4XtPLFDGHO&DUERQR 5(62/8&,Ï1'(&8(67,21(6 &XHVWLyQ &XHVWLyQ 5D]RQHVLVRQYHUGDGHUDVRIDOVDVODVVLJXLHQWHVDILUPDFLRQHV D5HFLEHHOQRPEUHGHJUXSRIXQFLRQDOXQiWRPRRJUXSRGHiWRPRVGLVWULEXLGRV GHWDOIRUPDTXHODPROpFXODDGTXLHUHXQDVSURSLHGDGHVTXtPLFDVFDUDFWHUtVWLFDV E 'RV FRPSXHVWRV RUJiQLFRV TXH SRVHHQ HO PLVPR JUXSR IXQFLRQDO VLHPSUH VRQ LVyPHURV F 'RV FRPSXHVWRV RUJiQLFRV FRQ OD PLVPD IyUPXOD PROHFXODU SHUR GLVWLQWD IXQFLyQQXQFDVRQLVyPHURV Solución D&LHUWR(VODGHILQLFLyQGHJUXSRIXQFLRQDO E)DOVD1RWLHQHSRUTXpWHQHUODPLVPDIyUPXODPROHFXODU F)DOVD3XHGHQVHULVyPHURVGHIXQFLyQ &XHVWLyQ &XHVWLyQ -XVWLILTXHODYHUDFLGDGRIDOVHGDGGHODVVLJXLHQWHVSURSRVLFLRQHV D/RVKLGURFDUEXURVVDWXUDGRVVRQPXFKRPiVUHDFWLYRVTXHORVLQVDWXUDGRV E *UXSR IXQFLRQDO HV XQ iWRPR R JUXSR GH iWRPRV TXH FRQILHUH D OD FDGHQD KLGURFDUERQDGDXQDVSURSLHGDGHVTXtPLFDVFDUDFWHUtVWLFDV F(QHOPHWDQRHOiWRPRGHFDUERQRSUHVHQWDKLEULGDFLyQVS Solución D )DOVR /RV KLGURFDUEXURV VDWXUDGRV VRQ SRFR UHDFWLYRV SRUTXH QR WLHQHQ JUXSRVIXQFLRQDOHV/RVKLGURFDUEXURVLQVDWXUDGRVVRQPiVUHDFWLYRVGHELGRDVXV GREOHVRWULSOHVHQODFHVVLHQGRODUHDFFLyQPiVIUHFXHQWHODGHDGLFLyQ E&LHUWR(VODGHILQLFLyQGHJUXSRIXQFLRQDO F&LHUWR (OFDUERQRGLVSRQHVXVHQODFHVVHQFLOORVWHWUDpGULFDPHQWHSRUORTXH XWLOL]DKLEULGDFLyQVS ,QLFLDFLyQDOD4XtPLFD &XHVWLyQ &XHVWLyQ 5D]RQHVLVRQYHUGDGHUDVRIDOVDVODVVLJXLHQWHVDILUPDFLRQHV D(OSXQWRGHHEXOOLFLyQGHOPHWDQRHVPHQRUTXHHOGHOEXWDQRO E/DPROpFXOD&+&OSRVHHXQDJHRPHWUtDWHWUDpGULFDFRQHOiWRPRGHFDUERQR RFXSDQGRODSRVLFLyQFHQWUDO F(OHWDQRHVPiVVROXEOHHQDJXDTXHHOHWDQRO Solución D &LHUWD (O EXWDQRO WLHQH FDUiFWHU SRODU \ IRUPD HQODFHV GH KLGUyJHQR D WUDYpV GHO JUXSR 2+ SRU OR TXH VX SXQWR GH HEXOOLFLyQ HV PD\RU $GHPiV DO WHQHUPD\RUWDPDxRTXHHOPHWDQRWDPELpQVRQPD\RUHVODVIXHU]DVGH9DQGHU :DDOVHQWUHVXVPROpFXODV(OPHWDQRHVXQDPROpFXODDSRODU\QRIRUPDHQODFHV GHKLGUyJHQR E &LHUWD (O iWRPR GH FDUERQR DGRSWD KLEULGDFLyQ VS \ SRU OR WDQWR OD JHRPHWUtDGHODPROpFXOD&+&OHVWHWUDpGULFD F )DOVD (O HWDQRO HV VROXEOH HQ DJXD SRU VHU XQD PROpFXOD SRODU \ FDSD] GH IRUPDU HQODFHV GH KLGUyJHQR D WUDYpV GHO JUXSR 2+ FRQ ODV PROpFXODV GH HVWH GLVROYHQWH PLHQWUDV TXH HO HWDQR HV XQD PROpFXOD QR SRODU \ SRU WDQWR QR VROXEOHHQDJXD &XHVWLyQ &XHVWLyQ /DV IyUPXODV PROHFXODUHV GH WUHV KLGURFDUEXURV OLQHDOHV VRQ &+ &+ &+ 5D]RQHVLVRQYHUGDGHUDVRIDOVDVODVVLJXLHQWHVDILUPDFLRQHV D/RVWUHVSHUWHQHFHQDODPLVPDVHULHKRPyORJD E/RVWUHVSUHVHQWDQUHDFFLRQHVGHDGLFLyQ F/RVWUHVSRVHHQiWRPRVGHFDUERQRFRQKLEULGDFLyQVS