.MCAD 303010000 1 74 16170 0 .CMD PLOTFORMAT 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 NO-TRACE-STRING 0 2 1 0 NO-TRACE-STRING 0 3 2 0 NO-TRACE-STRING 0 4 3 0 NO-TRACE-STRING 0 1 4 0 NO-TRACE-STRING 0 2 5 0 NO-TRACE-STRING 0 3 6 0 NO-TRACE-STRING 0 4 0 0 NO-TRACE-STRING 0 1 1 0 NO-TRACE-STRING 0 2 2 0 NO-TRACE-STRING 0 3 3 0 NO-TRACE-STRING 0 4 4 0 NO-TRACE-STRING 0 1 5 0 NO-TRACE-STRING 0 2 6 0 NO-TRACE-STRING 0 3 0 0 NO-TRACE-STRING 0 4 1 0 NO-TRACE-STRING 0 1 21 15 0 3 .CMD FORMAT rd=d ct=10 im=i et=3 zt=15 pr=3 mass length time charge temperature tr=0 vm=0 .CMD SET ORIGIN 0 .CMD SET TOL 0.001000000000000 .CMD SET PRNCOLWIDTH 8 .CMD SET PRNPRECISION 4 .CMD PRINT_SETUP 1.200000 0.983333 1.200000 1.200000 0 .CMD HEADER_FOOTER 1 1 *empty* *empty* *empty* 0 1 *empty* *empty* *empty* .CMD HEADER_FOOTER_FONT fontID=14 family=Arial points=10 bold=0 italic=0 underline=0 .CMD HEADER_FOOTER_FONT fontID=15 family=Arial points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE_NAME fontID=0 name=Variables .CMD DEFINE_FONTSTYLE_NAME fontID=1 name=Constants .CMD DEFINE_FONTSTYLE_NAME fontID=2 name=Text .CMD DEFINE_FONTSTYLE_NAME fontID=4 name=User^1 .CMD DEFINE_FONTSTYLE_NAME fontID=5 name=User^2 .CMD DEFINE_FONTSTYLE_NAME fontID=6 name=User^3 .CMD DEFINE_FONTSTYLE_NAME fontID=7 name=User^4 .CMD DEFINE_FONTSTYLE_NAME fontID=8 name=User^5 .CMD DEFINE_FONTSTYLE_NAME fontID=9 name=User^6 .CMD DEFINE_FONTSTYLE_NAME fontID=10 name=User^7 .CMD DEFINE_FONTSTYLE fontID=0 family=Times^New^Roman points=12 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=1 family=Times^New^Roman points=12 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=2 family=Times^New^Roman points=12 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=4 family=Arial points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=5 family=Courier^New points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=6 family=System points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=7 family=Script points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=8 family=Roman points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=9 family=Modern points=10 bold=0 italic=0 underline=0 .CMD DEFINE_FONTSTYLE fontID=10 family=Times^New^Roman points=10 bold=0 italic=0 underline=0 .CMD UNITS U=1 .CMD DIMENSIONS_ANALYSIS 0 0 .TXT 2 1 16129 0 Cg a26.600000,73.000000,46 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\b \fs28 Multi-}{\b \fs28 Component Systems}} } .TXT 4 8 16130 0 Cg a50.500000,53.600000,83 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i \fs24 Zoran Zdravkovski, Institute of Chemistry, Skopje, Macedoni}{\i \fs24 a}} } .TXT 4 -8 16131 0 Cg a72.400000,73.000000,1947 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain { In analytical chemistry, the concentration of a species can be determined from its absorbance at a specific wavelength of electromagnetic radiation. In such cases the Beer-Lambert law is applied:\par \par }{\i A = }{\i \f3 e}{\f2 \i l}{\i c\par }{ \par }{\i A}{ - absorbance at a specific wavelength\par }{\i \f3 e}{ - absorption coefficient characteristic of the species at a given frequency\par }{\i l}{ - thickness of the sample\par }{\i c}{ - concentration\par \par If the molar absorption coefficient is known (and thickness of the sample cell), the concentration can be determined for unknown samples. This relationship can be applied even for mixtures, since