?_jÿÿÿÿ¨¬ZQlpÝ"“ h ¥‚ …‚ÿ‚….ƒ1‚ ‡ €  †&ÿƒ‰Àÿ°ÿ`0 “& MathTypep…ú…-‡/‡Ú‡s ‡¥…ú…-‡C× ‡C¸ …-‡eá‡e·‡e3‡e_„û€þ‚ŽSymbol7…- ‡2 g1…aó ‡2 Ñ“ …aó ‡2 ¦ò…aó ‡2 òD…aó ‡2 ò…aó„û ÿ‚–Times New Romant4…-…ð ‡2 ÇF‡AB¢• ‡2 Ð)…A¢ ‡2 Ðó…A¢ ‡2 ñC…B• ‡2 ñó…B• ‡2 1¨ ‡AB¢• ‡2  ‡AB¢• ‡2 * Ñ …A¢ ‡2 1Ó…A¢ ‡2 RY‡AB¢• ‡2 R-‡AB¢• ‡2 RV…A¢ ‡2 Rx…A¢„û€þ‚’Times New Roman…-…ð ‡2 pU…y¹ ‡2 p…xÀ ‡2 ‘o…y¹ ‡2 ‘…xÀ ‡2 Ê ÷ …xÀ ‡2 Ñÿ…y¹ ‡2 ò‚…y¹ ‡2 ò¤…y¹„û€þ‚ŽSymbol…-…ð ‡2 gù…=Ó ‡2 È«…=Ó ‡2 ¦® …+Ó ‡2 ¦ …-Ó ‡2 Èý…=Ó ‡2 òÿ…-Ó ‡2 òu…+Ó„û€þ‚’Times New Roman…-…ð ‡2 pK…/i ‡2 ‘K…/i ‡2 g8 …=Ù ‡2 p™ …y¹ ‡2 p> ‡(1€À ‡2 p“ …-€ ‡2 pa…xÀ ‡2 p…)€ ‡2 ‘“ …xÀ ‡2 ‘> ‡(1€À ‡2 ‘“ …-€ ‡2 ‘g…y¹ ‡2 ‘…)€‡2 gæ Ÿ at equili`©i`©ÀÀiii‡2 g“brium.À€iÀ)`‡2 È:“Then yéÀ©À`¹‡2 Èø‘ or x`À€`À„û ÿ‚–Times New Romant4…-…ð ‡2 Ðm …A¢ ‡2 Ð;…A¢ ‡2 ñm …A¢ ‡2 ñ;…A¢ ‡2 (€…A¢ ‡2 (Ò…A¢„û€þ‚’Times New Roman…-…ð ‡2 ¹j …1À ‡2 ¦ …1À †& ÿ…û ‚¼"System…-…ðˆµi— & MrEd‚/&;)z4ÿÿ g ÿÿÿÿ|CONTEXTy¤|CTXOMAPÒ—|FONT4—|KWBTREE˜|KWDATAÝ—|KWMAP ˜|SYSTEM´x|TOPICæx|TTLBTREEJœ|bm0|bm1™ |bm2Õ|bm3Í<3lpL EÀ, ‹‚ …‚ÿ‚….ƒ1‚ ‡ à@ †&ÿƒ‰Àÿ·ÿ — “& MathTypeÀ…ú…-‡ý!‡ýü „û€þ‚’Times New Roman…- ‡2 `…1À„û€þ‚ŽSymbol…-…ð ‡2 `ø…+Ó ‡2 iÖ…-Ó„û€þ‚’Times New Roman…-…ð ‡2 i;…C ‡2 i…Té ‡2 iø…Té„û ÿ‚’Times New Roman…-…ð ‡2 ÉO‡PL|ˆ ‡2 ÉÒ…B• ‡2 ÉÂ…F|„û€þ‚’Times New Roman…-…ð ‡2 iw…(€ ‡2 iv …)€„û€þ‚ŽSymbol…-…ð ‡2 Š¡…lÓ †& ÿ…û ‚¼"System…-…ð 3 ‡2 ‘…xÀ ˆ2 Ê ÷ øïlpœ E¤ e‚ …‚ÿ‚….ƒ1‚ ‡ à  †&ÿƒ‰Àÿ·ÿ` — “& MathTypeÀ…ú…-‡ýJ‡ý\ „û€þ‚ŽSymbolP…- ‡2 `<…-Ó ‡2 i …-Ó„û€þ‚–Times New Roman+…-…ð ‡2 id…C ‡2 iR…Té ‡2 i,…Té„û ÿ‚–Times New Roman…-…ð ‡2 Éx‡PV|¢ ‡2 É…F| ‡2 Éö…D¢„û€þ‚–Times New Roman+…-…ð ‡2 iÁ…(€ ‡2 iÖ…)€„û€þ‚ŽSymbol…-…ð ‡2 Šæ…lÓ †& ÿ…û ‚¼"System…-…ð…F|„û€þ‚…û ‚¼Œ"SystemçfÞflp–$é:5~͉ *ƒ…‚  :bC Ì„:b„:b(‹b:ƒ¨ MM  …‰……‰……!!}€ƒ‡ˆ‰Š‹Žª®²µ¹º½¾¿™®È®²Ì²µÐµ¹Ò¹¹Ô¹½Ø½ÀÁÂÉÃÄÃÃÅÃőůÅÀÚÀÀÛÀÀÜÀÝâäçéìíñòõö÷øùûüÿ11111K1‚LL11d1‚11k1LL[L 1ZL1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL51aL1‚L1‚1‚L10LwL51_1L1‚L1‚1‚L10LwL51‚];‚1L1‚L1‚1‚L10LwL51‚];‚1L1‚L1‚1‚L10LDL (L51‚];‚1L1‚L1‚1‚L10LFL‚L‚'L51‚];‚1L1‚L1‚1‚L10LFL‚L‚&L51‚];‚1L1‚L1‚1‚L10LFL‚L‚&L51‚;;;;>;‚1L1‚L1‚1‚L10LFL‚L‚&L51‚;;;;;D;‚1L1‚L1‚1‚L10LFL‚L&L11ƒ11 