From 957b1fd814d421c03ea06be2e434c6ccdecb83b1 Mon Sep 17 00:00:00 2001 From: guyotv Date: Sat, 17 Sep 2022 19:00:18 +0200 Subject: [PATCH] Ajout des quatre premiers exos sur les gaz parfaits. --- Annexe-Exercices/Annexe-Exercices.tex | 64 ++++++++++++++++++++- Annexe-Exercices/Annexe-Exercices.tex.bak | 66 +++++++++++++++++++++- CoursMecaniqueOS.pdf | Bin 12836874 -> 12842810 bytes CoursMecaniqueOSDF.pdf | Bin 12836874 -> 12842810 bytes SolutionsOS.tex | 41 +++++++++++++- 5 files changed, 168 insertions(+), 3 deletions(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index f4fc40f..7a9ad5b 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2444,7 +2444,7 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \item \SI{20}{\gram} de glaçons à \SI{0}{\celsius} soit \item \SI{20}{\gram} de glaçons à \SI{-10}{\celsius}. \end{itemize} - Dans chaque cas, calculez la température d'équilibre. Réponse~: \SI{18,2}{\celsius}, \SI{11,0}{\celsius} et \SI{10,6}{\celsius} + Dans chaque cas, calculez la température d'équilibre. Réponses~: \SI{18,2}{\celsius}, \SI{11,0}{\celsius} et \SI{10,6}{\celsius} \begin{solos} Commençons par calculer toutes les grandeurs en jeu. @@ -2531,6 +2531,68 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r En consultant le tableau \ref{tabchaleurmassique}, page \pageref{tabchaleurmassique}, on constate qu'il s'agit de fer. \end{solos} \end{exos} + +\begin{exos} + Combien de molécules par centimètre cube contient un gaz parfait sous conditions normales de température (\SI{0}{\celsius}) et de pression (\SI{1}{atm}) ? Réponse~: \num{2,69e19} molécules. + \begin{solos} + La loi des gaz parfaits peut s'écrire~: + \[p\cdot V=N\cdot k\cdot T\] + où la constante de Boltzmann vaut~: + \[k=\frac{R}{N_A}=\frac{8,31}{6,022\cdot 10^{23}}=\SI{1,38e-23}{\joule\per\kelvin}\] + On a donc que le nombre de molécules vaut~: + \[N=\frac{p\cdot V}{k\cdot T}=\frac{1,013\cdot 10^5\cdot 10^{-6}}{1,38\cdot 10^{-23}\cdot 273,15}=\num{2,69e19}\] + \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~: + \[\rho =\frac{n\cdot M}{V}\] + \begin{solos} + Comme la masse molaire M est la masse m par mole, on a naturellement que \(M=m/n\;\Rightarrow\;m=n\cdot N\) et on peut écrire~: + \[\rho =\frac{m}{V}=\frac{n\cdot M}{V}\] + Ce qu'il fallait démontrer. + \end{solos} +\end{exos} + +\begin{exos} + Exprimez la loi des gaz parfait en fonction de la masse volumique et plus particulièrement la masse volumique en fonction de la température et de la pression. + + Puis, sous condition normale de température et de pression, calculez la masse volumique des gaz suivants~: hydrogène, azote et oxygène. Réponses~: \num{0,089}, \num{1,25} et \SI{1,43}{\kilo\gram\per\metre\cubed}. + \begin{solos} + De l'exercice \ref{exosmassemolvol}, on tire~: + \[n=\rho\cdot\frac{V}{M}\] + La loi des gaz parfaits nous donne alors~: + \begin{align*} + p\cdot V &=n\cdot R\cdot T\;\Rightarrow\\ + p\cdot V &=\rho\cdot\frac{V}{M}\cdot R\cdot T\;\Rightarrow\\ + \rho &=M\cdot\frac{p}{R\cdot T} + \end{align*} + En n'oubliant pas que la masse molaire doit être utilisée en \si{\kilo\gram\per\mol}, on a alors les masse volumique suivantes pour chaque gaz (diatomique)~: + \begin{align*} + \rho &=2\cdot 1\cdot 10^{-3}\cdot\frac{1,013\cdot 10^5}{8,31\cdot 273,15}=\SI{0,089}{\kilo\gram\per\metre\cubed}\\ + \rho &=2\cdot 14\cdot 10^{-3}\cdot 44,6=\SI{1,25}{\kilo\gram\per\metre\cubed}\\ + \rho &=2\cdot 16\cdot 10^{-3}\cdot 44,6=\SI{1,43}{\kilo\gram\per\metre\cubed} + \end{align*} + \end{solos} +\end{exos} + +\begin{exos} + Un gaz parfait comprenant trois moles de particules est à la pression de \SI{4}{\kilo\pascal} et à la température de \SI{20}{\celsius}. Calculez son volume. Puis recalculez-le si on augmente sa température à \SI{30}{\celsius} et diminue sa pression de 15\%. Réponses~: \SI{2,45}{\metre\cubed} et \SI{2,22}{\metre\cubed}. + \begin{solos} + De la loi des gaz parfaits, on tire~: + \[V=n\cdot R\cdot \frac{T}{p}=3\cdot 8,31\cdot \frac{293,15}{4\cdot 10^3}=\SI{2,45}{\metre\cubed}\] + Puis, de la même manière~: + \[V=3\cdot 8,31\cdot \frac{303,15}{0,85\cdot 4\cdot 10^3}=\SI{2,22}{\metre\cubed}\] + \end{solos} +\end{exos} + +%\begin{exos} +% Un énoncé de test. +% \begin{solos} +% Un autre corrigé de test. +% \end{solos} +%\end{exos} + } \subsection{Relatifs aux