From bda4e1b714c54af5a5ef3ece0d13264e851d8a5f Mon Sep 17 00:00:00 2001 From: guyotv Date: Tue, 16 Aug 2022 13:48:19 +0200 Subject: [PATCH] Encore deux exos de thermo. --- Annexe-Exercices/Annexe-Exercices.tex | 97 ++++++++++++++++++++++ Annexe-Exercices/Annexe-Exercices.tex.bak | 96 +++++++++++++++++++++ CoursMecaniqueOS.pdf | Bin 12807611 -> 12815398 bytes CoursMecaniqueOSDF.pdf | Bin 12807611 -> 12815398 bytes Macros.tex | 4 +- Macros.tex.bak | 4 +- SolutionsOS.tex | 80 ++++++++++++++++++ 7 files changed, 279 insertions(+), 2 deletions(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index f7b70ad..a7d84c0 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2400,6 +2400,103 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{sol} \end{ex} +\optv{OS}{ + +\begin{exos} + On veut refroidir un verre de \SI{2}{\deci\litre} de thé froid à \SI{20}{\celsius} pour en faire du thé glacé. Pour cela, on y introduit~: +\begin{itemize} +\item \SI{20}{\gram} d'eau à la température de la glace fondante, +\item \SI{20}{\gram} de glaçons à \SI{0}{\celsius}, +\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} + \begin{solos} + Commençons par calculer toutes les grandeurs en jeu. + + À deux décilitre d'eau correspond une masse de \SI{0,2}{\kilo\gram}. + + Pour que l'eau du thé descende sa température à \SI{0}{\celsius}, il faut lui transmettre une énergie de~: + \begin{align*} + Q_{eau\,chaude}&=m\cdot c_{eau}\cdot\Delta\theta=0,2\cdot 4180\cdot (0-20)\\ + &=\SI{-16720}{\joule} + \end{align*} + Cette énergie est négative, car on la retire à l'eau. + + Pour faire fondre \SI{20}{\gram} de glace, il faut une énergie de~: + \begin{align*} + Q_{glace}&=m\cdot L_f=0,02\cdot 3,3\cdot 10^5\\ + &=\SI{6600}{\joule} + \end{align*} + + Pour faire passer la glace de \SI{-10}{\celsius} à \SI{0}{\celsius}, il faut une énergie de~: + \begin{align*} + Q_{glace}&=m\cdot c_{glace}\cdot\Delta\theta=0,02\cdot 2060\cdot (0-(-10)\\ + &=\SI{412}{\joule} + \end{align*} + + \medskip + En cumulant l'énergie pour amener la glace à \SI{0}{\celsius} et pour la faire fondre, on obtient \(6'600+412=\SI{7012}{\joule}\). C'est l'énergie qu'il faut fournir à la glace pour qu'elle devienne intégralement de l'eau à \SI{0}{\celsius}. Or, elle est inférieure à ce que l'eau du thé peut donner. Donc, la température finale sera supérieure à \SI{0}{\celsius}, dans les trois cas. + + On peut maintenant calculer les températures finales~: + \begin{itemize} + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &=919,6\cdot\theta_{eq.}-16'720=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{16'720}{919,6}=\SI{18,2}{\celsius}\] + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}+Q_{glace}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &+6'600\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &+6'600\\ + &=919,6\cdot\theta_{eq.}-10'120=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{10'120}{919,6}=\SI{11,0}{\celsius}\] + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}+Q_{glace}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &+6'600+412\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &+7'012\\ + &=919,6\cdot\theta_{eq.