&RQFHSWRVEiVLFRV 5HVROXFLyQGH&XHVWLRQHV &XHVWLyQ &XHVWLyQ 6DELHQGRTXHODPDVDPROHFXODUGHKLGUyJHQRHV\ODGHOR[tJHQRFRQWHVWH UD]RQDGDPHQWHDODVVLJXLHQWHVFXHVWLRQHV D¢4XpRFXSDUiPiVYROXPHQXQPROGHKLGUyJHQRR XQPROGHR[tJHQRHQODV PLVPDVFRQGLFLRQHVGHSUHVLyQ\WHPSHUDWXUD" E¢4XpWHQGUiPiVPDVDXQPROGHKLGUyJHQRRXQPROGHR[tJHQR" F¢'yQGHKDEUiPiVPROpFXODVHQXQPROGHKLGUyJHQRRHQXQPROGHR[tJHQR" Solución D'HVSHMDQGRHOYROXPHQGHODHFXDFLyQGHORVJDVHVSHUIHFWRV QĊ5Ċ7 S (OYROXPHQGHKLGUyJHQR\GHR[tJHQRDOVHUQXQPROHQDPERVFDVRV\VHUS \7LJXDOHVVHUiHOPLVPR 9 E$SOLFDQGRHOFRQFHSWRGHPRO PRO+≡ĊPROpFXODV+≡J PRO2≡ĊPROpFXODV2≡J 3RUWDQWRODPDVDGHXQPROGHR[tJHQRHVVXSHULRUDODPDVDGHXQPROGH KLGUyJHQR F&RPRVHKDYLVWRHQHODSDUWDGRDQWHULRUHQDPERVKD\HOPLVPRQ~PHURGH PROpFXODV &XHVWLyQ &XHVWLyQ (Q P GH PHWDQR &+ PHGLGR HQ FRQGLFLRQHV QRUPDOHV GH SUHVLyQ \ WHPSHUDWXUDFDOFXOH D(OQ~PHURGHPROHVGHPHWDQR E(OQ~PHURGHPROpFXODVGHPHWDQR F(OQ~PHURGHiWRPRVGHKLGUyJHQR ,QLFLDFLyQDOD4XtPLFD Solución D$SOLFDQGRHOFRQFHSWRGHYROXPHQPRODU\FRPRP /FDOFXODPRVHO Q~PHURGHPROHVGHPHWDQRGHODVLJXLHQWHHFXDFLyQ /PHWDQR / = [ PROHVGHPHWDQR PROPHWDQR [ E $SOLFDQGR HO FRQFHSWR GH PRO HO Q~PHUR GH PROpFXODV GH PHWDQR SXHGH FDOFXODUVHGHODVLJXLHQWHH[SUHVLyQ PRO&+ PROHV = Ċ PROpFXODV&+ [ [ ·ĊPROpFXODVGHPHWDQR F(QFDGDPROpFXODGHPHWDQRKD\FXDWURiWRPRVGHKLGUyJHQRSRUWDQWR QGHiWRPRVGH+ [·ĊiW ·ĊiWGHKLGUyJHQR &XHVWLyQ &XHVWLyQ (Q·PROHVGH&2FDOFXOH D(OQ~PHURGHPROpFXODVGH&2 E/DPDVDGH&2 F(OQ~PHURWRWDOGHiWRPRV Datos.Masas atómicas: C = 12; O = 16. Solución D &RPR HQ XQ PRO GH GLy[LGR GH FDUERQR KD\ HO Q~PHUR GH $YRJDGUR GH PROpFXODVGHGLy[LGRGHFDUERQRHQ·PROHVKDEUi PRO&2 PROHV = Ċ PROpFXODV&2 [ [ ·ĊPROpFXODVGH&2 E/DPDVDGHGLy[LGRGHFDUERQRVHUi 0P&2 J &RQFHSWRVEiVLFRV PRO&2 PROHV = [ JGH&2 J&2 [ F&RPRHQFDGDPROpFXODKD\WUHViWRPRVXQRGHFDUERQR\GRVGHR[tJHQRHO Q~PHURWRWDOGHiWRPRVVHUi PRO&2 PROHV = [ Ċ iWRPRV [ [ ·ĊiWRPRV &XHVWLyQ &XHVWLyQ 5D]RQHTXpFDQWLGDGGHODVVLJXLHQWHVVXVWDQFLDVWLHQHPD\RUQ~PHURGHiWRPRV D·PROHVGH62 EJUDPRVGHQLWUyJHQRPROHFXODU F·OLWURVGHJDVKHOLRHQFRQGLFLRQHVQRUPDOHVGHSUHVLyQ\WHPSHUDWXUD Datos.Masas atómicas: N = 14; O = 16; S = 32. Solución &DOFXODPRVHOQ~PHURGHiWRPRVHQFDGDDSDUWDGR D$SOLFDQGRHOFRQFHSWRGHPRO\FRPRHQFDGDPROpFXODGHGLy[LGRGHD]XIUH KD\WUHViWRPRVXQRGHD]XIUH\GRVGHR[tJHQR 0P62 J PRO62 PROHV = [ Ċ iWRPRV [ Ċ iWRPRV [ E&RPRHQFDGDPROpFXODGHQLWUyJHQRKD\GRViWRPRVGHQLWUyJHQRHOQ~PHUR GHHVWRVVHUi 0P1 J J1 J [ < iWRPRV = [ Ċ iWRPRV [ F$SOLFDQGRHOFRQFHSWRGHYROXPHQPRODUHOQ~PHURGHiWRPRVGHKHOLRSXHGH FDOFXODUVHGHODVLJXLHQWHH[SUHVLyQ ,QLFLDFLyQDOD4XtPLFD /+H / = [ Ċ iWRPRV Ċ iWRPRV [ 3RUWDQWRODFDQWLGDGTXHWLHQHPDViWRPRVHV·/GHKHOLR &XHVWLyQ 5D]RQHVLVRQYHUGDGHUDVRIDOVDVODVVLJXLHQWHVDILUPDFLRQHV D 'RV PDVDV LJXDOHV GH ORV HOHPHQWRV $ \ % FRQWLHQHQ HO PLVPR Q~PHUR GH iWRPRV E/DPDVDDWyPLFDGHXQHOHPHQWRHVODPDVDHQJUDPRVGHXQiWRPRGHGLFKR HOHPHQWR F(OQ~PHURGHiWRPRVTXHKD\HQJGHR[tJHQRDWyPLFRHVLJXDODOQ~PHURGH PROpFXODVTXHKD\HQJGHR[tJHQRPROHFXODU Solución D )DOVD SRUTXH ODV PDVDV TXH FRQWLHQHQ HO PLVPR Q~PHUR GH iWRPRV GH DFXHUGRFRQHOFRQFHSWRGHPROVHULDQXQDPDVDGHOHOHPHQWR$LJXDOD0P$\ XQDPDVDGHOHOHPHQWR%LJXDOD0P%\VLHPSUHDOVHUGRVHOHPHQWRVGLVWLQWRV 0P$HVGLIHUHQWHGH0P% E )DOVD SRUTXH HV OD PDVD H[SUHVDGD