.MCAD 303010000 1 74 44 0 .CMD PLOTFORMAT 0 0 1 1 0 0 1 0 0 1 1 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 3 1 27 0 Cg a23.900000,73.000000,34 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\fs28 \b van der Waals Equation}} } .TXT 4 8 28 0 Cg a50.500000,53.600000,83 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\i \fs24 Zoran Zdravkovski, Institute of Chemistry, Skopje, Macedoni}{\i \fs24 a}} } .TXT 3 -7 32 0 Cg a69.600000,72.000000,315 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {For an industrial process, it is necessary to introduce 1400 mols of mesytylene (1,3,5-trihydroxybenzene) at a temperature of 560 K and pressure of 150 kPa in a steel cylinder. Calculate the volume of the cylinder using the van der Waals equation, if the }{\i \b a}{ and }{\i \b b}{ constants are known. }} } .TXT 10 0 33 0 Cg a57.500000,72.000000,77 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {The van der Waals formula is a third degree equation for the volume: }} } .EQN 3 18 38 0 { 5 25 896 0 8 56 252}{66} !1!*!!!$>{#e!!)1$W!!!!!!!q!!!!9S!!!!!!+u!!!#!1#e!q!!!!#gLVLV!!+u!!!O!26k!q!!!$#g!1!!!!.[!!!,!A!!!!!&!!!!#%&k!o!3#E!!!#1G#U!3!0LVLVIq!!6k!!#%LQG,<8(\!Q>{!!!G"#Iq#%!n1X-!/D:c23!!Aa!)!!!!KU&K!"!!!!!!!!!!!a!!!#DH!!!&!!!!$G'*!B)4!q!!!")3+E'+"$.[!!#`!I#c!!!! !!!!9Q#e!!!!!!&K#j-"3[/p&OH45;!s3{EJ3e!$!!+u!!!N!1#e"S!!!$&U<7&{!!#e!!!I!(G-"c!!!$&U<7'I"#&K!!!J'tG-6K!*!!!!)Q=y!C}6zG-!!!!!!"S!1!!!!&K!A!1/4:`1gK!!!!%!!!!(_#f!!+u!!#U !1&K"S!!!$&U<7(t!1#e!!!L!.)1"S!!!$&U=I$=!a#e!!#"C#e!!!N!.)1"S!!!$&U<7(^"c#e!!#}&ZIq!!!!!!"S!1!!!!!!!A!1/C:`2:)Q.A/t&P'\3[$u!!!%!a!!!#DH!A!%!!!!Iq#f!!9Q !!!S"r>{}$G!*!!!!!!#!!A!!!!!"!A&m/4:f4m/j!!+y!!!!(_#f!!+u!!#U!1&K!Q!!!!!!"34."34."34."34."34."34."34."34."34."34."34W  .TXT 9 -15 41 0 Cg a18.800000,69.000000,24 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain or in an expanded form:} } .EQN 2 3 40 0 { 3 46 1120 0 8 30 464}{66} !1!*!!!$&K&K!!&K$W!!!!!!!q!!!!9S!!!!!!+u!!!#!1#e!q!!!!#gLVLV!!+u!!!O!26k!q!!!$#g!1!!!!.[!!!,!A!!!!!&!!!!#%'-!M!C#E!!!#1G#U!3!0LVLVIq!!6k!!#%LR.Z6m)%!1>{!!!G"#Iq#%!n1X-!/D:c23!!.[!6!!!!Ke'mL7!!!!!!!*!"!1!!!!!#!"!u2yEN4Y!o2:4G/#K"1WG-!Q!%!!!!(_#e!!9Q!!!S "m!"11!"!!!!4'#%!!9Q!!!S"m!"I!#f!!!!/]#O!!9Q!!!S"m!"D'.\!!!!3e#%!!9Q!!!S"m!"5Y4(!!!!1a#%!!9Q!!!S"m!")a9R!!!!4'#%!!9Q!!!S"m!"7=>|!!!!3e#%!!9Q!!!S"m!"+UDH!!!!.{#O!!9Q!!!S"m!""CIr!!!!/=#|!!!!1Q#%!!9Q!!!S"m!"+&DH!!!!1a#%!!9Q!!!S"m!"EJG.!!!!3e#%!"!!!!#`!I#c!!!!!!!!9Q#e!!!#!!&K#j*L3['\3E!!!a!!!#DH!1!%!!!!Iq#e!!9Q!!!S"m!"1!#f!!!!EY"#!!9Q!!!S"m!"%9+v!!!!(_#8!!9Q!!!S"m!"Cu1B!!!!EY"#!!9Q!!!S"m!"2c6l !!!!EY"#!!9Q!!!S"m!"-I<8!!!!(?#8!!9Q!!!S"m!"7-Ab!!!!EY"#!!9Q!!!S"m!"3EG.!!!!EY"#!!9Q!!!S"m!"B$!"!!!!EY"#!!9Q!!!S"m!"8`)2!!!!(?#8!!9Q!!!S"m!"E*.\!!!!EY"#!!9Q!!!S"m!",x6l!!!!EY"#!!9Q!!!S"m!"-*<8!!!!(_#8!!9Q!!!S"m!"9BAb!!!!EY"#!!9Q!!!S"m!"(0G.!!!!EY"#!!9Q !!!S"m!"=+!"!!!!+E#8!".[!!#`!C#d!!!!!!!!9Q#e!!!!!!&K#j-"3[/p&OH45;!s3{EJ3e!$!!+u!!!N!1!!!a!!!0!"!1!*!!!!)Q>o!#1D!1!!!$)14'!*!!!!)Q>o!*#w!1!!!$&K4'!*!!!!)Q>o!')H!1!!!$&K4'!*!!!!)Q>o!,&j!1!!!$)14'!6!!!!Ke'mL7!!!!!!!*!"!!!!!!!#!"!u2yEN4Y!o2:4G/#K"1WG-!!&O !!!!(_#f!!+u!!#U!1!!"S!!!$&U<7&*&[#e!!!Q!-!!"S!!!$&U<7$I!q#e!!!I!(G-"S!!!$&U<7#w#e#e!!!J!(G-"c!!!#1G#U!+!0LVLVIr!!!!!!!1!!!!Ke&_!!9Q!!!!!,Ac!!!!!!#g!C'@5`*G29DG!a+u!!!N!1!!!a!!!0!"!1!$!!!!!!