}-10'120=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{9'708}{919,6}=\SI{10,6}{\celsius}\] + \end{itemize} + On constate que de l'eau froide ou de la glace dont la température diminue permettent un refroidissement relativement faible, alors que la fonte de la glace constitue le principal facteur d'abaissement de la température du thé. + \end{solos} +\end{exos} + +\begin{exos} + On remplit un calorimètre (récipient dont les parois sont isolées thermiquement) à \SI{20}{\celsius} de \SI{2}{\deci\litre} d'eau à la même température. Puis, on plonge dans celle-ci une masse de \SI{50}{\gram} de matière solide inconnue à \SI{80}{\celsius}. On laisse le système atteindre l'équilibre thermique. Sachant que la température atteinte alors est de \SI{21,55}{\celsius}, trouvez quelle est la matière qui a été plongée dans l'eau. On fait l'hypothèse que le calorimètre n'a pas de chaleur massique. + \begin{solos} + Le bilan thermique s'écrit~: + \begin{align*} + m_{eau}\cdot c_{eau}\cdot (\theta_{eq.}-20)+m_?\cdot c_?\cdot (\theta_{eq.}-80)&=0\\ + 0,2\cdot 4180\cdot (21,55-20)&+\\ + 0,05\cdot c_?\cdot (21,55-80)&=0\\ + 1295,8-2,92\cdot c_?&=0 + \end{align*} + Ainsi, la chaleur massique vaut~: + \[c_?=\frac{1295,8}{2,92}=\SI{444}{\joule\celsius\per\kilo\gram}\] + En consultant le tableau \ref{tabchaleurmassique}, page \pageref{tabchaleurmassique}, on constate qu'il s'agit de fer. + \end{solos} + +\end{exos} + +} + \Closesolutionfile{ansos} % ferme le fichier SolutionsOS.tex qui contient les solutions OS. \Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index f7b70ad..f2a3d0b 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2400,6 +2400,102 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{sol} \end{ex} +\optv{OS}{ + +\begin{exos} + On veut refroidir un verre de \SI{2}{\deci\litre} de thé froid à \SI{20}{\celsius} pour en faire du thé glacé. Pour cela, on y introduit~: +\begin{itemize} +\item \SI{20}{\gram} d'eau à la température de la glace fondante, +\item \SI{20}{\gram} de glaçons à \SI{0}{\celsius}, +\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} + \begin{solos} + Commençons par calculer toutes les grandeurs en jeu. + + À deux décilitre d'eau correspond une masse de \SI{0,2}{\kilo\gram}. + + Pour que l'eau du thé descende sa température à \SI{0}{\celsius}, il faut lui transmettre une énergie de~: + \begin{align*} + Q_{eau\,chaude}&=m\cdot c_{eau}\cdot\Delta\theta=0,2\cdot 4180\cdot (0-20)\\ + &=\SI{-16720}{\joule} + \end{align*} + Cette énergie est négative, car on la retire à l'eau. + + Pour faire fondre \SI{20}{\gram} de glace, il faut une énergie de~: + \begin{align*} + Q_{glace}&=m\cdot L_f=0,02\cdot 3,3\cdot 10^5\\ + &=\SI{6600}{\joule} + \end{align*} + + Pour faire passer la glace de \SI{-10}{\celsius} à \SI{0}{\celsius}, il faut une énergie de~: + \begin{align*} + Q_{glace}&=m\cdot c_{glace}\cdot\Delta\theta=0,02\cdot 2060\cdot (0-(-10)\\ + &=\SI{412}{\joule} + \end{align*} + + \medskip + En cumulant l'énergie pour amener la glace à \SI{0}{\celsius} et pour la faire fondre, on obtient \(6'600+412=\SI{7012}{\joule}\). C'est l'énergie qu'il faut fournir à la glace pour qu'elle devienne intégralement de l'eau à \SI{0}{\celsius}. Or, elle est inférieure à ce que l'eau du thé peut donner. Donc, la température finale sera supérieure à \SI{0}{\celsius}, dans les trois cas. + + On peut maintenant calculer les températures finales~: + \begin{itemize} + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &=919,6\cdot\theta_{eq.