HQ JUDPRV GH · Ċ iWRPRV GH GLFKRHOHPHQWRHVGHFLUGHXQPROGHiWRPRV F$SOLFDQGRHOFRQFHSWRGHPROFDOFXODPRVHOQ~PHURGHiWRPRVVDELHQGRTXH 0P2 J JGH2 J = [ Ċ iWRPRVGH2 Ċ iW2 [ &DOFXODPRVHOQ~PHURGHPROpFXODVGH2VDELHQGRTXH0P2 JGHOD VLJXLHQWHH[SUHVLyQ J2 J = [ Ċ PROpFXODVGH2 Ċ PROpFXODV2 [ 3RUWDQWRODDILUPDFLyQHVYHUGDGHUD &XHVWLyQ D¢&XiOHVODPDVDH[SUHVDGDHQJUDPRVGHXQiWRPRGHVRGLR" &RQFHSWRVEiVLFRV E¢&XiQWRViWRPRVGHDOXPLQLRKD\HQ·JGHHVWHHOHPHQWR" F¢&XiQWDVPROpFXODVKD\HQXQDPXHVWUDTXHFRQWLHQH·JGHWHWUDFORUXURGH FDUERQR" Datos.Masas atómicas: C = 12; Na = 23; Al = 27; Cl = 35’5. Solución D$SOLFDPRVHOFRQFHSWRGHPRO PRO1D≡ĊiWRPRV1D≡J 3RUWDQWR 1iW1D = J = Ċ JiW1D Ċ DW E(OQ~PHURGHiWRPRVGHDOXPLQLRSXHGHFDOFXODUVHGHODVLJXLHQWHH[SUHVLyQ 0P$O J J$O J$O [ Ċ iWRPRVGH$O = Ċ iW$O [ F$SOLFDPRVHOFRQFHSWRGHPRO 0P&O & J PRO&O &≡ĊPROpFXODV&O &≡J 3RUWDQWR J&O& J = [ Ċ PROpFXODV&O& Ċ PROpFXODV&O& [ &XHVWLyQ &XHVWLyQ 5D]RQHVLVRQYHUGDGHUDVRIDOVDVODVDILUPDFLRQHVVLJXLHQWHV D /D PDVD GH XQ LRQ PRQRYDOHQWH SRVLWLYR HV PHQRU TXH OD GHO iWRPR FRUUHVSRQGLHQWH E(OQ~PHURDWyPLFRGHXQLRQPRQRYDOHQWHSRVLWLYRHVPHQRUTXHHOGHOiWRPR FRUUHVSRQGLHQWH F (Q XQ JUDPR GH FXDOTXLHU HOHPHQWR KD\ PiV iWRPRV TXH KDELWDQWHV WLHQH OD 7LHUUD ,QLFLDFLyQDOD4XtPLFD Solución D9HUGDGHUDSRUTXHHOLRQPRQRYDOHQWHSRVLWLYRWLHQHXQHOHFWUyQPHQRVTXH HOiWRPRFRUUHVSRQGLHQWH\SRUWDQWRVXPDVDVHUiPHQRUHQ·Ċ.JTXH HV OD PDVD DWULEXLGD DO HOHFWUyQ $KRUD ELHQ GDGR TXH ODV PDVDV DWULEXLGDV DO SURWyQ \ DO QHXWUyQ VRQ GHO RUGHQ GH YHFHV VXSHULRUHV D OD PDVD GHO HOHFWUyQODPDVDGHXQLyQPRQRYDOHQWHSRVLWLYRHVDSUR[LPDGDPHQWHODPLVPD TXHODGHOiWRPRFRUUHVSRQGLHQWH E)DOVDSRUTXHHOQ~PHURDWyPLFRHVHOQ~PHURGHSURWRQHVGHOQ~FOHR\pVWH QR YDUtD HQ HO LyQ PRQRYDOHQWH SRVLWLYR TXH VH GLIHUHQFLD GHO iWRPR FRUUHVSRQGLHQWHHQTXHKDSHUGLGRXQHOHFWUyQ F(QXQPROGHiWRPRVGHFXDOTXLHUHOHPHQWRKD\·ĊiWRPRV&RPROD PDVD PRODU GHORV HOHPHQWRV HVWi FRPSUHQGLGD HQWUH \ PHQRV GH HQ XQ JUDPR GH HOHPHQWR KDEUtD HQWUH · Ċ iWRPRV \ [ 3RU WDQWR OD DILUPDFLyQHVYHUGDGHUD &XHVWLyQ &XHVWLyQ 8QYDVRFRQWLHQHP/GHDJXD&DOFXOH D&XiQWRVPROHVGHDJXDKD\HQHOYDVR E&XiQWDVPROpFXODVGHDJXDKD\HQHOYDVR F&XiQWRViWRPRVGHKLGUyJHQR\R[tJHQRKD\HQHOYDVR Datos.Masas atómicas: H = 1; O = 16. Solución D+D\TXHFRQVLGHUDUTXHDOVHUODGHQVLGDGGHODJXDJP/HOYDVRFRQWLHQH WDPELpQJGHDJXD&RPR0P+2 JHOQ~PHURGHPROHVVHUi J QPROHV+ = = PROHV+ JPRO E $SOLFDQGR HO FRQFHSWR GH PRO FDOFXODUHPRV HO Q~PHUR GH PROpFXODV HQ HO YDVR PRO+ PROHV [ Ċ PROpFXODVGH+ = Ċ PROpFXODV+ [ F &RPR FDGD PROpFXOD GH DJXD WLHQH XQ iWRPR GH R[tJHQR \ GRV iWRPRV GH KLGUyJHQR\FRPRWHQHPRV·ĊPROpFXODVGHDJXD &RQFHSWRVEiVLFRV QiW2 [·Ċ ·ĊiWRPRVGHR[tJHQR QiW+ [·Ċ ·ĊiWRPRVGHKLGUyJHQR &XHVWLyQ &XHVWLyQ 5D]RQHVLODVVLJXLHQWHVDILUPDFLRQHVVRQFRUUHFWDVRQR FJGH1+RFXSDQHQFRQGLFLRQHVQRUPDOHVXQYROXPHQGH·OLWURV E(QJ1+KD\·PROpFXODV F(QJGH2KD\·×iWRPRVGHR[tJHQR Datos.Masas atómicas: H = 1; N = 14; O = 16. Solución D'HDFXHUGRFRQODKLSyWHVLVGH$YRJDGURXQPROGHDPRQtDFRHQFRQGLFLRQHV QRUPDOHV RFXSD · / 3RU WDQWR OD DILUPDFLyQ HV FRUUHFWD \D TXH 0P 1+ J E$SOLFDQGRHOFRQFHSWRGHPRO PRO1+≡J1+≡ĊPROpFXODV1+ 3RUWDQWRODDILUPDFLyQHVFRUUHFWD F &RPR HQ XQD PROpFXOD GH 2 KD\ iWRPRV GH R[tJHQR HQ J GH 2 HO Q~PHURGHiWRPRVGHR[tJHQRVHUi