PortadaUNAMFACULTAD DE QUMICADEPARTAMENTO DE QUMICA ANALTICAQumica Analtica III - Semestre 2012/IICURVAS DE TITULACIN COMPLEJOMTRICAPLANTILLAS COMPLEMENTARIASPARA LA RESOLUCIN DE LA TAREA #3Libro de trabajo EXCELALAIN QUERE, Profesor - MARZO DE 2012

Plantilla titulacin PbPlantilla titulacin CuPlantilla titulacin mezcla Pb+CuTitulaciones complejomtricasM + L = MLTitulaciones complejomtricasmezcla de dos cationesEjemplo Titulacin Mg2+ conEDTA o EGTAEjemplo Titulacin Ca2+ conEDTA o EGTAEjemplo Titulacin Ca2+ Mg2+ con EDTA o EGTAAnexos1234

pX=f(x)_Ca A. QUERE - Fac. Qum. UNAMProfesorQUMICA ANALTICA III - SEMESTRE 2012/II - EJEMPLO TITULACINAlain QUERECurva terica de titulacin complejomtrica de Calcio con EGTA o EDTAEstudio del efecto de la variacin del pH y de CCaTHORENTCa2++X4-CaX2-lgKCaX=10.7lgaCa(OH)0.00Ca / OHX / HCaX/HpH10.00lgaX(H)0.48logb1M/OHlogb1X/Hlogb2X/Hlogb3X/Hlogb4X/HlogbCaX/H|H+|1.00E-10lgaCaX(H)0.001.010.316.619.321.43.2|OH-|1.00E-04Constante condicional1.00E+012.00E+103.98E+162.00E+192.51E+211.58E+03lgK'CaX=10.22El reactivo titulante es el EDTA143pCCa1.00E+0210.0010.2216,713,731,060.10280logK'CaX10.22CCa0.001lgK'CaXCCaPunto de equivalenciapX'inicial13.22pX'final3.007.22xequivpX'equivCuantitatividadxpX'=f(x)|X'|pH10.001.006.6199.9755%09.99E-0413.226.0E-14100011.61E-0313.019.7E-1422.61E-0312.811.6E-1334.21E-0312.602.5E-1346.79E-0312.394.1E-1351.09E-0212.186.6E-1361.75E-0211.971.1E-1272.81E-0211.761.7E-1284.46E-0211.552.8E-1297.02E-0211.354.5E-12101.09E-0111.147.3E-12111.65E-0110.931.2E-11122.42E-0110.721.9E-11133.40E-0110.513.1E-11144.55E-0110.305.0E-11155.74E-0110.098.1E-11166.85E-019.881.3E-10177.79E-019.682.1E-10188.51E-019.473.4E-10199.02E-019.265.5E-10209.37E-019.058.9E-10219.60E-018.841.4E-09229.75E-018.632.3E-09239.84E-018.423.8E-09pHCCaEn la curva de titulacin, este factor se mide por la distancia: pL'(x=0.5) - pL'(x=2) = logK'ZnXCZn249.90E-018.226.1E-0910.000.001259.94E-018.019.8E-0910.006.617.22269.96E-017.801.6E-08279.98E-017.592.6E-08289.99E-017.384.2E-08299.99E-017.176.7E-08301.00E+006.961.1E-07311.00E+006.761.8E-07321.00E+006.552.8E-07331.00E+006.344.6E-07341.00E+006.137.4E-07351.00E+005.921.2E-06361.00E+005.711.9E-06371.00E+005.503.1E-06381.01E+005.295.1E-06391.01E+005.098.2E-06401.01E+004.881.3E-05411.02E+004.672.1E-05421.03E+004.463.5E-05431.06E+004.255.6E-05441.09E+004.049.1E-05451.15E+003.831.5E-04461.24E+003.632.4E-04471.38E+003.423.8E-04481.62E+003.216.2E-04La distancia entre la lnea anaranjada y la lnea aguamarina es igual al factor de cuantitatividad. Si es < 3.5 unidades, la titulacin no es factible.492.00E+003.001.0E-03pH|H||OH|aM(OH)aX(H)aMX(H,OH)LogK'ML01.00E+001.00E-141.00E+002.53E+211.59E+03-7.500.53.16E-013.16E-141.00E+002.58E+195.02E+02-6.0111.00E-011.00E-131.00E+002.72E+171.59E+02-4.531.53.16E-023.16E-131.00E+003.18E+155.11E+01-3.0921.00E-021.00E-121.00E+004.91E+131.68E+01-1.762.53.16E-033.16E-121.00E+001.28E+126.01E+00-0.6331.00E-031.00E-111.00E+006.23E+102.58E+000.323.53.16E-043.16E-111.00E+004.64E+091.50E+001.2141.00E-041.00E-101.00E+004.20E+081.16E+002.144.53.16E-053.16E-101.00E+004.11E+071.05E+003.1151.00E-051.00E-091.00E+004.20E+061.02E+004.085.53.16E-063.16E-091.00E+004.62E+051.01E+005.0461.00E-061.00E-081.00E+005.98E+041.00E+005.926.53.16E-073.16E-081.00E+001.03E+041.00E+006.6971.00E-071.00E-071.00E+002.39E+031.00E+007.327.53.16E-083.16E-071.00E+006.72E+021.00E+007.8781.00E-081.00E-061.00E+002.05E+021.00E+008.398.53.16E-093.16E-061.00E+006.45E+011.00E+008.8991.00E-091.00E-051.00E+002.10E+011.00E+009.389.53.16E-103.16E-051.00E+007.31E+001.00E+009.84101.00E-101.00E-041.00E+003.00E+001.00E+0010.2210.53.16E-113.16E-041.00E+001.63E+001.00E+0010.49111.00E-111.00E-031.01E+001.20E+001.00E+0010.6211.53.16E-123.16E-031.03E+001.06E+001.00E+0010.66121.00E-121.00E-021.10E+001.02E+001.00E+0010.6512.53.16E-133.16E-021.32E+001.01E+001.00E+0010.58131.00E-131.00E-012.00E+001.00E+001.00E+0010.4013.53.16E-143.16E-014.16E+001.00E+001.00E+0010.08141.00E-141.00E+001.10E+011.00E+001.00E+009.66

&LFacultad de Qumica UNAM&RDr. Alain QUERE T., profesor&LDepartamento de Qumica Analtica&RSemestre 2012/IIEl valor de pH se hace variar mediante el control de nmero situado en la celda E15.La concentracin de calcio se hace variar mediante la barra de desplazamiento situada arriba de esta celda.

pX=f(x)_Ca

pX'=f(x)xpX'Titulacin Ca' + X' = CaX' - Grfica pX' = f(x)

pX=f(x)_Mg

pCCalogK'CaXLogK'MLpHlogK'Constante condicional Ca' + X' = CaX'7.22

pX=f(x)_Ca+Mg A. QUERE - Fac. Qum. UNAMProfesorQUMICA ANALTICA III - SEMESTRE 2012/II - EJEMPLO TITULACINAlain QUERECurva terica de titulacin complejomtrica de Magnesio con EGTA o EDTAEstudio del efecto de la variacin del pH y de CMgTHORENTMg2++X4-MgX2-lgKMgX=8.7lgaMg(OH)0.01Mg / OHX / HMgX/HpH10lgaX(H)0.48logb1M/OHlogb1X/Hlogb2X/Hlogb3X/Hlogb4X/HlogbMgX/H|H+|1.00E-10lgaMgX(H)0.002.510.316.619.321.47.6|OH-|1.00E-04Constante condicional3.16E+022.00E+103.98E+162.00E+192.51E+213.98E+07lgK'MgX=8.21El reactivo titulante es el EDTA143pCMg1.00E+0210.008.21162,821,601.17404logK'MgX8.21CMg0.001lgK'MgXCMgPunto de equivalenciapX'inicial11.21pX'final3.005.21xequivpX'equivCuantitatividadxpX'=f(x)|X'|pH10.001.005.6199.7525%09.99E-0411.216.1E-1278911.47E-0311.049.0E-1222.16E-0310.881.3E-1133.17E-0310.712.0E-1144.66E-0310.542.9E-1156.84E-0310.374.2E-1161.00E-0210.216.2E-1171.47E-0210.049.1E-1182.14E-029.871.3E-1093.12E-029.702.0E-10104.53E-029.542.9E-10116.52E-029.374.3E-10129.30E-029.206.3E-10131.31E-019.039.3E-10141.82E-018.871.4E-09152.46E-018.702.0E-09163.24E-018.532.9E-09174.14E-018.364.3E-09185.10E-018.206.4E-09196.04E-018.039.4E-09206.92E-017.861.4E-08217.68E-017.692.0E-08228.29E-017.523.0E-08238.77E-017.364.4E-08pHCMg249.13E-017.196.5E-0810.000.001259.39E-017.029.5E-0810.005.615.21269.58E-016.851.4E-07279.71E-016.692.1E-07289.80E-016.523.0E-07299.87E-016.354.4E-07309.91E-016.186.5E-07319.95E-016.029.6E-07329.97E-015.851.4E-06339.99E-015.682.1E-06341.00E+005.513.1E-06351.00E+005.354.5E-06361.01E+005.186.6E-06371.01E+005.019.7E-06381.01E+004.841.4E-05391.02E+004.682.1E-05401.03E+004.513.1E-05411.05E+004.344.6E-05421.07E+004.176.7E-05431.10E+004.019.9E-05441.15E+003.841.5E-04451.21E+003.672.1E-04461.31E+003.503.1E-04471.46E+003.344.6E-04481.68E+003.176.8E-04La distancia entre la lnea anaranjada y la lnea aguamarina es igual al factor de cuantitatividad. Si es < 3.5 unidades, la titulacin no es factible.492.00E+003.001.0E-03pKsMg/OH10.3pH|H||OH|aM(OH)aX(H)aMX(H,OH)LogK'ML-logSMg01.00E+001.00E-141.00E+002.53E+213.98E+07-5.10-17.700.53.16E-013.16E-141.00E+002.58E+191.26E+07-3.61-16.7011.00E-011.00E-131.00E+002.72E+173.98E+06-2.13-15.701.53.16E-023.16E-131.00E+003.18E+151.26E+06-0.70-14.7021.00E-021.00E-121.00E+004.91E+133.98E+050.61-13.702.53.16E-033.16E-121.00E+001.28E+121.26E+051.69-12.7031.00E-031.00E-111.00E+006.23E+103.98E+042.51-11.703.53.16E-043.16E-111.00E+004.64E+091.26E+043.13-10.7041.00E-041.00E-101.00E+004.20E+083.98E+033.68-9.704.53.16E-053.16E-101.00E+004.11E+071.26E+034.19-8.7051.00E-051.00E-091.00E+004.20E+063.99E+024.68-7.705.53.16E-063.16E-091.00E+004.62E+051.27E+025.14-6.7061.00E-061.00E-081.00E+005.98E+044.08E+015.53-5.706.53.16E-073.16E-081.00E+001.03E+041.36E+015.82-4.7071.00E-071.00E-071.00E+002.39E+034.98E+006.02-3.707.53.16E-083.16E-071.00E+006.72E+022.26E+006.23-2.7081.00E-081.00E-061.00E+002.05E+021.40E+006.53-1.708.53.16E-093.16E-061.00E+006.45E+011.13E+006.94-0.7091.00E-091.00E-051.00E+002.10E+011.04E+007.390.309.53.16E-103.16E-051.01E+007.31E+001.01E+007.841.30101.00E-101.00E-041.03E+003.00E+001.00E+008.212.2910.53.16E-113.16E-041.10E+001.63E+001.00E+008.453.26111.00E-111.00E-031.32E+001.20E+001.00E+008.504.1811.53.16E-123.16E-032.00E+001.06E+001.00E+008.375.00121.00E-121.00E-024.16E+001.02E+001.00E+008.075.6812.53.16E-133.16E-021.10E+011.01E+001.00E+007.666.26131.00E-131.00E-013.26E+011.00E+001.00E+007.196.7913.53.16E-143.16E-011.01E+021.00E+001.00E+006.707.30141.00E-141.00E+003.17E+021.00E+001.00E+006.207.80

