $ \hfill 2\textsuperscript{ème} loi \newline
|
||||||
|
$ \vec{F} = - \frac{G \cdot M \cdot m}{r^2} \cdot \vec{u_r} $ \hfill 3\textsuperscript{ème} loi \newline
|
||||||
|
$ T = 2 \cdot \pi \sqrt{\frac{R^3}{G \cdot M}} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Dérivées usuelles} \newline
|
||||||
|
$ v = \dot{r} $ \newline
|
||||||
|
$ a = \dot{v} = \ddot{r} $ \newline
|
||||||
|
$ \omega = \dot{\varphi} $ \newline
|
||||||
|
$ \alpha = \dot{\omega} = \ddot{\varphi} $ \newline
|
||||||
|
$ F = \dot{p} $ \newline
|
||||||
|
$ P = \dot{W} $ \newline
|
||||||
|
$ M = \dot{L} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{} \newline
|
||||||
|
\includegraphics[width=0.25\textwidth,keepaspectratio=true]{./sys_coord.png} \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Ressort / Pendule} \newline
|
||||||
|
$ \vec{F} = -k \cdot \vec{r} = -k \cdot (\vec{l} - \vec{l}_0) $ \hfill (ressort) \newline
|
||||||
|
$ T_0 = \frac{2 \cdot \pi}{\omega_0} $ \newline
|
||||||
|
$ f_0 = \frac{1}{T_0} = \frac{\omega_0}{2 \cdot \pi} $ \newline
|
||||||
|
$ \omega_0 = \sqrt{\frac{k}{m}} \text{ ou } \omega_0 = \sqrt{\frac{g}{l}} $ \newline
|
||||||
|
$ \ddot{x} + \omega_0^2 \cdot x = 0 $ \newline
|
||||||
|
$ x(t) = A_1 \cdot \cos(\omega_0 \cdot t + \Phi) $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Oscillateurs} \newline
|
||||||
|
$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = 0 \mid x = C \cdot e^{\gamma \cdot t} $ \newline
|
||||||
|
$ \gamma = - \lambda \pm \sqrt{\lambda^2 - \omega_0^2} $ \newline
|
||||||
|
$ \omega = \sqrt{| \omega_0^2 - \lambda^2 |} $ \newline
|
||||||
|
$ x(t) = A \cdot e^{- \lambda \cdot t} \cdot \cos(\omega \cdot t + \Phi), $ \hfill $ \lambda^2 < \omega_0^2 $ \newline
|
||||||
|
$ x(t) = e^{- \lambda \cdot t} \cdot (A_1 \cdot e^{\omega \cdot t} + A_2 \cdot e^{-\omega \cdot t}), $ \hfill $ \lambda^2 > \omega_0^2 $ \newline
|
||||||
|
$ x(t) = (A + B \cdot t) \cdot e^{- \lambda \cdot t}, $ \hfill $ \lambda^2 = \omega_0^2 $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Oscillateurs forcés} \newline
|
||||||
|
$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = f \cdot \cos(\Omega \cdot t) $ \newline
|
||||||
|
$ x = A(\Omega) \cdot \cos(\Omega \cdot t + \psi) $ \newline
|
||||||
|
$ \underline{x} = A(\Omega) \cdot e^{i \cdot \psi(\Omega)} \cdot e^{i \cdot \Omega \cdot t} = x_0 \cdot e^{i \cdot \Omega \cdot t} $ \newline
|
||||||
|
$ \omega_0 = \sqrt{\frac{k}{m}}, \lambda = \frac{\chi}{2 \cdot m}, f = \frac{F_e}{m} $ \newline
|
||||||
|
$ \omega = \sqrt{w_0^2 - \lambda^2}$ \newline
|
||||||
|
$ x_0 = A(\Omega) \cdot e^{i \cdot \psi(\Omega)} = \frac{f}{\omega_0^2 - \Omega^2 + i \cdot 2 \cdot \lambda \cdot \Omega} $ \newline
|
||||||
|
$ A(\Omega) = \|x_0\| = \frac{f}{\sqrt{(\omega_0^2 - \Omega^2)^2 + (2 \cdot \lambda \cdot \Omega)^2}} $ \newline
|
||||||
|
$ \psi(\Omega) = \arctan(\frac{\Im(x_0)}{\Re(x_0)}) = \arctan(\frac{-2 \cdot \lambda \cdot \Omega}{\omega_0^2 - \Omega^2}) $ \newline
|
||||||
|
