From eff363521fb7e19cd4f869ce0224f98c7e7d9ede Mon Sep 17 00:00:00 2001 From: guyotv Date: Mon, 19 Sep 2022 00:01:55 +0200 Subject: [PATCH] Suite des exos sur les gaz parfaits. --- Annexe-Exercices/Annexe-Exercices.tex | 70 ++++++++++++++++++++++ Annexe-Exercices/Annexe-Exercices.tex.bak | 70 ++++++++++++++++++++++ CoursMecaniqueOS.pdf | Bin 12842810 -> 12849987 bytes CoursMecaniqueOSDF.pdf | Bin 12842810 -> 12849987 bytes SolutionsOS.tex | 55 ++++++++++++++++- 5 files changed, 194 insertions(+), 1 deletion(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index 7a9ad5b..672e877 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2586,6 +2586,76 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + Un récipient contient deux kilogrammes d'hélium à une pression de deux bar. Déterminez la pression d'une même masse d'oxygène placée dans le même récipient~: + \begin{enumerate} + \item à la même température et + \item au double de la température. + \end{enumerate} + Réponses~: \SI{25000}{\pascal} et \SI{0,5e5}{\pascal}. + \begin{solos} + Par définition de la masse molaire, on a~: + \[M=\frac{m}{n}\;\Rightarrow\;n=\frac{m}{M}\] + On peut donc écrire la loi des gaz parfaits de la manière suivante~: + \[p\cdot V=\frac{m}{M}\cdot R\cdot T\;\Rightarrow\;p\cdot M=m\cdot R\cdot \frac{T}{V}\] + Comme la masse, la température et le volume sont constant, on a aussi~: + \[p\cdot M=const\;\;\text{ou}\;\;p_1\cdot M_1=p_2\cdot M_2\] + Ainsi, pour le premier point, on a finalement~: + \[p_2=p_1\cdot \frac{M_1}{M_2}=2\cdot 10^5\cdot \frac{4}{32}=\SI{25000}{\pascal}\] + car la masse molaire de l'hélium vaut quatre et que l'oxygène de masse atomique seize est diatomique. + + \smallskip + Si la température varie, l'équation des gaz parfaits devient~: + \[p\cdot \frac{M}{T}=\frac{m\cdot R}{V}\;\Rightarrow\;p\cdot \frac{M}{T}=const\] + ou sous la forme de deux états~: + \begin{align*} + p_1\cdot \frac{M_1}{T_1}=p_2\cdot \frac{M_2}{T_2}\;\Rightarrow\;p_2=p_1\cdot \frac{M_1\cdot T_2}{M_2\cdot T_1}\\ + p_2=2\cdot 10^5\cdot \frac{4\cdot 2\cdot T_1}{32\cdot T_1}=\SI{0,5e5}{\pascal} + \end{align*} + \end{solos} +\end{exos} + +\begin{exos} + Un mobil-home a un volume de \SI{200}{\metre\cubed}. + \begin{enumerate} + \item Trouvez la masse d'air dans celui-ci à une température de \SI{20}{\celsius} et une atmosphère. + \item En supposant le mobil-home étanche, calculez la pression intérieure si la température passe à \SI{25}{\celsius}. + \item Si il n'est pas étanche, calculez la masse d'air qui s'en échappe. + \end{enumerate} + On peut considérer que la masse molaire de l'air est de \SI{29}{\gram\per\mol}. Réponses~: \SI{238}{\kilo\gram}, \SI{1,02e5}{\pascal} et \SI{4}{\kilo\gram}. + \begin{solos} + On