.MCAD 303010000 1 68 16165 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 1.408333 1.200000 1.200000 1 .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 0 16129 0 Cg a53.500000,74.000000,151 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\b \fs28 Balancing Chemical Equations\par \par }{\b0 \fs28 \i \fs24 Zoran Zdravkovski, Institute of Chemistry, Skopje, Macedoni}{\b0 \fs28 \i \fs24 a}} } .TXT 11 1 16164 0 Cg b67.900000,67.900000,629 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { In mathematics, equations involving more than one variable in which the coefficients of the variables are integers and only integral solutions are sought, are known as }{\i Diophantine}{ equations.\par For the chemical reaction \par }{\i a}{Na}{\fs16 \dn8 2}{S}{\fs16 \dn8 4}{O}{\fs16 \dn8 6}{ + }{\i b}{H}{\fs16 \dn8 2}{O}{\fs16 \dn8 2}{ = }{\i c}{Na}{\fs16 \dn8 2}{SO}{\fs16 \dn8 4}{ + }{\i d}{H}{\fs16 \dn8 2}{O + }{\i e}{H}{\fs16 \dn8 2}{SO}{\fs16 \dn8 4}{\par a set of linear equation can be written in order to balance each of the elements involved in the reaction. They are entered in a }{\b \i Given/Find}{ block:}} } .TXT 18 1 16138 0 Cg a25.900000,39.800000,40 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { Initialization of the variables:}} } .EQN 3 1 16139 0 {0:a}NAME:1 .EQN 0 14 16140 0 {0:b}NAME:1 .EQN 0 11 16141 0 {0:c}NAME:1 .EQN 0 12 16142 0 {0:d}NAME:1 .EQN 0 11 16143 0 {0:e}NAME:1 .EQN 3 -49 16144 0 {0:Given}NAME .TXT 4 4 16145 0 Cg a14.300000,25.400000,22 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {t}{o balance sodium}} } .EQN 0 21 16146 0 2*{0:a}NAME÷2*{0:c}NAME .TXT 4 -21 16149 0 Cg a14.100000,25.400000,22 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {t}{o balance oxygen}} } .EQN 0 21 16150 0 6*{0:a}NAME+2*{0:b}NAME÷4*{0:c}NAME+{0:d}NAME+4*{0:e}NAME .TXT 4 -21 16147 0 Cg a13.100000,25.400000,22 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {t}{o balance sulfur}} } .EQN 0 21 16148 0 4*{0:a}NAME÷{0:c}NAME+{0:e}NAME .TXT 4 -21 16151 0 Cg a15.800000,27.800000,24 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {t}{o balance hydrogen}} } .EQN 0 21 16152 0 2*{0:b}NAME÷2*{0:d}NAME+2*{0:e}NAME .TXT 6 -25 16153 0 Cg a67.200000,66.900000,175 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { Certain restraints are added - all coefficients have to be greater than or equal to 1 (this speeds the calculation and usually simplifies finding the real solutions):}} } .EQN 6 2 16154 0 {0:a}NAMẸ1 .EQN 0 12 16155 0 {0:b}NAMẸ1 .EQN 0 10 16156 0 {0:c}NAMẸ1 .EQN 0 9 16157 0 {0:d}NAMẸ1 .EQN 0 11 16158 0 {0:e}NAMẸ1 .EQN 11 -42 16159 0 {0:Find}NAME({0:a}NAME,{0:b}NAME,{0:c}NAME,{0:d}NAME,{0:e}NAME)={0}?_n_u_l_l_ .TXT 5 21 16162 0 Cg a42.300000,73.000000,58 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {The result is given in vector form: a=1; b=7, c=1, etc.