El reactivo titulante es el EDTACaractersticas de la plantillaCon base en los datos de la reaccin principal de titulacin del calcio con el EGTA o con el EDTA, se calculan los coeficientes de reacciones parsitas y la constante condicional de la reaccin para 5< pH pCMg, la precipitacin de Mg(OH)2 tendr lugar, por lo que la titulacin del magnesio ya no podr realizarse.Se puede observar que para pH=10, logK'(MgEDTA) = 8.21 por lo que la titulacin es factible con EDTA pero si se emplea el EGTA, logK'(MgEGTA) = 5.09 por lo que la titulacin no es factible con EGTA. El factor de cuantitatividad es igual a 2.09 y la curva de titulacin no marca un punto de inflexin en el punto de equivalencia. Esta diferencia de comportamiento del EGTA y del EDTA con Mg2+ se aprovecha para determinar el Calcio en presencia de magnesio.Ver/ocultar comentariosPantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra de FrmulasCurva TitulacinGrafico Constante CondicionalRegresar a inicio de hojaTitulanteEGTATitulanteEDTATitulante EGTATitulante EDTA&L&"Arial,Negrita"Facultad de Qumica UNAM&R&"Arial,Negrita"Dr. Alain QUERE, profesor&LDepartamento de Qumica Analtica&RSemestre 2012/IIEl valor de a depende de los valores respectivos de CMg y de CCa ajustados en las celdas K14 y L14.Consultar:Schmid, R.W., and Reilley, C.N., New Complexon for Titration of Calcium in the Presence of Magnesium. Anal. Chem., 29, 264 (1957)

pY=f(x)_Pb

Titulacin Mezcla Ca + Mg con EDTApX'=f(x)XpX'

pY=f(x)_Cu310.2230734097310.22307340976.6115367048-7.5031689887-6.0099756968-4.5311012105-3.0942095813-1.7640910613-0.62830750940.31798816291.20959338842.14032619013.10765929924.08352020715.0376842295.92410060746.68770604167.32087360317.87280063258.3892936768.89047019229.37789446699.835734400310.223073409710.486175625410.616667442110.659906541810.650027299310.577937346110.398104336110.08039501719.6585206703

pCCalogK'CaXLogK'MLpHlogK'Constante condicional Ca' + X' = CaX'7.22

Ecuaciones_pL=f(x)_Mezcla A. QUERE-F.Q. UNAMI. TITULACIN DE UN CATION METLICO M CON UN LIGANTE LA continuacin se establece la ecuacin que permite efectuar la representacin de la curva de titulacin complejomtrica, pL' = f(x), para un catin M titulado con un ligante L. Se desprecia el efecto de dilucin.Se presenta un procedimiento simple para representar la curva de titulacin logartmica, pL' = f(x), mediante la hoja de clculo.Se considera la reaccin de titulacin de un catin M con un ligante L en condiciones especificadas que permiten definir y calcular la constante condicional, K'ML, de la reaccin. A continuacin, M' y L' representan los conjuntos respectivos de las especies M y L que permanecen sin reaccionar para formar el producto de la reaccin de titulacin (reaccin principal). ML' representa el conjunto de especies en las que la relacin de combinacin de M y L es 1:1.Reaccin de titulacin: M' + L' = ML'(1)Balance de M:|M'| + |ML'| = CM = |ML'|[1 + 1/(K'ML|L'|)](2)Balance de L:|L'| + |ML'| = CL = xCM(3)En la expresin (3), "x" representa la fraccin de titulante L agregado, relativa a la concentracin CM.Establecimiento de la ecuacin pL = f(x)De (2) se despeja:|ML'| = CM / [1 + 1 / (K'ML|L'|)]Este valor se sustituye en (3) :|L'| + CM / [1 + 1 / (K'ML|L'|)] = xCM(4)Si se ordenan los trminos se obtiene la ecuacin de segundo grado siguiente:K'ML|L'|2 + |L'|[1 + K'MLCM (1 x)] - x CM = 0(5)Cuya raz positiva se escribe :(6)Para obtener la ecuacin de la curva de titulacin pL' = f(x), de la expresin (6) se calcula pL' = -log|L'|.Sin embargo, si se aprovechan los recursos para graficacin de la hoja de clculo, es mucho ms sencillo escribir la relacin (4) en la forma:x = |L'| / CM + 1 / [1 + 1/(K'ML|L'|)](7)Para obtener la grfica pL' = f(x) sin resolver la ecuacin de segundo grado, se efectan los clculos de la funcin x = f(|L'|) para valores de pL' comprendidos entre (logK'ML + 3) y -logCM. Estos valores de pL' son adecuados cuando el parmetro de cuantitatividad definido como log(K'MLCM) es mayor que 2.La expresin (7) no involucra aproximacin alguna para calcular el valor de x.01 A. QUERE Facultad de Qumica UNAMClculo de pL en el punto de equivalencia de la reaccin de titulacinEn el punto de equivalencia, se tiene x = 1. Este valor se sustituye en la relacin (6):|L'| = [-1 + RAIZ(1 + 4CMK'ML)] / (2K'ML)(8)Representacin de la grfica pL' = f(x) con la hoja de clculoLos datos y clculos pueden presentarse en la forma siguiente:pL'inicial14.00pL'final3.00factor de cuantitatividadsemineutralizacinPunto de equivalencia (x = 1)1.00E+11logK'ML11.00CM0.001lgK'MLCMxpL'x = 0.5xequivpL'equivCuantitatividad11001008.000.5011.001.007.0099.9900%xpL'=f(x)|L'|02.00E+003.001.0E-0311.66E+003.186.6E-0421.43E+003.374.3E-0431.28E+003.552.8E-0441.18E+003.731.8E-0451.12E+003.921.2E-0461.08E+004.107.9E-0571.05E+004.285.2E-0581.03E+004.473.4E-0591.02E+004.652.2E-05101.01E+004.831.5E-05111.01E+005.029.6E-06121.01E+005.206.3E-06131.00E+005.384.1E-06141.00E+005.572.7E-06151.00E+005.751.8E-06161.00E+005.931.2E-06171.00E+006.127.6E-07181.00E+006.305.0E-07191.00E+006.483.3E-07201.00E+006.672.2E-07211.00E+006.851.4E-07221.00E+007.039.3E-08231.00E+007.226.1E-08241.00E+007.404.0E-08251.00E+007.582.6E-08269.99E-017.771.7E-08279.99E-017.951.1E-08289.99E-018.137.4E-09299.98E-018.324.8E-09309.97E-018.503.2E-09319.95E-018.682.1E-09329.93E-018.871.4E-09339.89E-019.058.9E-10349.83E-019.235.8E-10359.75E-019.423.8E-10369.62E-019.602.5E-10379.43E-019.781.6E-10389.15E-019.971.1E-10398.76E-0110.157.1E-11408.23E-0110.334.6E-11417.53E-0110.523.0E-11426.66E-0110.702.0E-11435.67E-0110.881.3E-11444.62E-0111.078.6E-12453.60E-0111.255.6E-12462.69E-0111.433.7E-12471.95E-0111.622.4E-12481.37E-0111.801.6E-12499.41E-0211.981.0E-12Esta hoja de clculo permite evaluar el efecto de cambiar el valor de la constante condicional de reaccin sobre la cuantitatividad de la reaccin de titulacin.506.38E-0212.176.8E-13514.28E-0212.354.5E-13522.85E-0212.532.9E-13531.88E-0212.721.9E-13Si el valor del factor de cuantitatividad, log(K'MLCo), no es mayor que 3.5, la localizacin precisa del punto de inflexin sobre la curva de titulacin se vuelve difcil o an imposible cuando el valor es menor que 2.5.541.24E-0212.901.3E-13558.19E-0313.088.3E-14565.38E-0313.275.4E-14573.54E-0313.453.5E-14582.32E-0313.632.3E-14591.52E-0313.821.5E-14 A. QUERE Facultad de Qumica UNAMRepresentacin de la grfica pL' = f(x) mediante ecuaciones aproximadasA continuacin, se presenta el procedimiento para calcular los puntos de la curva de titulacin pL' = f(x), empleando frmulas aproximadas (metodologa aplicable cuando se efectan clculos manuales).Tambin se calculan los valores exactos de pL' con base en la expresin (6) y stos se comparan con los valores aproximados correspondientes con el fin de evaluar a partir de qu valor del parmetro de cuantitatividad se presentan discrepancias apreciables.Tabla de variacin de concentraciones formalesReaccin principal:M'+L'ML'Ecuaciones aproximadas para graficar pL' = f(x)Conc. Inicial, x = 0Co = 0.001 M---------- pL' no est definido !Agregadox*Co-----Nota: La fraccin x agregada se refiere a CoAntes Pto. equiv. 0 < x < 1Co*(1 - x)-----x*Co(1)Pto. Equiv. x = 1----------CopL' = ()logK'ML -()logCo (2)Desp. Peq. x > 1-----Co*(x - 1)CopL' = -logCo(x - 1) (3)6.000.59.000.455.000000217160.53.000.456.9000003041000000000logK'ML9Co0.001lgK'MLCoCuantitatividad (Pto. Equiv.)9001006.0099.9000%xpL'aproxpL'exactolog(K ' Co) debe ser mayor que 3.500.00112.0012.000.00211.7011.700.00511.3011.300.0111.0011.000.0210.6910.690.0510.2810.280.19.959.950.29.609.600.39.379.370.59.009.000.78.638.630.88.408.400.98.058.050.957.727.720.987.317.310.997.007.0016.006.001.015.005.001.024.704.701.054.304.301.14.004.001.23.703.701.33.523.521.53.303.301.753.123.1223.003.00CARACTERSTICAS GENERALES DE LA CURVA DE TITULACINCuando logK'MLCo > 3.5:- La ecuacin aproximada (1) permite calcular satisfactoriamente pL' para valores de x < 0.98.- En la mitad de la titulacin (x = 0.5), se tiene pL' = logK'ML. - En el punto de equivalencia, se tiene: pL' = ()logK'ML - ()logCo.- Despus del punto de equivalencia (x > 1), la ecuacin (3) permite calcular satisfactoriamente pL' cuando se tiene x > 1.02.Para localizar satisfactoriamente el punto de inflexin correspondiente al punto de equivalencia, se requiere que el factor de cuantitatividad, logK'MLCo, sea mayor que 3.5 (cuantitatividad del 98.2%).La distancia medida como DpL' = pL'(x=2) - pL'(x=0.5), es aproximadamente igual al factor de cuantitatividad.