tire de la définition de la masse molaire~: + \[M=\frac{m}{n}\;\Rightarrow\;n=\frac{m}{M}\] + \begin{enumerate} + \item La loi des gaz parfaits devient alors~: + \[p\cdot V=\frac{m}{M}\cdot R\cdot T\] + La masse d'air qui se trouve dans le mobil-home est ainsi~: + \begin{align*} + m&=\frac{p\cdot V\cdot M}{R\cdot T}=\frac{1,013\cdot 10^5\cdot 200\cdot 29\cdot 10^{-3}}{8,31\cdot 293,15}\\ + &=\SI{241,183}{\kilo\gram} + \end{align*} + \item Avec une masse de \SI{241,183}{\kilo\gram} d'air, on a un nombre de moles de~: + \[n=\frac{241,183}{29\cdot 10^-3}=\SI{8316,652}{\mol}\] + 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}\\ + &=\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} + \end{align*} + \item À \SI{25}{\celsius} et une atmosphère, le nombre de moles dans le mobil-home vaut~: + \[n=\frac{p\cdot V}{R\cdot T}=\frac{1,013\cdot 10^5\cdot 200}{8,31\cdot 298,15}=\SI{8177,181}{\mol}\] + Ainsi, la différence~: + \[\Delta n=8316,652-8177,181=\SI{139,471}{\mol}\] + constitue le nombre de moles qui sont sorties du mobil-home. La masse correspondante est~: + \[m=M\cdot n=29\cdot 10^{-3}\cdot 139,471=\SI{4}{\kilo\gram}\] + \end{enumerate} + \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 7a9ad5b..06edcfd 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2586,6 +2586,76 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + Un récipient contient deux kilogrammes d'hélium à une pression de deux bar. Déterminez la pression d'une même masse d'oxygène placée dans le même récipient~: + \begin{enumerate} + \item à la même température et + \item au double de la température. + \end{enumerate} + Réponses~: \SI{25000}{\pascal} et \SI{0,5e5}{\pascal}. + \begin{solos} + Par définition de la masse molaire, on a~: + \[M=\frac{m}{n}\;\Rightarrow\;n=\frac{m}{M}\] + On peut donc écrire la loi des gaz parfaits de la manière suivante~: + \[p\cdot V=\frac{m}{M}\cdot R\cdot T\;\Rightarrow\;p\cdot M=m\cdot R\cdot \frac{T}{V}\] + Comme la masse, la température et le volume sont constant, on a aussi~: + \[p\cdot M=const\;\;\text{ou}\;\;p_1\cdot M_1=p_2\cdot M_2\] + Ainsi, pour le premier point, on a finalement~: + \[p_2=p_1\cdot \frac{M_1}{M_2}=2\cdot 10^5\cdot \frac{4}{32}=\SI{25000}{\pascal}\] + car la masse molaire de l'hélium vaut quatre et que l'oxygène de masse atomique seize est diatomique. + + \smallskip + Si la température varie, l'équation des gaz parfaits devient~: + \[p\cdot \frac{M}{T}=\frac{m\cdot R}{V}\;\Rightarrow\;p\cdot \frac{M}{T}=const\] + ou sous la forme de deux états~: + \begin{align*} + p_1\cdot \frac{M_1}{T_1}=p_2\cdot \frac{M_2}{T_2}\;\Rightarrow\;p_2=p_1\cdot \frac{M_1\cdot T_2}{M_2\cdot T_1}\\ + p_2=2\cdot 10^5\cdot \frac{4\cdot 2\cdot T_1}{32\cdot T_1}=\SI{0,5e5}{\pascal} + \end{align*} + \end{solos} +\end{exos} + +\begin{exos} + Un mobil-home a un volume de \SI{200}{\metre\cubed}. + \begin{enumerate} + \item Trouvez la masse d'air dans celui-ci à une température de \SI{20}{\celsius} et une atmosphère. + \item En supposant le mobil-home étanche, calculez la pression intérieure si la température passe à \SI{25}{\celsius}. + \item Si il n'est pas étanche, calculez la masse d'air qui s'en échappe. + \end{enumerate} + On peut considérer que la masse molaire de l'air est de \SI{29}{\gram\per\mol}. Réponses~: \SI{238}{\kilo\gram}, \SI{1,02e5}{\pascal} et \SI{4}{\kilo\gram}. + \begin{solos} + On tire de la définition de la masse molaire~: + \[M=\frac{m}{n}\;\Rightarrow\;n=\frac{m}{M}\] + \begin{enumerate} + \item La loi des gaz parfaits devient alors~: + \[p\cdot V=\frac{m}{M}\cdot R\cdot T\] + La masse d'air qui se trouve dans le mobil-home est ainsi~: + \begin{align*} + m&=\frac{p\cdot V\cdot M}{R\cdot T}=\frac{1,013\cdot 10^5\cdot 200\cdot 29\cdot 10^{-3}}{8,31\cdot 293,15}\\ + &=\SI{241,183}{\kilo\gram} + \end{align*} + \item Avec une masse de \SI{241,183}{\kilo\gram} d'air, on a un nombre de moles de~: + \[n=\frac{241,183}{29\cdot 10^-3}=\SI{8316,652}{\mol}\] + 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} + &=\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} + \end{align*} + \item À \SI{25}{\celsius} et une atmosphère, le nombre de moles dans le mobil-home vaut~: + \[n=\frac{p\cdot V}{R\cdot T}=\frac{1,013\cdot 10^5\cdot 200}{8,31\cdot 298,15}=\SI{8177,181}{\mol}\] + Ainsi, la différence~: + \[\Delta n=8316,652-8177,181=\SI{139,471}{\mol}\] + constitue le nombre de moles qui sont sorties du mobil-home. La masse correspondante est~: + \[m=M\cdot n=29\cdot 10^{-3}\cdot 139,471=\SI{4}{\kilo\gram}\] + \end{enumerate} + \end{solos} +\end{exos} + %\begin{exos} % Un énoncé de test. % \begin{solos} diff --git a/CoursMecaniqueOS.pdf b/CoursMecaniqueOS.pdf index f535c14c1b59b4625e76c5f321997c4376a9e9e7..ee358c8b4f01f951fdf4fd0ab6c69eb9ebd0a76c 100644 GIT binary patch delta 97378 zcmWjJLtrHe5(UuMwr$%sI<{@wd9iKVww(?;wr$&(KZ~=jO)Uzy>n0vzxGVvp!43pC zI@umX5RBNl2&}mgq#z#fU`f|0lSHpmO!YqW#@Lkd@*bd@0OkZ7bmhV6vvgb)v7bOB zM4-Dx`SF~C!cqRy|FV0T&dLMe@ZmRkUibQ(eg60UUA^8rraX=Blx4%+|6#wfHFPqG zVojZmYZP(vSe@S7ub}};8(}u8o=*q1ALFdGZ9vD3tjnb2*Qg0#At6ic@@~+C8Qe>{ zXbvS&Wv2b&mZPffdjY8Y9jIi)bc{_|5e{BKBJT(dr=w0xqb{rp|H+V@OsV==-1)(} zLDuGp#XZtxk@h?*y!4n`m=gvbzTlCv+Y}WR4PF`o>XvUROCI9J<)PL(BSWv58|w-z z%hqjdQG@WHruhXpJ-=TKE?>=GJ?qlYMAYqmD5JnK=Aqc__sV+KJQLpj46pZw8_eu} zx%#knzqWH{Z+{m5=6@Ra!Tj=l+5S|o2ZujO2Jfb4&fm%wx^%twsP5QL|g;|Dp#-%tFk@cSbNxLONanP2aZx1<~bT{1Wg}`R)gQL z0d?~wHcY;|rR1PHc^uz~Au_aL;utAd6L`LimLW=20h2e|xOUYVuyqj&8>$Wk*~-?