}} } .TXT 19 -24 3 0 Cg a27.100000,31.700000,38 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { Balance the chemical equation:}} } .EQN 4 0 4 0 {0:a}NAME*({0:HClO}NAME)[(3)+{0:b}NAME*{0:SnS}NAME+{0:c}NAME*({0:As}NAME)[(2)*({0:S}NAME)[(3)+{0:d}NAME*({0:H}NAME)[(2)*{0:O}NAME÷{0:e}NAME*({0:Sn}NAME)[(3)*((({0:AsO}NAME)[(4)))[(4)+{0:f}NAME*({0:H}NAME)[(2)*({0:SO}NAME)[(4)+{0:g}NAME*{0:HCl}NAME .TXT 6 0 5 0 Cg a26.300000,32.600000,39 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { Initialization of the unknowns:}} } .EQN 4 0 6 0 {0:a}NAME:1 .EQN 0 10 7 0 {0:b}NAME:1 .EQN 0 10 8 0 {0:c}NAME:1 .EQN 0 10 9 0 {0:d}NAME:1 .EQN 0 10 10 0 {0:e}NAME:1 .EQN 0 10 11 0 {0:f}NAME:1 .EQN 0 11 12 0 {0:g}NAME:1 .EQN 6 -61 13 0 {0:Given}NAME .TXT 2 35 14 0 Cg a18.200000,19.100000,26 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {Defining the equations:}} } .TXT 4 -35 15 0 Cg a16.500000,20.900000,22 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {To balance hydrogen}} } .EQN 0 36 16 0 {0:a}NAME+2*{0:d}NAME÷2*{0:f}NAME+{0:g}NAME .TXT 4 -36 17 0 Cg a15.800000,20.900000,22 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {To balance chlorine}} } .EQN 0 36 18 0 {0:a}NAME÷{0:g}NAME .TXT 4 -36 19 0 Cg a14.800000,19.100000,20 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {To balance oxygen}} } .EQN 0 36 20 0 3*{0:a}NAME+{0:d}NAME÷16*{0:e}NAME+4*{0:f}NAME .TXT 4 -36 21 0 Cg a11.300000,16.400000,17 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {To balance tin}} } .EQN 0 36 22 0 {0:b}NAME÷3*{0:e}NAME .TXT 4 -36 23 0 Cg a13.800000,15.500000,20 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {To balance sulfur}} } .EQN 0 36 24 0 {0:b}NAME+3*{0:c}NAME÷{0:f}NAME .TXT 4 -36 25 0 Cg a15.000000,20.000000,21 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {To balance arsenic}} } .EQN 0 36 26 0 2*{0:c}NAME÷4*{0:e}NAME .TXT 3 -34 16163 0 Cg a14.200000,71.000000,21 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {Adding restraints:}} } .EQN 3 2 28 0 {0:a}NAMẸ1 .EQN 0 11 29 0 {0:b}NAMẸ1 .EQN 0 11 30 0 {0:c}NAMẸ1 .EQN 0 10 31 0 {0:d}NAMẸ1 .EQN 0 10 32 0 {0:e}NAMẸ1 .EQN 0 9 33 0 {0:f}NAMẸ1 .EQN 0 9 34 0 {0:g}NAMẸ1 .EQN 4 -63 35 0 {0:y}NAME:{0:Find}NAME({0:a}NAME,{0:b}NAME,{0:c}NAME,{0:d}NAME,{0:e}NAME,{0:f}NAME,{0:g}NAME) .TXT 12 17 37 0 Cg a43.200000,43.500000,113 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { All coefficients need to be multiplied by a constant in order to get integers (in this case obviously by 3):}} } .EQN 6 -18 36 0 {0:y}NAME={0}?_n_u_l_l_ .EQN 8 30 38 0 {0:y}NAME*3={0}?_n_u_l_l_ .TXT 16 -30 16165 0 Cg a69.800000,71.000000,165 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\b References}{:\par \par 1. Zoran Zdravkovski, Mathcad in Chemistry Calculations, }{\i J. Chem. Ed.}{ }{\b 1991}{, }{\i 68}{, A95 and }{\b 1992}{, }{\i 69}{, A240.\par }} }