The evaluation of P, and T from these formulae indeed requires that the energy E be expressed as a function of the quantities N, V and S.
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1 d dq, dq d d d, d d d d, e evlutio of, d from tese formule ideed requires tt te eerg be epressed s fuctio of te qutities, d. f (,,) is sould, i priciple, be possible oce is kow s fuctio of, d. f (,, ) e rest of te termodmics follows strigtforwrdl. For istce, te Helmolt free eerg A, te Gibbs free eerg G d te etlp H re give b A = - G = - + º º º º º Ÿ H = +
2 (,, ) H (,, ) A (,, ) G (,, ) H = + dh = d + d + d d = d d + d dh = d + d + d d = d d + d dh = d + d + d (,, ) (,, ) A = - G = + da = d - d - d d = d d + d dg = d - d d +d +d d = d d + d da = -d - d + d dg = -d + d + d d = d d + d da = -d - d + d d = d d + d dg = -d + d + d (,, ) (,, ) (,, ) (,, ) A = - G = - + = + G = A = G = º º º º = + G = - + = A + =
3 e specific et t costt volume, v, d te oe t costt pressure, p, would be give b H= + G = - + H=G + = dq/d), d = dq d + d = dq/d), = d/d), d = d d + d = dq/d), = d/d), = dq/d), d = dq d + d d = dq d + d - d() + d + d d + d = dq + d + d H< < e clssicl idel gs d( + ) = dq + d + d dh = dq + d + d H = dq/d), = dh/d), = d/d), = dh/d),
4 ( ( % % o illustrte te pproc developed i te precedig sectios, we sll ow derive te vrious termodmic properties of clssicl idel gs composed of motomic molecules. e mi reso w we coose tis igl specilied sstem for cosidertio is tt it ffords eplicit, toug smptotic, evlutio of te umber (,, ). is emple becomes ll te more istructive we we fid tt its stud ebles us, i most strigtforwrd mer, to idetif te Boltm costt psicl costts; = k A. i terms of oter Moreover, te bevior of tis sstem serves s useful referece wit wic te bevior of oter psicl sstems, especill rel gses (wit or witout qutum effects), c be compred. Ad, ideed, i te limit of ig tempertures d low desities te idel-gs bevior becomes tpicl of most rel sstems. Before udertkig detiled stud of tis cse it ppers wort wile to mke remrk wic pplies to ll clssicl sstems composed of o-iterctig prti- prticles, irrespective of te iterl structure of te prticles. is remrk is relted to te eplicit depedece of te umber (,, ) o d ece to te equtio of stte of tese sstems. ow, if tere do ot eist sptil correltios mog te prticles, i.e. if te probbilit of oe of tem beig foud i prticulr regio of te vilble spce is completel idepedet of te loctio of te oter prticles, te te totl umber of ws i wic te prticles c be sptill distributed i te sstem will be simpl equl to te product of te umbers of ws i wic te idividul prticles c be ccommodted i te sme spce idepedetl of oe oter. (,, ) 0 # 0 0 &'& &'& &'& Wit d fied, ec of tese umbers will be directl proportiol to, te volume of te cotier; ccordigl, te totl umber of ws will be directl proportiol to te t power of : (,, )!. 0 = () = + ", 4
5 5 4 c b 5 4 ; 8 } ~ ~ } ] _^^ \ [[ _^^ o l p kjj mq ˆ Š : 9 wv u ] \ [[ i + 7 /. - c Q B b _^^ œ š J A ž = G ] \ [[ > ž Ž =? _^^ I ] \ [[ M U U ()!)" () = 0 () " Desit Of ttes (DO) () = + ", () = ( + " ) = +", () d 8 () d 6 d d K < ILK m ml ( ) D? ( ) F L / (m) H () d d L m L > () m : v L 0 =4 = = = K m ml WYX 4 ml m 8 6 ; L WYX 4 ml m 8 K m dfe dye d m g g 6 d ƒ () m Œ ml } { L : v rts / m ; L,, kl (,,) kl kl, k k Ak Ÿ k A 5
6 À À ¹ ¾½ ¼» º ¹ Û Ú Õ Ö Á Æ Î Ñ Ò Ú ß Þ Ý æå ä Õ Ä Ã Â Á ³ Õ æå ä ê ³ Ê É È Ç ß Þ Ý ê ² µ ê æå ä «ª Ù Ø For derivig oter termodmic properties of tis sstem, we require detiled kowledge of te w depeds o te prmeters, d. e problem essetill reduces to determiig te totl umber of ws i wic eqs = i i d = i c be mutull stisfied. I oter words, we ve to determie te totl umber of (idepedet) ws of stisfig te equtio ) were te r s re te eergies ssocited wit te vrious degrees of freedom of te prticles. r!,, H ml 8mL 8mL ±,, H,,,, µ 8m 8m 8m r r r r A importt result follows strigtforwrdl from eq., 8m 8m 8m r r À eve before te umber (,, ) is eplicitl evluted. From te ture of te epressio pperig o te rigtd side of tis equtio we coclude tt te volume d te eerg of te sstem eter ito te epressio for i te form of te combitio ( / ). osequetl, r Hece, for te costc of d, wic defies reversible dibtic process, Å Å ÌËÍ'Î, Ç Ï 5 / cost. Ð / 5 / / = cost. Ó Ô tt is, te pressure of sstem of oreltivistic, oiterctig prticles is precisel equl to two-tirds of its eerg desit. It sould be oted ere tt, sice eplicit computtio of te umber s ot et bee doe, results / = cost. d =/ old for qutum s well s clssicl sttistics. H cost.?,, k / è ë, 5 Ö / cost., / éè 5/ / 5 cost. / K, 5/ cost. 5/ K 6
7 ñ ð ï î Ö ñ ì Ö cost. ò í cost. cost. cost. cost. 5 / ð / 7
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