El titulante X es el EDTAEmpleo de la plantilla1. Ajustar el pH (barra en la celda J13).2. Ajustar las concentraciones CMg y CCa (barras en las celdas K13 y L13).3. Seleccionar el agente titulante (botones en las celdas M11 o N11).A pH=10 con EDTA se titula Ca+Mg.Con EGTA slo se titula el calcio solo.pCMglogK'MgXLogK'ML-logSMgpHlogK'pMg'Constante condicional Mg' + X' = MgX'-0.441239241410101010000.50.5111.51.5222.52.5333.53.5444.54.5555.55.5666.56.5777.57.5888.58.5999.59.5101010.510.5111111.511.5121212.512.5131313.513.5141438.211712021238.21171202125.6058560106-5.1034429127-17.7-3.6108413304-16.7-2.1338327057-15.7-0.7027892356-14.70.6093376539-13.70000000011.6926861903-12.70000000042.5055564676-11.70000000143.1331930176-10.70000000433.6765432188-9.70000001374.1867656907-8.70000004344.6777804764-7.70000013735.138948711-6.70000043435.5341857374-5.70000137345.8206798476-4.70000434296.0181142722-3.70001373346.2266387039-2.70004342736.5346942847-1.70013731436.9415446135-0.70043407757.39352030230.29862880727.83698343741.29567862628.21171202122.28647907798.44670051183.25860731488.50183062234.18066895198.37245213714.99897000438.0721062255.68066895197.65588117626.25860731487.18561513866.78647907796.69540523817.29567862626.19854233567.7986288072La determinacin del calcio y del magnesio tiene importancia en el anlisis de aguas naturales, de minerales como los carbonatos alcalinotrreos, de suelos y de lquidos fisiolgicos. El empleo del EDTA para determinar la dureza del agua fue establecido por G. Schwarzenbach en 1947 quien mostro que este reactivo permite realizar titulaciones complejomtricas de calcio o de magnesio en medio alcalino y descubrio sustancias qumicas, tiles para localizar visualmente el punto final de las reacciones de titulacin, que denomin indicadores metalocrmicos. Numerosos mtodos se desarrollaron para determinar por complejometra calcio y magnesio en disoluciones de mezclas de los dos cationes . El procedimiento ms comn consiste en titular una alcuota de la mezcla de los dos cationes con EDTA en medio alcalino de pH 10 con amortiguador amoniacal y empleando el negro de eriocromo T (NET) como indicador metalocr-mico. Luego, con una segunda alcuota, el magnesio se precipita en la forma de hidrxido por adicin de sosa hasta alcanzar pH = 13. El calcio solo puede entonces titularse con EDTA empleando murexida o calcon como indicadores metalocrmicos. El inconveniente de este procedimiento es que el hidrxido de magnesio retiene un poco de iones calcio y el resultado de la titulacin del calcio es menor que lo esperado. Otros procedimientos para analizar mezclas de calcio y magnesio consisten en precipitar el calcio mediante la adicin de oxalato de amonio, de sulfito o de sulfato en medio alcohlico. El magnesio que permanece en disolucin se puede titular con EDTA. Estos procedimientos no son ideales por implicar errores causados por un mayor nmero de manipulaciones que implican mayor tiempo de realizacin.En 1957, C. N. Reilley propuso el empleo del EGTA en lugar del EDTA para realizar la determinacin del calcio y del magnesio en disoluciones de los dos cationes, sin necesidad de efectuar separaciones previas por precipitacin (Anexo 3). Las tres hojas de clculo interactivas de este libro de trabajo, "pX=f(x)_Ca", "pX=f(x)_Mg" y "pX=f(x)_Ca+Mg", presentan el clculo de las constantes condicionales y las grficas de las curvas tericas de titulacin del calcio solo, del magnesio solo y de la mezcla de los dos empleando el EDTA o el EGTA con el fin de ilustrar este procedimiento que sigue vigente hoy en da.Ver / ocultar comentariosPantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra de FrmulasTitulante EGTATitulante EDTA&LFacultad Qumica UNAM&C&"Arial,Negrita"&14QUMICA ANALTICA III&RDr. Alain QUERE THORENT, profesor&LDepartamento de Qumica Analtica&RSemestre 2012/IIPara saber cmo se establece esta relacin, consultar las pginas de introduccin del libro de trabajo Excel "CLCULO DE CONSTANTE CONDICIONAL - Plantillas para la resolucin de las tareas # 1, 2 y 3" (A. QUERE - FQ UNAM).Mover el cursor de la barra de desplazamiento para ajustar el valor de log K'ML.Mover el cursor de la barra de desplazamiento para ajustar el valor de CM.El valor de log(K'ML CM) puede definirse como factor de cuantitatividad de la reaccin de titulacinObservar que en el punto de "semi-neutralizacin, el valor de pL' es igual a logK'ML excepto cuando el factor de cuantitatividad es menor que 2.6.El valor de pL'equiv se calcula con la ecuacin (8)Matemticamente, cuando |L'| tiende hacia cero, el valor pL' = -log|L'| tiende hacia +infinito.La disolucin contiene la mezcla ML' y M'. ML' desempea el papel de donador de partcula L' y M' es su receptor conjugado. El valor de pL' se calcula con la relacin:pL' = logK'ML + log(|Receptor|/|Donador|)La disolucin slo contiene ML' que desempea el papel de donador de partcula L' . El valor de pL' se calcula con la relacin:pL' = ()logK'ML - ()logCoLa disolucin contiene ML' que desempea el papel de donador dbil de partcula L' y L' que es el donador fuerte. El valor de pL' se calcula con la relacin:pL' = -log|L'| = -logCo(x-1)Los valores de pL' aproximado que difieren en ms de 0.1 unidad del valor exacto aparecen como texto de color rojo sobre fondo negro.

Ecuaciones_pL=f(x)_Mezcla

pL'=f(x)Punto de equivalencia (x = 1)pL'x = 0.5XpL'Titulacin M' + L' = ML' - Grfica pL' = f(x)

pY=f(x)_Pb+Cu A. QUERE-F.Q. UNAMNombre del alumnoQUMICA ANALTICA III - SEMESTRE 2012/II - RESOLUCIN DE LA TAREA #3Alfonso HFecha de entrega: 15 de Marzo de 2012OrtizCurva terica de titulacin complejomtrica de PLOMO con EDTAEstudio del efecto de la variacin del pH, de CTetren y de CPbRamrezPb2++Y4-PbY2-logKPbY =18logaPb(OH)0.310.3053454733M / OHY / HMY / HpH8logaT(H)3.173.1693223935logb1M/OHlogb2M/OHlogb3M/OHlogb1H/Llogb2H/Llogb3H/Llogb4H/Llogb1ML/H|H+|1.00E-08logaPb(T ')6.036.0296480165610.313.310.316.619.321.42.8|OH-|1.00E-06logaPb(glo)6.036.02964843021.00E+062.00E+102.00E+132.00E+103.98E+162.00E+192.51E+216.31E+02Constante condicionallogaY(H)2.312.3107088642M /TetrenTetren / HlogK'PbY=9.66logaPbY(H)0.000.0000027402log bM/Tlog b1H/Llog b2H/Llog b3H/Llog b4H/Llog b5H/LConcent. AnalticaNota: Cuando las frmulas introducidas son las correctas, el color de fondo de las celdas cambia a verde claro y el texto a verde oscuro. Si el fondo es gris la frmula es incorrecta.9.6610.59.718.826.931.634.6CTetren3.16E+105.01E+096.31E+187.94E+263.98E+313.98E+340.0500143pCPb1.00E+0289.665004,567,151,791.41822logK'PbY9.66CPb0.001lgK'MLCMPunto de equivalencia6.32992433180.9995321834pY'inicial12.66pY'final3.006.66xequivpY'equivCuantitatividad12.65964544586.6596454458xpY'=f(x)|Y'|pH8.001.006.3399.9532%00.000999001212.662.2E-138009.99E-0412.662.2E-1310.001571999712.463.4E-131.57E-0312.463.4E-1320.002472840312.275.4E-132.47E-0312.275.4E-1330.003887900812.078.5E-133.89E-0312.078.5E-1340.006107758911.871.3E-126.11E-0311.871.3E-1250.009582886411.672.1E-129.58E-0311.672.1E-1260.015005403411.483.3E-121.50E-0211.483.3E-1270.02342370711.285.3E-122.34E-0211.285.3E-1280.036390348511.088.3E-123.64E-0211.088.3E-1290.056122417510.891.3E-115.61E-0210.891.3E-11100.085603384710.692.0E-118.56E-0210.692.0E-11110.128462930210.493.2E-111.28E-0110.493.2E-11120.188360831110.295.1E-111.88E-0110.295.1E-11130.267611499510.108.0E-112.68E-0110.108.0E-11140.36520271399.901.3E-103.65E-019.901.3E-10150.47528731819.702.0E-104.75E-019.702.0E-10160.58782728479.513.1E-105.88E-019.513.1E-10170.69187787799.314.9E-106.92E-019.314.9E-10180.77951409479.117.7E-107.80E-019.117.7E-10190.84771148218.911.2E-098.48E-018.911.2E-09200.89758700578.721.9E-098.98E-018.721.9E-09210.9324309518.523.0E-099.32E-018.523.0E-09220.95600267018.324.8E-099.56E-018.324.8E-09230.97160432088.137.5E-09CTetrenpHCPb9.72E-018.137.5E-09240.9817831047.931.2E-080.058.000.0019.82E-017.931.2E-08250.98836341447.731.9E-088.006.338.009.666.669.88E-017.731.9E-08260.9925950527.532.9E-089.93E-017.532.9E-08270.99531149437.344.6E-089.95E-017.344.6E-08280.99706020927.147.2E-089.97E-017.147.2E-08290.99819881126.941.1E-079.98E-016.941.1E-07300.9989623356.751.8E-079.99E-016.751.8E-07310.99950935286.552.8E-071.00E+006.552.8E-07320.99995392986.354.5E-071.00E+006.354.5E-07331.00038901256.157.0E-071.00E+006.157.0E-07341.00090570055.961.1E-061.00E+005.961.1E-06351.00161225615.761.7E-061.00E+005.761.7E-06361.00265676265.562.7E-061.00E+005.562.7E-06371.00425814955.374.3E-061.00E+005.374.3E-06381.00675207635.176.8E-061.01E+005.176.8E-06391.01066128634.971.1E-051.01E+004.971.1E-05401.01680517724.771.7E-051.02E+004.771.7E-05411.02647155144.582.6E-051.03E+004.582.6E-05421.04168654874.384.2E-051.04E+004.384.2E-05431.06563933974.186.6E-051.07E+004.186.6E-05441.10335059733.991.0E-041.10E+003.991.0E-04451.16272486643.791.6E-041.16E+003.791.6E-04461.25620741073.592.6E-041.26E+003.592.6E-04471.40339282843.394.0E-041.40E+003.394.0E-04481.63513221713.206.4E-04La distancia entre la lnea anaranjada y la lnea aguamarina es igual al factor de cuantitatividad. Si es < 3.5 unidades, la titulacin no es factible.1.64E+003.206.4E-04491.9999997813.001.0E-032.00E+003.001.0E-03LogK'MLpH|H||OH|aM(OH)aX(H)aM(X')aM(global)aY(H)aML(H,OH)Sin/TetrenCon/Tetren01.00E+001.00E-141.00E+003.99E+341.00E+001.00E+002.53E+216.32E+02-0.60-0.600.53.16E-013.16E-141.00E+001.26E+321.00E+001.00E+002.58E+192.01E+020.890.8911.00E-011.00E-131.00E+004.02E+291.00E+001.00E+002.72E+176.41E+012.372.371.53.16E-023.16E-131.00E+001.30E+271.00E+001.00E+003.18E+152.10E+013.823.8221.00E-021.00E-121.00E+004.38E+241.00E+001.00E+004.91E+137.31E+005.175.172.53.16E-033.16E-121.00E+001.66E+221.00E+001.00E+001.28E+123.00E+006.376.3731.00E-031.00E-111.00E+008.04E+191.00E+001.00E+006.23E+101.63E+007.427.423.53.16E-043.16E-111.00E+005.49E+171.00E+001.00E+004.64E+091.20E+008.418.415.354.47E-062.24E-091.00E+008.68E+101.02E+001.02E+008.85E+051.00E+0012.0512.054.53.16E-053.16E-101.00E+006.62E+131.00E+001.00E+004.11E+071.02E+0010.3910.3951.00E-051.00E-091.00E+001.20E+121.00E+001.00E+004.20E+061.01E+0011.3811.385.53.16E-063.16E-091.00E+002.92E+101.05E+001.06E+004.62E+051.00E+0012.3412.3161.00E-061.00E-081.01E+008.40E+082.88E+002.89E+005.98E+041.00E+0013.2212.766.53.16E-073.16E-081.03E+002.61E+076.15E+016.15E+011.03E+041.00E+0013.9712.2071.00E-071.00E-071.10E+008.62E+051.84E+031.84E+032.39E+031.00E+0014.5811.367.53.16E-083.16E-071.32E+003.16E+045.00E+045.00E+046.72E+021.00E+0015.0510.4781.00E-081.00E-062.02E+001.48E+031.07E+061.07E+062.05E+021.00E+0015.389.668.53.16E-093.16E-064.36E+001.05E+021.50E+071.50E+076.45E+011.00E+0015.559.0191.00E-091.00E-051.30E+011.31E+011.21E+081.21E+082.10E+011.00E+0015.568.609.53.16E-103.16E-055.32E+013.24E+004.88E+084.88E+087.31E+001.00E+0015.418.45101.00E-101.00E-043.20E+021.57E+001.01E+091.01E+093.00E+001.00E+0015.028.5210.53.16E-113.16E-042.94E+031.16E+001.36E+091.36E+091.63E+001.00E+0014.328.65111.00E-111.00E-034.09E+041.05E+001.50E+091.50E+091.20E+001.00E+0013.318.7411.53.16E-123.16E-038.34E+051.02E+001.56E+091.56E+091.06E+001.00E+0012.058.78121.00E-121.00E-022.20E+071.01E+001.57E+091.60E+091.02E+001.00E+0010.658.7912.53.16E-133.16E-026.51E+081.00E+001.58E+092.23E+091.01E+001.00E+009.188.65131.00E-131.00E-012.02E+101.00E+001.58E+092.17E+101.00E+001.00E+007.697.6613.53.16E-143.16E-016.33E+111.00E+001.58E+096.35E+111.00E+001.00E+006.206.20141.00E-141.00E+002.00E+131.00E+001.58E+092.00E+131.00E+001.00E+004.704.70

Para obtener la grfica pL' = f(x), se introduce el valor de logK ' ML en la celda C70 y el de CM en la celda E70. En la celda C69 se calcula pL' inicial = (logK ' ML + 3) y en la celda E69 se calcula pL' final = -logCM. A continuacin, en el rango C73:C132 se calcula una serie de 60 valores de pL' con igual espaciamiento entre pL' inicial y pL' final. Luego, en el rango D73:D132, se calculan los valores correspondientes de |L'|. Finalmente, en el rango B73:B132, se calculan los valores de x empleando la expresin (7). Se selecciona el rango B73:C132 y se representa la grfica .Adicionalmente, en el rango K71:L71, se presenta el resultado del clculo de cuantitatividad de la reaccin de titulacin en su punto de equivalencia y en la celda J71 se calcula pL' en el punto de equivalencia empleando la relacin (8). El valor de pL' en x = 0.5 (50% titulado), se calcula empleando la relacin (6).GRFICO DINMICOLas barras de desplazamiento en la fila 71 se emplean para ajustar el valor de logK ' ML en el intervalo comprendido entre -2 y 25 y el valor de CM en el intervalo 1.00 E-05 M y 0.1 M.pL'exactopL'aproxxpL'Titulacin M' + L' = ML' Empleo de ecuaciones aproximadasDpL'=6.00CLCULO DE LOS VALORES [x, pL'aprox]:1. Se construye la serie de valores de x que permite obtener un espaciamiento equilibrado entre valores correspondientes de pL'2. Para 0.001< x < 0.99, los valores de pL' se calculan con la expresin (1) de la tabla anterior3. Para x = 1, pL' se calcula con la expresin (2)4. Para x > 1, pL' se calcula con la expresin (3)

GRFICO DINMICOLas barras de desplazamiento ubicadas en la fila 155 permiten cambiar los valores del logaritmo de la constante condicional y la concentracin del ion metlico en forma independiente.