1‚;;;;;C;‚1L1‚L1‚1‚L10LFL‚L'L1111 1…;;;;;;B;‚1L1‚L1‚1‚L10LFL)L11111…;;;;;;A;‚1L1‚L1‚1‚L10LFL‚L‚(L1ƒ1111…;;;;;;@;‚1L1‚L1‚1‚L10L8LLL‚L‚'L1111…;;;;;;?;‚1L1‚L1‚1‚L10L9LL‚L‚L‚'L1ƒ111 1…;;;;;;>;‚1L1‚L1‚1‚L10L:LL‚L‚L‚'L111 1…;;;;;„;;;>;‚1L1‚L1‚1‚L10L&L…LLL‚L‚'L1111 1‚;;;;;;>;‚1L1‚L1‚1‚L10L&L„LL‚L‚(L11„1111‚;;;;;;?;‚1L1‚L1‚1‚L10L'L‚L )L%111‚; ;;@;‚1L1‚L1‚1‚L10L&L9L%11 1‚];‚1L1‚L1‚1‚L10L;L„L8L%1 1‚];‚1L1‚L1‚1‚L10L:L‚L7L51‚];‚1L1‚L1‚1‚L10L9L‚L7L51‚];‚1L1‚L1‚1‚L10L8L‚LL5L51^‚1L1‚L1‚1‚L10LwL51`L1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10LwL1%1‚L1‚1‚L10L L LLLLL‚LL‚LLL‚L L1%1‚L1‚1‚L10L"L‚L‚L„LLLLLLLLLL„LLL‚L‚L‚L‚L L1%1‚L1‚1‚L10L"L‚L‚L‚LL‚L‚L‚LL‚L‚L‚LL L1%1‚L1‚1‚L1/L"L‚L‚L‚L†LLL‚L‚L†LLL‚L‚ L‚ L1%1‚L1‚1‚L1(L‚LL‚L‚L‚L†LLL‚L‚L†LLL‚L‚L L1%1‚L1‚1‚L1&LL‚LL‚L†LLL‚L‚L†LLL‚L‚L L1%1‚L1‚1‚L1!L‚LL‚LL‚L†LLL‚L‚L†LLL‚L‚LL1%1‚L1‚1‚L1!L LLL‚L†LLL‚L‚L†LLL‚L‚L‚L1%1‚L1‚1‚L1L L L‚L‚LL‚L‚L‚LL‚L‚L‚L‚LL L1%1‚L1‚1‚L1LLL‚L‚L‚L‚L‚L‚L‚L„LLL„LLLL‚ L1%1‚L1‚1‚L1 LLLL‚L‚LLLLL„LLLL‚L L1%1‚L1‚1‚L1 L L!LL‚L‚L‚L‚4L1%1‚L1‚1‚L1 L L%LL‚L‚L‚L‚4L1%1‚L1‚1‚L1 LL'LL‚L‚LL4L1%1‚L1‚1‚L1LL'L L ML1%1‚L1‚1‚L1LL'LcL1%1‚L1‚1‚L1LL'LcL1%1‚L1‚1‚L1 LL'LcL1%1‚L1‚1‚L1 L L'LcL1%1‚L1‚1‚L1'LcL1%1‚L1‚1‚L1'LcL1%1‚L1‚1‚L1'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1LL L LL‚LcL1%1‚L1‚1‚L1L LLcL1%1‚L1‚1‚L1LLL LLcL1%1‚L1‚1‚L1LLL LL LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL51aL1‚L1‚1‚L1L'LcL51_1L1‚L1‚1‚L1L'LcL51‚];‚1L1‚L1‚1‚L1L'LcL51‚];‚1L1‚L1‚1‚L1L'LcL51‚];‚1L1‚L1‚1‚L1L L L LcL,11‚];‚1L1‚L1‚1‚L1LL LcL,11‚];‚1L1‚L1‚1‚L1LL ‚LL LcL,11‚;;;I;‚1L1‚L1‚1‚L1LL L ‚LL LcL,11‚; ;;H;‚1L1‚L1‚1‚L1L'LcL(1‚11‚; ;;H;‚1L1‚L1‚1‚L1L'LcL'11‚1‚; ;;G;‚1L1‚L1‚1‚L1L'LcL&111‚; ;;G;‚1L1‚L1‚1‚L1L'LcL&111‚; ;;G;‚1L1‚L1‚1‚L1L'LcL&111‚; ;;G;‚1L1‚L1‚1‚L1L'LcL&111‚; ;;G;‚1L1‚L1‚1‚L1L'LcL&111‚; ;;G;‚1L1‚L1‚1‚L1L'LcL'11‚1‚; ;;H;‚1L1‚L1‚1‚L1L'LcL(1‚11‚; ;;H;‚1L1‚L1‚1‚L1L LL LcL51‚; ;I;‚1L1‚L1‚1‚L1LLL LLcL51‚];‚1L1‚L1‚1‚L1LLL LL