incertitudes} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index e26bfb8..7a9ad5b 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2444,7 +2444,7 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \item \SI{20}{\gram} de glaçons à \SI{0}{\celsius} soit \item \SI{20}{\gram} de glaçons à \SI{-10}{\celsius}. \end{itemize} - Dans chaque cas, calculez la température d'équilibre. Réponse~: \SI{18,2}{\celsius}, \SI{11,0}{\celsius} et \SI{10,6}{\celsius} + Dans chaque cas, calculez la température d'équilibre. Réponses~: \SI{18,2}{\celsius}, \SI{11,0}{\celsius} et \SI{10,6}{\celsius} \begin{solos} Commençons par calculer toutes les grandeurs en jeu. @@ -2531,6 +2531,68 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r En consultant le tableau \ref{tabchaleurmassique}, page \pageref{tabchaleurmassique}, on constate qu'il s'agit de fer. \end{solos} \end{exos} + +\begin{exos} + Combien de molécules par centimètre cube contient un gaz parfait sous conditions normales de température (\SI{0}{\celsius}) et de pression (\SI{1}{atm}) ? Réponse~: \num{2,69e19} molécules. + \begin{solos} + La loi des gaz parfaits peut s'écrire~: + \[p\cdot V=N\cdot k\cdot T\] + où la constante de Boltzmann vaut~: + \[k=\frac{R}{N_A}=\frac{8,31}{6,022\cdot 10^{23}}=\SI{1,38e-23}{\joule\per\kelvin}\] + On a donc que le nombre de molécules vaut~: + \[N=\frac{p\cdot V}{k\cdot T}=\frac{1,013\cdot 10^5\cdot 10^{-6}}{1,38\cdot 10^{-23}\cdot 273,15}=\num{2,69e19}\] + \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~: + \[\rho =\frac{n\cdot M}{V}\] + \begin{solos} + Comme la masse molaire M est la masse m par mole, on a naturellement que \(M=m/n\;\Rightarrow\;m=n\cdot N\) et on peut écrire~: + \[\rho =\frac{m}{V}=\frac{n\cdot M}{V}\] + Ce qu'il fallait démontrer. + \end{solos} +\end{exos} + +\begin{exos} + Exprimez la loi des gaz parfait en fonction de la masse volumique et plus particulièrement la masse volumique en fonction de la température et de la pression. + + Puis, sous condition normale de température et de pression, calculez la masse volumique des gaz suivants~: hydrogène, azote et oxygène. Réponses~: \num{0,089}, \num{1,25} et \SI{1,43}{\kilo\gram\per\metre\cubed}. + \begin{solos} + De l'exercice \ref{exosmassemolvol}, on tire~: + \[n=\rho\cdot\frac{V}{M}\] + La loi des gaz parfaits nous donne alors~: + \begin{align*} + p\cdot V &=n\cdot R\cdot T\;\Rightarrow\\ + p\cdot V &=\rho\cdot\frac{V}{M}\cdot R\cdot T\;\Rightarrow\\ + \rho &=M\cdot\frac{p}{R\cdot T} + \end{align*} + En n'oubliant pas que la masse molaire doit être utilisée en \si{\kilo\gram\per\mol}, on a alors les masse volumique suivantes pour chaque gaz (diatomique)~: + \begin{align*} + \rho &=2\cdot 1\cdot 10^{-3}\cdot\frac{1,013\cdot 10^5}{8,31\cdot 273,15}=\SI{0,089}{\kilo\gram\per\metre\cubed}\\ + \rho &=2\cdot 14\cdot 10^{-3}\cdot 44,6=\SI{1,25}{\kilo\gram\per\metre\cubed}\\ + \rho &=2\cdot 16\cdot 10^{-3}\cdot 44,6=\SI{1,43}{\kilo\gram\per\metre\cubed} + \end{align*} + \end{solos} +\end{exos} + +\begin{exos} + Un gaz parfait comprenant trois moles de particules est à la pression de \SI{4}{\kilo\pascal} et à la température de \SI{20}{\celsius}. Calculez son volume. Puis recalculez-le si on augmente sa température à \SI{30}{\celsius} et diminue sa pression de 15\%. Réponses~: \SI{2,45}{\metre\cubed} et \SI{2,22}{\metre\cubed}. + \begin{solos} + De la loi des gaz parfaits, on tire~: + \[V=n\cdot R\cdot \frac{T}{p}=3\cdot 8,31\cdot \frac{293,15}{4\cdot 10^3}=\SI{2,45}{\metre\cubed}\] + Puis, de la même manière~: + \[V=3\cdot 8,31\cdot \frac{303,15}{0,85\cdot 4\cdot 10^3}=\SI{2,22}{\metre\cubed}\] + \end{solos} +\end{exos} + +%\begin{exos} +% Un énoncé de test. +% \begin{solos} +% Un autre corrigé de test. +% \end{solos} +%\end{exos} + } \subsection{Relatifs aux incertitudes} @@ -2547,6 +2609,8 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{itemize} \begin{solos} Procédons simplement au calcul de l'incertitude absolue. + + \smallskip Pour l'accélération centripète, on a~: \begin{align*} I(a)&=I(v^2/R)=v^2/R\cdot i(v^2/R)\\ diff --git a/CoursMecaniqueOS.pdf b/CoursMecaniqueOS.pdf index 751c2664918bd76eb275d7f7b5b5a629f94e27ea..f535c14c1b59b4625e76c5f321997c4376a9e9e7 100644 GIT binary patch delta 278047 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