COMPARACIN pL'aprox con pL'exactoEl alumno debe observar que cuando logK'MLCo (parmetro de cuantitatividad) se vuelve menor que 4, los valores de pL' calculados con las frmulas aproximadas desvan de los valores exactos. La discrepancia es tanto mayor cuanto ms pequeo sea logK'MLCo.La grfica muestra el valor de DpL' = pL'x=0.5 - pL'x=2 logKMLCoEl valor de DpL' es prcticamente igual al factor de cuantitatividad, esto es logKMLCo, cuando logKMLCo 1.3 .Para un valor de Co = 0.001 M, el alumno puede hacer variar el valor de logK'ML desde 8 hasta 1 y observar como crecen las discrepancias entre los valores de pL' aproximado y exacto.Ver/ocultar comentariosVer/ocultar comentariosPantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra Frmulas&LFacultad de Qumica UNAM&RDr. Alain QUERE T., profesor&LDepartamento de Qumica Analtica&RSemestre 2010/IILa concentracin de plomo se hace variar mediante la barra de desplazamiento situada arriba de esta celda.El valor de pH se hace variar mediante el control de nmero situado en la celda E19.La concentracin de tetren se hace variar mediante la barra de desplazamiento situada debajo de la celda H15.Introducir los valores de logbi en el rango B10:D10. En el rango B11:D11 se introducen las frmulas que proporcionan los valores de bi.

pY=f(x)_Pb+Cu

pY'=f(x)XpY'Titulacin Pb' + Y' = PbY' - Grfica pY' = f(x)

Anexo1

Con/TetrenSin/TetrenpCPblogK'PbYpHlogK'Constante condicional Pb' + Y' = PbY'6.66

Anexo2 A. QUERE-F.Q. UNAMNombre del alumnoQUMICA ANALTICA III - SEMESTRE 2012/II - RESOLUCIN DE LA TAREA #3Alfonso HFecha de entrega: 15 de Marzo de 2012OrtizCurva terica de titulacin complejomtrica de COBRE con EDTAEstudio del efecto de la variacin del pH, de CTetren y de CCuRamrezCu2++Y4-CuY2-logK =18.8pH8.00M/OHY/HMY/ HMY/OH|H+|1.00E-08log b1M/OHlog b2M/OHlog b3M/OHlog b4M/OHlog b1H/Llog b2H/Llog b3H/Llog b4H/Llogb1ML/Hlogb1ML/OH|OH-|1.00E-065.51013.615.610.316.619.321.432.5logaCu(OH)0.120.1226311539316227.7660168381000000000039810717055349.8398107170553497019952623149.6889398107170553499001995262314968890000025118864315095843000001000316.2277660168logaT(H)3.173.1693223935M/TetrenMTetren/HTetren/H0logaCu(T ')16.6318.3303354197log bM/Tlogb1MT/Hlogb2MT/Hlog b1H/Llog b2H/Llog b3H/Llog b4H/Llog b5H/LConcent. AnalticalogaCu(glo)16.6318.330335419722.85.299.718.826.931.634.6CTetrenlogaY(H)2.312.310708864263095734448019930000000158489.31924611210000000005011872336.27272630957344480197000079432823472428620000000000039810717055350155000000000000000398107170553501440000000000000000000.05lgaCuY(H,OH)0.000.0001416558143.00pCCu1008.00-0.14Constante condicional0.7212193602169030logK'CuY-0.14CCu0.001lgK'MLCMPunto de equivalencia (x = 1)logK'CuY=-0.14-1.840.7212193602169030pY'inicial6.00pY'final3.00-3.14xequivpY'equivCuantitatividad6-4.840902628xpY'=f(x)|Y'|pH8.001.003.000.0720%01.00E-036.001.0E-061.00E-036.001.00E-0611.15E-035.941.2E-061.15E-035.941.15E-0621.33E-035.881.3E-061.33E-035.881.33E-0631.53E-035.821.5E-061.53E-035.821.53E-0641.76E-035.761.8E-061.76E-035.761.76E-0652.03E-035.692.0E-062.02E-035.692.02E-0662.33E-035.632.3E-062.33E-035.632.33E-0672.68E-035.572.7E-062.68E-035.572.68E-0683.09E-035.513.1E-063.09E-035.513.09E-0693.56E-035.453.6E-063.56E-035.453.56E-06104.10E-035.394.1E-064.09E-035.394.09E-06114.72E-035.334.7E-064.71E-035.334.71E-06125.43E-035.275.4E-065.43E-035.275.43E-06136.26E-035.206.3E-066.25E-035.206.25E-06147.20E-035.147.2E-067.20E-035.147.20E-06158.29E-035.088.3E-068.29E-035.088.29E-06169.55E-035.029.5E-069.54E-035.029.54E-06171.10E-024.961.1E-051.10E-024.961.10E-05181.27E-024.901.3E-051.26E-024.901.26E-05191.46E-024.841.5E-051.46E-024.841.46E-05201.68E-024.781.7E-051.68E-024.781.68E-05211.93E-024.711.9E-051.93E-024.711.93E-05222.22E-024.652.2E-052.22E-024.652.22E-05232.56E-024.592.6E-05CTetrenpHCCu2.56E-024.592.56E-05242.95E-024.532.9E-050.058.000.0012.95E-024.532.95E-05253.40E-024.473.4E-058.001.438.00-0.14-3.143.39E-024.473.39E-05263.91E-024.413.9E-053.91E-024.413.91E-05274.50E-024.354.5E-054.50E-024.354.50E-05285.18E-024.295.2E-055.18E-024.295.18E-05295.97E-024.226.0E-055.96E-024.225.96E-05306.87E-024.166.9E-056.87E-024.166.87E-05317.91E-024.107.9E-057.91E-024.107.91E-05329.11E-024.049.1E-059.10E-024.049.10E-05331.05E-013.981.0E-041.05E-013.981.05E-04341.21E-013.921.2E-041.21E-013.921.21E-04351.39E-013.861.4E-041.39E-013.861.39E-04361.60E-013.801.6E-041.60E-013.801.60E-04371.84E-013.731.8E-041.84E-013.731.84E-04382.12E-013.672.1E-042.12E-013.672.12E-04392.44E-013.612.4E-042.44E-013.612.44E-04402.81E-013.552.8E-042.81E-013.552.81E-04413.24E-013.493.2E-043.24E-013.493.24E-04423.73E-013.433.7E-043.73E-013.433.73E-04434.30E-013.374.3E-044.29E-013.374.29E-04444.95E-013.314.9E-044.94E-013.314.94E-04455.69E-013.245.7E-045.69E-013.245.69E-04466.56E-013.186.6E-046.55E-013.186.55E-04477.55E-013.127.5E-047.54E-013.127.54E-04488.69E-013.068.7E-04La distancia entre la lnea anaranjada y la lnea aguamarina es igual al factor de cuantitatividad. Si es < 3.5 unidades, la titulacin no es factible.8.69E-013.068.69E-04491.00E+003.001.0E-031.00E+003.001.00E-03LogK'MLpHaM(OH)aM(X')aM(global)aML(H,OH)Sin/TetrenCon/Tetren01.00E+001.00E+001.00E+001.00E+030.400.400.51.00E+001.00E+001.00E+003.17E+021.891.8911.00E+001.08E+001.08E+001.01E+023.373.341.51.00E+003.44E+003.44E+003.26E+014.814.2721.00E+007.42E+017.42E+011.10E+016.154.282.51.00E+002.00E+032.00E+034.16E+007.314.0131.00E+004.55E+044.55E+042.00E+008.313.653.51.00E+008.68E+058.68E+051.32E+009.253.315.351.00E+006.28E+106.28E+101.00E+0012.852.064.51.00E+003.34E+083.34E+081.03E+0011.202.6851.00E+007.08E+097.08E+091.01E+0012.182.335.51.00E+001.63E+111.63E+111.00E+0013.141.9261.00E+004.35E+124.35E+121.00E+0014.021.396.51.01E+001.27E+141.27E+141.00E+0014.780.6871.03E+003.72E+153.72E+151.00E+0015.41-0.157.51.10E+001.00E+171.00E+171.00E+0015.93-1.0381.33E+002.14E+182.14E+181.00E+0016.37-1.848.52.10E+003.00E+193.00E+191.00E+0016.67-2.4995.20E+002.41E+202.41E+201.00E+0016.76-2.909.52.23E+019.73E+209.73E+201.01E+0016.59-3.05101.73E+022.02E+212.02E+211.03E+0016.10-2.9710.52.40E+032.71E+212.71E+211.10E+0015.25-2.80115.41E+043.00E+213.00E+211.32E+0014.11-2.6411.51.76E+063.11E+213.11E+212.00E+0012.83-2.42128.06E+073.14E+213.14E+214.16E+0011.50-2.0912.55.25E+093.15E+213.15E+211.10E+0110.12-1.66134.38E+113.15E+213.15E+213.26E+018.67-1.1913.54.11E+133.15E+213.15E+211.01E+027.19-0.69144.02E+153.15E+213.15E+213.17E+025.70-0.20

Instrucciones de trabajo:1. En las celdas del rango M7:M12, introducir las frmulas que calculan los coeficientes de equilibrios parsitos.2. En la celda K12, introducir la frmula que calcula la constante condicional.3. En H19 e I19 se introducen las frmulas que calculan pY' y la cuantitatividad en el Punto Equiv.4. En la celda C20 se introduce la frmula: "=$C$18-($C$18-$E$18)*A20/49. La frmula se copia por arrastre del controlador de relleno hasta la celda C69.

5. En la celda D20 se introduce la frmula que calcula |Y'|.6. En la celda B20 se introduce la frmula que calcula x [con la ecuacin (7) en la fila 50 de la hoja "Ecuaciones_pL=f(x)"].La magnitud del salto de pY' en la vecindad del punto de equivalencia depende esencialmente del factor de cuantitatividad, log(K'PbYCPb). En la curva de titulacin, este factor se mide por la distancia: pY'(x=0.5) - pY'(x=2) = logK'PbYCPb

Instrucciones de utilizacin:Los tres parmetros, pH, CTetren y CPb, que influyen sobre el valor de la constante condicional y, de ah sobre la factibilidad de la titulacin, pueden hacerse variar mediante las barras de desplazamiento.Por ejemplo, para conocer el intervalo de pH en el que la titulacin es factible, cuando CPb = 1E-4 M, se hace variar el pH en el intervalo en que se tenga log K' 7.50. Esta condicin se cumple en el intervalo: 3.04 < pH < 13.06Validar UsuarioPantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra FrmulasEnfocar Curva de titulacinEnfocar Constante Condi-cionalVer/ocultar comentariosRegresar a Inicio de hojaRegresar a Inicio de hoja&L&"Arial,Negrita"Facultad de Qumica UNAM&R&"Arial,Negrita"Dr. Alain QUERE T., profesor&LDepartamento de Qumica Analtica&RSemestre 2010/IILa concentracin de tetren se hace variar mediante la barra de desplazamiento situada debajo de la celda J15.La concentracin de cobre se hace variar mediante la barra de desplazamiento situada arriba de esta celda.El valor de pH se hace variar mediante el control de nmero situado en la celda E19.