LcL51‚];‚1L1‚L1‚1‚L1LL‚LLL‚LLLcL51‚];‚1L1‚L1‚1‚L1L'LcL51‚];‚1L1‚L1‚1‚L1L'LcL51^‚1L1‚L1‚1‚L1L'LcL51`L1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1LL ƒLL L L L=L1%1‚L1‚1‚L1LL ‚LL L"L‚?L1%1‚L1‚1‚L1LL L ‚LL L"L‚?L1%1‚L1‚1‚L1L‚LL L ‚LL L"L‚?L51aL1‚L1‚1‚L1L'L"L‚;L51_1L1‚L1‚1‚L1L'L"L‚L7LƒL51‚];‚1L1‚L1‚1‚L1L'L"L‚L7LƒL51‚];‚1L1‚L1‚1‚L1L'L"L7LƒL51‚];‚1L1‚L1‚1‚L1L'L"L‚L7LƒL51‚];‚1L1‚L1‚1‚L1L'LLLL‚L LLL‚LLLƒL L„LL51‚];‚1L1‚L1‚1‚L1L'LLL‚L‚L„LL…LLL‚L‚LLL‚LL51‚;;;;@;‚1L1‚L1‚1‚L1L'LLL‚L‚L‚L‚L‚L‚L‚L‚LLƒL51‚;;;;;;?;‚1L1‚L1‚1‚L1L'LL…LLL‚LL‚L‚L‚L‚L‚L‚L‚LƒL11„111‚;;;;;;?;‚1L1‚L1‚1‚L1L'LL„LL‚L‚ L†LLL‚L‚L‚L‚L‚LƒL1111…;;;;;„;;;>;‚1L1‚L1‚1‚L1LL L LL‚LL‚L L…LL‚L‚L‚L‚L‚LƒL1111…;;; ;„;;;>;‚1L1‚L1‚1‚L1L LLLLL‚L‚L‚L‚L‚LƒL1ƒ111…;;;;;„;;;>;‚1L1‚L1‚1‚L1LLL LLL„LL‚L‚L‚L‚L‚L‚L‚LƒL 111…;;;;‚;;;>;‚1L1‚L1‚1‚L1LLL LL LL‚LL‚L‚L‚L‚L‚L‚ L‚LƒL1ƒ111…;;;;‚;;;>;‚1L1‚L1‚1‚L1L'LL‚LL‚L„LL„LLLL LLL1111…;;;;;;>;‚1L1‚L1‚1‚L1L'LL‚LLLL„LLLƒL LƒLL1111‚;;; ;;?;‚1L1‚L1‚1‚L1L'LOL‚L11„111‚;;; ;;?;‚1L1‚L1‚1‚L1L'LOL‚L&11‚;;;@;‚1L1‚L1‚1‚L1L'LOL‚L&11‚];‚1L1‚L1‚1‚L1L'LNLL&11‚];‚1L1‚L1‚1‚L1L'LOL‚L51‚];‚1L1‚L1‚1‚L1L L L LcL51‚];‚1L1‚L1‚1‚L1LL LcL51^‚1L1‚L1‚1‚L1LL ‚LL LcL51`L1‚L1‚1‚L1LL L ‚LL LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'L&L LLLƒLL1%1‚L1‚1‚L1L'L(L‚ LLL‚LLL1%1‚L1‚1‚L1L'L(L‚ L„LLL„LLL‚L‚L1%1‚L1‚1‚L1L'L(L‚ LLƒLLL…LLL‚L1%1‚L1‚1‚L1L'L(L‚ L‚L‚L‚L‚L1%1‚L1‚1‚L1L'L(L‚LL‚L‚L‚L‚L1%1‚L1‚1‚L1LL L LL‚L(L‚LL‚L‚L‚L‚L1%1‚L1‚1‚L1L LL(LL‚LL„LLL‚L1%1‚L1‚1‚L1LLL LL(L‚LLL‚LL‚L‚L‚L1%1‚L1‚1‚L1LLL LL L(L‚LL†LLL†LLLLL1%1‚L1‚1‚L1L'L(L‚ LLLƒLL1%1‚L1‚1‚L1L'L(L‚%L‚L1%1‚L1‚1‚L1L'L(L‚LL‚L1%1‚L1‚1‚L1L'L(L‚L‚L‚L1%1‚L1‚1‚L1L'L&L LL1%1‚L1‚1‚L1L'LOL‚L1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L L L LcL1%1‚L1‚1‚L1LL LcL1%1‚L1‚1‚L1LL ‚LL LcL1%1‚L1‚1‚L1LL L ‚LL LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L LL LcL1%1‚L1‚1‚L1LLL LLcL1%1‚L1‚1‚L1LLL LL LcL1%1‚L1‚1‚L1LL‚LLL‚LLLcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1LL ƒLL L LcL1%1‚L1‚1‚L1LL ‚LL LcL1%1‚L1‚1‚L1LL L ‚LL LcL1%1‚L1‚1‚L1L‚LL L ‚LL LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'L0L )L1%1‚L1‚1‚L1L'L2L‚L'L1%1‚L1‚1‚L1L'L2L‚L‚&L1%1‚L1‚1‚L1L'L2L‚L‚%L1%1‚L1‚1‚L1L'L2L‚L‚%L1%1‚L1‚1‚L1L'L2L‚ L‚$L1%1‚L1‚1‚L1L'L2L‚ L‚$L1%1‚L1‚1‚L1L'L2L‚ L‚$L1%1‚L1‚1‚L1LL L LL‚L2L‚ L‚$L1%1‚L1‚1‚L1L LL%LLL‚ L‚$L1%1‚L1‚1‚L1LLL LL&LL‚L‚L‚%L1%1‚L1‚1‚L1LLL LL L'LL‚L‚L‚%L1%1‚L1‚1‚L1L'L(L…LLL‚L‚&L1%1‚L1‚1‚L1L'L(L„LL‚L'L1%1‚L1‚1‚L1L'L)L‚L )L1%1‚L1‚1‚L1L'L(L8L1%1‚L1‚1‚L1L'L(L„L7L51aL1‚L1‚1‚L1L'L'L‚L6L51_1L1‚L1‚1‚L1L'L&L‚L6L51‚];‚1L1‚L1‚1‚L1L L L L%L‚LL4L51‚];‚1L1‚L1‚1‚L1LL LcL51‚];‚1L1‚L1‚1‚L1LL ‚LL LcL51‚];‚1L1‚L1‚1‚L1LL L ‚LL LcL51‚];‚1L1‚L1‚1‚L1L'LcL51‚;;;;@;‚1L1‚L1‚1‚L1L'LcL51‚;;;;;;?;‚1L1‚L1‚1‚L1L'LcL11ƒ11 1‚;;;;„;;;>;‚1L1‚L1‚1‚L1L'LcL1111‚ 1…;;; ;„;;;>;‚1L1‚L1‚1‚L1L'LcL1111 1…;;;;…;;;;>;‚1L1‚L1‚1‚L1L'LcL1ƒ1111…;;;;;‚;;?;‚1L1‚L1‚1‚L1L'LcL1111…;;;;;;@;‚1L1‚L1‚1‚L1L'LcL1ƒ1111…;;;;;;;?;‚1L1‚L1‚1‚L1L'LcL11111…;;;;;„;;;>;‚1L1‚L1‚1‚L1L LL LcL11111‚;;;;„;;;>;‚1L1‚L1‚1‚L1LLL LLcL11„1111‚;;;;;;?;‚1L1‚L1‚1‚L1LLL LL L L L„LLLL‚LL‚LL‚LLLLLL L$11 1‚; ;;@;‚1L1‚L1‚1‚L1LL‚LLL‚LLL L‚LL‚L‚LŒLLLLLLL‚L‚L„LLŠLLLLLL L$11‚ 1‚];‚1L1‚L1‚1‚L1L'L L‚L‚L‚LL…LLL‚L‚L‚L‚L‚L‚L„LL L$1 1‚];‚1L1‚L1‚1‚L1L'L L‚L‚L‚L†LLL‚L‚L‚L‚L‚L‚LL L51‚];‚1L1‚L1‚1‚L1L'L L‚L‚L‚L„LLL‚L‚L‚L†LLL‚L‚L51‚];‚1L1‚L1‚1‚L1L'L L‚ L‚L‚LL‚L‚L‚L‚L…LL‚L‚L51^‚1L1‚L1‚1‚L1L'L L‚ L‚L‚LL‚L‚L‚L‚LL‚L‚L51`L1‚L1‚1‚L1L'L L‚ L‚L‚L‚L‚L‚L‚L‚L‚L‚L‚L L1%1‚L1‚1‚L1L'L L‚ L‚L‚LL…LLL‚L‚L‚L‚L‚L‚LL‚ L1%1‚L1‚1‚L1L'L L‚ L„LLLL†LLLL‚L‚L‚L‚L‚L†LL L1%1‚L1‚1‚L1L'L L‚L‚L‚LƒLLL‚L‚L‚LLL L1%1‚L1‚1‚L1L'L L‚L‚L‚ L‚L‚ L‚L1%1‚L1‚1‚L1L'L L‚L‚L‚ L‚L‚ L‚L1%1‚L1‚1‚L1L'L L‚LL L‚L‚LL1%1‚L1‚1‚L1L'L L L‚L‚LL$L1%1‚L1‚1‚L1L'LL‚L‚L‚L‚$L1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1L'LcL1%1‚L1‚1‚L1 L'LcL1%1‚L1‚1‚L1 L'LcL1%1‚L1‚1‚L1 L'LHL L1%1‚L1‚1‚L1 L'LJL‚LL1%1‚L1‚1‚L1 L L'LJL‚L‚L1%1‚L1‚1‚L1 L L'LJL‚L‚ L1%1‚L1‚1‚L1 L L'LJL‚L‚ L1%1‚L1‚1‚L1 L L'LJL‚ L‚ L1%1‚L1‚1‚L1 L L'LJL‚ L‚ L1%1‚L1‚1‚L1 L L#LJL‚ L‚ L1%1‚L1‚1‚L1 L LLJL‚ L‚ L1%1‚L1‚1‚L1 L LL LLLL‚LLL LL„LL L‚ L1%1‚L1‚1‚L1 LLL L‚L„LLŒLLLLLLL„LL L‚LL‚L‚ L1%1‚L1‚1‚L1 LL L L‚L‚L‚L‚L‚L‚L L‚L‚L‚L‚ L1%1‚L1‚1‚L1 LLL‚L‚L‚L‚L‚L‚L‚L‚LL‚L‚L1%1‚L1‚1‚L1 LLL‚L†LLL‚L‚L‚L‚L‚L‚L‚LL1%1‚L1‚1‚L1 LLL‚L…LL‚L‚L‚L‚L‚L‚L