From 4d009de98cd540129fbd98261719addec7e55e03 Mon Sep 17 00:00:00 2001 From: guyotv Date: Mon, 19 Sep 2022 18:44:41 +0200 Subject: [PATCH] Encore des exos sur la loi de gaz parfaits. --- Annexe-Exercices/Annexe-Exercices.tex | 62 +++++++++++++++++++++ Annexe-Exercices/Annexe-Exercices.tex.bak | 64 +++++++++++++++++++++- CoursMecaniqueOS.pdf | Bin 12849987 -> 12856471 bytes CoursMecaniqueOSDF.pdf | Bin 12849987 -> 12856471 bytes SolutionsOS.tex | 56 +++++++++++++++++-- 5 files changed, 174 insertions(+), 8 deletions(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index 672e877..c00b824 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2544,6 +2544,19 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + Une boite cubique de \SI{20}{\centi\metre} d'arête et remplie d'azote à \SI{0}{\celsius} sous une pression d'un bar. On la chauffe à \SI{20}{\celsius}. Quelle force s'exerce sur chaque paroi ? Réponse~: \SI{4293}{\newton}. + \begin{solos} + Le volume de la boite est \(V=0,2^3=\SI{0,008}{\metre\cubed}\). Il ne change pas, pas plus que la quantité d'azote qu'il contient. Ainsi, la loi des gaz parfaits peut s'écrire~: + \begin{align*} + \frac{p_1}{T_1}&=\frac{p_2}{T_2}\;\Rightarrow\;p_2=p_1\cdot \frac{T_2}{T_1}\\ + p_2&=10^5\cdot \frac{293,15}{273,15}=\SI{107,322}{\kilo\pascal} + \end{align*} + La force sur chaque parois est donc de~: + \[F=p\cdot S=107 322\cdot 0,2^2=\SI{4293}{\newton}\] + \end{solos} +\end{exos} + \begin{exos}\label{exosmassemolvol} Démontrez que la masse volumique \(\rho\) d'un gaz parfait de molécules de masse molaire M dans un volume V remplit de n moles de celui-ci s'écrit~: \[\rho =\frac{n\cdot M}{V}\] @@ -2656,6 +2669,55 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + On gonfle, à une température de \SI{20}{\celsius}, un pneu de vingt pouces de rayon dont le diamètre de la chambre à air vaut \SI{5}{\centi\metre} à une pression manomètrique de \SI{4}{\bar}. Après un long voyage, la température de l'air qu'il contient vaut \SI{25}{\celsius}. Quelle masse d'air doit-on retirer pour rétablir la pression à sa valeur d'origine ? + + La pression manométrique est la pression absolue à laquelle on a retiré une atmosphère. On peut considérer que la masse molaire de l'air est de \SI{29}{\gram\per\mol}. Réponses~: \SI{5}{\gram}. + \begin{solos} + La première chose à calculer est le volume de la chambre à air. Elle peut être assimilée à un cylindre de longueur L donnée par~: + \[L=2\cdot\pi\cdot R=2\cdot\pi\cdot 20\cdot 2,54\cdot 10^{-2}=\SI{3,192}{\metre}\] + et donc la base a une surface de~: + \[S=\pi\cdot R^2=\pi\cdot 0,05^2=\SI{7,854e-3}{\metre\squared}\] + Son volume vaut donc finalement~: + \[V=L\cdot S=3,192\cdot 7,854\cdot 10^{-3}=\SI{25,07e-3}{\metre\cubed}\] + + \smallskip + Il faut ensuite déterminer le nombre de moles contenues dans le pneu aux pression et température données. La pression étant manométrique, on obtient sa valeur absolue par~: + \begin{align*} + p&=p_{mano.