PRO2≡J2≡ĊPROpFXODV2≡[ĊiWRPRV2 3RUWDQWRODDILUPDFLyQQRHVFRUUHFWD &XHVWLyQ &XHVWLyQ (QOLWURVGHKLGUyJHQR\HQOLWURVR[tJHQRDPERVHQODVPLVPDVFRQGLFLRQHV GHSUHVLyQ\WHPSHUDWXUDKD\ D(OPLVPRQ~PHURGHPROHV E,GpQWLFDPDVDGHDPERV F(OPLVPRQ~PHURGHiWRPRV ,QGLTXHVLVRQFRUUHFWDVRQRHVWDVDILUPDFLRQHVUD]RQDQGRODVUHVSXHVWDV Solución D'HVSHMDQGRHOQ~PHURGHPROHVGHODHFXDFLyQGHORVJDVHVSHUIHFWRV SĊ9 Q= 5Ċ7 ,QLFLDFLyQDOD4XtPLFD $SOLFDQGRHVWDH[SUHVLyQDODVFRQGLFLRQHVGHOHQXQFLDGR SĊ/ SĊ/ Q+ = Q2 = ; 57 57 /XHJR Q+ = Q2 Por tanto, hay el mismo número de moles y la afirmación es correcta. b) Como la 0P+HVGLVWLQWDGHla 0P2 QJUDPRV+ QPROHVGH+[0P+ QJUDPRV2 QPROHVGH+[0P2 /XHJRQJUDPRV+≠QJUDPRV23RUWDQWRODDILUPDFLyQHVLQFRUUHFWD F&DOFXOHPRVHOQ~PHURGHiWRPRVHQFDGDFDVRFRPRHQDPEDVPROpFXODVKD\ GRViWRPRV Q2 PROHV PROGH2 = [ Q2 [[ Ċ iW2 [ Ċ iW2 [ Q+ PROHV PRO+ = [ Q+ [[ Ċ iW+ [ Ċ iW+ [ &RPR HO Q~PHUR GH PROHV GH 2 HV LJXDO DO Q~PHUR GH PROHV GH + WHQHPRV TXH[ [/XHJRKD\HOPLVPRQ~PHURGHiWRPRV\ODDILUPDFLyQHVFRUUHFWD &XHVWLyQ &XHVWLyQ &DOFXOH F/DPDVDHQJUDPRVGHXQDPROpFXODGHDJXD E(OQ~PHURGHiWRPRVGHKLGUyJHQRTXHKD\HQJGHDJXD F (O Q~PHUR GH PROpFXODV TXH KD\ HQ / GH + TXH HVWiQ HQ FRQGLFLRQHV QRUPDOHVGHSUHVLyQ\WHPSHUDWXUD Datos.Masas atómicas: H = 1; O = 16. Solución D$SOLFDQGRHOFRQFHSWRGHPRO\FRPR0P+2 J PRO+2≡ĊPROpFXODV+2≡J /XHJR &RQFHSWRVEiVLFRV PROpFXOD+2 = J = Ċ J Ċ PROpFXODV E 'H OD VLJXLHQWH H[SUHVLyQ FDOFXODPRV ORV JUDPRV GH KLGUyJHQR TXH KD\ HQ JUDPRVGHDJXD J+2 J = [ JGH+ J+ [ $SOLFDQGRGHQXHYRHOFRQFHSWRGHPRO PRO+≡ĊiWRPRV+≡J J+ J = [ Ċ iWGH+ Ċ iW+ [ F$SOLFDQGRODKLSyWHVLVGH$YRJDGUR /GH+ / = [ Ċ PROpFXODVGH+ Ċ PROpFXODVGH+ [ &XHVWLyQ &XHVWLyQ &DOFXOHHOQ~PHURGHiWRPRVTXHKD\HQ DJGH&2 E/GHJDV+HPHGLGRVHQFRQGLFLRQHVQRUPDOHV F·PROHVGH2 Datos.Masas atómicas: C = 12; O = 16 Solución D (O Q~PHUR GH PROHV GH GLy[LGR GH FDUERQR FRQWHQLGR HQ J GH HVWD VXVWDQFLDFRPR0P&2 JVHUi J&2 QPROHV&2 = = PRO&2 JPRO &RPRHQFDGDPROpFXODGHGLy[LGRGHFDUERQRKD\WUHViWRPRVGRVGHR[tJHQR\ XQRGHFDUERQRDSOLFDQGRHOFRQFHSWRGHPRO QiWRPRV [·ĊiW ·ĊiWRPRV E$SDUWLUGHODKLSyWHVLVGH$YRJDGURFDOFXODPRVORViWRPRVGHKHOLRHQ/ PHGLDQWHODVLJXLHQWHHFXDFLyQ /+H / [ Ċ iW+H = Ċ iW+H [ ,QLFLDFLyQDOD4XtPLFD F &RPR HQ FDGD PROpFXOD GH 2 KD\ GRV iWRPRV GH R[tJHQR HO Q~PHUR GH iWRPRVVHUi PRO2 PROHV = [ Ċ iWRPRVGH2 [ Ċ iW2 [ &XHVWLyQ &XHVWLyQ /DV PDVDV DWyPLFDV GHO KLGUyJHQR \ GHO KHOLR VRQ \ UHVSHFWLYDPHQWH ,QGLTXHUD]RQDGDPHQWHVLODVVLJXLHQWHVDILUPDFLRQHVVRQYHUGDGHUDVRIDOVDV D8QPROGH+HFRQWLHQHHOPLVPRQ~PHURGHiWRPRVTXHXQPROGH+ E/DPDVDGHXQiWRPRGHKHOLRHVJUDPRV F(QXQJUDPRGHKLGUyJHQRKD\·ĊiWRPRV Solución D)DOVDSRUTXHHQXQPROGHKHOLRKD\·ĊiWRPRVGHKHOLRPLHQWUDV TXHHQXQPROGH+KD\·ĊPROpFXODVGH+\FRPRHQFDGDPROpFXOD GH+KD\GRViWRPRVKDEUi[·ĊiWRPRVGH+ E)DOVDSRUTXHODPDVDGHXQiWRPRGHKHOLRHV Ċ iW+H iW+H [ < J = J+H [ F9HUGDGHUDSRUTXH PROGHiWRPRVGH+≡ĊiWRPRV+≡J &XHVWLyQ &XHVWLyQ /DHVWULFQLQDHVXQSRWHQWHYHQHQRTXHVHKDXVDGRFRPRUDWLFLGDFX\DIyUPXOD HV&+123DUDPJGHHVWULFQLQDFDOFXOH D(OQ~PHURGHPROHVGHFDUERQR E(OQ~PHURGHPROpFXODVGHHVWULFQLQD F(OQ~PHURGHiWRPRVGHQLWUyJHQR Datos.Masas atómicas: C = 12; H = 1; N = 14; O = 16. Solución &RQFHSWRVEiVLFRV D&DOFXOHPRVHOQ~PHURGHPROHVGHHVWULFQLQDHQPJGHHVWDVXVWDQFLD 0P&+12 J JHVWULFQLQD J = [ Ċ PROHVGHHVWULFQLQD PROGHHVWULFQLQD [ (OQ~PHURGHPROHVGHFDUERQRVHUi [Ċ ·ĊPROHVGHFDUERQR E(OQ~PHURGHPROpFXODVGHHVWULFQLQDTXHKD\HQORVPROHVGHHVWDVXVWDQFLD FDOFXODGRVHQHODSDUWDGRDVHUi PROGHHVWULFQLQD Ċ PROHV = [ Ċ PROpFXODV Ċ PROpFXODV [ F&RPRHQXQDPROpFXODGHHVWULFQLQDKD\GRViWRPRVGHQLWUyJHQRHOQ~PHUR GHiWRPRVVHUi PROHVWULFQLQD Ċ PROHV [ Ċ iWRPRVGH1 = [ Ċ iW1 [ &XHVWLyQ &XHVWLyQ 'H XQ UHFLSLHQWH TXH FRQWLHQH J GH PHWDQR VH H[WUDHQ ⋅ PROpFXODV &DOFXOH D/RVPROHVGHPHWDQRTXHTXHGDQ E/DVPROpFXODVGHPHWDQRTXHTXHGDQ F/RVJUDPRVGHPHWDQRTXHTXHGDQ Datos0DVDVDWyPLFDV: H = 1; C = 12. Solución D&RPR0P&+ JHOQ~PHURGHPROHVGHPHWDQRHQJVHUi J QPROHV&+ = = PROHV JPRO &DOFXOHPRVORVPROHVGHPHWDQRTXHTXLWDPRVDOH[WUDHU⋅PROpFXODV PRO&+ [ = [ PROHVGH&+ Ċ PROpFXODV&+ Ċ 3RUWDQWRHOQ~PHURGHPROHVTXHTXHGDVHUi QPROHV&+ PROHV²·PROHV ·PROHV E/DVPROpFXODVGHPHWDQRTXHTXHGDQODVFDOFXODPRVGHODVLJXLHQWHHFXDFLyQ ,QLFLDFLyQDOD4XtPLFD PRO&+ PROHV = [ Ċ PROpFXODVGH&+ Ċ PROpFXODV&+ [ F/DPDVDGHPHWDQRVHUi PRO&+ PROHV = [ JGH&+ J&+ [ &XHVWLyQ &XHVWLyQ D ¢&XiQWRV iWRPRV GH R[tJHQR KD\ HQ OLWURV GH R[tJHQR PROHFXODU HQ FRQGLFLRQHVQRUPDOHV" E8QDSHUVRQDEHEHDOGtDOLWURGHDJXD6XSRQLHQGRTXHODGHQVLGDGGHODJXD HV GH JP/ ¢FXiQWRV iWRPRV GH KLGUyJHQR LQFRUSRUD D VX FXHUSR SRU HVWH SURFHGLPLHQWR" Datos.0DVDVDWyPLFDV: H = 1; O = 16. Solución D$SOLFDQGRHOFRQFHSWRGHYROXPHQPRODUFDOFXODPRVORVPROHVGH2TXHKD\ HQORV/HQFRQGLFLRQHVQRUPDOHV /2 / = [ PROHV PRO2 [ (OQ~PHURGHPROpFXODVFRQWHQLGRVHQHOORVVHUi ·PRO[·ĊPROpFXODVPRO ·ĊPROpFXODVGH2 (OQ~PHURGHiWRPRVGHR[tJHQRHQORV/VHUi [·Ċ ·ĊiWRPRVGH2 E/DPDVDGHDJXDTXHEHEHDOGtDVHUi P G[9 JP/[P/ JGH+2 $SOLFDQGRHOFRQFHSWRGHPRO\FRPRHQFDGDPROpFXODGHDJXDKD\GRViWRPRV GHKLGUyJHQR\0P+2 JORViWRPRVGHHVWHHOHPHQWRTXHLQFRUSRUDOD SHUVRQDDVXFXHUSRVHUiQ QiWRPRV+ [QPROpFXODVGH+2 [QPROHVGH+2[1$ J+2 QiW+ [ [ Ċ 23 = [ iWRPRVGH+ JPRO &RQFHSWRVEiVLFRV 5HVROXFLyQGH3UREOHPDV 3UREOHPD 3UREOHPD &XDQGR VH FDOLHQWD GH PRGR FRQYHQLHQWH FORUXUR DPyQLFR VH GHVFRPSRQH HQ FORUXURGHKLGUyJHQR\DPRQtDFR6LVHOLEHUDQJGHDPRQtDFRFDOFXOH D /DPDVDREWHQLGDGHFORUXURGHKLGUyJHQR E /DVIUDFFLRQHVPRODUHVGHFDGDJDVJHQHUDGR Datos. Masas atómicas: H = 1; N = 14; Cl = 35’5 Solución D /DUHDFFLyQHV1+&O+&O1+ 3XHVWR TXH OD PDVD PROHFXODU GHO DPRQtDFR 1+ HV LJXDO D Ċ JĊPRO ORVJGHDPRQtDFRFRUUHVSRQGHDPROGH1+ (QFRQVHFXHQFLDVHGHEHKDEHUIRUPDGRPROGH+&O3XHVWRTXHODPDVD PROHFXODU GHO +&O HV LJXDO D · · J Ċ PRO /D PDVD GH +&O IRUPDGDHVLJXDODPROĊ·JĊPRO ·JGH+&O E /DV IUDFFLRQHV PRODUHV GH FDGD JDV YLHQH GDGD SRU IPRODU PROHV GH JDV PROHVWRWDOHV3RUWDQWR IPRODU1+ PRO1+>PRO1+PRO+&O@ · IPRODU+&O PRO+&O>PRO1+PRO+&O@ · 3UREOHPD 3UREOHPD 8QDPH]FODGHGRVJDVHVHVWiFRQVWLWXLGDSRUJGH62\RWURVJGH62\HVWi FRQWHQLGDHQXQUHFLSLHQWHD&\DDWPGHSUHVLyQ&DOFXOH D (OYROXPHQTXHRFXSDODPH]FOD E /DIUDFFLyQPRODUGHFDGDJDV Datos.R = 0’082 atm L K-1 mol-1. Masas atómicas: O = 16; S = 32. Solución D &DOFXODPRVSULPHURHOYROXPHQTXHRFXSDQFDGDJDV 'HODHFXDFLyQGHORVJDVHVSHUIHFWRV3Ċ9 QĊ5Ċ7GHDTXtGHVSHMDQGR 9 WHQGUHPRV 9 Q Ċ 5 Ċ 7 3 SRU WDQWR WHQHPRV TXH FDOFXODU SULPHUR ,QLFLDFLyQDOD4XtPLFD Q PROHVGHJDV 0DVDGHJDVJ0DVDPROHFXODUGHJDVJĊPRO (QHO FDVRGHO62UHVXOWDQ62 JĊJĊPRO PROHV ·PROHV(QHOFDVRGHO62UHVXOWDQ62 JĊJĊPRO PROHV ·PROHV 7 . .3RUFRQVLJXLHQWH 9/ ·PROĊ·DWPĊ/Ċ.