L1%1‚L1‚1‚L1 LLL‚ LL‚L‚L‚L‚L†LL"L1%1‚L1‚1‚L1 L L L‚L‚L‚L‚L‚L‚L‚L!L1%1‚L1‚1‚L1 L L L‚L‚L‚L‚L‚LL‚ L‚L L1%1‚L1‚1‚L1 L LL†LLL‚L‚LL‚L‚ L‚L‚L1%1‚L1‚1‚L1 L L LƒL‚LLLL‚L L‚L‚L1%1‚L1‚1‚L1 L L1L‚L‚L‚L1%1‚L1‚1‚L1 L L1L‚L‚L‚L1%1‚L1‚1‚L1 L L2LL‚L‚ L1%1‚L1‚1‚L1 L L8L‚L !L1%1‚L1‚1‚L1 L L8L‚=L1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL1%1‚L1‚1‚L1 L LwL51aL1‚L1‚1‚L1 L L'LLLL‚LLL…LLLLL51_1L1‚L1‚1‚L1 L L)L‚LLL‚L‚L‚LƒLLL‚L51‚];‚1L1‚L1‚1‚L1 L L)L‚L‚L„LLL‚L‚L‚L‚LL‚L51‚];‚1L1‚L1‚1‚L1 L L*L‚LLLL‚L‚L‚L‚L…LLL51‚];‚1L1‚L1‚1‚L1 L!L‚*L‚L‚L‚ L‚L‚L‚L‚L„LL51‚];‚1L1‚L1‚1‚L1]L‚L‚L‚ L‚L‚L‚L‚L‚L51‚];‚1L1‚L1‚1‚L1]L†LLL‚ L‚L‚L‚L‚LL51‚;;;I;‚1L1‚L1‚1‚L1]LLL‚L‚L‚L‚L„LL51‚;;;;H;‚1L1‚L1‚1‚L1]L‚LLL‚L‚L‚L‚L‚L‚LL111 1…;;;;;G;‚1L1‚L1‚1‚L1]L‚L‚L†LLL‚L‚L‚L‚L‚LL1111‚ 1…;;;;;G;‚1L1‚L1‚1‚L1]L‚L‚LL„LLLL‚LL‚LLL1111 1‚; ;G;‚1L1‚L1‚1‚L1]L‚L‚L‚L‚,L11111‚;;;G;‚1L1‚L1‚1‚L1]L‚L‚L‚L‚,L11111‚;;‚;H;‚1L1‚L1‚1‚L1]L‚L‚L‚L,L1111‚;;‚;I;‚1L1‚L1‚1‚L1[L LL,L11111…;;;;M;‚1L1‚L1‚1‚L1xLƒL,L11111…;;;;M;‚1L1‚L1‚1‚L1J11111‚;;;M;‚1L1‚L1‚1‚L1J1111 1‚;;G;‚1L1‚L1‚1‚L1J1111‚ 1‚];‚1L1‚L1‚1‚L1J11 1‚];‚1L1‚L1‚1‚L1g1‚];‚1L1‚L1‚1‚L1g1‚];‚1L1‚L1‚1‚L1g1^‚1L1‚L1‚1‚L1g1`L1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L 111„111111„111„11111 11‚111‚11D1‚L1‚1‚L 1111„111„1111111‡11111„111111111‚111111E1‚L1‚1‚L 111111111„111‡11111„111„111ƒ1111111111E1‚L1‚1‚L 111111111„111‡11111„111„111 11„1111111E1‚L1‚1L1‰11111111111ƒ111ƒ111„111111111LBL‚L1‚11‰11111 1„111 11 1„111„111„11111111 11CL1‚1ƒ11ƒ1 1„111 1 1„111„111„111ƒ111111 11K1‚1ƒ11ƒ111„111„1111111‡11111…111111111„1111111K1‚1‚1‚111111111‚111 111‚11‚111L1‚1‚1‚!11 1*11 11U1‚11!11 1*11‚ 11U1‚11!1 1*111U1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚1 L191LL1‚1 ^1‚1111O1L/L1‚1‚L~11111‚M;‚1L.1‚L1‚1‚L11111‚M;‚1L.1‚L1‚1‚L11ƒ1111‡111111111111111111111ˆ11111111‚11„111…11111‚M;‚1L.1‚L1‚1‚L11„111‡11111‡11111ƒ1111ƒ111‡1111111111111111ƒ111ˆ11111„111„1111„1111111‚M;‚1L.1‚L1‚1‚L11„111„111‡11111ƒ11 