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Thermodynamics and Statistical Mechanics Camille Bélanger-ChampagneLuis Anchordoqui LehmanMcGill College University City University of New York ThermodynamicsCharged Particle and StatisticalCorrelations Mechanics in Minimum Bias Events at ATLAS • Thermodynamic● Physics motivation potentials Enthalpy • ● Minbias event and track selection Thermodynamics IV Helmholtz free energy th • ● Azimuthal correlation results 18 SeptemberFebruary 2014 26 2012,• Gibbs free energy WNPPC 2012 • Chemical● Forward-Backward potential correlation results • Landau free energy Thursday, September 18, 14 1 THERMODYNAMIC POTENTIALS Thermodynamic systems may do work on their environments Under certain constraintsOverview ☛ work done may be bounded from above by change in appropriately defined thermodynamic potential Imagine creating thermodynamic system from scratch in thermally insulated box of volume V Work to assemble system ☛ = U W After bringing together all particles from infinity system has internal energy U System however ☛ may not be in thermal equilibrium Spontaneous processes will then occur so as to maximize system’s entropy but internal energy remains @ U dU = δQ δW ☛ U(S, V ) First Law − Combining Main thermodynamics identity dU = TdS PdV − @U @U it follows that T = and P = { @S V − @V S ✓ ◆ ✓ ◆ Second Law in form δQ TdS yields ☛ dU TdS PdV − C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 2 FREE WORK For porcess @ fixed ☛ (S, V ) dU 0 Overview) This says spontaneous processes in system with dS = dV =0 always lead to reduction of internal energy In systems @ fixed (S, V ) spontaneous processes drive internal energy U to a minimum Allowing for other work processes ☛ δW TdS dU − Work done by system under thermodynamic conditions of constant entropy is bounded above by dU and maximum δ W is achieved− for reversible processes It is useful to define ☛ δW = δW PdV free − which is differential work done by system other than required to change V It follows that ☛ δW TdS PdV dU free − − CONCLUSION: for systems @ fixed ( S, V ) ☛ δ W dU free C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 3 THE LEGENDRE TRANSFORMATION Consider function Z =Overview Z ( x, y ) and compute differential dZ(x, y)=Xdx + Ydy (A) x, X and y, Y are (by definition) canonically conjugate pairs To replace (x, y) by ( X, Y ) as independent variables transform Z = Z ( x, y ) according to M(X, Y )=Z xX yY − − it follows that ☛ dM = dZ Xdx Ydy xdX ydY (B) − − − − Substituting (A) into (B) ☛ dM = xdX ydY (C) − − (A) and (C) give reciprocity relations: @Z @Z @M @M = X, = Y, = x, = y @x @y @X − @Y − C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 4 THE LEGENDRE TRANSFORMATION II To replace only one Overviewof the variables tansform according to N(x, Y )=Z yY (D) − it follows that dN = dZ Ydy ydY (E) − − Substituting (A) into (E) ☛ dN = Xdx ydY − with reciprocity relations: @N @N = X, = Y @x @y − C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 5 THE LEGENDRE TRANSFORMATION III Curve in a plane can Overviewbe equally represented by: pairs of coordinates (point geometry) envelope of a family of tangent lines (line geometry) Z y C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 6 THE LEGENDRE TRANSFORMATION IV Overview For given Z = Z ( y ) consider tangent line that goes through point (y, Z) and has slope dZ/dy = Y If Z intercept is N ☛ equation of line is given by (D) Z N Y = − N = Z yY y 0 ) − − it follows that ☛ dN = dZ ydY Ydy − − but ☛ dZ = Ydy dN = ydY dN ) − which is reciprocal relation ☛ y = − dY Legendre transformation is mapping from ( y, Z ) space to ( Y,N ) space ☛ point representation of curve into tangent line representation C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 7 ENTHALPY Suppose ☛ we spontaneouslyOverview create system while it is thermally insulated but in constant mechanical contact with “volume bath” @ pressure P = U + PV W quantity U + PV ☛ enthalpy Legendre transformation with respect to pairs of variables P, V H = U + PV (123) so that in equilibrium ☛ dH = dU + PdV + VdP Enthalpy should be considered as a function of S and P that are its native variables ☛ dH = TdS + VdP (124) @H @H It follows that ☛ T = and V = (125) @S @P !P !S @T @V = (126) and Maxwell relation ☛ @P @S !S !P C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 8 ENTHALPY II Overview In general we have ☛ dH TdS + VdP In systems @ fixed (S, P ) spontaneous processes drive enthalpy H to a minimum For general systems dH TdS δW + PdV + VdP − δW TdS + VdP dH free − CONCLUSION: for systems @ fixed ( S, P ) ☛ δ W dH free C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 9 HELMHOLTZ FREE ENERGY Suppose ☛ we spontaneously create system while it is in contact Overviewwith thermal reservoir at