ĊPRO Ċ.DWP ·/62 9/ ·PROĊ·DWPĊ/Ċ.ĊPRO Ċ.DWP ·/62 9WRWDO 962962 ·/·/ ·/ 6HSRGUtDKDEHUUHVXHOWRWHQLHQGRHQFXHQWDORVPROHVWRWDOHVGHJDVHV QWRWDO Q62Q62 ·· ·PROHV\DSOLFDQGROD IyUPXODWHQGUtDPRV 9/ ·PROĊ·DWPĊ/Ċ.ĊPRO Ċ.DWP ·/ F /DV IUDFFLRQHV PRODUHV GH FDGD JDV YLHQH GDGD SRU IPRODU PROHV GH JDV PROHVWRWDOHV3RUWDQWR IPRODU62 ·PRO·PRO · IPRODU62 ·PRO·PRO · 3UREOHPD 3UREOHPD (QXQUHFLSLHQWHGH/GHFDSDFLGDGTXHHVWiD&KD\JGHXQDPH]FOD HTXLPRODUGHKLGUyJHQR\KHOLR&DOFXOH D /DSUHVLyQWRWDOGHOUHFLSLHQWH E /DVSUHVLRQHVSDUFLDOHVHMHUFLGDVSRUORVJDVHV Datos.R = 0’082 atm L K-1 mol-1. Masas atómicas: H = 1; He = 4. Solución D &DOFXOHPRVORVPROHVH[LVWHQWHVGHFDGDJDV JĊPRO Ċ[PROHV+JĊPRO Ċ\PROHV+H J [PROHV+ \PROHV+H 5HVROYLHQGRHVWHVLVWHPDUHVXOWD[ \ PROHV (OQWRWDOGHPROHVHV++H PROHVWRWDOHV$SOLFDQGROD IyUPXODGHORVJDVHVSHUIHFWRVOOHJDPRV &RQFHSWRVEiVLFRV 3DWP PROĊ·DWPĊ/Ċ.ĊPRO Ċ./ DWP E 3DUD HOOR SRGHPRV FDOFXODU ODV IUDFFLRQHV PRODUHV \ GH DTXt OOHJDPRV D 3SDUFLDO IPRODUĊ3WRWDO IPRODU+ IPRODU+H ·\DTXHVRQHTXLPROHFXODUHV3RUWDQWR 3SDUFLDO+ 3SDUFLDO+H ·Ċ DWP 6HSRGUtDKDEHUHVWLPDGRWDPELpQSRUPHGLRGHODHFXDFLyQGHORVJDVHV 3SDUFLDO+ 3SDUFLDO+H Ċ·Ċ DWP 3UREOHPD 3UREOHPD 6HGLVSRQHGHXQUHFLSLHQWHGH/GHFDSDFLGDGTXHVHPDQWLHQHVLHPSUHDOD WHPSHUDWXUDGH&\VHLQWURGXFHQHQHOPLVPR/GH&2DDWP\/GH &2DDWPDPERVD&&DOFXOH D /DFRPSRVLFLyQHQSRUFHQWDMHGHODPH]FOD E /DSUHVLyQGHOUHFLSLHQWH Datos.R = 0’082 atm L K-1 mol-1. Solución D /RVPROHVGHFDGDJDVTXHVHYDQDLQWURGXFLUVHFDOFXODQFRQODHFXDFLyQGH ORVJDVHV Q 3Ċ95Ċ7 DWPĊ/·DWP/.PRO Ċ. · PROHVGH&2 Q 3Ċ95Ċ7 DWPĊ/·DWP/.PRO Ċ. · PROHVGH&2 /RVPROHVWRWDOHVLQWURGXFLGRVVRQ·· ·PROHV (OSRUFHQWDMHGHFDGDJDVHQODPH]FODYLHQHGDGRSRU 3RUFHQWDMHGHJDV PROGHJDVPROHVWRWDOHVĊ 3RUFHQWDMHGH&2 ··Ċ · 3RUFHQWDMHGH&2 ··Ċ · E &RQORVGDWRVDQWHULRUHVOOHJDPRVDSDUWLUGHODHFXDFLyQGHORVJDVHV 3DWP ·PROĊ·DWPĊ/Ċ.ĊPRO Ċ./ ·DWP ,QLFLDFLyQDOD4XtPLFD 3UREOHPD 3UREOHPD (Q XQ UHFLSLHQWH GH / GH FDSDFLGDG TXH VH PDQWLHQH D OD WHPSHUDWXUD FRQVWDQWHGH&VHLQWURGXFHQ/GH+DDWPGHSUHVLyQ\/GH1D DWPGHSUHVLyQDPERVD&&DOFXOH D /DSUHVLyQWRWDOTXHHMHUFHODPH]FODJDVHRVDHQHOUHFLSLHQWH E /DVSUHVLRQHVSDUFLDOHVGHORVJDVHVGHVSXpVGHH[WUDHUGHOUHFLSLHQWH/GH OD PH]FOD JDVHRVD D DWP GH SUHVLyQ SHUR PDQWHQLHQGR OD WHPSHUDWXUD FRQVWDQWH Datos.R = 0’082 atm L K-1 mol-1. Solución D (VWHSUREOHPDHVVLPLODUDODQWHULRUSHURFRQRWURVJDVHV /RVPROHVGHFDGDJDVTXHVHYDQDLQWURGXFLUVHFDOFXODQFRQODHFXDFLyQGH ORVJDVHV Q 3 Ċ 9 5 Ċ 7 DWP Ċ / · DWP / . PRO Ċ . ·PROHVGH+ Q 3Ċ95Ċ7 DWPĊ/·DWP/.PRO Ċ. ·PROHVGH1 /RVPROHVWRWDOHVLQWURGXFLGRVVRQ·· ·PROHV 3DWP ·PROĊ·DWPĊ/Ċ.ĊPRO Ċ./ ·DWP E /RVPROHVGHJDVPH]FODTXHVHH[WUDHQVRQ Q 3Ċ95Ċ7 DWPĊ/·DWP/.PRO Ċ. ·PROHVGH PH]FOD /RVPROHVTXHTXHGDQHQHOUHFLSLHQWH ·²·PROHV ·PROHV\ GHDTXtVHOOHJDDODSUHVLyQWRWDOUHVWDQWH 3DWP ·PROĊ·DWPĊ/Ċ.