$ \Omega_r = \sqrt{w_0^2 - 2 \cdot \lambda^2} $ \hfill $ \frac{\mathrm{d}A(\Omega)}{\mathrm{d}\Omega} = 0 $ \newline
|
||||||
|
$ Q = \frac{\Omega_r}{\Delta \Omega} = \frac{\Omega_r^2}{2 \cdot \lambda \cdot \omega} $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Coordonnées polaires $ (O,\vec{e_r},\vec{e}_{\varphi}) $} \newline
|
||||||
|
$ \vec{r} = r \cdotbis \vec{e_r} $ \newline
|
||||||
|
$ \vec{v} = \dot{r} \cdotbis \vec{e_r} + r \cdotbis \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \vec{a} = (\ddot{r} - r \cdotbis \dot{\varphi}^2) \cdotbis \vec{e_r} + (r \cdotbis \ddot{\varphi} + 2 \cdotbis \dot{r} \cdotbis \dot{\varphi}) \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e_r} = \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \vec{e_r} $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Coord. cylindriques $ (O,\vec{e}_{\rho},\vec{e}_{\varphi},\vec{e}_z) $} \newline
|
||||||
|
$ \vec{r} = \rho \cdotbis \vec{e}_{\rho} + z \cdotbis \vec{e}_z $ \newline
|
||||||
|
$ \vec{v} = \dot{\rho} \cdotbis \vec{e}_{\rho} + \rho \cdotbis \dot{\varphi} \cdotbis \vec{e}_{\varphi} + \dot{z} \cdotbis \vec{e}_z $ \newline
|
||||||
|
$ \vec{a} = (\ddot{\rho} - \rho \cdotbis \dot{\varphi}^2) \cdotbis \vec{e}_{\rho} + (\rho \cdotbis \ddot{\varphi} + 2 \cdotbis \dot{\rho} \cdotbis \dot{\varphi}) \cdotbis \vec{e}_{\varphi} + \ddot{z} \cdotbis \vec{e}_z $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\rho} = \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \vec{e}_{\rho} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_z = 0 $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Coord. sphériques $ (O,\vec{e}_{r},\vec{e}_{\theta},\vec{e}_{\varphi}) $} \newline
|
||||||
|
$ \vec{r} = r \cdotbis \vec{e_r}$ \newline
|
||||||
|
$ \vec{v} = \dot{r} \cdotbis \vec{e_r} + r \cdotbis \dot{\theta} \cdotbis \vec{e}_{\theta} + r \cdotbis \dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \vec{a} = \begin{pmatrix}
|
||||||
|
\ddot{r} - \dot{r} \cdotbis \dot{\theta}^2 - r \cdotbis \dot{\varphi}^2 \cdotbis \sin^2(\theta) \\
|
||||||
|
2 \cdotbis \dot{r} \cdotbis \dot{\theta} + r \cdotbis \ddot{\theta} - r \cdotbis \dot{\varphi}^2 \cdotbis \sin(\theta) \cdotbis \cos(\theta) \\
|
||||||
|
2 \cdotbis \dot{r} \cdotbis \dot{\varphi} \cdotbis \sin(\theta) + r \cdotbis \ddot{\varphi} \cdotbis \sin(\theta) + 2 \cdotbis r \cdotbis \dot{\varphi} \cdotbis \dot{\theta} \cdotbis \cos(\theta) \\
|
||||||
|
\end{pmatrix} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e_r} = \dot{\theta} \cdotbis \vec{e}_{\theta} + \dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\theta} = -\dot{\theta} \cdotbis \vec{e_r} + \dot{\varphi} \cdotbis \cos(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
|
||||||
|
$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e_r} - \dot{\varphi} \cdotbis \cos(\theta) \cdotbis \vec{e}_{\theta} $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Équations de base} \newline
|
||||||
|
$ \sum \vec{F} = m \cdot \vec{a} $ \newline
|
||||||
|
$ \sum \vec{M}_O = \frac{\mathrm{d}}{\mathrm{d}t} \vec{L}_O $ \newline
|