}+\SI{1}{atm}\\ + &=4\cdot 10^5+1,013\cdot 10^5=\SI{5,013e5}{\pascal} + \end{align*} + La loi des gaz parfaits nous donne alors~: + \[n=\frac{p\cdot V}{R\cdot T}=\frac{5,013\cdot 10^5\cdot 25,07\cdot 10^{-3}}{8,31\cdot 293,15}=\SI{5,176}{\mol}\] + + \smallskip + Pour revenir à la pression avant le voyage, on va retirer de l'air. Cette opération se fait à température constante, parce qu'elle est très rapide. La température du gaz qui reste dans la chambre est donc de \SI{25}{\celsius}. Or, à cette température, le nombre de moles qui donnent une pression manométrique de \SI{4}{\bar} vaut~: + \[n=\frac{p\cdot V}{R\cdot T}=\frac{5,013\cdot 10^5\cdot 25,07\cdot 10^{-3}}{8,31\cdot 298,15}=\SI{5,072}{\mol}\] + Il faut donc retirer la différence~: + \[\Delta n=5,176-5,072=\SI{0,104}{\mol}\] + Soit en terme de masse~: + \[m=\Delta n\cdot M=0,104\cdot 29=\SI{5}{\gram}\] + \end{solos} +\end{exos} + +\begin{exos} + La pression atmosphérique sur la lune est de l'ordre de \SI{e-15}{\bar}. Combien de particules par centimètre cube contient son atmosphère à \SI{0}{\celsius} ? Réponse~: \SI{26529}{molécules}. + \begin{solos} + Très simplement, on a~: + \begin{align*} + N&=\frac{P\cdot V}{k\cdot T}=\frac{10^{-15}\cdot 10^5\cdot 10^{-6}}{1,38\cdot 10^{-23}\cdot 273,15}\\ + &=\SI{26529}{molécules} + \end{align*} + \end{solos} +\end{exos} + +%\begin{exos} +% Un énoncé de test. +% \begin{solos} +% Un autre corrigé de test. +% \end{solos} +%\end{exos} + %\begin{exos} % Un énoncé de test. % \begin{solos} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index 06edcfd..8886afa 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2544,6 +2544,17 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + Une boite cubique de \SI{20}{\centi\metre} d'arête et remplie d'azote à \SI{0}{\celsius} sous une pression d'un bar. On la chauffe à \SI{20}{\celsius}. Quelle force s'exerce sur chaque parois ? Réponse~: + \begin{solos} + Le volume de la boite est \(V=0,2^3=\SI{0,008}{\metre\cubed}\). Il ne change pas, pas plus que la quantité d'azote qu'il contient. Ainsi, la loi des gaz parfaits peut s'écrire~: + \begin{align*} + \frac{p_1}{T_1}&=\frac{p_2}{T_2}\;\Rightarrow\;p_2=p_1\cdot \frac{T_2}{T_1}\\ + p_2&=10^5\cdot \frac{293,15}{273,15}=\SI{}{\pascal} + \end{align*} + \end{solos} +\end{exos} + \begin{exos}\label{exosmassemolvol} Démontrez que la masse volumique \(\rho\) d'un gaz parfait de molécules de masse molaire M dans un volume V remplit de n moles de celui-ci s'écrit~: \[\rho =\frac{n\cdot M}{V}\] @@ -2638,13 +2649,13 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \[n=\frac{241,183}{29\cdot 10^-3}=\SI{8316,652}{\mol}\] Ainsi, par la loi des gaz parfaits, on a~: \begin{align*} - p&=n\cdot R\frac{T}{V}=8316,652\cdot 8,31\cdot \frac{298,15}{200} + p&=n\cdot R\frac{T}{V}=8316,652\cdot 8,31\cdot \frac{298,15}{200}\\ &=\SI{103027,78}{\pascal} \end{align*} ou, parce que le volume ne change pas~: \begin{align*} \frac{P_1}{T_1}&=\frac{P_2}{T_2}\;\Rightarrow\;p_2=p_1\cdot \frac{T_2}{T_1}\\ - p_2=1,013\cdot 10^5\cdot \frac{298,15}{293,15}=\SI{103027,78}{\pascal} + p_2&=1,013\cdot 10^5\cdot \frac{298,15}{293,15}=\SI{103027,78}{\pascal} \end{align*} \item À \SI{25}{\celsius} et une atmosphère, le nombre de moles dans le mobil-home vaut~: \[n=\frac{p\cdot V}{R\cdot T}=\frac{1,013\cdot 10^5\cdot 200}{8,31\cdot 298,15}=\SI{8177,181}{\mol}\] @@ -2656,6 +2667,55 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + On gonfle, à une température de \SI{20}{\celsius}, un pneu de