Anexo2

pY'=f(x)XpY'Titulacin Cu' + Y' = CuY' - Grfica pY' = f(x)

Anexo3

Con/TetrenSin/TetrenlogK'CuYpCCupHlogK'Constante condicional Cu' + Y' = CuY'-3.14

Anexo4 A. QUERE-F.Q. UNAMII. TITULACIN DE UNA MEZCLA DE CATIONES M Y N CON EL LIGANTE LA continuacin se establece la ecuacin que permite efectuar la representacin de la curva de titulacin complejomtrica, pL' = f(x), para una mezcla de dos cationes M y N titulados con un ligante L. Como en el caso de la titulacin de un cation M, se desprecia el efecto de la dilucin.Para representar la curva de titulacin pL' = f(x) con la hoja de clculo y sin resolver una ecuacin de tercer grado, se emplea el mismo procedimiento simple que en el caso de la titulacin de un solo cation.Se considera la reaccin de titulacin de la mezcla de los cationes M y N con un ligante L en condiciones especificadas que permiten definir y calcular las constantes condicionales, K'ML y K'NL. A continuacin, M', N' y L' representan los conjuntos respectivos de las especies M, N y L que permanecen sin reaccionar para formar los productos de las reacciones de titulacin (reacciones principales). ML' y NL' representan los conjuntos de especies en las que la relacin de combinacin de M o N y L es 1:1.Reaccin de titulacin 1:M'+L'=ML'(1)Reaccin de titulacin 2:N'+L'=NL'(2)Balance de M:|M'| + |ML'| = CM = |ML'|[1 + 1/(K'ML|L'|)](3)Balance de N:|N'| + |NL'| = CN = |NL'|[1 + 1/(K'NL|L'|)] = aCM(4)Nota: el parmetro "a" es igual al cociente CN/CMBalance de L:|L'| + |ML'| + |NL'| = CL = xCM(5)La resolucin de este sistema de ecuaciones permite obtener las expresiones pL', pM' o pN' = f(x), las cuales son de tercer grado. Dichas expresiones pueden resolverse por aproximaciones sucesivas.Por ejemplo, para calcular |L'| = f(x), la ecuacin de tercer grado es:K'MLK'NL|L'|3 + [(K'ML +K'NL) + CMK'MLK'NL(1+ax)]|L'|2 + [1 + CM(K'ML+aK'NLx(K'ML+K'NL))]|L'| - xCM = 0(6)Sin embargo, como en el caso de la titulacin de un catin M con un ligante L examinado antes, la hoja de clculo permite obtener la grfica pL' = f(x) sin que sea necesario resolver la ecuacin de tercer grado.Para ello se sustituyen |ML'| y |NL'| obtenidos de las expresiones (3) y (4) en la expresin (5):|L'| + CM / [1 + 1/(K'ML|L'|] + aCM / [1 + 1/(K'NL|L'|] = xCM(7)de all se obtiene:x = |L'| / CM + 1 / [1 + 1/(K'ML|L'|] + a / [1 + 1/(K'NL|L'|]a=CN/CM(8)La relacin (8) permite calcular x para valores de pL' comprendidos entre (logK'ML + 3) y -logCN cuando logK'MLCM es mayor que logK'NLCN. Si se tiene logK'NLCN > logK'MLCM, x se calcula para valores de pL' comprendidos entre (logK'NL + 3) y -logCM.La expresin (7) no involucra aproximacin alguna para calcular el valor de x.REPRESENTACIN GRFICA DE pL' = f(x) A. QUERE Facultad de Qumica UNAM9701000pL' ini.pL' fin.1 Pto. Equivalenciaf(|L'|)2 Pto. Equivalenciaf(|L'|)5011872336.27272logK'ML9.7CM0.0115.702.00xequivpL'equiv0.00E+00xequivpL'equiv4.07E-125011872336272.72logK'NL12.7a=CN/CM1lgK'MLCMlgK'NLCN111.20f'(|L'|)25.85f'(|L'|)12701007.7010.70|L'|equiv6.31E-123169786418.57663|L'|equiv1.41E-061.00E+11xpL'|L'|00.00115.702.00E-1610.00215.433.75E-1620.00415.157.05E-1630.00714.881.32E-1540.01214.602.49E-1550.02314.334.68E-1560.04214.068.79E-1570.07713.781.65E-1480.13513.513.10E-1490.22713.235.83E-14100.35512.961.10E-13110.50912.692.06E-13120.66212.413.87E-13130.78812.147.28E-13140.87911.861.37E-12150.94111.592.57E-12160.98411.324.83E-12171.02211.049.08E-12181.06710.771.71E-11191.13210.493.21E-11201.22910.226.03E-11211.3609.951.13E-10221.5159.672.13E-10231.6679.404.00E-10241.7909.127.52E-10251.8768.851.41E-09261.9308.582.65E-09271.9628.304.99E-09281.9798.039.38E-09Los valores numricos que aparecen sobre el grfico indican los valores de pL' en los puntos de equivalencia.291.9897.751.76E-08301.9947.483.31E-08311.9977.216.22E-08CARACTERSTICAS GENERALES DE LA CURVA DE TITULACIN321.9986.931.17E-07331.9996.662.20E-07Si K'ML > K'NL, la primera reaccin de titulacin es M' + L' = ML'.A la mitad de la primera titulacin (x = 0.5), se tiene pL' = logK'ML. En el primer punto de equivalencia, se tiene el equilibrio: ML' + N' = M' + NL'y se tiene aproximadamente: pL' = ()log[K'ML*K'NL*a]342.0006.384.13E-07352.0006.117.76E-07362.0005.841.46E-06372.0005.562.74E-06382.0005.295.15E-06392.0015.019.68E-06La magnitud del salto de pL' en el primer punto de equivalencia depende de la diferencia DlogK' = logK'ML - logK'NL. Este valor debe ser mayor que 3.5 unidades de pL' para localizar con precisin satisfactoria el punto de inflexin.A la mitad de la segunda titulacin (x = 1 + (a/2)), se tiene pL' = logK'NL.402.0024.741.82E-05412.0034.473.42E-05422.0064.196.43E-05432.0123.921.21E-04442.0233.642.27E-04452.0433.374.27E-04En el segundo punto de equivalencia, el equilibrio dominante es: NL' = N' + L' y se tiene aproximadamente pL' = ()logK'NL - ()log(a*CM) .La magnitud del salto de pL' en el segundo punto de equivalencia depende de la diferencia DpL' = pL'[x=1+(a/2)] - pL'[x=1+2a].462.0803.108.02E-04472.1512.821.51E-03482.2832.552.83E-03492.5322.275.32E-03503.0002.001.00E-02Si DpL' es menor que 3.5 unidades, el punto de inflexin del segundo punto de equivalencia no podr localizarse con precisin satisfactoria o no podr apreciarse si DpL' < 2.5.Si DlogK' y DpL' son ambos mayores que 3.5 unidades, los dos puntos de inflexin podrn localizarse satisfactoriamente y ser posible determinar las concentraciones individuales de M y de N.Si DlogK' < 3.5 y DpL' > 3.5, slo podr determinarse la suma de las concentraciones de M y N.Representacin de la grfica pL' = f(x) mediante ecuaciones aproximadasEl procedimiento para calcular los puntos de la curva de titulacin pL' = f(x) de la mezcla M + N, mediante frmulas aproximadas (metodologa aplicable cuando se efectan clculos manuales), se presenta en el archivo "Titul_M+N_aprox.xls".Puesto que para cada punto calculado de la curva de titulacin se calculan los valores exactos de pL' por resolucin de la ecuacin (6) de tercer grado por el mtodo de Newton-Raphson , los clculos por aproximaciones sucesivas hacen lento el conjunto de clculos de todo el libro de trabajo. Por este motivo, estos clculos se presentan en un archivo por separado.

Instrucciones de trabajo:La metodologa es la misma que en el caso de la hoja "pY=f(x)_Pb":1. Se establece el intervalo de variacin de pY' (rango C20:C69) y se calculan los valores correspon-dientes de |Y'| (rango D20:D69).2. Se calculan los valores de X, fraccin de titulante agregado, correspondientes a los de |Y'| (rango B20:B69).3. Se calcula pY' equivalencia (celda H19).4. Se calcula la cuantitatividad (celda I19).

TRABAJE CON INTELIGENCIA PARA GANAR TIEMPO!El trabajo realizado con anterioridad en otra plantilla puede utilizarse en esta y otras plantillas.Instrucciones de utilizacin:Los dos parmetros, pH y CTetren que influyen sobre el valor de la constante condicional, as como CCu, influyen sobre la factibilidad de la titulacin. Estos parmetros pueden hacerse variar mediante las barras de desplazamiento.Para conocer el intervalo de pH en el que la titulacin es factible en ausencia de Tetren, cuando CCu = 1E-4 M, se hace variar el pH en el intervalo en que se tenga log K' 7.50. Esta condicin se cumple en el intervalo: 2.60 < pH < 13.40.No debe olvidarse que el pH tambin influye sobre la precipitacin de hidrxidos metlicos. Es as que, en teora, no se debe rebasar pH = 5 para no provocar la precipitacin de Cu(OH)2. Otros factores tambin deben tomarse en cuenta.Enfocar Curva de titulacinEnfocar Constante Condi-cionalVer/ocultar comentariosPantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra FrmulasRegresar a Inicio de hojaRegresar a Inicio de hoja&LFacultad de Qumica UNAM&C&"Arial,Negrita"&14QUMICA ANALTICA III&RDr. Alain QUERE T., profesor&LDepartamento de Qumica Analtica&RSemestre 2012/II

Anexo4

pL'1 Pto. Equivalencia2 Pto. EquivalenciaXpLTitulacin de una mezcla M + N con L

HojaLibre A. QUERE-F.Q. UNAMNombre del alumnoQUMICA ANALTICA III - SEMESTRE 2012/II - RESOLUCIN DE LA TAREA #3Alfonso HFecha de entrega: 15 de marzo 2012OrtizCurva terica de titulacin de la mezcla de Cu(II) y Pb(II) con EDTAEstudio del efecto de la variacin del pH, de CTetren , de CPb y de CCuRamrezCu2++Y4-CuY2-lgKCuY =18.8Esta plantilla no cuenta, a propsito, con indicadores que ayudan para saber si las frmulas introducidas son correctas.Para ver las instrucciones de trabajo, pulsar el botn "Ver comentarios".lgaCu(OH)0.12lgaCu(T ')16.63lgaCu(glo)16.63lgaCuY(H)0.00lgK'CuY=-0.14lgaT(H)3.17lgaY(H)2.31Pb2++Y4-PbY2-lgKPbY =18lgaPb(OH)0.31lgaPb(T ')6.03lgaPb(glo)6.03lgaPbY(H)0.00lgK'PbY=9.66pY'inicial12.66a=CPb/CcupHCPbCCuCTetrenpY'final3.001lgK'CuCCulgK'PbCPbxpY'=f(x)|Y'|-3.146.668.000.0010.0010.050001.0E-0312.662.2E-1311.6E-0312.463.4E-1322.5E-0312.275.4E-1333.9E-0312.078.5E-1346.1E-0311.871.3E-12100359.6E-0311.672.1E-1261.5E-0211.483.3E-1272.3E-0211.285.3E-1283.6E-0211.088.3E-1295.6E-0210.891.3E-11108.6E-0210.692.0E-11111.3E-0110.493.2E-11121.9E-0110.295.1E-11132.7E-0110.108.0E-11143.7E-019.901.3E-10154.8E-019.702.0E-10165.9E-019.513.1E-10176.9E-019.314.9E-10187.8E-019.117.7E-10198.5E-018.911.2E-09209.0E-018.721.9E-09219.3E-018.523.0E-09229.6E-018.324.8E-09239.7E-018.137.5E-09249.8E-017.931.2E-08259.9E-017.731.9E-08269.9E-017.532.9E-08271.0E+007.344.6E-08281.0E+007.147.2E-08291.0E+006.941.1E-07301.0E+006.751.8E-07311.0E+006.552.8E-07321.0E+006.354.5E-07331.0E+006.157.0E-07341.0E+005.961.1E-06351.0E+005.761.7E-06361.0E+005.562.7E-06371.0E+005.374.3E-06381.0E+005.176.8E-06391.0E+004.971.1E-05401.0E+004.771.7E-05411.0E+004.582.6E-05421.0E+004.384.2E-05431.1E+004.186.6E-05441.1E+003.991.0E-04451.2E+003.791.6E-04461.3E+003.592.6E-04471.4E+003.394.0E-04481.6E+003.206.4E-04492.0E+003.001.0E-03En presencia de Tetren 0.05 M

Pantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra Frmulas&LFacultad de Qumica UNAM&RDr. Alain QUERE T., Profesor&LDepartamento de Qumica Analtica&RSemestre 2012/IIEl valor de a depende de los valores respectivos de CPb y de CCu ajustados en las celdas H15 y I15.Instrucciones 3: En la celda B16 introducir la frmula que permite calcular x con la ecuacin (8) ubicada en la fila 43 de la hoja "Ecuaciones_pL=f(x)_Mezcla"Instrucciones 1: En la celda C16 introducir la frmula que permite establecer los valores de pY'. Arrastrar el cuadro de relleno hasta la celda C65.La frmula debe adaptarse de la que se emple de conformidad con el numeral 3 de Instrucciones de trabajo de la hoja pY=f(x)_Pb.Instrucciones 2: En la celda D16 introducir la frmula que permite calcular |Y'| a partir del valor contenido en C16.