# z1;eu9bQX7+{sHzw%{5BPi+AW%n;%k|`g%TWT&F;0J>Q=ng}P1vPBt(`$pM!FV#$^e z0ebYOU}-=$z6@lsscm;N1#b;wfqi(hMEJ@2*9VM)}@J%3>@1+HUYZ8r^_aoV1NxtY|6ixyJW}KjT+!zuD^8PMk4aLTq zm^#T-9qCG-4i|3cx7{L*2iB+t)}>@l(YaxvmIocVVb15P9@yrzt13L!T#K4_ri%$qa173{hCVVlF55g72{WO7$=OFO{MD+&_d7QYUJ@=AxwBIT=z!lgkvHK#miA3`lJ7RAa*lHfXbn-P zpC-S{73!wIT;XpE!|)Mvs$pr_B(Nt=ST3ye&q_`?pXOGrw3LVzFlP7=E>ga?Zfv>x zNEt^zav8WXK^o(3wgIP(nz|UbojQ|Qn%%RY3f6T+OkdJitg6?RGM%jDCGSy=at{8% zG!S3%CVjlsct7{8?((-d724eaDA|)WlD`}wAK)GFIQFW84U7G1W-FZ0nG@m2j;Cg4 zNVTfEffjMdhb1&h__=2pwGY;VhHrWIa0K=_4cf>A zrKhHV|J5#e@Use(tQbY-E8qRMK?th_XDRUoQH>TSQ?&ILDZ}0*+j~R=P%yn==N9V3 z<%VGKSV6xhrr(&by`A;AV-=lt>NK<9wk35I-z7D%7|az!8KG3DambBOBrP4Lbg zEQ$L#YH7I&e>BuR?Y!?t!}#ryP)Ws^6hvXnn0a@~q=#^$BVfi7ddUc)N8wwpoKAI; zcwnHN*-w*vl|cxIWz5(YD^&0l_JUk=Bsm}EPsz5AoiuI)L?xPlNN=O$NKZ?7T0VGM z&^x8e#C6kzYsDXuO}5JC6y8K{5h>Yf-UPc+nd!FvCpI6|fRr^o`%I*zNR1B$a!RlF1n zI@0IiG=f|?V!Tdgz%+S!Wn*!f)?Uw)aefjjO(Hfa|LQ=hm*>(YXLDenV5$N- zcL98+8p$hJ)WEAr*;v3@$=KPHz)i`*+1S9r$vWBIRIe;H2z^{;E~Q?G=k8jU3Yh+L z=gKkgeL72{9Y?8)ERLV&!2mEq}CCqum;a?yI&-ePrqCF^^=#($mp zf!Op5=PRbwl`0B3AGx-TqY=q<%$ssqaYfHE_?KSD;_d*4vNOns>rZ$jL*cy(`*0edAp$j~9>bdpqghNat%D+->}>5t2G*_%#Xd=(YkfPJp!T zDUeqz>$6kVxy}LipDj?7dF|4YC=L1r3ar8xW zFRrsOVx%Xiz|Ixct6E{wXwB&ybr-fn=MviY>YT&fgPDzT*`(NRo-7+@VzS$}R-PC> zJK(trbYi&8Ty+_S~wjR z$cNR)0V-IPG~4mvU*hyyrhAeV-PzTfU&B@F>{r+Ie1~Q^)>?}aB>%vgek%9UsoTuJ z@=S!7F1~VxNd2t+F3kqa-K%c$3}AC?I78=f$z!?KVlcrPvKKMfLf%laEy;=Oi+mlH z(v%qmd*U|Khb#WWUhIVRICs0vl{s?pKpu7oM_kgJ616CkN+`M!xgXTAV_$w6@C0?{1~D?B0^s`yw~88z zQ`R9&IaaG1TzDz_F6lnFH8P{Ha5dj`x0a};!O z_5+E3hL%e$Nh#93$NlH?0QRqIh!ruQ#Z4$i89{{<;X*l|iJs8ohL_LKZwRX9?R6up zxo()ThahE*0(?Ua@B%RJV>!FA@hoyP9JTK>*#%!?1y}z&Oh+}`Lb`#u{Um506x-m=j2sil|V z^UwHR;(P(tVXngYy)A-9k%yko(?Gjv$gs1A%D*q1-(M7XHx8#6^S#d9@yHV+D{LldI6-5C^_lT&`W1enSejvwlJq}nqIKW z9>I?2cGf%iaggZ_+@e&a+?ts)o7&JLm3YGuMC34^fQUC1DEmfo)G}|J&2oy4$K%1j z-N1|#o*uHJh979lHUN9Ob?SPhPZ>~?P`!5l(2X)1=#n9*kxyB&Wm#L#!T3aGrC|Nun=|UN&_-3Ha*VCQBKaC*zCV6 zK%{vGX_n!WpY6=QlXl?_;YW2FeZ0eTwCABYi=GQ%4y>?lwmt&7j9`vxTI6^Gl; zBd^v(%+I_Ehh|G=9&kDdN&xcPhocBR;7kt~VGAl>u4`#92mXI_$_*UaJ zq?@W?To9MwPk>Vi*S{IX?K%tJOJc)3^Yg>o%`DRhCqGc_Sf-SyoCJVSA8~=7<5jFn z4O~<&oL)?c>9k=5Xg67w1VeR#KjV)9T`uk(6>32rZXO%m*@jg23$0i^v6~?cFf0iQ z!_&a6$sFy z0z;;5C-M^a+yW~x3d-ZzJ|9a7bs=Nzpc)Mhe`enf`6K0Emv!^-gz0`e2UZxJK~Ooe~MZX@%|lhd$h ze#pZFmgTk_J3F-uH+v~63}d6vYW;t(-N!zdZ)p{37LoFUsna;(r>YI=FAxmqHqJT-KYR&Pw1AQBy z*Im566@_HHv;{3i@^4@N5(#sjE*W=|o*b<{pQLeNqMV|~= z)(IyRs`{w?>Cs{A*tB=YI4>IFZl|zBwgTf5c720O?2NHL<>|@-Rx?3QZBi>Kv&)j- zaQ*V)4B(~;(H3Dv5Wr>iOm!v#vhbNBp~?v=ck?K4++hX8#QU`U4&6gpIdUfph2#0Tt>LxTgwYyu9o0~LvnzOz9WGh_vSwdUn35} zzS~JcWw-}KVC4}KD;WNXz>^HSq>P%4h#cFrbKL~r^n)UhgW8cYgZHUXTlF%$eFF

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