Solución D)DOVD3RUGHILQLFLyQGHVHULHKRPyORJDORVKRPyORJRVVRORVHGLIHUHQFLDQHQ HO Q~PHUR GH JUXSRV –CH2 – (O &+ WLHQH XQD LQVWDXUDFLyQ SRU OR TXH QR SHUWHQHFH D OD PLVPD VHULH KRPyORJD TXH ORV KLGURFDUEXURV VDWXUDGRV &+ \ &+TXHVtVRQPLHPEURVGHODPLVPDVHULH E )DOVD /RV KLGURFDUEXURV VDWXUDGRV QR H[SHULPHQWDQ UHDFFLRQHV GH DGLFLyQ PLHQWUDVTXHORVKLGURFDUEXURVDOLIiWLFRVLQVDWXUDGRVGDQUHDFFLRQHVGHDGLFLyQ DOGREOHRWULSOHHQODFH 4XtPLFDGHO&DUERQR F&LHUWD 7RGRVORVFDUERQRVFRQHQODFHVVHQFLOORVSUHVHQWDQKLEULGDFLyQVS \ WRGDVODVPROpFXODVWLHQHQDOPHQRVXQR &XHVWLyQ &XHVWLyQ /DV IyUPXODV PROHFXODUHV GH WUHV KLGURFDUEXURV OLQHDOHV VRQ &+ &+ &+ 5D]RQHVLVRQYHUGDGHUDVRIDOVDVODVVLJXLHQWHVDILUPDFLRQHV D/RVWUHVSHUWHQHFHQDODPLVPDVHULHKRPyORJD E/RVWUHVH[SHULPHQWDQUHDFFLRQHVGHVXVWLWXFLyQ F6yORXQRGHHOORVWLHQHiWRPRVGHFDUERQRFRQKLEULGDFLyQVS Solución D)DOVD 3RUGHILQLFLyQGHVHULHKRPyORJDORVKRPyORJRVVRORVHGLIHUHQFLDQHQ HO Q~PHUR GH JUXSRV PHWLOHQR –CH2– VLQ HPEDUJR HO &+ WLHQH XQD LQVDWXUDFLyQ PLHQWUDV TXH HO &+ \ HO &+ Vt SHUWHQHFHQ D OD PLVPD VHULH KRPyORJD\VRQKLGURFDUEXURVVDWXUDGRV E)DOVD(O&+HVXQDOTXHQR\ODUHDFFLyQFDUDFWHUtVWLFDGHOGREOHHQODFHHVOD DGLFLyQ F &LHUWD 6yOR HO &+ \D TXH ORV FDUERQRV FRQ GREOH HQODFH SUHVHQWDQ KLEULGDFLyQVS &XHVWLyQ &XHVWLyQ D,QGLTXHORVJUXSRVIXQFLRQDOHVSUHVHQWHVHQODVVLJXLHQWHVPROpFXODV L &+&+&+2+&+ LL &+&+2+&+2 LLL &+&+1+&22+ E(VFULEDXQLVyPHURGHIXQFLyQGHODPROpFXODGHODSDUWDGRL F(VFULEDXQLVyPHURGHSRVLFLyQGHODPROpFXODGHODSDUWDGRLL Solución D L &+&+&+2+&+ JUXSR IXQFLRQDO DOFRKRO 52+ (O JUXSR ²2+ JUXSR KLGUR[LOR FRQILHUH D OD PROpFXOD XQDV SURSLHGDGHV TXtPLFDV FDUDFWHUtVWLFDV LL&+&+2+&+2JUXSRIXQFLRQDODOFRKRO52+\DOGHKtGR5&+2 ,QLFLDFLyQDOD4XtPLFD LLL &+&+1+&22+ JUXSR IXQFLRQDO DPLQD 51+ \ iFLGR FDUER[tOLFR5&22+ E&+&+2&+&+ F&+2+&+&+2 &XHVWLyQ &XHVWLyQ ,QGLTXH VL OD HVWUXFWXUD GH FDGD SDUHMD UHSUHVHQWD HO PLVPR FRPSXHVWR R FRPSXHVWRVGLIHUHQWHVLGHQWLILFDQGRORVJUXSRVIXQFLRQDOHVSUHVHQWHV D&+&+2&+\&+2&+&+ E&+&+2&+\&+&+2+&+ F&+&+&+2+\&+&+2+&+ Solución D(VHOPLVPRFRPSXHVWR6LJLUDPRVODPROpFXODYHPRVTXHHVODPLVPD HVWUXFWXUD(OJUXSRIXQFLRQDOHVpWHU525· E&RPSXHVWRVGLIHUHQWHV&+&+2&+HVXQpWHU525·&+&+2+&+HVXQ DOFRKRO52+ F &RPSXHVWRV GLIHUHQWHV 6RQ GRV LVyPHURV GH SRVLFLyQ DPERV VRQ DOFRKROHV 52+ &XHVWLyQ &XHVWLyQ ,QGLTXHORVJUXSRVIXQFLRQDOHVGHODVVLJXLHQWHVPROpFXODV D&+&+&2&+&+ E&+&+&+2+&22+ F&+&+&+1+&+2 Solución D*UXSRIXQFLRQDOFHWRQD&RQWLHQHHOJUXSRFDUERQLOR5&25· E *UXSR IXQFLRQDO iFLGR &RQWLHQH HO JUXSR FDUER[LOR 5&22+ *UXSR IXQFLRQDODOFRKRO&RQWLHQHHOJUXSRKLGUR[LOR52+ F*UXSRIXQFLRQDODOGHKtGR&RQWLHQHHOJUXSRFDUERQLORHQFDUERQRWHUPLQDO 5&+2 *UXSRIXQFLRQDODPLQR&RQWLHQHHOJUXSR1+ 4XtPLFDGHO&DUERQR &XHVWLyQ &XHVWLyQ 'HILQDVHULHKRPyORJDHLQGLTXHFXiOHVGHORVVLJXLHQWHVFRPSXHVWRVSHUWHQHFHQ DODPLVPDVHULHTXH&+2+ D&+&+&+&+2+ E&+&+2+ F&+&22+ Solución (V XQ FRQMXQWR GH FRPSXHVWRV RUJiQLFRV TXH FRQWLHQH XQ PLVPR JUXSR IXQFLRQDO\GLILHUHQVRORHQODORQJLWXGGHODFDGHQDHVGHFLUHQHOQ~PHUR GH–CH2 – D 3HUWHQHFH D OD PLVPD VHULH TXH &+2+ &XPSOH OD GHILQLFLyQ GH VHULH KRPyORJD E 3HUWHQHFH D OD PLVPD VHULH TXH &+2+ &XPSOH OD GHILQLFLyQ GH VHULH KRPyORJD F 1R SHUWHQHFH D OD PLVPD VHULH TXH &+2+ 1R FXPSOH OD GHILQLFLyQ HV XQ iFLGRFDUER[tOLFRFRQWLHQHXQJUXSRIXQFLRQDOGLIHUHQWH &XHVWLyQ &XHVWLyQ Q 'DGRV ORV FRPSXHVWRV EXWDQRO &+&+2+&+&+ \ PHWLOEXWDQRO &+&+&+&+&+2+UHVSRQGDUD]RQDGDPHQWHDODVVLJXLHQWHVFXHVWLRQHV D¢6RQLVyPHURVHQWUHVt" E¢3UHVHQWDDOJXQRGHHOORVLVRPHUtDySWLFD" Solución D 1R 1R WLHQHQ OD PLVPD IyUPXOD PROHFXODU SDUD HO EXWDQRO HV &+2 \ SDUDHOPHWLOEXWDQROHV&+2 E 6t (O EXWDQRO SUHVHQWD LVRPHUtD ySWLFD \D TXH HO FDUERQR Q~PHUR GRV HV TXLUDOHVGHFLUHVXQiWRPRGHFDUERQRFRQFXDWURVXVWLWX\HQWHVGLVWLQWRV &XHVWLyQ &XHVWLyQ VWLyQ D'HILQDVHULHKRPyORJD ,QLFLDFLyQDOD4XtPLFD E(VFULEDODIyUPXODGHXQFRPSXHVWRTXHSHUWHQH]FDDODPLVPDVHULHKRPyORJD GH FDGD XQR GH ORV TXH DSDUHFHQ D FRQWLQXDFLyQ &+&+ &+&+&+2+ &+&+1+ Solución D (V XQ FRQMXQWR GH FRPSXHVWRV RUJiQLFRV TXH FRQWLHQH XQ PLVPR JUXSR IXQFLRQDO\GLILHUHQVRORHQODORQJLWXGGHODFDGHQDHVGHFLUHQHOQ~PHURGH JUXSRV–CH2 – E (O &+&+&+ SHUWHQHFH D OD PLVPD VHULH KRPyORJD TXH HO &+&+ (O &+&+2+ SHUWHQHFH D OD PLVPD VHULH TXH HO &+&+&+2+ (O &+&+1+ SHUWHQHFHDODPLVPDTXHHO&+&+&+1+ &XHVWLyQ &XHVWLyQ 'HILQDORVVLJXLHQWHVFRQFHSWRV\SRQJDXQHMHPSORGHFDGDXQRGHHOORV D6HULHKRPyORJD E,VRPHUtDGHFDGHQD F,VRPHUtDJHRPpWULFD Solución D (V XQ FRQMXQWR GH FRPSXHVWRV RUJiQLFRV TXH FRQWLHQH XQ PLVPR JUXSR IXQFLRQDO\GLILHUHQVRORHQODORQJLWXGGHODFDGHQDHVGHFLUHQHOQ~PHURGH JUXSRV–CH2 –3RUHMHPSOR&+&+\&+&+&+ E (VWRV LVyPHURV SRVHHQ HO PLVPR JUXSR IXQFLRQDO SHUR OD HVWUXFWXUD GH OD FDGHQDHVGLIHUHQWHSXHGHVHUOLQHDORUDPLILFDGD3RUHMHPSOR&+&+&+&+ \&+&+&+&+ F6HSUHVHQWDHQORVDOTXHQRVGHELGRDOLPSHGLPHQWRGHJLURGHOGREOHHQODFH (O LVyPHUR FLV WLHQH ORV VXVWLWX\HQWHV HQ HO PLVPR ODGR GHO GREOH HQODFH \ HO LVyPHURWUDQVORVWLHQHHQODGRVRSXHVWRV3RUHMHPSOR 