ĊPRO Ċ./ ·DWP /DFDSDFLGDGGHOUHFLSLHQWHVHPDQWLHQHFRQVWDQWHD/ 3DUD HYDOXDU ODV SUHVLRQHV SDUFLDOHV HVWLPDPRV SULPHUR ODV IUDFFLRQHV PRODUHVFRQORTXHOOHJDPRVD3SDUFLDO IPRODUĊ3WRWDO /DVIUDFFLRQHVPRODUHVGHORVJDVHVQRVHDOWHUD\DTXHYLHQHQGDGDVSRUORV PROHVGHJDVGLYLGLGRVSRUORVPROHVWRWDOHV(QFRQVHFXHQFLDSXHVWRTXHOD PH]FODGHJDVHVVRQPH]FODVKRPRJpQHDVODH[WUDFFLyQGHJDVHVGHODPLVPD HVKDFHTXHVHPDQWHQJDQLQYDULDEOHVHVWDVIUDFFLRQHVPRODUHV3RUWDQWR IPRODU+ ·· · IPRODU1 ·· · 3SDUFLDO+ ·Ċ· ·DWP 3SDUFLDO1 ·Ċ· DWP &RQFHSWRVEiVLFRV 3UREOHPD 3UREOHPD 8QUHFLSLHQWHGH/FRQWLHQHXQDPH]FODGH&2\&2FX\DVIUDFFLRQHVPRODUHV VRQ·\·UHVSHFWLYDPHQWHHMHUFLHQGRODPH]FODXQDSUHVLyQGHDWPD ODWHPSHUDWXUDGH&&DOFXOH D /DSUHVLyQSDUFLDOHMHUFLGDSRUFDGDJDVHQHOUHFLSLHQWH E (OQ~PHURGHJUDPRVGHFDGDFRPSXHVWR Datos.R = 0’082 atm L K-1 mol-1. Masas atómicas: C = 12; O = 16. Solución D &RPR3SDUFLDO IPRODU3WRWDO3RUWDQWRGHORVYDORUHVGHOHQXQFLDGRWHQHPRV 3SDUFLDO&2 ·ĊDWP ·DWP 3SDUFLDO&2 ·ĊDWP ·DWP E (OQGHJUDPRVGHFDGDFRPSXHVWRGHSHQGHGHOQGHPROHVGHFDGDXQRGH HOORV 3RU HOOR DSOLFDQGR OD HFXDFLyQ GH ORV JDVHV OOHJDPRV 39 Q57 P570 GRQGHPVRQORVJUDPRVGHFRPSXHVWR\0VXPDVDPROHFXODU P 39057 DWPĊ/ĊJĊPRO ·DWP/.PRO Ċ. ·JSDUD&2 P 39057 DWPĊ/ĊJĊPRO ·DWP/.PRO Ċ. ·JSDUD&2 3UREOHPD 3UREOHPD &DOFXOHODFRPSRVLFLyQFHQWHVLPDOGHOiFLGRVXOI~ULFR Datos.Masas atómicas: H = 1; S = 32; O = 16 Solución &DOFXODPRVODPDVDPROHFXODUGHOiFLGRVXOI~ULFR 0+62 [[[ XPD 3DUD FDOFXODU OD FDQWLGDG GH FDGD HOHPHQWR FRQWHQLGR HQ J GH FRPSXHVWR HVWDEOHFHPRVODVUHODFLRQHV = [ + [ ,QLFLDFLyQDOD4XtPLFD = [ 6 [ = [ 2 [ 3UREOHPD 3UREOHPD 6HGLVXHOYHQJGHiFLGRQtWULFRHQ/GHDJXD&DOFXOH D /DPRODULGDGGHODGLVROXFLyQDQWHULRU E /D PRODULGDG GH OD GLVROXFLyQ UHVXOWDQWH GH GLOXLU P/ GH OD GLVROXFLyQ RULJLQDOKDVWD/ Datos.0DVDVDWyPLFDV+ 1 2 Solución D &DOFXODPRVODPDVDPROHFXODUGHOiFLGRQtWULFR 0+12 Ċ XPD /DPRODULGDGYLHQHGDGDSRU 0+12 PROHV/ ·0 E /Ċ0PROHVĊ/ ·Ċ·PROHV ·PROHV 3XHVWRTXHODPDVDHVFRQVHUYDWLYDHQHVWHWLSRGHUHDFFLRQHVWHQHPRVTXH / Ċ 0 PROHV Ċ / Ċ 0 PROHV · PROHV \ GH DTXt 0 PROHV · ·ĊFRQORTXHODFRQFHQWUDFLyQPRODUGHODGLVROXFLyQ GLOXLGDVHUi·0≅·0 3UREOHPD 3UREOHPD 6HWLHQHXQUHFLSLHQWHFRQP/GHXQDGLVROXFLyQTXHFRQWLHQHJGH1D2+ VLHQGRVXGHQVLGDGGH·JP/&DOFXOH D /DPRODOLGDGGHODGLVROXFLyQ E /DVIUDFFLRQHVPRODUHVGHVROXWR\GLVROYHQWH Datos.0DVDVDWyPLFDV+ 2 1D . &RQFHSWRVEiVLFRV Solución D /DPRODOLGDGYLHQHGDGDSRUODH[SUHVLyQ0 PROHVGHVROXWRJGH GLVROYHQWH3RUWDQWR 0ROHVGHVROXWR JJĊPRO ·PROHV 3HVRGHGLVROYHQWH 3HVRGLVROXFLyQ²SHVRGHVROXWR ·JĊP/ĊP/ ²J J 0 ·PROHVVROXWR·NJGLVROYHQWH ·PRODO E /DVIUDFFLRQHVPRODUHVGHVROXWR\GLVROYHQWHVRQ IPRODU PROHVVROXWRPROHVWRWDOHV ·PROHV>·JJĊ PRO @ ··· · IPRODU PROHVGLVROYHQWHPROHVWRWDOHV ·PROHV>·J JĊPRO @ ··· · 3UREOHPD 3UREOHPD 8QKLGURFDUEXURWLHQHODVLJXLHQWHFRPSRVLFLyQFHQWHVLPDOGHKLGUyJHQR GH FDUERQR 6DELHQGR TXH OD PDVD PROHFXODU GHO FRPSXHVWR HV &DOFXOH D/DIRUPXODHPStULFD b) /DIRUPXODPROHFXODU 'DWRV0DVDVDWyPLFDV+ & Solución D/DUHODFLyQGHPROHVGHFDGDHOHPHQWRSUHVHQWHHQHOKLGURFDUEXURVHREWLHQH GLYLGLHQGRODFDQWLGDGGHFDGDXQRHQWUHVXPDVDPRODU J + = PRO JPRO J = PRO & JPRO /DUHODFLyQHQPROHVFRLQFLGHFRQODUHODFLyQHQiWRPRV6LGLYLGLPRVHQWUHHO PHQRUGHHOORVTXHGD = iW & + = iW ,QLFLDFLyQDOD4XtPLFD &RPRODUHODFLyQGHFRPELQDFLyQWLHQHTXHVHUVHJ~QQ~PHURVHQWHURVVH PXOWLSOLFDQSRUGRV\GDODIRUPXODHPStULFDVLJXLHQWH&+ E&RPRVHFRQRFHHOYDORUGHODPDVDPROpFXOD [[Q[[Q Q GHVSHMDQGRQ FRQORTXHODIRUPXODPROHFXODUHV&+ 3UREOHPD 3UREOHPD (OVXOIDWRGHFREUHKLGUDWDGRDOFDOHQWDUVHD&VHWUDQVIRUPDHQVXOIDWRGH FREUHDQKLGUR&DOFXOH D /D IyUPXOD GHO VXOIDWR GH FREUH DQKLGUR VDELHQGR TXH VX FRPSRVLFLyQ FHQWHVLPDOHV6·2·\&XUHVWRKDVWD E (O Q~PHUR GH PROpFXODV GH DJXD TXH WLHQH HO