||||||
|
$ \sum \vec{p} = cte $ \newline
|
||||||
|
$ E_i - E_f = 0 $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Signes} \newline
|
||||||
|
$ r, v, a, \omega, \alpha, F $ \hfill avec \newline
|
||||||
|
$ M, L, p $ \hfill sans \newline
|
||||||
|
&
|
||||||
|
\textbf{Angles} \newline
|
||||||
|
$ \cos(\pi \pm \alpha) = - \cos(\alpha) $ \newline
|
||||||
|
$ \cos(\frac{\pi}{2} + \alpha) = - \sin(\alpha) $ \newline
|
||||||
|
$ \cos(\frac{\pi}{2} - \alpha) = \sin(\alpha) $ \newline
|
||||||
|
|
||||||
|
$ \sin(\pi + \alpha) = - \sin(\alpha) $ \newline
|
||||||
|
$ \sin(\pi - \alpha) = \sin(\alpha) $ \newline
|
||||||
|
$ \sin(\frac{\pi}{2} \pm \alpha) = \cos(\alpha) $ \newline
|
||||||
|
\\ \hline
|
||||||
|
% &
|
||||||
|
% \textbf{Configurabilité} \newline
|
||||||
|
% $ a \oldcdot b $ ou $ a b$ \newline
|
||||||
|
% $ \frac{a}{b} $ ou $ a/b$ \newline
|
||||||
|
% $ \vec{a} \oldbullet \vec{b} $ ou $ \vec{a} \circ \vec{b} $ \newline
|
||||||
|
% $ \oldvec{a} $ ou $ \overrightarrow{a} $ ou $ \mathbf{a} $ ou $ \oldvec{\mathbf{a}} $ \newline
|
||||||
|
% $ \dot{x} $ ou $ \frac{\mathrm{d}x}{\mathrm{d}t} $ \newline
|
||||||
|
% $ \ddot{x} $ ou $ \frac{\mathrm{d^2}x}{\mathrm{d}t^2} $ \newline
|
||||||
|
% &
|
||||||
|
% \\ \hline
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\end{document}
|
||||||
267
physique-II.tex
Normal file
267
physique-II.tex
Normal file
@ -0,0 +1,267 @@
|
|||||||
|
\documentclass[a4paper,10pt]{article}
|
||||||
|
%\documentclass[a4paper,10pt]{scrartcl}
|
||||||
|
|
||||||
|
\usepackage{xltxtra}
|
||||||
|
\usepackage{pbox}
|
||||||
|
\usepackage{mathtools}
|
||||||
|
\usepackage{amssymb}
|
||||||
|
\usepackage{tabularx}
|
||||||
|
\usepackage{siunitx}
|
||||||
|
\usepackage[top=13pt, bottom=12pt, left=13pt, right=12pt]{geometry}
|
||||||
|
% \setromanfont[Mapping=tex-text]{Linux Libertine O}
|
||||||
|
% \setsansfont[Mapping=tex-text]{DejaVu Sans}
|
||||||
|
% \setmonofont[Mapping=tex-text]{DejaVu Sans Mono}
|
||||||
|
|
||||||
|
\title{}
|
||||||
|
\author{}
|
||||||
|
\date{}
|
||||||
|
|
||||||
|
\let\oldcdot\cdot
|
||||||
|
\let\oldbullet\bullet
|
||||||
|
\let\oldvec\vec
|
||||||
|
\let\olddot\dot
|
||||||
|
\let\oldddot\ddot
|
||||||
|
|
||||||
|
% \renewcommand{\cdot}{ }
|
||||||
|
\renewcommand{\bullet}{\circ}
|
||||||
|
% \renewcommand{\vec}{\mathbf}
|
||||||
|
% \renewcommand{\dot}[1]{\frac{\mathrm{d}#1}{\mathrm{d}t}}
|
||||||
|
% \renewcommand{\ddot}[1]{\frac{\mathrm{d^2}#1}{\mathrm{d}t^2}}
|
||||||
|
% \renewcommand{\frac}[2]{#1 / #2}
|
||||||
|
\newcommand{\cdotbis}{ }
|
||||||
|
|
||||||
|
\setlength{\parindent}{0pt}
|
||||||
|
\setlength{\parskip}{0pt}
|
||||||
|
|
||||||
|
\begin{document}
|
||||||
|
% \maketitle
|
||||||
|
|
||||||
|
\begin{tabularx}{\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} $ \newline
|
||||||
|
&
|
||||||
|
\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{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial}{\partial \dot{q_j}} L \right) - \frac{\partial}{\partial q_j} L = 0 $ \newline
|
||||||
|