the absorbance of radiation by one species is usually unaffected by the presence of other species in the solution (however, if the species interact in any way, this is not true). In other words, the absorbance at a specific wavelength is an additive value: it is the sum of the absorbances of all species in the mixture. This can be expressed as:\par \par }{\i A}{\fs16 \dn8 \i total}{\i = }{\i \f3 e}{\fs16 \dn8 \i0 \f3 1}{\f2 \i l}{\i c}{\i \fs16 \dn8 \i0 \f3 1 }{\fs24 \f3 \i0 + }{\i \f3 e}{\fs16 \dn8 \i0 \f3 2}{\f2 \i l}{\i c}{\i \fs16 \dn8 \i0 \f3 2 }{\fs24 \f3 \i0 +}{ }{\i \f3 e}{\fs16 \dn8 \i0 \f3 3}{\f2 \i l}{\i c}{\i \fs16 \dn8 \i0 \f3 3 }{\fs24 \f3 \i0 +}{ . . . + }{\i \f3 e}{\f2 \i \f2 \fs16 \i0 \dn8 n}{\f2 \i l}{\i c}{\f2 \fs16 \i0 \dn8 n}{\f3 \fs16 \i0 \dn8 }{or }{\i A}{\fs16 \dn8 \i total}{\i = }{\f3 \i S}{\i \f3 e}{\fs16 \dn8 \f2 \i i}{\f2 \i l}{\i c}{\i \fs16 \dn8 \i0 \f3 i}{\par \par Apparently, for an }{\i n}{ component mixture, }{\i n}{ wavelengths must be selected to obtain }{\i n}{ absorbance values. Then a set of }{\i n}{ linear equations is obtained, which in Mathcad can be solved either in a }{\b \i Given/Find}{ block, or by using matrices.}} } .TXT 56 1 16133 0 Cg a71.700000,72.000000,115 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain { For the infrared spectrophotometric analysis of a four-component system, the following data were obtained:}} } .TXT 6 4 5 0 Cg a50.800000,84.200000,115 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain { }{\i p}{-xylene }{\i m}{-xylene }{\i o}{-xylene ethylbenzene }{\i A}{\fs16 \dn8 mixture}} } .TXT 3 0 6 0 Cg a42.600000,82.900000,123 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain { }{\f3 l}{ }{\f3 e}{l }{\f3 e}{l }{\f3 e}{l }{\f3 e}{l }} } .TXT 4 0 7 0 Cg a52.000000,83.000000,84 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {12.5 1.502 0.0514 0 0.0408 0.1013 }} } .TXT 3 0 8 0 Cg a53.500000,85.400000,85 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {13.0 0.0261 1.1516 0 0.0820 0.09943 }} } .TXT 3 0 9 0 Cg a52.000000,83.000000,80 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {13.4 0.0342 0.0355 2.532 0.2933 0.2194 }} } .TXT 3 0 10 0 Cg a53.000000,84.200000,84 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {14.3 0.0340 0.0684 0 0.3470 0.03396 }} } .TXT 9 -5 16135 0 Cg a50.200000,70.000000,96 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\f4 \fs24 1. System of linear equations in a }{\f4 \fs24 \i \b Given/Find}{\f4 \fs24 block:}} } .TXT 4 2 16136 0 Cg a54.000000,71.000000,74 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Initialization of the variables - the concentrations of the components:}} } .EQN 5 1 12 0 {0:cp}NAME:0.1 .TXT 0 17 13 0 Cg a17.900000,23.000000,32 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i p}{-xylene}{ concentration}} } .EQN 3 -17 14 0 {0:cm}NAME:0.1 .TXT 0 17 15 0 Cg a18.300000,23.000000,32 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i m}{-xylene}{ concentration}} } .EQN 3 -17 16 0 {0:co}NAME:0.1 .TXT 0 17 17 0 Cg a17.900000,23.000000,32 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i o}{-xylene}{ concentration}} } .EQN 3 -17 18 0 {0:ce}NAME:0.1 .TXT 0 17 19 0 Cg a21.200000,25.400000,29 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {ethylbenzene concentration}} } .TXT 10 -17 16138 0 Cg a56.200000,70.000000,75 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Definition of the four