4XtPLFDGHO&DUERQR &XHVWLyQ &XHVWLyQ 'DGRV ORV VLJXLHQWHV FRPSXHVWRV &+&22&+&+ &+&21+ &+&+2+&+ \ &+&+2+&22+ D,GHQWLILTXHORVJUXSRVIXQFLRQDOHVSUHVHQWHVHQFDGDXQRGHHOORV E¢$OJXQRSRVHHiWRPRVGHFDUERQRDVLPpWULFR"5D]RQHVXUHVSXHVWD Solución D &+&22&+&+ JUXSR pVWHU 5&225· &+&21+ JUXSR DPLGD 5 &21+&+&+2+&+DOFRKRO52+&+&+2+&22+iFLGRFDUER[tOLFR5 &22+\DOFRKRO52+ E 6t (O &+&+2+&22+ FRQWLHQH XQ FDUERQR DVLPpWULFR \D TXH HO FDUERQR Q~PHURGRVWLHQHWRGRVORVVXVWLWX\HQWHVGLVWLQWRV &XHVWLyQ &XHVWLyQ ([SOLTXH XQR GH ORV WLSRV GH LVRPHUtD TXH SXHGHQ SUHVHQWDU ORV VLJXLHQWHV FRPSXHVWRV\UHSUHVHQWHORVFRUUHVSRQGLHQWHVLVyPHURV D&+&2&+ E&+&+&+&+ F&+&+)&22+ Solución D 3XHGH SUHVHQWDU LVRPHUtD GH IXQFLyQ 6RQ LVyPHURV FRQ GLVWLQWRV JUXSRV IXQFLRQDOHV(OLVyPHURVHUi&+&+&+2 E 3XHGH SUHVHQWDU LVRPHUtD GH FDGHQD /RV LVyPHURV SRVHHQ HO PLVPR JUXSR IXQFLRQDO SHUR OD HVWUXFWXUD GH OD FDGHQD HV GLIHUHQWH SXHGH VHU OLQHDO R UDPLILFDGD(OLVyPHURVHUi&+&+&+&+ F 3XHGHSUHVHQWDULVRPHUtDGHSRVLFLyQ/RVLVyPHURVWLHQHQHOPLVPRJUXSR IXQFLRQDO HQ SRVLFLyQ GLIHUHQWH HQ XQD PLVPD FDGHQD (O LVyPHUR VHUi &+)&+&22+ 3XHGH SUHVHQWDU WDPELpQ LVRPHUtD ySWLFD (Q HVWH FRPSXHVWR HO FDUERQR Q~PHURGRVHVTXLUDOORVHQDQWLyPHURVVHUHSUHVHQWDQDFRQWLQXDFLyQ ,QLFLDFLyQDOD4XtPLFD &XHVWLyQ &XHVWLyQ 'HILQDORVVLJXLHQWHVFRQFHSWRV\SRQJDXQHMHPSORGHFDGDXQRGHHOORV D,VRPHUtDGHIXQFLyQ E,VRPHUtDGHSRVLFLyQ F,VRPHUtDySWLFD Solución D,VyPHURVGHIXQFLyQVRQFRPSXHVWRVFRQODPLVPDIyUPXODPROHFXODUSHURFRQ GLVWLQWRVJUXSRVIXQFLRQDOHV3RUHMHPSOR&+&+&+2+\&+2&+&+ E,VyPHURVGHSRVLFLyQVRQFRPSXHVWRVTXHWLHQHQHOPLVPRJUXSRIXQFLRQDOHQ SRVLFLyQ GLIHUHQWH HQ XQD PLVPD FDGHQD 3RU HMHPSOR &+&+2+&+ \ &+&+&+2+ F ,VyPHURV ySWLFRV VRQ FRPSXHVWRV TXH WLHQHQ WRGDV VXV SURSLHGDGHV LJXDOHV \ TXHVRORVHGLIHUHQFLDQHQVXGLVWLQWRFRPSRUWDPLHQWRIUHQWHDODOX]SRODUL]DGD \DTXHODGLVSRVLFLyQHVSDFLDOGHORVVXVWLWX\HQWHVHVGLIHUHQWH6HSUHVHQWDHVWD LVRPHUtD FXDQGR H[LVWH HQ OD PROpFXOD XQ FDUERQR TXLUDO R DVLPpWULFR FDUERQR XQLGRDFXDWURJUXSRVGLIHUHQWHV3RUHMHPSOR &XHVWLyQ &XHVWLyQ ([SOLTXHSRUTXpHO&+&+&+2+HVPiVVROXEOHHQDJXDTXHHO&+&+&+&+ Solución (O &+&+&+2+ HV PiV VROXEOH HQ DJXD SRUTXH HV SRODU \ IRUPD FRQ IDFLOLGDG HQODFHVGHKLGUyJHQRFRQODVPROpFXODVGHDJXDGHELGRDODSUHVHQFLDGHOJUXSR KLGUR[LOR /RVDOFRKROHVVRQPROpFXODVSRODUHV\SRUWDQWRVROXEOHVHQGLVROYHQWHVSRODUHV FRPRHODJXD/DVROXELOLGDGHQDJXDGLVPLQX\HDODXPHQWDUODPDVDPROHFXODU GHODOFRKRO (O&+&+&+&+HVXQDOFDQRLQVROXEOHHQDJXDSRUVHUXQDPROpFXODDSRODU 