FRPSXHVWR KLGUDWDGR FRQRFLHQGRTXH·JGHOKLGUDWRVHWUDQVIRUPDQDOFDOHQWDUHQ·J GHOFRPSXHVWRDQKLGUR 'DWRV0DVDVDWyPLFDV+ 2 6 &X · Solución D /DUHODFLyQGHPROHVGHFDGDHOHPHQWRSUHVHQWHHQHOFRPSXHVWRVHREWLHQH GLYLGLHQGRODFDQWLGDGGHFDGDXQRHQWUHVXPDVDDWyPLFD 6 ·JJĊPRO ·PRO 2 ·JJĊPRO ·PRO &X ·J·JĊPRO ·PRO /DUHODFLyQHQPROHVFRLQFLGHFRQODUHODFLyQHQiWRPRV6LGLYLGLPRVHQWUH HOPHQRUGHHOORVTXHGD 6 ··DW DW 2 ··DW DW &X ··DW DW &RPR OD UHODFLyQ GH FRPELQDFLyQ WLHQH TXH VHU VHJ~Q Q~PHURV HQWHURV OD IRUPXODHPStULFDHVODVLJXLHQWH&X62 E *UDPRVGHDJXD·²· ·J 3DUDKDOODUODUHODFLyQHQWUHORVPROHVGHVXOIDWR\DJXD ·· ·· · 'LYLGLHQGRHQWUHHOPHQRU ·· ·· 3RUWDQWRODIyUPXODHV&X62Ċ+2 &RQFHSWRVEiVLFRV 3UREOHPD 3UREOHPD 8QDVXVWDQFLDRUJiQLFDHVWiIRUPDGD~QLFDPHQWHSRU&·+·\2 UHVWRKDVWD(QFRQGLFLRQHVQRUPDOHVJGHOFRPSXHVWRJDVHRVRRFXSDQ ·FP&DOFXOH D/DIyUPXODHPStULFD E/DIyUPXODPROHFXODU 'DWRV0DVDVDWyPLFDV+ & 2 Solución D6HFDOFXODHOWDQWRSRUFLHQWRGHR[LJHQRFRQWHQLGDHQODPXHVWUD 2 ·· · $FRQWLQXDFLyQVHGLYLGHHOWDQWRSRUFLHQWRGHFDGDHOHPHQWRSRUODPDVDGHXQ PROGHiWRPRVPDVDDWyPLFD &= = + = = 2 = = 3DUD HQFRQWUDU OD UHODFLyQ GH FRPELQDFLyQ VH GLYLGHQ WRGRV HVRV YDORUHV SRU HO PHQRU & = + = 2 = /DIyUPXODHPStULFDVHUi C4 H10 O E 3DUDFDOFXODUODIyUPXODPROHFXODUHVQHFHVDULRFRQRFHUODPDVDGHXQPRO TXHVHFDOFXODDSOLFDQGRHOFRQFHSWRGHYROXPHQPRODU J 0 = 0 JPRO / / [[Q[[Q[Q Q Q /DIyUPXODPROHFXODUVHUi C4 H10 O ,QLFLDFLyQDOD4XtPLFD 3UREOHPD 3UREOHPD &XDQGR VH TXHPDQ · J GH XQ FRPSXHVWR RUJiQLFR VyOR FRQWLHQH & + 2 VH SURGXFHQ ·J GH &2 \ · J GH +2 ¢&XiO VHUi OD IyUPXOD HPStULFD GHO FRPSXHVWR" Datos.0DVDVDWyPLFDV+ & 2 Solución 6H FDOFXODQ ORV JUDPRV GH FDUERQR H KLGUyJHQR TXH KD\ HQ OD PXHVWUD \ SRU GLIHUHQFLDODGHR[tJHQR J&2 J [ J = J [ J+ 2 J = [ J J+ [ &DUERQR ·J+LGUyJHQR ·J2[LJHQR ·²·²· ·J $FRQWLQXDFLyQVHRSHUDLJXDOTXHDQWHV & = + = ; 2 = = = = 3DUDREWHQHUXQDFRPELQDFLyQVHJ~QQ~PHURVHQWHURVVHPXOWLSOLFDQWRGRVSRU FXDWURGDQGRODVLJXLHQWHIyUPXODHPStULFD &7 + 624 3UREOHPD 3UREOHPD (O VXOIDWR GH DPRQLR 1+62 VH XWLOL]D FRPR IHUWLOL]DQWH HQ DJULFXOWXUD &DOFXOH D(OWDQWRSRUFLHQWRHQSHVRGHQLWUyJHQRHQHOFRPSXHVWR E/DFDQWLGDGGHVXOIDWRGHDPRQLRQHFHVDULDSDUDDSRUWDUDODWLHUUDNJGH QLWUyJHQR 'DWRV0DVDVDWyPLFDV+ 1 2 6 &RQFHSWRVEiVLFRV Solución D&RPRHQFDGDPROGHVXOIDWRGHDPRQLRKD\JGHQLWUyJHQRHOHQSHVR GHQLWUyJHQRVHUi 0P [1+ 62 ] = J J1+ 2 62 = [ 1 J1 [ E&RPRVDEHPRVTXHDODERQDUFRQJGHVXOIDWRGHDPRQLRDSRUWDPRVJ GHQLWUyJHQRDODWLHUUDSDUDDSRUWDU.JGHQLWUyJHQRODFDQWLGDGQHFHVDULD GHVXOIDWRGHDPRQLRVHUi J1+ 62 [ [ JGH1+ 62 = J1 J 3UREOHPD 3UREOHPD 6H WRPDQ P/ GH XQ iFLGR VXOI~ULFR GH GHQVLGDG · J FP \ GHO GH ULTXH]DHQSHVR\VHOHDGLFLRQDDJXDKDVWDP/ D&DOFXOHODPRODULGDGGHODGLVROXFLyQUHVXOWDQWH E ,QGLTXH HO PDWHULDO QHFHVDULR \ HO SURFHGLPLHQWR D VHJXLU SDUD SUHSDUDU OD GLVROXFLyQ 'DWRV0DVDVDWyPLFDV+ 2 6 Solución D&DOFXODPRVODPDVDGHP/GHGLVROXFLyQ P GĊ9 ·JP/[P/ J &RPRODULTXH]DHVGHOODFDQWLGDGGHVROXWRiFLGRVXOI~ULFRVHUi QJ+62 J[· ·JGH+62 7UDQVIRUPDPRVODPDVDHQPROHV 0P+62 J J PROHV+62 = JPRO ,QLFLDFLyQDOD4XtPLFD 3RUWDQWRODPRODULGDGVHUi 0 PROHV = / E 0DWHULDO QHFHVDULR 3LSHWD GH P/ PDWUD] DIRUDGR GH P/ \ IUDVFR ODYDGRU 3URFHGLPLHQWR(QHOPDWUD]DIRUDGRGHP/VHDxDGHQDSUR[LPDGDPHQWH P/GHDJXDGHVWLODGDRGHVPLQHUDOL]DGD\DFRQWLQXDFLyQVHDxDGHQORVP/GH