#dLVLVLVLVLVLVLVLVLVLVLVLVLVLVLVLVLVLVLVLVLJ!Q  .TXT 10 -6 34 0 Cg a71.100000,72.300000,260 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {Higher degree equations are handled by Mathcad with the }{\b root}{ function. The syntax is: }{\i \b root(f(x),x)}{. \par It returns the value of }{\i \b x}{ that makes the function}{\i \b f}{ equal to zero. }} } .TXT 10 0 17 0 Cg a30.000000,71.000000,41 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {defining the constants and parameters:}} } .EQN 4 3 3 0 {0:R}NAME:8.3136 .EQN 0 13 1 0 {0:a}NAME:3.56 .EQN 0 10 2 0 {0:b}NAME:1.979*(10)^(-4) .EQN 0 15 5 0 {0:p}NAME:150000 .EQN 0 13 6 0 {0:n}NAME:1400 .EQN 0 10 4 0 {0:T}NAME:560 .TXT 4 -65 35 0 Cg a72.800000,72.600000,433 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {Since an iterative algorithm is used for the calculation, an initial guess value is necessary. }{It must be borne in mind that for a third degree equation there are three solutions and which root is found depends on the initialization value. Of the three roots, only one will have physical meaning. In other words a negative or a complex value for the volume, although mathematically correct, has no physical meaning. }} } .TXT 14 0 15 0 Cg a9.100000,72.000000,14 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {guess value}} } .EQN 0 14 8 0 {0:V}NAME:10000 .TXT 0 16 23 0 Cg a39.900000,40.000000,113 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {a good guess value always speeds the calculation, although in this case, even a bad estimate converges fast.}} } .EQN 13 -28 9 0 {0:root}NAME(({0:V}NAME)^(3)-({0:n}NAME*{0:b}NAME+{0:n}NAME*{0:R}NAME*({0:T}NAME)/({0:p}NAME))*({0:V}NAME)^(2)+{0:a}NAME*({0:n}NAME)^(2)*({0:V}NAME)/({0:p}NAME)-{0:a}NAME*{0:b}NAME*(({0:n}NAME)^(3))/({0:p}NAME),{0:V}NAME)={0}?_n_u_l_l_ .TXT 8 -2 18 0 Cg a43.600000,67.000000,59 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {or the equation can be written in the recognizable form:}} } .EQN 6 3 13 0 {0:V}NAME:{0:root}NAME(({0:p}NAME+({0:a}NAME*({0:n}NAME)^(2))/(({0:V}NAME)^(2)))*({0:V}NAME-{0:n}NAME*{0:b}NAME)-{0:n}NAME*{0:R}NAME*{0:T}NAME,{0:V}NAME) .TXT 6 36 19 0 Cg a18.300000,35.000000,26 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {the answer is the same:}} } .EQN 0 20 20 0 {0:V}NAME={0}?_n_u_l_l_ .TXT 7 -58 25 0 Cg a65.400000,69.000000,93 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {However, a guess value in the "wrong direction" gets the }{\b \i not converging}{ message:}} } .TXT 3 0 16 0 Cg a9.100000,72.000000,14 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {guess value}} } .EQN 0 15 10 0 {0:V}NAME:.1 .EQN 6 -12 11 0 {0:root}NAME(({0:V}NAME)^(3)-({0:n}NAME*{0:b}NAME+{0:n}NAME*{0:R}NAME*({0:T}NAME)/({0:p}NAME))*({0:V}NAME)^(2)+{0:a}NAME*({0:n}NAME)^(2)*({0:V}NAME)/({0:p}NAME)-{0:a}NAME*{0:b}NAME*(({0:n}NAME)^(3))/({0:p}NAME),{0:V}NAME)={0}?_n_u_l_l_ .TXT 13 -3 42 0 Cg a68.800000,71.000000,128 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain Instead of making {\i wild guesses}, the function can be graphed, and the intercepts with the apscissa easily recognized:} } .EQN 8 39 43 0 {0:V}NAME:0;100 .EQN 4 -35 44 0 &&(_n_u_l_l_&_n_u_l_l_)&(({0:V}NAME)^(3)-({0:n}NAME*{0:b}NAME+{0:n}NAME*{0:R}NAME*({0:T}NAME)/({0:p}NAME))*({0:V}NAME)^(2)+{0:a}NAME*({0:n}NAME)^(2)*({0:V}NAME)/({0:p}NAME)-{0:a}NAME*{0:b}NAME*(({0:n}NAME)^(3))/({0:p}NAME)),0@&&(_n_u_l_l_&_n_u_l_l_)& {0:V}NAME 0 1 0 1 10 0 1 0 0 0 1 0 0 1 0 0 0 5 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 25 21 0 3 .TXT 32 -4 30 0 Cg a70.300000,71.000000,337 {\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 \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 }} }