4XtPLFDGHO&DUERQR &XHVWLyQ &XHVWLyQ 'DGRV ORV VLJXLHQWHV FRPSXHVWRV RUJiQLFRV &+&+&+ &+2+ &+ &+&+ ,QGLTXHUD]RQDGDPHQWH D¢&XiOHVVROXEOHHQDJXD" E¢&XiOHVVRQKLGURFDUEXURV" F¢&XiOSUHVHQWDUHDFFLRQHVGHDGLFLyQ" Solución D&+&+&+HVXQDOFDQRDSRODU\SRUWDQWRQRHVVROXEOHHQDJXD&+2+HV XQDOFRKROSRODU\SRUWDQWRVROXEOHHQDJXDIRUPDHQODFHVGHKLGUyJHQRFRQ ODV PROpFXODV GH HVWH GLVROYHQWH &+ &+&+ QR HV VROXEOH HQ DJXD \D TXH QR SRVHHJUXSRVSRODUHV E +LGURFDUEXURV VRQ ORV FRPSXHVWRV TXH HVWiQ IRUPDGRV SRU FDUERQR H KLGUyJHQRVRODPHQWHSRUWDQWRVyORORVRQ&+&+&+\&+ &+&+ F/RVDOTXHQRVWLHQHQGREOHHQODFH\H[SHULPHQWDQODVUHDFFLRQHVGHDGLFLyQD HVWHHQODFH6yORH[SHULPHQWDUiDGLFLyQHO&+ &+&+ &XHVWLyQ &XHVWLyQ /RV FRPSXHVWRV &+&+2+ \ &+&+&+ WLHQHQ PDVDV PROHFXODUHV VLPLODUHV ,QGLTXHMXVWLILFDQGRODUHVSXHVWD D&XiOWLHQHPD\RUSXQWRGHIXVLyQ E&XiOGHHOORVSXHGHH[SHULPHQWDUXQDUHDFFLyQGHHOLPLQDFLyQ\HVFUtEDOD Solución D(O&+&+2+HVPiVSRODUTXHHO&+&+&+ \VXVPROpFXODVSXHGHQ IRUPDU HQODFHV GH KLGUyJHQR D WUDYpV GH ORV JUXSRV KLGUR[LOR SRU WDQWR WLHQH PD\RU SXQWRGHIXVLyQ +2 624 E(O&+&+2+ + − &+2 − &+2 − 2+ ⎯⎯⎯→ &+2 = &+2 + +2 2 FDORU /D UHDFFLyQ GH HOLPLQDFLyQ VH FDUDFWHUL]D SRU OD SpUGLGD GH DOJXQRV iWRPRV GH XQDFDGHQDFDUERQDGD(QODUHDFFLyQGHVFULWDVHHOLPLQDXQDPROpFXODGHDJXD \VHJHQHUDXQGREOHHQODFHHQODFDGHQD ,QLFLDFLyQDOD4XtPLFD &XHVWLyQ &RPSOHWHODVVLJXLHQWHVUHDFFLRQHVHLQGLTXHGHTXpWLSRVRQ D &+3&+ = &+2 + +%U → Solución: &+3&+ = &+2 + +%U → &+3&+%U&+3 (V XQD UHDFFLyQ GH DGLFLyQ GH +%U XQD KLGUREURPDFLyQ GH XQ DOTXHQR &XDQGR XQ UHDFWLYR DVLPpWULFR VH DGLFLRQD D XQ DOTXHQR R DOTXLQR DVLPpWULFR HO IUDJPHQWRPiVSRVLWLYRJHQHUDOPHQWHHO+VHXQHDOiWRPRGHFDUERQRDOTXH \D HVWiQ XQLGRV PD\RU Q~PHUR GH iWRPRV GH + (VWH UHVXOWDGR VH FRUUHVSRQGH FRQODUHJODHPStULFDSURSXHVWDSRU0DUNRYQLNRY &+3&+2 &+2 2+ + +2 624 → E Solución: &+3&+2&+22+ + +2624 → &+3&+ = &+2 + +22 (V XQD UHDFFLyQ GH HOLPLQDFLyQ GH +2 XQD GHVKLGUDWDFLyQ GH XQ DOFRKRO TXH UHTXLHUHXQFDWDOL]DGRUiFLGR &RQ DOFRKROHV SULPDULRV PD\RUHV TXH HO SURStOLFR VH REWLHQHQ PH]FODV GH DOTXHQRV FRPR FRQVHFXHQFLD GH OD LVRPHUL]DFLyQ ´FDPELR GH SRVLFLyQµ GHO GREOH HQODFH (VWH WLSR GH LVRPHUL]DFLyQ HV FRP~Q EDMR ODV FRQGLFLRQHV GH OD GHVKLGUDWDFLyQFDWDOL]DGD SRUiFLGRV SRUORTXHQR HVXQWLSRGHUHDFFLyQ PX\ DGHFXDGDSDUDODSUHSDUDFLyQGHDOTXHQRVDSDUWLUGHDOFRKROHVSULPDULRV +2 624 &6+6 ( EHQFHQR ) + +123 ⎯⎯⎯→ F Solución: +2 624 &6+6 ( EHQFHQR ) + +123 ⎯⎯⎯→ &6+5122 + +22 (V XQD UHDFFLyQ GH VXVWLWXFLyQ DURPiWLFD HOHFWURItOLFD WtSLFD GH ORV DQLOORV DURPiWLFRV (VWH FDVR FRQVLVWH HQ OD QLWUDFLyQ GHO EHQFHQR \ HO UHDFWLYR HOHFWUyILORHVHOLRQQLWURQLR (12 +2 ) TXHSURFHGHGHODPH]FODGHiFLGRQtWULFR\ iFLGRVXOI~ULFRFRQFHQWUDGRV G 4XtPLFDGHO&DUERQR +2 624 &+2 = &+2 + +2 2 ⎯⎯⎯→ Solución: +2 624 &+2 = &+2 + +22 ⎯⎯⎯→ &+3&+22+ (VXQDUHDFFLyQGHDGLFLyQGH+2XQDKLGUDWDFLyQGHXQDOTXHQR &+2 = &+2 + +&O → H Solución: &+2 = &+2 + +&O → &+3&+2&O (VXQDUHDFFLyQGHDGLFLyQGH+&OXQDKLGURFORUDFLyQGHXQDOTXHQR $O&O3 I &6+6 ( EHQFHQR ) + &O 2 ⎯⎯⎯ → Solución: $O&O3 &6+6 ( EHQFHQR ) + &O 2 ⎯⎯⎯ → &6+5&O + +&O (V XQD UHDFFLyQ GH VXVWLWXFLyQ DURPiWLFD HOHFWURItOLFD WtSLFD GH ORV DQLOORV DURPiWLFRVXQDFORUDFLyQGHEHQFHQR &+3&22+ + &+3&+2 2+ → J Solución: &+3&22+ + &+3&+22+ → &+3&22&+2&+3 + +22 (V XQD UHDFFLyQ GH HVWHULILFDFLyQ /RV iFLGRV FDUER[tOLFRV UHDFFLRQDQ UiSLGDPHQWHFRQORVDOFRKROHVHQSUHVHQFLDGHFDQWLGDGHVFDWDOtWLFDVGHiFLGRV PLQHUDOHVSDUDSURGXFLUpVWHUHV ,QLFLDFLyQDOD4XtPLFD K &+2 = &+2 + %U2 → Solución: &+2 = &+2 + %U2 → %U&+2&+2%U (VXQDUHDFFLyQGHDGLFLyQGH %U2 XQDEURPDFLyQGHXQDOTXHQR &4+10 + 2 2 → L Solución: 2 &4+10 + 13 22 → 8 &22 + 10 +2 2 (V XQD UHDFFLyQ GH FRPEXVWLyQ /RV KLGURFDUEXURV FXDQGR VH TXHPDQ HQ SUHVHQFLD GH FDQWLGDG VXILFLHQWH GH R[tJHQR SURGXFHQ GLy[LGR GH FDUERQR \ DJXD (QODFRPEXVWLyQGHXQPROGHEXWDQRVHUHTXLHUHQPROHVGHR[tJHQR3DUD HOLPLQDUHVWHFRHILFLHQWHIUDFFLRQDULRVHKDQPXOWLSOLFDGRWRGRVORVFRHILFLHQWHV SRU OX]( Kν ) &+4 + &O 2 ⎯⎯⎯⎯ → M Solución: OX]( Kν ) &+4 + &O 2 ⎯⎯⎯⎯ → &+3&O + &+2&O 2 + &+&O3 + &&O 4 + +&O (VXQDUHDFFLyQGHVXVWLWXFLyQODFORUDFLyQGHXQDOFDQR0HGLDQWHHVWDUHDFFLyQ VHVXVWLWX\HXQiWRPRGHKLGUyJHQRGHOKLGURFDUEXURSRUXQiWRPRGHKDOyJHQR (VWD VXVWLWXFLyQ TXH VH SURGXFH D DOWDV WHPSHUDWXUDV \ HVSHFLDOPHQWH HQ SUHVHQFLDGHODOX]WLHQHOXJDUPHGLDQWHXQDUHDFFLyQHQ FDGHQDSRUUDGLFDOHV OLEUHVTXHVXHOHSURSRUFLRQDUPH]FODGHSURGXFWRVPRQR\SROLKDORJHQDGRV &+2 = &+2 + 22 → N Solución: &+2 = &+2 + 3 22 → 2 &22 + 2 +2 2 (VXQDUHDFFLyQGHFRPEXVWLyQ O 4XtPLFDGHO&DUERQR &+2 = &+&+3 + +, → Solución: &+2 = &+&+3 + +, → &+3&+,&+3 (VXQDUHDFFLyQGHDGLFLyQGH+,DXQDOTXHQRTXHWUDQVFXUUHGHDFXHUGRFRQOD UHJODGH0DUNRYQLNRY +2624 P &+2 = &+&+3 + +22 ⎯⎯⎯→ Solución: +2624 &+2 = &+&+3 + +22 ⎯⎯⎯→ &+3&+2+&+3 (VXQDUHDFFLyQGHDGLFLyQGH+2XQDKLGUDWDFLyQGHXQDOTXHQR7UDQVFXUUHGH DFXHUGRFRQODUHJODGH0DUNRYQLNRY Q &+3&+2 &+&O&+2 + .2+ → .&O + Solución: &+3&+2 &+&O&+2 + .2+ → .