( ) = H ( T 0 ) + C p. ( )dt. ! carnot. Cycles Carnot Engine 1) Isothermal compression at T c

Transcription

1 Equatons hermal hyscs Cycles Carnot Engne Isothermal compresson at c Adabatc compresson 3 Isothermal expanson at H Adabatc expanson carnot C H H Q C C Otto Cycle: Heat Engne Effcency: rev W e engne Q C s always less than. what you get what you pay ump Effcency: rev pump W n rev engne hs s greater than by defnton Refrgerator Effcency: rev Q C Q frdge C what you pay W e Q C what you get hs s usually greater than. Expansons Adabatc: const const const Isothermal Constant temperature: d W nr const Isothermal Compressablty: Energes Energy: E F k B ln Z,N Internal Energy per molecule/partcle: U n k E kn Helmholtz Free Energy: F E S df Sd d + µdn F k B ln Z Gbbs Free Energy: Specfc Gbbs Free Energy when g G m, where m s the mass: G E S + dg Sd + d + µdn Chemcal otental Can be seen as the Gbbs Free Energy per molecule: µ E N F S, N G, N g, Enthalpy: H E + dh du + d + d H dq + d ds + d H 0 + C p 0 d Entropy If not sothermal, consder the start and end states to obtan S S dq ncd ds S d + S d S k B ln S F S E,N S E Sackur-etrode Equaton S Nx B ln N + 3 ln 3 ln Mk B * -, Heat Capactes C s the overall capacty, whle c s the specfc heat capacty, and s per mole or kg c dqrev d c Q H S

2 Equatons hermal hyscs c v Q E S n R hermal dffusvty: D k C p c p c v nr c p c v > monatomc 7 3 datomc Low-temperature specfc heat: c erms from: electron gas, dsturbances n magnetc order, and Debye model Avalablty A E o S + o 0 hs s maxmzed n equlbrum Maxwell Relatons: S S S S S S Gases Clausus-Clapeyron Equaton L s the Latent heat d d L dp d S S L S L S Conducton: 3 n d c v v Densty of State: D k k Effuson: n n after n n before Gbb s hase Law: F C + F of degrees of freedom C of components of phases Heat flow: j k Dffuson equaton: D m m Enrchment Factor t D v Ideal Gas Law: nr Nk B F Z m R N, for deal gas Ideal Gas ressure: 3 mn d v 3 v Isobarc hermal Expansvty: Mean Free ath: n d d Quantum Concentraton s the de Brogle wavelength for a partcle wth thermal energy k B and mass m n Q 3 h Mk B Speed Mean: v mean Speed Most probable: v probable k m R M 8k m 8R M Speed RMS: v rms 3k m 3R M an der Waal s Law: + an nb nr Lenard-Jones otental for an der Waal s solds: a U r 4 r a 6, r * + -. scosty: 3 n d mv Work: W F dx pd hermal volume expanson coeffcent: d d *+ k, - p

3 Equatons hermal hyscs Compressblty: X d dp * a,bar +, B B bulk modulus enson coeffcent: d d Fluds Bernoull s Equaton conservaton of flow along a ppe: p + gh + v p + gh + v oseulle s equaton flow of flud n a ppe: d R 4 a b dt 8 L Stoke s Law lamnar flow: 6rv Solds ypcal equaton of state for a sold: Sold State hyscs Number of states: g d kdk g k dk d D: k Ak 3D: k k k B + m + n ML Sngle artcle artton Functon: e all k states k k B Grand Sngle-state partton functon: G + e µ e N s µ N s 0 N partcle partton functon: Z e k B N N all mcrostates E Grand Canoncal artton Functon Z G e µn s E s G N s,s all sngle partcle states E f gd 0 N f gd 0 Energy: artcles: k M hotons: E pc ck Bosons: f Fermons: f µ k e B µ k e B + Number of partcles fxed: N all states ± e µ Ferm wavenumber etc: N k f 3 f Mv f p f k f f k f f f k B E n + Mean number of excted quanta: n e Debye frequency 3 D 6N D k D D v 3 v Laws of hermodynamcs Zeroth Law: If two systems are separately n equlbrum wth a thrd system, then they must be n thermal equlbrum wth each other. 3

4 Equatons hermal hyscs Frst Law E Q + W de dq + dw de E d + E d Q s heat added to the system. W s work done on the system. Asdes: Joule s Law: dq cd Work Done: dw rev d Stretched strng: dw dl tenson n strng Stretched surface: dw da where surface tenson de ds d + µdn Second Law: It s mpossble to construct an engne whch, operatng n a cycle, wll produce no other effect than the extracton of heat from a reservor and the performance of an equvalent amount of work. Kelvn- lanck It s mpossble to construct a refrgerator whch, operatng n a cycle, wll produce no other effect than the transfer of heat from a cooler body to a hotter one. Clausus dq 0 hrd Law Absolute Zero, 0, s unobtanable. ara-magnets Energy: de ds MdB Work Done dw rev µ o M db M dh s olume M s Magnetc Moment per unt volume Radaton lanck Dstrbuton: E hc hc e k, +. * + -. lanck Dstrbuton Functon: hc I d d hc * 5 k -, e / +,./ Stefan s Law: I 4 dq Ae 4 dt Wen s Law: m.9x0 3 k m Statstcs Macrostates hs s the bulk moton of the system,.e. an overall vew. Calculated by averagng over all mcrostates, e.g. x p X X, for an solated system. Equlbrum s when the macrostate has the maxmum possble number mcrostates. Mcrostates hs s a descrpton of the system at a mcroscopc level, where the poston and momentum or quantum state of each partcle s specfed. otal number n C r. Average: Q Bnomal: r n ; p Qf QdQ f QdQ n C r p r p nr n r n r pr p nr Boltzmann Dstrbuton: EdE Ae E Gaussan Dstrbuton: x x Generally: p x e For gases: 4 f v m k B 3 v e mv k B 4 3 M R v e Mv R Gbbs Dstrbuton k de e µn hs s the same as the Boltzmann dstrbuton, except t ncludes the number of partcles. 4

5 Equatons hermal hyscs Grand otental: G F µn k ln Mean: x x x xdx N x Normalsaton: x xdx xdx artton Functon g s the degeneracy of that energy Z N,dst s the total partton functon for dstngushable partcles, whle Z N,ndst s for ndstngushable partcles. Z g e Z n Q Z N,dst Z N N Z N,ndst Z N Grand artton Functon: e µn osson Dstrbuton: p r; r e r Scale Heght: z + z o e z Standard Devaton: x x Sterlng s Approxmaton: ln N N ln N N emperatures Centgrade System: centgrade lm p 0 trple pont 73.6k 5