iFLGRVXOI~ULFRFRQFHQWUDGRTXHKHPRVWRPDGRFRQODSLSHWDGHOIUDVFRGHOiFLGR VXOI~ULFR FRQFHQWUDGR 6H DJUHJD OXHJR DJXD FRQ HO IUDVFR ODYDGRU KDVWD ODV SUR[LPLGDGHV GHO HQUDVH $ FRQWLQXDFLyQ VH HQUDVD DxDGLHQGR FRQ OD SLSHWD DJXD JRWD D JRWD KDVWD TXH OD SDUWH LQIHULRU GH OD VXSHUILFLH FyQFDYD VHD WDQJHQWHFRQODUD\DGHOHQUDVH3RU~OWLPRVHFLHUUDHOPDWUD]DIRUDGRFRQVX WDSyQ \ VH LQYLHUWH YDULDV YHFHV SDUD XQLIRUPL]DU OD FRQFHQWUDFLyQ GH OD GLVROXFLyQSUHSDUDGD 3UREOHPD 3UREOHPD D &DOFXOH OD PDVD GH 1D2+ VyOLGR GHO GH SXUH]D HQ SHVR QHFHVDULD SDUD SUHSDUDUP/GHGLVROXFLyQDFXRVD·0 E ([SOLTXH HO SURFHGLPLHQWR SDUD SUHSDUDU OD GLVROXFLyQ LQGLFDQGR HO PDWHULDO QHFHVDULR 'DWRV0DVDVDWyPLFDV+ 2 1D Solución D /RV PROHV QHFHVDULRV SDUD SUHSDUDU P/ GH GLVROXFLyQ DFXRVD · 0 VHUiQ QPROHV 0RODULGDGĊ9 QPROHV ·PROHV/[·/ ·ĊPROHVGH1D2+ 7UDQVIRUPDQGRORVPROHVDJUDPRV 0P1D2+ J QJ1D2+ ·Ċ[ ·J1D2+SXUR &RPRODVRVDHVGHOGHSXUH]DHQSHVRGH1D2+ &RQFHSWRVEiVLFRV QJ1D2+ ·[ ·J1D2+ E 0DWHULDO QHFHVDULR %DODQ]D YLGULR GH UHORM YDVR GH SUHFLSLWDGRV HPEXGR PDWUD]DIRUDGRGHP/IUDVFRODYDGRU\SLSHWD 3URFHGLPLHQWR6HSHVDQ·JGHKLGUy[LGRGHVRGLRVREUHXQYLGULRGHUHORM\ VHDxDGHQDOYDVRGHSUHFLSLWDGRVTXHFRQWLHQHXQRVP/GHDJXDGHVWLODGDR GHVPLQHUDOL]DGD6HDJLWDKDVWDTXHVHGLVXHOYDFRPSOHWDPHQWH$FRQWLQXDFLyQ VHYLHUWHPHGLDQWHHOHPEXGRODGLVROXFLyQHQHOPDWUD]DIRUDGRODYDQGRFRQXQ SRFR GH DJXD WDQWR HO YDVR GH SUHFLSLWDGRV FRPR HO HPEXGR SDUD GLVROYHU ODV SHTXHxDVFDQWLGDGHVGHKLGUy[LGRGHVRGLRTXHSXGLHUDQKDEHUTXHGDGR(ODJXD GHOODYDGRVHYLHUWHWDPELpQHQHOPDWUD]DIRUDGR6HDJUHJDOXHJRDJXDKDVWD ODVSUR[LPLGDGHVGHOHQUDVH$FRQWLQXDFLyQVHHQUDVDDxDGLHQGRFRQODSLSHWD DJXD JRWD D JRWD KDVWD TXH OD SDUWH LQIHULRU GH OD VXSHUILFLH FyQFDYD VHD WDQJHQWHFRQODUD\DGHOHQUDVH3RU~OWLPRVHFLHUUDHOPDWUD]DIRUDGRFRQVX WDSyQ \ VH LQYLHUWH YDULDV YHFHV SDUD XQLIRUPL]DU OD FRQFHQWUDFLyQ GH OD GLVROXFLyQSUHSDUDGD 3UREOHPD 3UREOHPD &DOFXODU OD PRODOLGDG GH XQD GLVROXFLyQ DFXRVD GH KLGUy[LGR GH VRGLR DO HQ SHVR 'DWRV0DVDVDWyPLFDV+ 2 1D Solución /DGLVROXFLyQGHKLGUy[LGRGHVRGLRDOHQSHVRLQGLFDTXHKD\JGHKLGUy[LGR GHVRGLRVROXWRHQJGHGLVROXFLyQ PDVDGH1D2+ J 0DVDGHGLVROYHQWH PDVDGLVROXFLyQ²PDVDGHVROXWR J²J J+2 /DPRODOLGDGPGHODGLVROXFLyQVHUi 0P1D2+ J J1D2+ PROHVGHVROXWR JPRO P = = PRODO J+2 .JGLVROYHQWH J ,QLFLDFLyQDOD4XtPLFD 3UREOHPD 3UREOHPD (Q HO ODERUDWRULR VH GLVSRQH GH XQ iFLGR FORUKtGULFR FX\D GHQVLGDG HV GH JP/\GHULTXH]DHQSHVR&DOFXOH D6XIUDFFLyQPRODU E6XPRODOLGDG 'DWRV0DVDVDWyPLFDV+ 2 &O Solución D&DOFXODPRVHOQ~PHURGHPROHVGHiFLGRFORUKtGULFR\GHDJXDTXHKD\HQXQ OLWURGHGLVROXFLyQ 0DVDGH/GHGLVROXFLyQ GĊ9 ·JP/[P/ J &RPRODGLVROXFLyQWLHQHXQGHiFLGRFORUKtGULFR 0DVDGH+&OSXUR [· J+&O 0DVDGH+2 PDVDGLVROXFLyQ²PDVDVROXWR ² J+2 (OQ~PHURGHPROHVGHVROXWR\GLVROYHQWHVHUiQ 00+&O ·J00+2 J J = PROHV QPROHV+&O JPRO J = PROHV QPROHV+2 JPRO 3RUWDQWRODIUDFFLyQPRODUVHUi PROHV+&O [+&O = PROHV+&O PROHV+2 PROHV+ 2 [+ 2 = PROHV+&O PROHV+2 &RPRVHFXPSOH[VROXWR[GLVROYHQWH &RQFHSWRVEiVLFRV &RPSUREDPRVTXH·· E 'H ORV GDWRV GHO DSDUWDGR DQWHULRU VDEHPRV TXH WHQHPRV J GH iFLGR FORUKtGULFRVROXWR\JGHDJXDGLVROYHQWHOXHJRODPRODOLGDGVHUi J+&O PROHVGHVROXWR JPRO P = = PRODO 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