&O + &+3&+2 &+2+&+3 (VXQDUHDFFLyQGHVXVWLWXFLyQGHOKDOyJHQRGHXQKDOXURGHDOTXLORSRUXQJUXSR KLGUR[LOR(VWDUHDFFLyQVHOOHYDDFDERFDOHQWDQGRHQGLVROXFLyQDFXRVDEiVLFD\ SURGXFHDOFRKROHV 5HDOPHQWH HQ HO WUDWDPLHQWR GH KDOXURV GH DOTXLOR FRQ EDVHV HQWUDQ HQ FRPSHWHQFLDODUHDFFLyQ GHVXVWLWXFLyQ SDUDGDUDOFRKROHV\ODGHHOLPLQDFLyQ SDUD GDU DOTXHQRV $XQTXH HO SUHGRPLQLR GH XQD X RWUD GHSHQGH GH YDULRV IDFWRUHV ODV FRQGLFLRQHV TXH VH HVSHFLILFDQ HQ FDGD XQR GH ORV HMHPSORV TXH YHUHPRVFRQGXFHQDODREWHQFLyQPD\RULWDULDGHORVSURGXFWRVTXHVHGHVFULEHQ HQFDGDFDVR &+3&+ = &+&+3 + +%U → R ,QLFLDFLyQDOD4XtPLFD Solución: &+3&+ = &+&+3 + +%U → &+3&+%U&+2&+3 (VXQDUHDFFLyQGHDGLFLyQGH+%UDXQDOTXHQR &+3&+2+&+2 &+3 + +2 62 4 → +2 2 + S Solución: &+3&+2+&+2 &+3 + +2 624 → +2 2 + &+3&+ = &+&+3 ( PD\RULWDULR ) + &+2 = &+&+2 &+3 (V XQD UHDFFLyQ GH HOLPLQDFLyQ GH +2 XQD GHVKLGUDWDFLyQ GH XQ DOFRKRO 6H SURGXFHXQDPH]FODGHDOTXHQRVVLHQGRHOSURGXFWRPD\RULWDULRHOEXWHQR .2+ ( WDQ RO &+3&+%U&+3 ⎯⎯⎯⎯⎯ → +%U + T Solución: .2+ ( WDQ RO &+3&+%U&+3 ⎯⎯⎯⎯⎯ → +%U + &+3&+ = &+2 (VXQDUHDFFLyQGHHOLPLQDFLyQGH+%UXQDGHVKLGURKDORJHQDFLyQGHXQKDOXUR GH DOTXLOR (VWD UHDFFLyQ VH OOHYD D FDER FDOHQWDQGR FRQ KLGUy[LGR SRWiVLFR GLVXHOWR HQ DOFRKRO 6L HO PHGLR EiVLFR QR HV IXHUWH QL DOFRKyOLFR SXHGH SURGXFLUVHODUHDFFLyQGHVXVWLWXFLyQYpDVHODUHDFFLyQµQµ &+3&+2 &22+ + &+3&+2 2+ → U Solución: &+3&+2 &22+ + &+3&+2 2+ → &+3&+2 &22&+2 &+3 + +2 2 (VXQDUHDFFLyQGHHVWHULILFDFLyQ FDWDOL]DGRU V &+3&+ = &+2 + +2 ⎯⎯⎯⎯⎯ → Solución: FDWDOL]DGRU &+3&+ = &+2 + +2 ⎯⎯⎯⎯⎯ → &+3&+2 &+3 4XtPLFDGHO&DUERQR (VXQDUHDFFLyQGHDGLFLyQGH+ODKLGURJHQDFLyQGHXQDOTXHQR &+ ≡ &+ + +&O → W Solución: +&O &+ ≡ &+ + +&O → &+2 = &+&O ⎯⎯⎯ → &+2 &&O 2 (V XQD UHDFFLyQ DGLFLyQ GH +&O XQD KLGURFORUDFLyQ GH XQ DOTXLQR /D VROXFLyQ TXH VH RIUHFH FRUUHVSRQGH SULPHUR D OD DGLFLyQ GH XQ PRO GH FORUXUR GH KLGUyJHQRSRUPROGHDFHWLOHQRODDGLFLyQGHXQVHJXQGRPROGH+&ODODOTXHQR REWHQLGRHQODSULPHUDHWDSDGDUtDHOGHULYDGRGLKDORJHQDGRFRUUHVSRQGLHQWHD ODRULHQWDFLyQ0DUNRYQLNRY .2+ H WDQ RO &+3&+%U&+%U&+3 ⎯⎯⎯⎯⎯ → 2 +%U + X Solución: .2+ H WDQ RO &+3&+%U&+%U&+3 ⎯⎯⎯⎯⎯ → 2+%U + &+3& ≡ &&+3 (V XQD UHDFFLyQ GH HOLPLQDFLyQ GH GRV PROpFXODV GH +%U GHO GLEURPXUR GH DOTXLOR /D GHVKLGURKDORJHQDFLyQ GH XQ GLKDOXUR YHFLQDO GD OXJDU D XQ KDOXUR YLQtOLFR \ pVWH HQ FRQGLFLRQHV IXHUWHPHQWH EiVLFDV FRPR HV HVWH FDVR SXHGH H[SHULPHQWDUXQDVHJXQGDGHVKLGURKDORJHQDFLyQSDUDGDUXQDOTXLQR Kν Y &+3&+2 &+3 + &O 2 ⎯⎯⎯ → +&O + Solución: Kν &+3&+2&+3 + &O 2 ⎯⎯⎯ → +&O + &+3&+2&+2&O + &+3&+&O&+3 + (VXQDUHDFFLyQGHVXVWLWXFLyQODKDORJHQDFLyQGHXQDOFDQR/DFORUDFLyQGHORV DOFDQRVVXSHULRUHVHVVLPLODUDODGHOPHWDQRYpDVHUHDFFLyQ´MµREWHQLpQGRVH PH]FODVGHSURGXFWRVPiVFRPSOHMDV3RUHVWRODFORUDFLyQGHDOFDQRVQRHVHQ JHQHUDOXQDUHDFFLyQDGHFXDGDSDUDSUHSDUDUFORUXURVGHDOTXLOR ,QLFLDFLyQDOD4XtPLFD 3W 3G Z &+ ≡ &+ + +2 ⎯⎯⎯→ Solución: 3W 3G &+ ≡ &+ + 2+2 ⎯⎯⎯→ &+3&+3 (VXQDUHDFFLyQGHDGLFLyQGH+ODKLGURJHQDFLyQGHXQDOTXLQR6HSUHVHQWDOD HFXDFLyQ TXH FRUUHVSRQGH D OD KLGURJHQDFLyQ FRPSOHWD GHO DFHWLOHQR TXH UHTXLHUHGRVPROHVGH+SRUPROGH &+ ≡ &+ Catalizador CH2 = CH2 + H2 ⎯⎯⎯⎯⎯ → [ Solución: Catalizador CH2 = CH2 + H2 ⎯⎯⎯⎯⎯ → CH3CH3 (VXQDUHDFFLyQGHDGLFLyQGH+ODKLGURJHQDFLyQGHXQDOTXHQR Luz CH3 CH3 + Cl2 ⎯⎯⎯ → \ Solución: Luz CH3CH3 + Cl2 ⎯⎯⎯ → CH3CH2Cl + CH3CHCl2 + ClCH2CH2Cl + ..... + HCl (VXQDUHDFFLyQGHVXVWLWXFLyQODKDORJHQDFLyQGHXQDOFDQR/DFORUDFLyQGHORV DOFDQRV FRQGXFH D PH]FODV FRPSOHMDV GH SURGXFWRV KDORJHQDGRV YpDVH UHDFFLRQHV´Mµ\´Zµ CH3 OH + O2 ⎯⎯ → ] Solución: 2 CH3 OH + 3 O2 ⎯⎯→ 2 CO2 + 4 H2O (VXQDUHDFFLyQGHFRPEXVWLyQ DD CH3 CH 2 CH = CH 2 + HI ⎯⎯→ 4XtPLFDGHO&DUERQR Solución: &+3&+2&+ = &+2 + +, ⎯⎯→ &+3&+2&+,&+3 (VXQDUHDFFLyQGHDGLFLyQGH+,DXQDOTXHQRTXHWUDQVFXUUHGHDFXHUGRFRQOD UHJODGH0DUNRYQLNRY &+3&+ = &+&+3 + +2 → EE Solución: &+3&+ = &+&+3 + +2 → &+3&+2&+2&+3 (VXQDUHDFFLyQGHDGLFLyQGH+ODKLGURJHQDFLyQGHXQDOTXHQR FF &+3&+3 + 22 → Solución: 2 &+3&+3 + 7 22 → 4 &22 + 6 +2 2 (VXQDUHDFFLyQGHFRPEXVWLyQ .2+ ( WDQ RO GG &+3&+2&+2%U ⎯⎯⎯⎯⎯ → Solución: .2+ ( WDQ RO &+3&+2&+2%U ⎯⎯⎯⎯⎯ → &+3&+ = &+2 + +%U (V XQD UHDFFLyQ GH HOLPLQDFLyQ GH +%U HQ XQ KDOXUR GH DOTXLOR SDUD GDU XQ DOTXHQR &XHVWLyQ ,QGLTXHHOWLSRDTXHSHUWHQHFHFDGDXQDGHODVVLJXLHQWHVUHDFFLRQHV D &+3&+2%U + 1D2+ → &+3&+2 2+ + 1D%U ,QLFLDFLyQDOD4XtPLFD E &+3&+2 &+ = &+2 + +&O → &+3&+2 &+&O&+3 → &+3&+ = & ( &+3 ) 2 + 1D%U + +2 2 F &+3&+%U&+ ( &+3 ) 2 + 1D2+ ⎯⎯⎯⎯ H WDQ RO Solución: D5HDFFLyQGHVXVWLWXFLyQQXFOHyILOD E5HDFFLyQGHDGLFLyQKLGURFORUDFLyQGHXQDOTXHQR F5HDFFLyQGHHOLPLQDFLyQ &XHVWLyQ 3RQJDXQHMHPSORGHFDGDXQDGHODVVLJXLHQWHVUHDFFLRQHV D$GLFLyQDXQDOTXHQR E6XVWLWXFLyQHQXQDOFDQR F'HVKLGUDWDFLyQGHXQDOFRKRO G5HDFFLyQGHHOLPLQDFLyQGH+&OHQXQFORUXURGHDOTXLOR Solución: FDWDOL]DGRU → &+3&+2 &+3 a) &+3&+ = &+2 + +2 ⎯⎯⎯⎯⎯ OX] → &+3%U + &+2%U2 + &+%U3 + &%U4 + +%U b) &+4 + %U2 ⎯⎯⎯ +2624 &+2 = &+2 + +22 c) &+3&+22+ ⎯⎯⎯→ .2+ ( WDQ RO d) &+3&+&O&+3 ⎯⎯⎯⎯⎯ → &+3&+ = &+2 + +&O