1ƒ111‡1111111111111111ƒ111‡11111„111„1111111111‚M;‚1L.1‚L1‚1‚L11ƒ11†11111Š1111111ƒ1111„111‡11111“111111111111111‡11111„111„111111111‚M;‚1L.1‚L1‚1‚L111„111‰1111111„11111„111„111’11111111111111†11111„11111111111‚M;‚1L.1‚L1‚1‚L1111‡11111‰111111ƒ1111ƒ11‡11111’11111111111111ˆ11111‹1111111111111„;;ƒ;;;=;‚1L.1‚L1‚1‚L11111‚11„111ƒ1111ƒ1111‚11‰111111111…1111„1111ƒ11„11111…;; ;;<;‚1L.1‚L1‚1‚L11 1 1&1 1111 111ƒ111„111…;; ;<;‚1L.1‚L1‚1‚L11 1 1&1 1111 11111„111‚; ;<;‚1L.1‚L1‚1‚L1 1 1#1 1‡11111 1111ƒ111‚;;;<;‚1L.1‚L1‚1‚L1W1‚;;=;‚1L.1‚L1‚1‚L1W1‚;;@;‚1L.1‚L1‚1‚L1W1…;;;;@;‚1L.1‚L1‚1‚L1W1…;;;;@;‚1L.1‚L1‚1‚L1W1‚;;<;‚1L.1‚L1‚1‚L1W1‚M;‚1L.1‚L1‚1‚L1W1‚M;‚1L.1‚L1‚1‚L1W1‚M;‚1L.1‚L1‚1‚L1W1‚M;‚1L.1‚L1‚1‚L1W1N‚1L.1‚L1‚1‚L1W1PL.1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚LF1qL1qL1‚L1‚1‚LF1o1L1o1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1„;;;;;ƒ;;;;;;9;‚1L1‚;;;;„;;;;;<;‚1L1‚L1‚1‚LF1…;;;„;;;„;;;„;;;;„;;;„;;;8;‚1L1‚;;;Ž;;;;;;;;;„;;;„;;;;;‚1L1‚L1‚1‚LF1…;;;„;;;;;;;;„;;=;‚1L1‚;;;‡;;;;;„;;;;„;;@;‚1L1‚L1‚1‚LF1…;;;ƒ;;ƒ;;;;;ƒ;;;ƒ;;8;‚1L1‚;;;‡;;;;;ƒ;;ƒ;;;ƒ;;;;‚1L1‚L1‚1‚LF1…;;;„;;;„;;;;;;„;;;„;;;8;‚1L1‚;;;‡;;;;;„;;;„;;;„;;;;;‚1L1‚L1‚1‚LF1„;;;;ˆ;;;;;„;;;‹;;;;;;;8;‚1L1‚;;;‡;;;;;„;;;‹;;;;;;;;;‚1L1‚L1‚1‚LF1…;;;;;„;;;;„;;9;‚1L1‚;;;„;;;;;„;;<;‚1L1‚L1‚1‚LF1…;;;d;‚1L1‚; ;Y;‚1L1‚L1‚1‚LF1…;;;d;‚1L1‚; ;Y;‚1L1‚L1‚1‚LF1„;;e;‚1L1ƒ; ;Y;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1‚m;‚1L1‚m;‚1L1‚L1‚1‚LF1n‚1L1n‚1L1‚L1‚1‚LF1pL1pL1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚Lh111d1‚L1‚1‚Lh111d1‚L1‚1‚Lh111d1‚L1‚1‚Lh111d1‚L1‚1‚LJ11111…11111111111 1*11111…11111111111(1‚L1‚1‚LH1‚1‚11111…1111111111111 1(1‚1‚11111…11111111111111'1‚L1‚1‚LH11„111„1111„111„111„111„111„111111)11„111„1111„111„111„111„111„111111&1‚L1‚1‚LG11ƒ11„1111„111„111„111„111111 (11ƒ11„1111„111„111„111„1111111&1‚L1‚1‚LG1 11„1111„111„111„111ƒ11ƒ111111)1 11„1111„111„111„111ƒ11ƒ111111&1‚L1‚1‚LG1 11„1111„111„111„111„111„111111)1 11„1111„111„111„111„111„111111'1‚L1‚1‚LG1 11„1111„111„111„111„111„1111 11*1 11„1111„111„111„111„111„11111(1‚L1‚1‚LG1 111Š111111‚1111‚1111‚11 11*1 111Š111111‚1111‚1111‚1111'1‚L1‚1‚LG1111„11‚1‚111‚111‚111ƒ1+1111„11‚1‚111‚111‚1111&1‚L1‚1‚LH11F1 