\\ \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) (V -n \cdot b) = n \cdot R \cdot T $ \hfill Van der Waals \newline
|
||||||
|
&
|
||||||
|
\textbf{Maxwell-Boltzmann} \newline
|
||||||
|
$ P_i = Cst \cdot e^{-\frac{E_i}{k_B \cdot T}} $ \newline
|
||||||
|
$ \sum P_i = 1 $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Lois thermodynamiques} \newline
|
||||||
|
$ \mathrm{d} U = \delta W + \delta Q $ \hfill 1\textsuperscript{ère} \newline
|
||||||
|
$ \mathrm{d} S = \delta S_{ext} + \delta S_{int} = \frac{\delta Q}{T} + \delta S_{int} $ \hfill 2\textsuperscript{ème} \newline
|
||||||
|
&
|
||||||
|
\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 $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Isentropie} \newline
|
||||||
|
$ P \cdot V^\gamma = cte $ \newline
|
||||||
|
$ T \cdot V^{\gamma - 1} = cte $ \newline
|
||||||
|
&
|
||||||
|
\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 \mathrm{d}V = -W_{ext}$ \newline
|
||||||
|
\\ \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} $ \newline
|
||||||
|
&
|
||||||
|
\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 \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Cycle} \newline
|
||||||
|
$ \circlearrowright $ Cycle moteur \newline
|
||||||
|
$ \circlearrowleft $ Cycle récepteur \newline
|
||||||
|
&
|
||||||
|
\textbf{Cycle} \newline
|
||||||
|
$ \Delta U = 0 = W + Q $ \newline
|
||||||
|
$ \Delta S = 0 = \int \frac{\delta Q_f}{T} + \int \frac{\delta Q_c}{T} + S_{int}$ \newline
|
||||||
|
$ W = - (Q_c + Q_f) $ \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\textbf{Condutibilité} \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} = \mathrm{d}A \cdot \kappa \cdot \Delta T $ \hfill $ \lambda \gg d $ \newline
|
||||||
|
&
|
||||||
|
\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 $ \newline
|
||||||
|
\\ \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) $ \newline
|
||||||
|
&
|
||||||
|
\textbf{Lennard-Jones II} \newline
|
||||||
|
\includegraphics[width=0.2\textwidth,keepaspectratio=true]{./lennard-jones.png} \newline
|
||||||
|
\\ \hline
|
||||||
|
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\offinterlineskip
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |X|X|X| }
|
||||||
|
\textbf{Diagramme de phase} \newline\newline
|
||||||
|
\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./diagramme-phase.png} \newline
|
||||||
|
&
|
||||||
|
\textbf{Diagramme P-V} \newline\newline
|
||||||
|
\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./diagramme-PV.png} \newline
|
||||||
|
&
|
||||||
|
\textbf{Diagramme P-T} \newline\newline
|
||||||
|
\includegraphics[width=0.3\textwidth,keepaspectratio=true]{./diagramme-PT.png} \newline
|
||||||
|
\\\hline
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\begin{tabularx}{\textwidth}{ |l|X|X|X|X| }
|
||||||
|
\hline
|
||||||
|
& 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} \Delta T \\
|
||||||
|
&= \frac{p_0}{\gamma - 1} \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} \Delta T \\
|
||||||
|