linear equations with four unknowns in a "block":}} } .EQN 3 0 21 0 {0:Given}NAME .EQN 4 0 22 0 1.502*{0:cp}NAME+0.0514*{0:cm}NAME+0.0408*{0:ce}NAME¸0.1013 .EQN 4 0 23 0 0.0261*{0:cp}NAME+1.1516*{0:cm}NAME+0.0820*{0:ce}NAME¸0.09943 .EQN 4 0 24 0 0.0342*{0:cp}NAME+0.0355*{0:cm}NAME+2.532*{0:co}NAME+0.2933*{0:ce}NAME¸0.2194 .EQN 4 0 25 0 0.0340*{0:cp}NAME+0.0684*{0:cm}NAME+0.3470*{0:ce}NAME¸0.03396 .EQN 9 1 27 0 ({4,1}÷{0:ce}NAME÷{0:co}NAME÷{0:cm}NAME÷{0:cp}NAME):{0:Find}NAME({0:cp}NAME,{0:cm}NAME,{0:co}NAME,{0:ce}NAME) .EQN 2 30 28 0 {0:cp}NAME={0}?_n_u_l_l_ .TXT 3 16 16140 0 Cg a16.000000,21.000000,51 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {computed molar concentration of the components}} } .EQN 1 -16 29 0 {0:cm}NAME={0}?_n_u_l_l_ .EQN 4 0 31 0 {0:co}NAME={0}?_n_u_l_l_ .EQN 4 0 32 0 {0:ce}NAME={0}?_n_u_l_l_ .EQN 10 -28 33 0 {0:xp}NAME:({0:cp}NAME)/({0:cp}NAME+{0:cm}NAME+{0:co}NAME+{0:ce}NAME) .EQN 0 31 34 0 {0:xp}NAME={0}?_n_u_l_l_ .TXT 0 16 16141 0 Cg a18.000000,20.000000,52 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {molar fraction of }{\i p}{-xylene in the mixture}} } .EQN 4 -16 36 0 {0:xp}NAME={0}?{0:%}NAME .TXT 9 -37 16142 0 Cg b73.000000,73.000000,70 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\f4 \fs24 2. System of linear equations using matrices}{\f4 \fs24 :}} } .TXT 4 1 16160 0 Cg a67.800000,72.000000,142 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {The matrix }{\i \b C}{ is formed of the coefficients of the variables. The vector }{\i \b A}{ contains the absorbancis of the mixtures. }} } .EQN 11 5 16145 0 {0:C}NAME:({4,4}÷0.3470÷0.2933÷0.0820÷0.0408÷0÷2.532÷0÷0÷0.0684÷0.0355÷1.1516÷0.0514÷0.0340÷0.0342÷0.0261÷1.5020) .EQN 0 40 16146 0 {0:A}NAME:({4,1}÷0.03396÷0.21940÷0.09943÷0.10130) .TXT 13 -42 16161 0 Cg a4.400000,69.000000,8 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {Since}} } .EQN 0 11 16150 0 {0:C}NAME*{0:c}NAME:{0:A}NAME .TXT 0 12 16162 0 Cg a15.700000,46.000000,31 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i \b c}{ can be defined as:}} } .EQN 0 25 16148 0 {0:c}NAME:({0:C}NAME)^(-1)*{0:A}NAME .EQN 7 -31 16153 0 ({0:c}NAME)[(0)={0}?_n_u_l_l_ .TXT 0 15 16163 0 Cg a17.900000,23.000000,32 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i p}{-xylene}{ concentration}} } .EQN 3 -15 16167 0 ({0:c}NAME)[(1)={0}?_n_u_l_l_ .TXT 0 15 16164 0 Cg a18.300000,23.000000,32 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i m}{-xylene}{ concentration}} } .EQN 1 -32 16152 0 {0:c}NAME={0}?_n_u_l_l_ .EQN 2 17 16168 0 ({0:c}NAME)[(2)={0}?_n_u_l_l_ .TXT 0 15 16165 0 Cg a17.900000,23.000000,32 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\i o}{-xylene}{ concentration}} } .EQN 3 -15 16155 0 ({0:c}NAME)[(3)={0}?_n_u_l_l_ .TXT 0 15 16166 0 Cg a21.200000,25.400000,29 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {ethylbenzene concentration}} } .TXT 14 -35 16170 0 Cg a70.300000,71.000000,334 {\rtf1\ansi \deff2 {\fonttbl {\f0\fnil Arial;} {\f1\fnil Courier;} {\f2\fnil Times New Roman;} {\f3\fnil Symbol;} {\f4\fnil Century Gothic;} } {\plain {\b References}{:\par \par 1. Z. Zdravkovski, Mathcad in Chemistry Calculations, }{\i J. Chem. Ed.}{ }{\b 1991}{, }{\i 68}{, A95 and }{\b 1992}{, }{\i 69}{, A240.\par \par 2. T. R. Dickson, The Computer and Chemistry, W. H. Freeman, San Francisco, 1968 .\par \par 3. P. W. Atkins, Physical Chemistry, W. H. Freeman, New York, 4th Ed., 1990.\par \par }} }