1ƒ1,11F111&1‚L1‚1‚LH1‚1‚F1 1-1‚1‚F111'1‚L1‚1‚LJ1H1 1/1H11(1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L11W1‚L1‚1‚L)111-1‚L1‚1‚L)111-1‚L1‚1‚L)111-1‚L1‚1‚L)111-1‚L1‚1‚L 11111…111111111111‚111‚11U1‚L1‚1‚L 1‚1‚11111…11111111111111‚111111V1‚L1‚1‚L 11„111„1111„111„111„111„111„111111111111V1‚L1‚1‚L11ƒ11„1111„111„111„111„1111111„1111111V1‚L1‚1L1 11„1111„111„111„111ƒ11ƒ1111111111LSL‚L1‚11 11„1111„111„111„111„111„11111111 11TL1‚1 11„1111„111„111„111„111„11111111 11\1‚1 111Š111111‚1111‚1111‚1111„1111111\1‚1111„11‚1‚111‚111‚11111‚11‚111]1‚11F111 11f1‚1‚1‚F111‚ 11f1‚1H1111f1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1‚11d1ƒ,;2)l!gGÎ5  NEÿÿÿÿ S"ÿÿÿÿ_1v ÿÿÿÿÿÿÿÿÿÿÿÿ_‚ Distillation column design programV3 µ# €f€€‚ÿDistillation Column Design, McCabe-Thiele Method'_Ü$ €€@Ö€‚ÿ†Xµb. *€±€€€‚€‚‚‚‚ÿThis is a distillation column design program to supplement Levinson, PE Readiness™: Chemical, which is available from the MGI Management Institute (http://www.mgi.com).Disclaimer: results from this program do not constitute engineering advice or engineering services. Such services must be delivered personally by a qualified engineer.æ±ÜH5 8€c€@Ö‚H€‚‚‚ƒ‚ƒãš#øÐ‰‚ƒ‚ÿTo use the program, enter:1. The names of the components (usually the more volatile one is component A)2. Their relative volatility (alpha)3. The design conditions::çb‚S t€Ï€P‘€Ö‚H€ƒ€‚€ƒ€‚€ƒ€‚€ƒ€ãm1‰‚€ƒ€‚ÿ· Reflux ratio RD · Required minimum overhead composition, molar fraction of A (xD)· Feed composition, molar fraction of A (xF)· Feed condition q· Required maximum bottom composition, molar fraction of A (xB))H«& €€@Ö‚H€‚ÿ`;‚ % €v€‚H€‚ÿTo draw the equilibrium curve, click the button for it. R-«]% €Z€‚H€‚ÿThe relative volatility must be available.) †& €€@Ö‚H€‚ÿ`;]æ% €v€‚H€‚ÿTo draw the operating lines, click the button for them. P+†6% €V€‚H€‚ÿThe design conditions must be available.)