&= \frac{V_0}{\gamma - 1} \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} \Delta T \\
|
||||||
|
&= \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \Delta (V^{1-\gamma})
|
||||||
|
\end{aligned}
|
||||||
|
$
|
||||||
|
\\\hline
|
||||||
|
Chaleur &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
Q &= n \cdot R \cdot T_0 \cdot \ln \frac{V_1}{V_0} \\
|
||||||
|
&= n \cdot R \cdot T_0 \cdot \ln \frac{p_1}{p_0} \\
|
||||||
|
\end{aligned}
|
||||||
|
$ &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
Q &= C_p \cdot \Delta T \\
|
||||||
|
&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \Delta T \\
|
||||||
|
&= \frac{\gamma \cdot p_0}{\gamma - 1} \Delta V \\
|
||||||
|
\end{aligned}
|
||||||
|
$ &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
Q &= C_v \cdot \Delta T \\
|
||||||
|
&= \frac{n \cdot R}{\gamma - 1} \Delta T \\
|
||||||
|
&= \frac{V_0}{\gamma - 1} \Delta p \\
|
||||||
|
\end{aligned}
|
||||||
|
$ &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
Q &= 0
|
||||||
|
\end{aligned}
|
||||||
|
$
|
||||||
|
\\\hline
|
||||||
|
Travail &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
W &= -n \cdot R \cdot T_0 \cdot \ln \frac{V_1}{V_0} \\
|
||||||
|
&= -n \cdot R \cdot T_0 \cdot \ln \frac{p_1}{p_0} \\
|
||||||
|
\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} \Delta T \\
|
||||||
|
&= \frac{p_0 \cdot V_0^\gamma}{\gamma - 1} \Delta (V^{1-\gamma})
|
||||||
|
\end{aligned}
|
||||||
|
$
|
||||||
|
\\\hline
|
||||||
|
Entropie &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
\Delta S &= n \cdot R \cdot \ln \frac{V_1}{V_0} \\
|
||||||
|
&= n \cdot R \cdot \ln \frac{p_1}{p_0} \\
|
||||||
|
\end{aligned}
|
||||||
|
$ &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
\Delta S &= C_p \cdot \ln \frac{V_1}{V_0} \\
|
||||||
|
&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \frac{V_1}{V_0} \\
|
||||||
|
&= C_p \cdot \ln \frac{T_1}{T_0} \\
|
||||||
|
&= \frac{\gamma \cdot n \cdot R}{\gamma - 1} \cdot \ln \frac{T_1}{T_0}
|
||||||
|
\end{aligned}
|
||||||
|
$ &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
\Delta S &= C_v \cdot \ln \frac{p_1}{p_0} \\
|
||||||
|
&= \frac{n \cdot R}{\gamma - 1} \cdot \ln \frac{p_1}{p_0} \\
|
||||||
|
&= C_v \cdot \ln \frac{T_1}{T_0} \\
|
||||||
|
&= \frac{n \cdot R}{\gamma - 1} \cdot \ln \frac{T_1}{T_0}
|
||||||
|
\end{aligned}
|
||||||
|
$ &
|
||||||
|
$
|
||||||
|
\begin{aligned}
|
||||||
|
\Delta S &= 0
|
||||||
|
\end{aligned}
|
||||||
|
$
|
||||||
|
\\\hline
|
||||||
|
\end{tabularx}
|
||||||
|
|
||||||
|
\end{document}
|
||||||
BIN
relations.png
Normal file
BIN
relations.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 148 KiB |
BIN
sys_coord.png
Normal file
BIN
sys_coord.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 88 KiB |
BIN
titrage-faible.png
Normal file
BIN
titrage-faible.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 109 KiB |
BIN
titrage-fort.png
Normal file
BIN
titrage-fort.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 62 KiB |
Loading…
Reference in New Issue
Block a user