}-16'720=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{16'720}{919,6}=\SI{18,2}{\celsius}\] + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}+Q_{glace}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &+6'600\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &+6'600\\ + &=919,6\cdot\theta_{eq.}-10'120=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{10'120}{919,6}=\SI{11,0}{\celsius}\] + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}+Q_{glace}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &+6'600+412\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &+7'012\\ + &=919,6\cdot\theta_{eq.}-10'120=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{9'708}{919,6}=\SI{10,6}{\celsius}\] + \end{itemize} + On constate que de l'eau froide ou de la glace dont la température diminue permettent un refroidissement relativement faible, alors que la fonte de la glace constitue le principal facteur d'abaissement de la température du thé. + \end{solos} +\end{exos} + +\begin{exos} + On remplit un calorimètre (récipient dont les parois sont isolées thermiquement) à \SI{20}{\celsius} de \SI{2}{\deci\litre} d'eau à la même température. Puis, on plonge dans celle-ci une masse de \SI{50}{\gram} de matière solide inconnue à \SI{80}{\celsius}. On laisse le système atteindre l'équilibre thermique. Sachant que la température atteinte alors est de \SI{21,55}{\celsius}, trouvez quelle est la matière qui a été plongée dans l'eau. On fait l'hypothèse que le calorimètre n'a pas de chaleur massique. + \begin{solos} + Le bilan thermique s'écrit~: + \begin{align*} + m_{eau}\cdot c_{eau}\cdot (\theta_{eq.}-20)+m_?\cdot c_?\cdot (\theta_{eq.}-80)&=0\\ + 0,2\cdot 4180\cdot (21,55-20)&\\ + +0,05\cdot c_?\cdot (21,55-80)&=0\\ + 1295,8-2,92\cdot c_?&=0 + \end{align*} + Ainsi, la chaleur massique vaut~: + \[c_?=\frac{1295,8}{2,92}=\SI{444}{\joule\celsius\per\kilo\gram}\] + \end{solos} + En consultant le tableau \ref{chaleurmassique}, page \pageref{chaleurmassique}, on constate qu'il s'agit de fer. +\end{exos} + +} + \Closesolutionfile{ansos} % ferme le fichier SolutionsOS.tex qui contient les solutions OS. \Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions diff --git a/CoursMecaniqueOS.pdf b/CoursMecaniqueOS.pdf index b4d4f5c85c050f3802cd01c4c59dbfc6f7f613f9..10a0829596da1c07c5125d402db380612889521d 100644 GIT binary patch delta 294165 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zkxXJGapkTF#+5hI-VPxccj&Pt=V%Tfl@&J+f)S;qNob@&u0AmrfBN|&aEWX@R|gF9 z>hTUgok&6>^(q3RKz;1yQOAgk+hG{+B*rPv073S_br z(2l%14kACDEEs*c8AjhkGj1e5lE)EkdP-XplLE+GQ=V%Xx`}vEvZ~geY YKl*X}-mFXr7hZS&-Mi1d{QUj@2ghUabN~PV diff --git a/Macros.tex b/Macros.tex index 99c640f..58bf24e 100644 --- a/Macros.tex +++ b/Macros.tex @@ -18,7 +18,9 @@ Glace & \num{2,06e3}\\ %\hline Glycérine & \num{2,4e3}\\ %\hline -Mercure & \num{0,14e3}\\ +Mercure & \num{0,14e3}\\ +%\hline +Fer & \num{0,444e3}\\ \hline \end{tabular} \end{table} diff --git a/Macros.tex.bak b/Macros.tex.bak index 99c640f..58bf24e 100644 --- a/Macros.tex.bak +++ b/Macros.tex.bak @@ -18,7 +18,9 @@ Glace & \num{2,06e3}\\ %\hline Glycérine & \num{2,4e3}\\ %\hline -Mercure & \num{0,14e3}\\ +Mercure & \num{0,14e3}\\ +%\hline +Fer & \num{0,444e3}\\ \hline \end{tabular} \end{table} diff --git a/SolutionsOS.tex b/SolutionsOS.tex index 4a2803d..c4fd34f 100644 --- a/SolutionsOS.tex +++ b/SolutionsOS.tex @@ -767,3 +767,83 @@ \[E=\frac{F_n/A}{\alpha\cdot\Delta\theta}=\frac{8,8\cdot 10^7}{11\cdot 10^{-6}\cdot 40}=\SI{2e11}{\newton\per\metre\squared}\] \end{Solution OS} +\begin{Solution OS}{23} + Commençons par calculer toutes les grandeurs en jeu. + + À deux décilitre d'eau correspond une masse de \SI{0,2}{\kilo\gram}. + + Pour que l'eau du thé descende sa température à \SI{0}{\celsius}, il faut lui transmettre une énergie de~: + \begin{align*} + Q_{eau\,chaude}&=m\cdot c_{eau}\cdot\Delta\theta=0,2\cdot 4180\cdot (0-20)\\ + &=\SI{-16720}{\joule} + \end{align*} + Cette énergie est négative, car on la retire à l'eau. + + Pour faire fondre \SI{20}{\gram} de glace, il faut une énergie de~: + \begin{align*} + Q_{glace}&=m\cdot L_f=0,02\cdot 3,3\cdot 10^5\\ + &=\SI{6600}{\joule} + \end{align*} + + Pour faire passer la glace de \SI{-10}{\celsius} à \SI{0}{\celsius}, il faut une énergie de~: + \begin{align*} + Q_{glace}&=m\cdot c_{glace}\cdot\Delta\theta=0,02\cdot 2060\cdot (0-(-10)\\ + &=\SI{412}{\joule} + \end{align*} + + \medskip + En cumulant l'énergie pour amener la glace à \SI{0}{\celsius} et pour la faire fondre, on obtient \(6'600+412=\SI{7012}{\joule}\). C'est l'énergie qu'il faut fournir à la glace pour qu'elle devienne intégralement de l'eau à \SI{0}{\celsius}. Or, elle est inférieure à ce que l'eau du thé peut donner. Donc, la température finale sera supérieure à \SI{0}{\celsius}, dans les trois cas. + + On peut maintenant calculer les températures finales~: + \begin{itemize} + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &=919,6\cdot\theta_{eq.}-16'720=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{16'720}{919,6}=\SI{18,2}{\celsius}\] + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}+Q_{glace}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &+6'600\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &+6'600\\ + &=919,6\cdot\theta_{eq.}-10'120=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{10'120}{919,6}=\SI{11,0}{\celsius}\] + \item Le bilan thermique est le suivant~: + \begin{align*} + &\sum Q=Q_{eau\,chaude}+Q_{eau\,froide}+Q_{glace}\\ + &=m_{eau\,c}\cdot c_{eau}\cdot (\theta_{eq.}-20)\\ + &+m_{eau\,f}\cdot c_{eau}\cdot (\theta_{eq.}-0)\\ + &+6'600+412\\ + &=0,2\cdot 4180\cdot (\theta_{eq.}-20)+0,02\cdot 4180\cdot\theta_{eq.}\\ + &+7'012\\ + &=919,6\cdot\theta_{eq.}-10'120=0 + \end{align*} + La température d'équilibre est donc~: + \[\theta_{eq.}=\frac{9'708}{919,6}=\SI{10,6}{\celsius}\] + \end{itemize} + On constate que de l'eau froide ou de la glace dont la température diminue permettent un refroidissement relativement faible, alors que la fonte de la glace constitue le principal facteur d'abaissement de la température du thé. + +\end{Solution OS} +\begin{Solution OS}{24} + Le bilan thermique s'écrit~: + \begin{align*} + m_{eau}\cdot c_{eau}\cdot (\theta_{eq.}-20)+m_?\cdot c_?\cdot (\theta_{eq.}-80)&=0\\ + 0,2\cdot 4180\cdot (21,55-20)&+\\ + 0,05\cdot c_?\cdot (21,55-80)&=0\\ + 1295,8-2,92\cdot c_?&=0 + \end{align*} + Ainsi, la chaleur massique vaut~: + \[c_?=\frac{1295,8}{2,92}=\SI{444}{\joule\celsius\per\kilo\gram}\] + En consultant le tableau \ref{tabchaleurmassique}, page \pageref{tabchaleurmassique}, on constate qu'il s'agit de fer. + +\end{Solution OS}

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