Lecture 2 Grand Canoncal Ensemble GCE 2.1 hermodynamc Functons Contnung on from last day we also note that thus, dω = df dµ µd = Sd P dv dµ (2.1) P = V = S = From the expresson for the entropy, we therefore fnd E µ = Ω + S = Ω ( Ω But recall that,,µ V,µ (2.2) (2.3) (2.4) ) V,µ. whch therefore mples that, ( ) ln Ξ E µ = k 2 ( ) k ln Ξ = µ,v µ,v k ln Ξ Ω = k ln Ξ (2.5) or P V k = ln Ξ (2.6)
LECURE 2. GRAD CAOICAL ESEMBLE GCE 7 2.2 Partcle Fluctuatons 2.2.1 Mean-square devaton n ( ) 2 = 2 2 (2.7) On your assgnment you wll show that, ( ) 2 = k = z z ( ) 2 Ω = k 2 (2.8) and the relatve fluctuatons are then, = ( ) 1/2 k 2 Ω ( 2 ). (2.9) Ω ow the Grand Potental s an extensve quantty (recall that Ω Ω(, V, µ), but the chemcal potental s clearly an ntensve quantty, ( ) ( ) F G µ = = snce F and G are extensve. hus the relatve fluctuatons n partcle number must scale as: 2.2.2 Gbbs-Duhem Equaton,P 1/2 1 1/2 We wll now do an alternatve dervaton of the above result 2.9 Start wth the Gbbs free energy, G = µ = µ More generally, for a multcomponent system, snce G s extensve we have, G(λ 1, λ 2,..., λ m,, P ) = λg( 1, 2,..., m,, P ) (2.10) where λ s an arbtrary scale factor.
LECURE 2. GRAD CAOICAL ESEMBLE GCE 8 Dfferentate 2.10 w.r.t. λ to fnd, L.H.S.: α λ α,p, j (α = λ ) = = = λ,p, j (forλ = 1),P, j µ R.H.S.: λg λ = G( 1,..., m,, P ) herefore, most generally for a multcomponent system G ( 1, 2,..., m,, P ) = ow take dfferental of the above expresson µ (2.11) dg = [µ d + dµ ] (2.12) But f G = G ( 1,..., m,, P ) then, dg = d + dp P, 1,..., m = Sd + V dp + + P, 1,..., m d,p µ d (2.13) Comparng 2.12 and 2.13 gves us the famous Gbbs-Duhem equaton: dµ = Sd + V dp (2.14) For a sngle component system, ths becomes: dµ = Sd + V dp (2.15)
LECURE 2. GRAD CAOICAL ESEMBLE GCE 9 At constant temperature, ths reduces to dµ = V dp dµ = V dp = vdp where v s the volume per molecule ( ) ( ) P = v = v ( P = v2 Invertng the above expresson gves us, = v 2 ) where κ s the Isothermal Compressblty. ( P = P (v, )) ( ) 2 = 2 2 = k κ v v κ (2.16) (2.17).B. ( ) 2 = 2 2 > 0, κ t > 0 Once agan, from 2.17 t s clear that the relatve fluctuatons n partcle number scale as, ( ) 1/2 k = κ 1! (2.18) v For a macroscopc system, where 10 11, the relatve fluctuatons are therefore extremely small. Pcture: P() <> where the probablty of fndng the system wth partcles s P () = z Q Ξ
LECURE 2. GRAD CAOICAL ESEMBLE GCE 10 Small fluctuatons means that P () should be sharply peaked around herefore, wth Ξ = =0 z Q, the sum wll be domnated by terms wth whch mples that, Ξ z Q whch agrees wth our earler result 2.6 k ln Ξ k ln z k ln Q = µ + F (, V, ) = P V hs demonstrates consstency between the Grand Canoncal and and Canoncal ensembles. 2.3 Comments about κ : Partcle fluctuatons here are exceptons when the scalng arguments 2.18 breakdown because the sothermal compressblty dverges and one obtans large partcle fluctuatons. hs can happen at a phase transton or when system s n a two-phase state (where system s a mxture of two phases wth dfferent v s, but exstng at the same pressure). For such cases, = 0