HojaLibre

pY'=f(x)XpX'Titulacin Mezcla Pb + Cu con EDTA

Informe de compatibilidad0.39695676730.39699115253-0.1419326237-0.14193262371.429033688131.88944575071.89052982363.3376344673.37048854524.27401506864.81073129814.28050205356.1507291114.01153544617.31201335493.64867046988.30657418313.31386743159.25248523452.057110720612.8545706452.675961839611.19989843522.33122695312.18087527421.923616740713.13645055541.38514585914.0224767090.6848461914.7833045521-0.149480313315.4072993361-1.028236082215.9310495838-1.84090262816.3668016378-2.486801971216.668437545-2.901934755316.763127782-3.048123779316.5926111446-2.967412712316.09940523-2.803772372315.2487760041-2.637151237314.10705022-2.417656651112.8294302913-2.086045122111.5042834731-1.659620907910.1185011869-1.18609712178.6711633761-0.6948537457.1905159413-0.19766424425.6969620944

En presencia de Tetren 0.05 MCon/TetrenSin/TetrenpCPblogK'PbYpHlogK'Const. Condic. Pb' + Y' = PbY'6.66008148880.50.5111.51.5222.52.5333.53.55.355.354.54.5555.55.5666.56.5777.57.5888.58.5999.59.5101010.510.5111111.511.5121212.512.5131313.513.51414-0.6027551624-0.60275516246.329822722939.659645445839.65964544580.89132981360.89132981362.37299627012.37299627013.8184486843.8184486845.1732281635.1732281636.36911623016.36911623017.417983827.417983828.41215454118.412154542412.045374250512.053193172110.394853212610.394863582811.378415907511.378988601312.312157367912.335007439412.762607895113.219364588412.198723589113.974045893111.35706294714.579361151610.473875596415.05279980339.659645445815.38394840289.012980932615.55075366158.596762672115.56348613538.44757651615.40990805878.519073096115.01770806598.654836173614.31869052218.743506596513.30919918288.781081010512.05244534728.788604328410.64982944898.64904582969.18372637277.66202272387.69481080296.19727129666.19835465534.69944487824.6994792563Con/TetrenSin/TetrenpCCulogK'CuYpHlogK'Const. Condic. Cu' + Y' = CuY'-3.14

PROCEDIMIENTOS RECOMENDADOSPARA TITULACIONES COMPLEJOMTRICAS CON EDTAhttp://www.marcel-benoist.org/http://www.marcel-benoist.org/d/pre/d1963sch.htmEn 1963, el profesor G. Schwarzenbach recibi el premio Marcel Benoist por sus estudiossobre los complejos metlicos y la complejometra. A continuacin se presenta el contenidode una pgina de Internet sobre el laureado, traducida al espaol.(http://www.marcel-benoist.org/d/pre/d1963sch.htm)1963 Gerold Schwarzenbach"La Comisin Administrativa tiene as un representante asignado de qumica inorgnica,establecido con base en El trabajo sobre complejos de metales para interpretar elcomportamiento de muchos elementos traza en sistemas biolgicos. El ProfesorSchwarzenbach tambin ha proporcionado a la qumica analtica una distinta y verstiltcnica de medicin".Nacio en 1904 en Horgen (ZH). EPF Zurich: estudios de Qumica, 1928 Dr. (Prof. WD Treadwell). UNIV. Zurich: 1929 Asistente (Prof. P. Karrer), 1930 Privat Docent, 1943 Profesor ordinario, 1947 Prof. titular de qumica analtica e inorgnica especial. 1955 Prof. de Qumica Inorgnica de la EPF Zurich(-1973). Distinciones: miembro de la Leopoldina Halle, de la Academia Americana de las Artes y las Ciencias. Fallece 1978 Qumica AnalticaLos trabajos de Schwarzenbach sobre los agentes complejantes se han considerado porunanimidad como decisivos. El objetivo principal fue la explicacin cuantitativa de los equilibriosentre los iones metlicos y los ligandos. Mediante equipos de flujo mejorado, Schwarzenbachlogr seguir la formacin y descomposicin de enlaces de muy corta duracin. En su informelos expertos consideraron que la bioqumica y la biologa estaban relacionados con loslos resultados de sus investigaciones por el papel de los iones metlicos en los procesosenzimticos. Los ferricromos, por ejemplo, son complejos de hierro biolgicamente activos(son promotores de crecimiento en relacin con la produccin de antibiticos), cuyasconstantes de estabilidad han sido determinadas por Schwarzenbach y sus colaboradores.De acuerdo con los expertos, la investigacin bsica realizada por Schwarzenbach sobre lascomplexonas se volvi rpidamente importante en qumica analtica ya que permite determinarcuantitativamente muchos cationes de forma simple, lo que facilita el anlisis de alimentos,piensos, medicamentos y otras preparaciones.

Empleo de la plantilla despus de completarla:1. Fijar CPb = CCu = 0.001M y CTetren = 0. Hacer variar pH desde 2 hasta 14. Para 2.5 < pH < 13.5, slo se observa un salto de pH en x = 2 . A partir de pH = 2.8, hasta pH 13.2, la magnitud del salto de pY' en x = 2 es suficiente para localizar satisfactoriamente el punto de inflexin. Tericamente podra determinarse la composicin de una mezcla de los dos cationes con una sola titulacin a 2.8 < pH < 13.2. Sin embargo en la prctica no se puede trabajar a pH mayor que 5 porque los cationes metlicos empiezan a precipitar en la forma de hidrxidos a menos que se empleen agentes enmascarantes de la precipitacin (consultar la determinacin de Cu con EDTA en la hoja Anexo1). (Nota: Los datos anteriores son tiles para que corroboren si las frmulas introducidas en la hoja de clculo son las correctas). 2. Con las mismas concentraciones de cationes metlicos y a pH = 5.5, fijar la concentracin deTetren en 0.01 M. Observar que en estas condiciones se tiene slo un punto de inflexin en x = 1. Las grficas de variacin de las constantes condicionales de formacin de PbY y de CuY permiten saber qu ion metlico fue titulado en estas condiciones. (Nota la presencia de la Tetren impide la precipitacin del hidrxido de Cu(II) an a pH 14).3. Las observaciones anteriores deben ser suficientes para disear un procedimiento de determinacin de Pb y Cu.Tip:Para no cometer errores en la ecuacin (8) de la hoja "Ecuaciones_pL=f(x)_Mezcla" que proporciona el valor de x, es preciso identificar cul de los dos metales es M y cul es N.Ver/ocultar comentariosPantalla CompletaEncabezadosEtiquetas de HojasBarra de estadoBarra FrmulasEnfocar Curva de titulacin&LFacultad de Qumica UNAM&C&"Arial,Negrita"&16QUMICA ANALTICA III&RDr. Alain QUERE, profesor&LDepartamento de Qumica Analtica&RSemestre 2012/II31.1 Direct titration of copper with murexide as indicator

REAGENTS: Standard solution of 0.01 M EDTA; murexide; approx. 1 M NH4OH; approx. 1 M NH4Cl.

PROCEDURE: The test-solution should not contain more than about 20 mg of Cu per 100 mL. Acid solutions are neutralized with ammonia and only a slight excess is used to give a pH of about 8. With only weakly acidic solutions about 10 mL of NH4Cl is added as well for every 100 mL of solution to ensure that the ammonia does not produce too high a pH. Indicator is then added and the titration with EDTA carried out to the colour change from yellow to violet.

REMARKS: ....In complete contrast to the corresponding titration of Ni it is better with Cu to work at low pH values. This arise from the fact that ammine complexes of copper are appreciably more stable than those of nickel, so that the copper-murexide complex can be broken down again by too much ammonia (...). High concentration of ammonia must therefore be avoided. Some ammonia must, of course, be present to keep the metal in solution. The danger of precipitating basic salts, e.g. 3Cu(OH)2.CuCl2, is further diminished by the presence of much NH4+, which keeps the pH down. Naturally ammonia is a poor buffer for the region pH 7-8; but trietanolamine cannot be used since its complexes with Cu are appreciably more stable than those with ammonia.

Referencia: G. Schwarzenbach, H. Flaschka, Complexometric Titrations. Methuen, London, 1969. p. 256.44.1 Direct titration of lead using Erio T

REAGENTS: Standard 0.01 M EDTA; Erio T; pH 10 buffer; 1M Na-K tartrate; if necessary KCN as a masking agent.

PROCEDURE: The sample solution, which should not contain more than about 30 mg of Pb per 100 mL is treated with 5 mL of tartrate or 1:4 triethanolamine (or the PbSO4 that is to be titrated is taken up in tartrate or triethanolamine) and made approximately neutral with NaOH. If necessary a masking agent such as KCN is added at this stage. Then 2 mL of buffer and indicator are added and the titration carried on to the colour change from red to blue. The solution must not be too cold (approx. 40C), and the last trace of a red blue vanishes with the final drops of titrant.

44.2 Substitution titration for leadREAGENTS: Standard 0.01 M EDTA; Erio T; pH 10 buffer; approx. 0.1 M MgY2- ; approx. 1 M NaOH.

PROCEDURE: The solution to be analysed which should not contain more than about 30 mg of Pb per 100 mL is treated with 5 mL MgY (in the case of PbSO4 this is disolved directly in MgY) and made roughly neutral with NaOH. 2 mL of buffer and some indicator are then added (and masking agent if required) and titration is carried out to the colour change from red to blue.REMARKS: Back titration is also a good method for lead. In this case a small excess of EDTA is added, and after neutralization and addition of indicator titration is carried out with ZnSO4 or MgSO4.

Referencia: G. Schwarzenbach, H. Flaschka, Complexometric Titrations. Methuen, London, 1969. p. 299.44.3 Direct titration of lead using xylenol orange

REAGENTS: 0.01 M EDTA; acetate buffer pH 5; 20% urotropine;0.1% xylenol orange.

PROCEDURE: The sample solution which should not contain more than about 30-50 mg Pb per 100 mL is first adjusted to pH 2-3 with NaOH, should this be necessary, before adding 3 mL of acetate buffer or better 10 mL of urotropine buffer. At this stage the pH should be about 5. After adding a few drops of indicator solution titration wit EDTA is carried out to the colour change from reddish-violet to pure yellow.

REMARKS: If the lead content is highthe pH can drop substantially during the titration. On this account it is advisable to check the acidity in the neighbourhood of the end-point and correct it, if necessary, by adding more urotropine. High concentrations of acetate buffer must be avoided, otherwise the end-point becomes long drawn out. The procedure can be modified for a back-titration in which PbSO4 dissolves relatively quickly in EDTA, The excess of EDTA in the amount of complexone solution added can be kept quite small; in this way any large change in pH in the back-titration can be avoided.