vingt pouces de rayon dont le diamètre de la chambre à air vaut \SI{5}{\centi\metre} à une pression manomètrique de \SI{4}{\bar}. Après un long voyage, la température de l'air qu'il contient vaut \SI{25}{\celsius}. Quelle masse d'air doit-on retirer pour rétablir la pression à sa valeur d'origine ? + + La pression manométrique est la pression absolue à laquelle on a retiré une atmosphère. On peut considérer que la masse molaire de l'air est de \SI{29}{\gram\per\mol}. Réponses~: \SI{5}{\gram}. + \begin{solos} + La première chose à calculer est le volume de la chambre à air. Elle peut être assimilée à un cylindre de longueur L donnée par~: + \[L=2\cdot\pi\cdot R=2\cdot\pi\cdot 20\cdot 2,54\cdot 10^{-2}=\SI{3,192}{\metre}\] + et donc la base a une surface de~: + \[S=\pi\cdot R^2=\pi\cdot 0,05^2=\SI{7,854e-3}{\metre\squared}\] + Son volume vaut donc finalement~: + \[V=L\cdot S=3,192\cdot 7,854\cdot 10^{-3}=\SI{25,07e-3}{\metre\cubed}\] + + \smallskip + Il faut ensuite déterminer le nombre de moles contenues dans le pneu aux pression et température données. La pression étant manométrique, on obtient sa valeur absolue par~: + \begin{align*} + p&=p_{mano.}+\SI{1}{atm}\\ + &=4\cdot 10^5+1,013\cdot 10^5=\SI{5,013e5}{\pascal} + \end{align*} + La loi des gaz parfaits nous donne alors~: + \[n=\frac{p\cdot V}{R\cdot T}=\frac{5,013\cdot 10^5\cdot 25,07\cdot 10^{-3}}{8,31\cdot 293,15}=\SI{5,176}{\mol}\] + + \smallskip + Pour revenir à la pression avant le voyage, on va retirer de l'air. Cette opération se fait à température constante, parce qu'elle est très rapide. La température du gaz qui reste dans la chambre est donc de \SI{25}{\celsius}. Or, à cette température, le nombre de moles qui donnent une pression manométrique de \SI{4}{\bar} vaut~: + \[n=\frac{p\cdot V}{R\cdot T}=\frac{5,013\cdot 10^5\cdot 25,07\cdot 10^{-3}}{8,31\cdot 298,15}=\SI{5,072}{\mol}\] + Il faut donc retirer la différence~: + \[\Delta n=5,176-5,072=\SI{0,104}{\mol}\] + Soit en terme de masse~: + \[m=\Delta n\cdot M=0,104\cdot 29=\SI{5}{\gram}\] + \end{solos} +\end{exos} + +\begin{exos} + La pression atmosphérique sur la lune est de l'ordre de \SI{e-15}{\bar}. Combien de particules par centimètre cube contient son atmosphère à \SI{0}{\celsius} ? Réponse~: \SI{26529}{molécules}. + \begin{solos} + Très simplement, on a~: + \begin{align*} + N&=\frac{P\cdot V}{k\cdot T}=\frac{10^{-15}\cdot 10^5\cdot 10^{-6}}{1,38\cdot 10^{-23}\cdot 273,15}\\ + &=\SI{26529}{molécules} + \end{align*} + \end{solos} +\end{exos} + +%\begin{exos} +% Un énoncé de test. +% \begin{solos} +% Un autre corrigé de test. +% \end{solos} +%\end{exos} + %\begin{exos} % Un énoncé de test. % \begin{solos} diff --git a/CoursMecaniqueOS.pdf b/CoursMecaniqueOS.pdf index ee358c8b4f01f951fdf4fd0ab6c69eb9ebd0a76c..8b5758f552a559506eeaf5645d259032ba427a7e 100644 GIT binary patch delta 295527 zcmYJ)Q+Os#w+7%?6Wg|JeX(uZw%^#+8{4*RbE1iDbHac2(LT7l)>)r)UDdVfso}Cj zDBwmSRD(SjXhMnum=FXhN)cppBUoUL92oD+a^l@>D}tqdgrS!Jb)!67OZE5F&y3&V z_S&7;rd58;@dL9i0QZJ( z-7I^9QR&L^Ffhdei{&{S5@qN&&3H`?VV+z-bwJK608*9KOP^roBu9ES_jyH;ZLnoc zM?IoZ-Pbs^l5plk2A0yh;$|NHP{IoXkKqu^BS;AMq~LsCj#uQeY9)`X`9dHZM?4Id#xGP z<({o3W<2-We=Hvn$uhHJ1s&_p%3UC>( 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