æ_& €€@Ö‚H€‚ÿP+6¯% €V€‚H€‚ÿTo design the column, click the button. ±‹_`& €€‚H€‚ÿThe McCabe-Thiele diagram will appear in the square picture box, with the rectification plates in blue and the stripping plates in red. º”¯& €)€‚H€‚ÿA detailed report of the design (plates, location of the feed plate, liquid and vapor composition on each plate) appears in the table beneath it.)`C& €€@Ö‚H€‚ÿ¶ù& €!€‚H€‚ÿOnce the column has been designed, you can save the entire project as an HTML file that is suitable for printing or display on the Internet. ª„C£ & € €‚H€‚ÿIf you select COLUMN as the project name, the HTML file will be saved as COLUMN.HTM and the McCabe-Thiele diagram as COLUMN.BMP. sNù % €œ€‚H€‚ÿ(Caution: if you use an existing file name, the file will be overwritten.) |W£ ’ % €®€‚H€‚ÿNetscape Navigator won't, apparently, display a bitmap; you need Internet Explorer. pK  % €–€‚H€‚ÿOf course, you also need Internet Explorer to run this ActiveX document.)’ + & €€@Ö‚H€‚ÿW, ‚ + &€X€‚H€ãR“x™‰‚ÿSee an example of the program operationD+ Æ 1AÿÿÿÿÿÿÿÿÿÿÿÿÆ Ã Relative volatility9‚ ÿ # €,€€‚ÿRelative Volatility€ZÆ  & €´€@Ö€‚‚‚ÿRelative volatility applies to ideal solutions. When A and B are the two components, =ÿ ¼ 5 :€€@Ö€ƒ€†"€€‚‚ÿâ à % €Å€€‚‚ÿSubsequent releases of this program will give the user the option of using the Antione equation for vapor pressures. The mixture must obey Raolt's Law; that is, fugacity and activity must not be significant considerations.?¼ 1ÿÿÿÿÿÿÿÿÿÿÿÿäDFeed condition4à 6# €"€€‚ÿFeed ConditionR&ˆ, (€L€@Ö€‚€‚€‚ÿCalculation of feed condition qf6îU#z€"ɾ l €€¨X˜€‚ÿ€€¨X˜‚ÿ€€¨X˜‚ÿÿÿFeedqNotes×TˆÑ@ƒ#Ô€«É¾ l €€¨X˜€‚ÿ0€€¨X˜€†"€€‚ÿ€"€¨X˜ÿ4€$€¨X€‚‚€€‚‚‚‚€‚ÿÿÿSubcooledCPL is the liquid heat capacity. TB is the bubble point, and TF is the feed temperature. l is the heat of vaporization. îÑ@à q>1 means that the feed condenses some vapor. For example, if q=1.5, each entering mole of feed condenses 0.5 moles of vapor, and causes 1.5 moles of liquid to flow to the stripping section.•4îfAa#’€hɾ l €€¨Xÿ€€¨X˜€‚ÿ€^€¨X˜‚ÿ€d€¨X˜‚ÿÿÿSaturated liquid, at the bubble temperature1u Ñ@ÛAU#z€@ɾ l €€¨X˜€‚ÿ€.€¨X˜‚ÿ€<€¨X˜‚ÿÿÿLiquid-vapor mixture0