.MCAD 303010000 1 74 16133 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 kg m s K 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 2 0 Cg a14.400000,37.100000,25 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\b \fs28 Heat Capacity}} } .TXT 4 6 16132 0 Cg a50.500000,54.600000,103 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\b0 \fs28 \i \fs24 Zoran Zdravkovski, Institute of Chemistry, Skopje, Macedoni}{\b0 \fs28 \i \fs24 a}} } .TXT 3 -6 3 0 Cg a72.900000,72.700000,302 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { The heat capacity of solids usually is computed by the Dulong-Petit rule:\par \par Cv = 3*R (R - the ideal gas constant). \par \par This expression gives acceptable values only at high temperatures. Debay has given a more general expression based on the vibrational modes of the crystal:}} } .EQN 17 12 4 0 {0:Cv}NAME:9*{0:L}NAME*{0:k}NAME*((1)/(({0:x0}NAME)^(3)))*(0&{0:x0}NAME`(({0:x}NAME)^(4)*({0:e}NAME)^({0:x}NAME))/(((({0:e}NAME)^({0:x}NAME)-1))^(2))&{0:x}NAME) .TXT 10 -12 5 0 Cg a41.000000,46.100000,56 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {L - Avogadro's constant, k - Boltzman's constant, \par }} } .TXT 8 0 7 0 Cg a3.600000,4.700000,8 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {x0 - }} } .EQN 0 4 6 0 ({0:h}NAME*{0:vm}NAME)/({0:k}NAME*{0:T}NAME) .TXT 0 12 8 0 Cg a13.900000,16.400000,21 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {h - Plank constant}} } .TXT 12 -16 10 0 Cg a3.600000,3.800000,7 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {vm -}} } .EQN 0 5 9 0 ({0:\q}NAME*{0:k}NAME)/({0:h}NAME) .TXT 0 11 11 0 Cg a47.100000,47.000000,263 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain {\f1 q}{ - characteristic temperature or Debay temperature (function of the characteristic frequency vm); the values for different metals are as follows:\par \par Na K Cu Ag Au Be Mg Ca Hg\par 159 100 315 215 180 1000 290 230 96}} } .TXT 18 -16 12 0 Cg a25.700000,29.900000,40 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { Initialization of the constants:}} } .EQN 4 0 13 0 {0:N}NAME:6.02217*(10)^(23) .EQN 0 23 14 0 {0:h}NAME:6.6262*(10)^(-34) .EQN 0 23 15 0 {0:k}NAME:1.38062*(10)^(-23) .TXT 8 -46 16 0 Cg a39.700000,46.100000,55 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { The temperature is defined as a range variable:}} } .EQN 4 0 17 0 {0:T}NAME:10,30;400 .EQN 0 21 18 0 {0:\q}NAME:315 .EQN 0 17 19 0 {0:vm}NAME:({0:\q}NAME*{0:k}NAME)/({0:h}NAME) .EQN 0 17 20 0 {0:x0}NAME({0:T}NAME):({0:h}NAME*{0:vm}NAME)/({0:k}NAME*{0:T}NAME) .EQN 11 -54 21 0 {0:Cv}NAME({0:T}NAME):9*{0:N}NAME*{0:k}NAME*((1)/(({0:x0}NAME({0:T}NAME))^(3)))*(0&{0:x0}NAME({0:T}NAME)`(({0:x}NAME)^(4)*({0:e}NAME)^({0:x}NAME))/(((({0:e}NAME)^({0:x}NAME)-1))^(2))&{0:x}NAME) .TXT 9 -1 22 0 Cg a49.400000,56.900000,69 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { The heat capacity is plotted as a function of the temperature:}} } .EQN 3 10 23 0 &&(_n_u_l_l_&_n_u_l_l_)&{0:Cv}NAME({0:T}NAME)@&&(_n_u_l_l_&_n_u_l_l_)&{0:T}NAME 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 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 40 20 0 3 .TXT 28 -10 24 0 Cg a72.100000,71.900000,250 {\rtf1\ansi \deff0 {\fonttbl {\f0\fnil Times New Roman;} {\f1\fnil Symbol;} {\f2\fnil Century Gothic;} {\f3\fnil Arial;} } {\plain { It's obvious that the Dulong-Petit rule is applicable at higher temperatures. \par It can also be shown that for temperatures below 30 K, the heat capacity is proportional to the temperature and the expression }{\i C}{v = a*T can be used.}} } .TXT 10 0 16133 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 }} }