temperature T = U TS W − quantity U TS ☛ Helmholtz free energy Legendre transformation− with respect to pairs of variables T, S F = U TS (127) − so that in equilibrium ☛ dF = dU TdS SdT − − F should be considered as a function of T and V that are its native variables ☛ dF = SdT PdV (128) − − @F @F It follows that ☛ S = and P = (129) − @T − @V !V !T @S @P = and Maxwell relation ☛ @V @T (130) !T !V C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 10 HELMHOLTZ FREE ENERGY II Overview In general the Second Law tells us ☛ dF SdT PdV − Equality holds for reversible processes and inequality for spontaneous processes In systems @ fixed (T,V ) spontaneous processes drive Helmholtz free energy F to a minimum We may also write work done by a thermodynamic system under conditions of constant T is bounded above by dF and− maximum δ W is achieved for reversible processes δW SdT PdV dF free − − CONCLUSION: for systems @ fixed ( T,V ) ☛ δ W dF free C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 11 GIBBS FREE ENERGY If we create thermodynamicOverview system at conditions of constant T it absorbs heat energy Q = TS from reservoir and we must expend work energy PV in order to make room for it = U TS + PV Gibbs free energy is obtainedW by a second− Legendre transformation G = U TS + PV (131) − For equilibrium systems dG = SdT + VdP − (132) so that G = G ( T,P ) in native variables @G @G (133) It follows that ☛ S = and V = − @T @P !P !T @S @V and Maxwell relation ☛ = (134) − @P @T !T !P C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 12 GIBBS FREE ENERGY II Second Law says thatOverview dG SdT + VdP In systems @ fixed (T,P) spontaneous processes drive Gibbs free energy G to a minimum For general systems δW SdT + VdP dG free − CONCLUSION: for systems @ fixed ( T,P ) ☛ δ W dG free C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 13 HOMEWORK ON THERMODYNAMIC POTENTIALS For given F ☛ obtain:Overview entropy from (129) @F internal energy U = F + TS = F T (135) − @T !V @S @2F heat capacity CV = T = T @T − @T2 (136) !V !V Show that for perfect gas ☛ combination of (58), (127), (115) yields γ 1 F = C T + U C T ln (TV − ) TS (137) V 0 − V − 0 Check that (135) and (136) yield familiar results for perfect gas From (129) with the use of (50) obtain equation of state @F C T (γ 1) nRT P = = V − = (138) − @V V V !T C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 14 OPEN SYSTEMS Mass of system was Overviewconsidered as constant and dropped from arguments of thermodynamic functions If system can exchange mass with environment or there are chemical reactions in system masses of components change and can cause change of other quantities If mass is added to system with a constant volume pressure typically increases Throughout we will consider number of particles N instead of mass or number of kilomoles n Connection between N and n has been given in (6) Using N is preferable in statistical physics while n is more convenient in chemistry C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 15 CHEMICAL POTENTIAL With account of massOverview changes ☛ internal energy becomes , U = U(S, V, N) and main thermodynamic identity (84) should be modified accordingly dU = TdS PdV + µdN (139) − ☟ @U µ = chemical potential per particle ☛ @N (140) !S,V Chemical potential per kilomole ☛ substitute last term in (139) by µdn For multicomponent system such as mixture of different gases described by numbers of particles N i ☛ µdN µ dN ) i i i X C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 16 EULER’S THEOREM Since all arguments in OverviewU ( S, V, N ) are extensive quantities multiplying them all by a parameter λ means simply increasing whole system by λ that leads to increase of U by λ Mathematically ☛ for any function f this property can be expressed as λf(x, y, z)=f(λx, λy,λz) (141) Differentiate (141) with respect to λ and then set λ =1 to obtain @f @f @f f = x + y + z (142) @x @y @z Applying Euler’s theorem @U @U m @U U = S + V + nj @S @V @nj ✓ ◆V,ni ✓ ◆S,ni j=1 ✓ ◆S,V,ni From the differential of U : X @U @U @U = T, = P, = µj @S @V − @nj ✓ ◆V,ni ✓ ◆S,ni ✓ ◆S,V,ni C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 17 GIBBS-DUHEM RELATION So that one obtains ☛ U = TS PV + µN (143) Overview− Using definition of Gibbs free energy (131) we rewrite this relation in the form ☛ G = µN (144) ☟ Chemical potential is Gibbs free energy per particle For open systems ☛ differentials of thermodynamic potentials become: (124), (128), (132) dH = TdS + VdP + µdN (145) dF = SdT PdV + µdN (146) − − (147) dG = SdT + VdP + µdN − From (144) follows ☛ dG = µdN + Ndµ (148) which combined with (147) yields the Gibbs-Duhem equation SdT VdP + Ndµ =0 (149) − C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14
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