Referencia: G. Schwarzenbach, H. Flaschka, Complexometric Titrations. Methuen, London, 1969. p. 299-300.http://www.marcel-benoist.org/http://www.marcel-benoist.org/d/pre/d1963sch.htm

http://aslo.org/lo/toc/vol_27/issue_2/0375.pdfLimnol. Oceanogr., 27(2), 1982, 375-380 1982, by the American Society of Limnology and Oceanography, Inc.Interference in the determination of calcium in seawaterAbstractThe interference of magnesium and strontium with the EGTA complexometric titration of calcium was insignificant at thelevel of accuracy used for the molar ratios found in seawater. Interference effects were found, however, for samples containinghigher magnesium or strontium to calcium ratios. An average value of 0.02103 (SD 0.00001) was obtained for Ca:Cl fornine bottles of IAPSO Standard Seawater (P79, 19.376 Cl).The determination of the calcium concentration of seawater is important to various investigations in chemical oceano-graphy (Krumgalz and Holzer 1980). Studies of calcium distributions on a worldwide basis make a convenientshipboard procedure desirable for rapid analysis of samples, ideally with precision and accuracy at the 0.1% level.That this has not been available explains in part the scarcity of worldwide calcium data.Table 1. Recent reported studies on determination of calcium using EGTA titration.ReferenceMethodConclusionSchmid and Reilley 1957Hg-electrodeNo Mg interference at seawater ratiosRingbom et al. 1958Zn-ZinconPositive Mg interference from Mg:Ca 1:5 and higherSadek et al. 1959Zn-ZinconNo Mg interference at seawater ratiosWanninen 1961Theoretical, Zn-ZinconTitration error if Mg > CaDate and Toei 1963Various chemical visualNo Mg interference at seawater ratios when end-pointsindicatorssharpCulkin and Cox 1966Zn-ZinconMg interference of +0.729% on Ca titre; Sr interferenceof +0.388% on Ca titreTsunogai et al. 1968GHAMg interference of -0.23% on Ca titre; Sr interferenceof +0.77% on Ca titreSchwarzenbach and FlaschkaStability constantsSr interference1969conditional constantsIncreased titration error at seawater ratios-dependenton end-point sensitivityHoribe et al. 1974Cal-RedSr interference of +0.37% on Ca titreJagner 1974Computer simulated curves of Zn-ZinconMg interference at seawater ratiosLebel and Poisson 1976Amalgamated-Ag electrodeNo Mg interference; Sr interference of = 0.9% on Ca titreKrumgalz and Holzer 1980Ca-ion selectrodeNo Mg interferenceKanamori and Ikegami 1980Ca-ion selectrodeNo Mg interference; no Sr interferenceThe complexometric titration, at present, is the best such method, but investigators have differed in the end-pointdetection technique used and in their evaluation of interference by other alkaline earth elements. Studies using differentend-point techniques, some of which also considered magnesium to calcium ratios in seawater, do not agree on theeffect of magnesium on the titration of calcium with EGTA [ethylene glycol bis-(b aminoethyl ether)-N-N-tetraaceticacid]. Table 1 lists many of these studies; the references cited report that magnesium has no effect, causes a positiveinterference, and, in one case, has a negative interference.In most cases where strontium interference was evaluated, a positive interference was found, but the degree ofcorrection (of the calcium titre) varies from about -0.38% in several studies to -0.77% and -0.88% in other investi-gations which claim all or nearly all strontium is cotitrated. With this variability in the literature, we decided to determinea correction factor for use in our visual end-point calcium analysis. We acknowledge the comments by F. Culkin,R. Feely, and J. P. Riley.Our titration is essentially unchanged from that of Tsunogai et al. (1968). The reagents were prepared as they describedexcept that larger volumes of the bulk titrant were prepared and used in a continuously stirred automatic delivery systemmade to facilitate routine analysis. Air pumped into the container to replace the titrant being taken out was water-saturated. The titrant solutions were standardized against calcium carbonate of primary standard quality (JMCPuratronic, 99.9975% purity) rather than zinc, and the EGTA (Eastman Chemicals) was used without furtherpurification. The CaCO3 used in the calcium solutions with a seawater matrix was Fisher Scientific ChelometricStandard (99.96% purity); all other calcium solutions were made with primary standard calcium carbonate. All theother chemicals used (GHA, propanol, amyl-alcohol, NaCl, MgSO4, and SrCl2,) were Fisher Scientific reagent grade.Twenty-five milliliters of a titrant strong enough to complex about 98% of the dissolved calcium were added tosamples of 25 mL (30 mL in the lower salinity determinations and 22 mL in the higher salinity determinations).Table 2. Study of net interference on calcium determination in artificial seawater. Major groupings of calciumconcentration approximate that found at 30, 0,35, and 40 salinity.cCa*cNa:cCacMg:cCa103 cSr:cCatcCa*measuredcCacorrectedD(%)8,8450008,8418,841-0.058,84145.6008,8388,834-0.088,84145.65.28.88,8388,833-0.098,9430008,9468,9460.038,93545.1008,9348,930-0.068,93545.15.18.78,9348,929-0.0710,32100010,32210,3220.0110,30545.60010,31410,3100.0510,30545.65.28.810,31610,3110.0610,42300010,41910,419-0.0410,41545.10010,42610,4220.0710,41545.15.18.710,42610,4210.0611,78200011,78511,7850.0311,78245.60011,78611,782011,78245.65.28.811,78411,775-0.0611,90200011,90611,9060.0311,90245.10011,91011,9060.0311,90245.15.18.711,90611,897-0.04* Calcium concentrations reported in mM, actual concentration based on in vacua mass. All ratios are molar: ratios approximate those of seawater. Correction by subtraction of appropriate blank solution calcium concentration, 4 mM from solutions containing sodium, 5 from thelower two salinitv matrices and the high salinity matrix. D = 100 (corrected - actual)/actual.GHA propanol reagent (4 mL) and the borate buffer (4 mL) were added to this solution. This was stirred rapidly forabout 3 min and the amyl-alcohol (5 mL) added. The solution was then stirred vigorously and titrated with dilute EGTAunder fluorescent lighting via a micrometer piston-buret (Gilmont, 2.5 mL capacity) until a faint pink color remained.At this point the titration became a series of small additions with vigorous stirring followed by periods in which theimmiscible layers separated and the organic layer was checked for remaining red color. This process was continueduntil all the red color was gone. Reagent blanks, analyzed with each batch of samples, had 50 mL of distilled-deionizedwater in place of sample and initial titrant. The blank volume was subtracted from the dilute titrant volume incalculating calcium concentration. Reagent blanks were typically < 1 mM.Table 2 shows the actual and measured concentrations for solutions of varied calcium content and salinity. Table 3shows the amount of calcium measured in solutions containing calcium and each individual alkaline earth in variousratios. Table 4 shows the amount of calcium measured in solutions of the salinity matrix and of the individual salts.The solutions of the high purity CaCO3, were accurate to about 4 mM and those of salinity matrix are probablyaccurate to 6 mM. The corrected values for calcium have had the appropriate amount of calcium subtracted, asindicated in Table 4; calcium impurities were consistent with those listed for the reagents used. Each calciumconcentration listed in the tables is the average of two trials, which typically differed by 2 or 3 mM at most unlessotherwise indicated. This is similar to the precision (1 s) in the data of Tsunogai et al. (1968).Each calcium concentration listed in the tables is the average of two trials, which typically differed by 2 or 3 mM atmost unless otherwise indicated. This is similar to the precision (1 s) in the data of Tsunogai et al. (1968).It is apparent from Table 2 that the presence of normal concentrations of sodium, magnesium, and strontium have nonet effect on our determination of calcium above the approximate level of accuracy of about 0.1% so that no correctionfactor seems necessary. A sufficient amount of titrant must be added to complex at least 98% of dissolved calciumbefore the buffer is added; this apparently reduces the loss of calcium by coprecipitation with Mg(OH)2.Although our results differ from those of Culkin and Cox (1966) and of Tsunogai et al. (1968), they do not disprovethe possible cancellation of positive and negative corrections as indicated in the data of Tsunogai et al. (1968). Thedata of Table 3 show no effect above 0.1%.Table 3. Study of separate alkaline earth elements interferenceSolution composition (mM)cCa measuredcCa correctedD*(%)10,399 Ca10,40210,4020.0310,399 Ca, 54,000 Mg10,396( 5)10,395-0.0410,399 Ca, 54,000 Mg,470,000 Na10,40110,396-0.0310,399 Ca, 108,000 Mg10,340( 10)10,338-0.5910,399 Ca, 108,000 Mg, 470,000 Na10,39610,390-0.0910,399 Ca, 216,000 Mg10,180( 20)10,176-2.110,399 Ca, 216,000 Mg, 470,000 Na10,300( 10)10,292-110,399 Ca10,40210,4020.0310,399 Ca, 91 Sr10,40510,4040.0510,399 Ca, 91 Sr, 470,000 Na10,40010,395-0.0410,399 Ca, 182 Sr10,430( 10)10,4290.2810,399 Ca, 182 Sr, 470,000 Na10,41710,4110.12* D = 100 (corrected - actual)/actual. End-point color change different-less sharp End-point much less sharp--organic layer clear, bulk solution orange-pink. End-point much less sharp-nearly no color extracted into organic layer after bulk titrant addition. End-point color change slightly different-greenish.Interference effects begin to appear at higher magnesium or strontium molar ratios. Tsunogai et al. found the inter-ference of magnesium to be negative and, for strontium, related to the extraction into the organic layer of the calcium-GHA complex. They found a positive interference for strontium at twice the seawater molar ratios. Therefore theinterferences of the individual alkaline earth elements on the calcium titration that we found are consistent in direction,though clearly not in magnitude, with those that were reported by Tsunogai et al. (1968). The presence of sodium(chloride) in the solutions also seems to diminish these interference effects in both cases. Although we have noexplanation for the reduced interference effect when sodium is present, it does suggest the advantage of eitherstandardizing the titrant against a seawater matrix calcium standard or of having some matrix available to evaluateindividual interference effects with a procedure to be used for seawater.Table 4. Study of calcium impurities in reagents used to prepare salt matrix.Solution composition (mM)*cCa measured (mM)403,000 Na, 46,000 Mg, 78 Sr6470,000 Na, 53,000 Mg, 91 Sr4537,000 Na, 60,800 Mg, 104 Sr991 Sr153,000 Mg1470,000 Na4* The three mixed salt solutions approximate 30, 35, and 40 salinity.Other types of interference (e.g. with the indicator: Goldstein and Stark-Mayer 1958; Goldstein 1959) are notapparent in our data and examination of the titrant-cation stability constants is of limited use.Table 5. Ca:Cl values determined for Standard Seawater using EGTA titrations.g Ca: ClBatchCorrectionReference0.02122P33Negative Mg, SrCulkin and Cox 19660.02127UnspecifiedImplicit in titre standardizationJagner 19740.02109P61UnspecifiedShiller and Gieskes 19800.0213P73NoneKrumgalz and Holzer 19800.02122P67NoneKanarnori and Ikegami 19800.02114P78NoneThis work0.02103P79NoneThis work0.02106P86NoneThis workThe magnitude of the stability constants (1011, 108.5, 105.2 for Ca, Sr, and Mg: Schwarzenbach and Flaschka 1969)suggests the possible interference of strontium even at its approximate 1% molar ratio to calcium in seawater. A moreappropriate evaluation of conditional constants (Schwarzenbach and Flaschka 1969) or at least a first approxima-tion of these constants (calculated considering only the effect of the other two cations on each individual at the concen-trations found in 35 salinity seawater) yields values of 106.5, 100.5, and 10-3.8 for Ca, Sr, and Mg. This wouldseem to indicate sufficient separation for titration, although the reduction in the calcium-titrant stability constant resultsin reduced titration accuracy (Schwarzenbach and Flaschka 1969). This is a simple evaluation of the system anddoes not consider all the conditions of the titration (e.g. magnesium precipitation, alcohol extraction). The empiricalapproach, as taken here and by others, would seem most useful at present. In determining the calcium concentrationin seawater samples, we have used IAPSO Standard Seawater as a running standard for intercomparison purposes.For example, P79 (19.376 Cl) was used as the standard for 7 months. The calcium concentration determined forthis seawater was 10,408 3 (2 s) mM (9 determinations), which was equivalent to 10,167 mmol. kg-1 or 0.02103 gCa per Cl. Table 5 shows values reported for Standard Seawater. Our values are generally lower, without negativecorrection, than those reported elsewhere. It should be noted, however, that the varied origin of different batches maycontribute to offsets in addition to analytical errors. Until more intercomparison is done with identical samples, assuggested by Culkin and Cox (1966), little can be said to explain the variation in data. Our study, however, indicatesthat the procedure of Tsunogai et al. (1968) basically meets the analytical requirements mentioned.Eric J. OlsonChen-Tung A. ChenSchool of OceanographyOregon State UniversityCorvallis 9733 1This research was supported by NSF grants OCE78-20449 and 80-18770.ReferencesCulkin, F., and R. A. Cox. 1966. Sodium, potassium, magnesium, calcium, and strontium in seawater. Deep-Sea Res. 13: 789-804.Date, Y., and K. Toei. 1963. The selective titrations of calcium in the presence of magnesium. Bull. Chem. Soc. Japan 36: 518-521.Goldstein, D. 1959. A new indicator for the complexometric determination of calcium. Anal. Chim. Acta 21: 339340., and C. Stark-Mayer. 1958. New specific test for calcium. Anal. Chim. Acta 19: 437-439.Horibe,Y., K. Endo, and H. Tsubota. 1974. Calcium in the South Pacific, and its correlation with carbonate alkalinity. Earth Planet.Sci. Lett. 23: 136-140.Jagner, D. 1974. High precision determination of calcium in the presence of higher concentrations of magnesium by means of acomputerized photometric titration. Application to seawater. Anal. Chim. Acta 68: 83-92.Kanamofu, S., and H. Ikegami. 1980. Computerprocessed potentiometric titration for the determination of calcium and magnesiumin sea water. J. Oceanogr. Soc. Japan 36: 177-184.Krumgalz, d., and r. Holzer. 1980. On the determination of Ca2+ ion concentration in seawater. Limnol. Oceanogr. 25: 367370.Lebel, j., and a. Poisson. 1976. Potentiometric determination of calcium and magnesium in seawater. Mar. Chem. 4: 321332.Ringbom, A., G. Pensar, and E. Wanninen. 1958. A complexometric titration method for determining calcium in the presence ofmagnesium. Anal. Chim. Acta 19: 525-531.Sadek, F. S., R. W. Schmid, and C. N. Reilley. 1959. Visual EGTA titration of calcium in the presence of magnesium. Talanta 2: 38-51.Schmid, R. W., and C. N. Reilley. 1957. New complexon for titration of calcium in the presence of magnesium. Anal. Chem. 29: 264-268.Schwarzenbach, G., and H. Flaschka. 1969. Complexometric titrations, 2nd English ed. Methuen.Shiller, A. M., and J. M. Gieskes. 1980. Processes affecting the oceanic distributions of dissolved calcium and alkalinity.J. Geophys. Res. 85: 2719-2727.Tsunogai, S., M. Nishimura, and S. Nakaya. 1968. Complexometric titration of calcium in the presence of larger amounts ofmagnesium. Talanta 15: 385-390.Wanninen, E. 1961. Selection of complexing agents for complexometric titrations. Talanta 8: 355-363.Submitted: 25 February 1981Accepted: 22 July 1981

&LFacultad de Qumica UNAM&C&"Arial,Negrita"&14QUMICA ANALTICA III&RDr. Alain QUERE T., profesor&LDepartamento de Qumica Analtica&RSemestre 2012/Ihttp://aslo.org/lo/toc/vol_27/issue_2/0375.pdf

EGTAcido bis (aminoetil)glicolter-N,N,N',N'-tetracticoOtras denominaciones:cido 3,6-dioxaoctametilendinitrilotetracticocido etilenglicolbis(2-aminoetilter)-N,N,N',N'-tetracticocido etilenbis(oxietilenonitrilo)-tetracticocido 2,2'-etilendioxibis[etiliminodi(actico)]cido glicoleterdiaminotetracticoPGTE, AEGT, DGlTE, GlAeDTE, ChelDEFrmula bruta: C14H24N2O10Masa molar: 380.35 g mol-1.Es un slido blanco dificilmente soluble en agua pura pero muy soluble en disoluciones alcalinasLa solubilidad del cido en agua es de aproximadamente 500 mg/L (0.0013 M)En 1957, C. N. Reilley y R. W. Schmid propusieron el empleo del EGTA para la determinacin selectiva del calcio(II) enpresencia de magnesio. El punto final de la valoracin puede determinarse mediante indicadores metalocrmicos comola calmagita o por tcnicas potenciomtricas con electrodo de plata o con electrodos selectivos.La determinacin directa del calcio en presencia de magnesio no puede realizarse con el EDTA como reactivo titulantedebido a que las estabilidades de los dos complejos CaY2- (logK = 10.7) y MgY2- (logK = 8.7) son demasiado cercanas.Con el EGTA, las estabilidades son suficientemente diferentes logK = 11 para el calcio y logK = 5.2 para el magnesio.El calcio en muestras biolgicas puede determinarse en presencia de un exceso 100 veces mayor de magnesio, cincy fosfato mediante enmascaramiento con tartrato.En la hoja "Anexo1" se reproduce un artculo cientfico que trata de las interferencias en la determinacin de calcio enagua de mar. El documento puede imprimirse.

&LFacultad de Qumica UNAM&C&"Arial,Negrita"&14QUMICA ANALTICA III&RDr. Alain QUERE T., profesor&LDepartamento de Qumica Analtica&RSemestre 2012/IIndustriestrasse 25 CH-9471 Buchs

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03777, 03778 Ethylene glycol-bis(2-aminoethyl)-N,N,N',N'-tetraacetic acid (Ethylenebis(oxyethylenenitrilo)tetraacetic acid, 'Glycol ether diamine tetraacetic acid', EGTA, Chel-DE)

CAS number: 67-42-5Product Description:Appearance: powder, whiteMolecular formula: C14H24N2O10Formula weight: 380.4 g/molMelting Point: 241C, with decompositionSolubility: 0.1 M in 1 M NaOH at 20 C, clear, colorlesspKa1: < 2.1pKa2: 2.71pKa3: 8.81pKa4: 9.51Store at Room Temperature03778 BioChemika Ultra03777 BioChemika Ultra for molecular biologyThe Biochemika Ultra grades are suitable for different applications like purification, precipitation, crystallization and other applications which require tight control of elemental content. Trace elemental analyses have been performed on the BioChemika Ultra qualities and the molecular biology quality is additionally tested for absence of nucleases. The Certificate of Analysis provides lot-specific results.

Applications:EGTA is a reagent that is used to chelate Ca2+ in the presence of Mg2+.2 EGTA chelates Ca2+ at a ratio of 1:1.The log (stability constants) for several cations are as follows:1Mg2+ = 5.2Ca2+ = 11.0Mn2+ = 12.1Fe2+ = 11.8Co2+ = 12.3Ni2+ = 11.8Cu2+ = 17.7Zn2+ = 12.9A protocol for the determination of free calcium in calcium-EGTA solutions has been reported.3 A procedure for making a calibration standard for calcium ion concentration, with detection accurate to 10 M in a mixture of EGTA, HEDTA, and NTA has been reported.4EGTA can be used as an anti-coagulant when dissolved at 1 g per 100 ml of blood. EDTA is more commonly used for the same purpose; either agent chelates the calcium ion from blood.EGTA serves as an effective inhibitor of metalloproteinases.5,8It can induce the rapid and extensive release of calcium from energised mitochondria.6EGTA activates Ca2+-stimulated ATPase of rat liver plasma membranes.7It is used as a buffer component in dideoxy sequencing of RNA using reverse transcriptase.9

Precautions and DisclaimerFor Laboratory Use Only. Not for drug, household or other uses.Preparation InstructionsThis product is soluble in 1 M NaOH (38 mg/ml, or 0.1 M), yielding a clear, colorless solution. A saturated solution at room temperature was found to be 2 mM in EGTA and had a pH of 2.72. This product has the following maximal solubilities in aqueous media at the respective pH values:pH 8.48 > 0.52 MpH 5.4 > 0.48 MpH 4.5 = 0.45 MpH 4.2 = 0.42 MpH 4.0 = 0.31 M

References:1. Data for Biochemical Research, 3rd ed., Dawson, R.M.C., et al., Oxford University Press (New York, NY: 1986), pp. 404-405.2. Schmid, R.W., and Reilley, C.N., New Complexon for Titration of Calcium in the Presence of Magnesium. Anal. Chem., 29, 264 (1957)3. Bers, D.M., A simple method for the accurate determination of free [Ca] in Ca-EGTA solutions. Am. J. Physiol., 242, C404-408 (1982)4. May, P.M., et al., Calibration of ionized calcium and magnesium with ligand mixtures for intracellular ion-selective electrode measurements. Anal. Chem., 57, 1511-1517 (1985)5. Brauner, P., Fridlender B., Use of chelating agents as terminators of alkaline phosphatase activity in enzyme-linked immunosorbent assay (ELISA) tests. J. Immunol. Meth. 42, 375 (1981)6. Riley, W.W., Pfeiffer, D.R., Rapid and extensive release of Ca2+ from energized mitochondria induced by EGTA. J. Biol. Chem. 261, 28 (1986)7. Birch-Machin, M.A., Dawson, A.P., Effects of chelating agents on the Ca2+-stimulated ATPase of rat liver plasma membranes. Biochim. Biophys. Acta 855, 277 (1986)8. Vidal, H., et al., A test to evaluate the effect of individual components of ethylene glycol bis(betaaminoethylether)-N,N,N',N'-tetraacetic acid buffers on enzymatic activity. Anal. Biochem. 193, 135 (1991)9. C.S. Hahn et al., Meth. Enzymol. 180, 123 (1989)10. Merck13, 356011. Beil.4,IV,2433

No. 235/2 eApplication BulletinOf interest to:Food analysis, Special milk productsA 1, 7Potentiometric titration of Ca (Mg) in milk productsSummaryA simple method to determine the calcium content in milk products is described.Use of Cu-EGTA and the ion selective Cu electrode (Cu-ISE) allows the determinationto be performed without time-consuming sample preparation. If EDTA isemployed as a complexing agent in place of EGTA, the sum of the Ca and Mg contentscan be determined. The amount of magnesium can then be calculated as thedifference between the two titrations.TheoryIf the Cu-ISE is used to indicate the complexometric determination of calcium, additionof Cu-ions is necessary. As these Cu ions would also react with the complexingagent (titrant), they are added to the sample in the form of a Cu complex. Thecomplex reacts with Ca-ions as follows:CuEGTA + Ca2+ CaEGTA + Cu2+Apparatus and accessories Titrino 702 or 716 or 736 or 751 or 785 or Titroprocessor 726 with Dosino 700or Dosimat 685 Magnetic Swing-out stirrer 2.728.0040 Exchange unit(s) 6.3014.XX3 Cu-ISE 6.0502.140 and electrode cable 6.2104.020 Ag/AgCl - reference electrode 6.0726.107 (sat. KNO3 external electrolyte) withelectrode cable 6.2106.020Reagents Titrant: c(EGTA) = 0.1 mol/LDissolve 38.04 g ethylene glycol-0,0-bis-(2-aminoethyl)-N,N,N,N-tetraaceticacid in 250 mL c(NaOH) = 1 mol/L and, after cooling, fill up to 1 L with dist. H2O. Buffer solution pH = 10Dissolve 54 g NH4Cl in ca. 400 mL dist. H2O. Add 300 mL w(NH3) = 25% and fillup to 1 L with dist. H2O.Application BulletinNo. 235/2 ePotentiometric titration of Ca (Mg) in milk productsPage 2 Cu complexMix 100 mL c(EGTA) = 0.1 mol/L with 100 mL of a solution containing 0.2 mol/LNH4Cl and exactly 0.1 mol/L Cu(NO3)2. Titration can be used to check that thissolution contains no excess Cu(II) or EGTA. Sulphuric acid: c(H2SO4) = 0.05 mol/L Sodium hydroxide: c(NaOH) = 0.1 mol/LSample preparationMilk, milk drinks, yoghurt, brine bath, etc.Weigh ca. 10 g of sample exactly into a beaker and dilute with ca. 90 mL dist. H2O.CheeseWeigh approx. 1 g finely grated cheese (ca. 1% Ca) or an equivalent amount of adifferent type of cheese into a beaker. Add 10 mL sulphuric acid and ca. 50 mLdist. H2O. Heat at 40 C and stir at this temperature for 10 min. After cooling, neutraliseto pH = 7 with NaOH.AnalysisAdd 1 mL Cu-EGTA and 10 mL buffer solution pH = 10 to the prepared sample solution.Stirring, allow between 10 - 30 s time for reaction, then titrate with EGTA inthe MET-mode of the Titrator.Calculation:1 mL c(EGTA) = 0.1 mol/L = 4.008 mg Ca% Ca = EP1 * C01 * C02 / C00C00 = Sample weight in gC01 = 4.008C02 = 0.1 (for %)Remarks If the magnesium content is also required, a second titration has to be performedwith c(EDTA) = 0.1 mol/L and Cu-EDTA. This determines the sum of Caand Mg. The Mg content can then be determined by calculation:mL EDTA - mL EGTA (use the same sample quantity)!1 mL c(EDTA) = 0.1 mol/L = 2.4305 mg Mg If for cheese, work is done without heating and without addition of H2SO4 (similarto the milk procedure), the Ca can not be completely determined (resultsfound too low).Literature Metrohm Application Bulletin No. 101 Tschager,E. / Jager,H.Milchwirtschaftliche Berichte 87, (1986) 91 - 95

Informe de compatibilidad para CurvTitul_2012_II_Plantilla.xlsEjecutar el 12/03/2012 15:12Las siguientes caractersticas de este libro no son compatibles con versiones anteriores de Excel. Estas caractersticas podran perderse o degradarse si abre el libro con una versin anterior de Excel o si lo guarda con un formato de archivo anterior.Prdida significativa de funcionalidadN de aparicionesVersinUna o ms celdas de este libro contienen reglas de validacin de datos que hacen referencia a valores en otras hojas de clculo. Estas reglas de validacin de datos no se guardarn.1Ecuaciones_pL=f(x)'!C154Excel 97-2003

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