• These were composite variables – Introduced for their usefulness – Not fundamental • Thermodynamics can be expressed through U,T and S • h is an example of a variable that is introduced for its utility • There are two other composite variables that are useful – The Helmholtz function f=u-Ts, (F=U-TS=mf) – The Gibbs function g=h-Ts, (G=H-TS=mg)
Slide 1 Chemical equilibrium Chemical equilibrium • In previous sections, we introduced the non-flow and steady flow availabilities • These were composite variables – Introduced for their usefulness – Not fundamental • Thermodynamics can be expressed through U,T and S • h is an example of a variable that is introduced for its utility • There are two other composite variables that are useful – The Helmholtz function f=u-Ts, (F=U-TS=mf) – The Gibbs function g=h-Ts, (G=H-TS=mg) Helmholtz function (free energy) Defined as TSUF −= (Note how similar it looks to availability A=U+P0V-T0S) 11222121 11221212 STSTWQ STSTUUFF +−−= +−−=− −− Consider the change of Helmholtz function: Availability breaks this into W1-2=(WU)1-2+p0(V2-V1) For an isothermal process (remembering Q1-2/T< s2-s1)="" )(="" )(="" 2121="" 2121="" reversibleffw="" leirreversibffw="" −="−" −="" −="" ‘free’="" energy="" refers="" to="" the="" amount="" of="" energy="" ‘free’="" to="" do="" work="" gibbs="" free="" energy="" defined="" as="" tshg="" −="Note" how="" similar="" it="" looks="" to="" availability="" b="H-T0S" (="" )="" 11222121="" 11221212="" ststwq="" ststhhgg="" s="" +−−="+−−=−" −−="" consider="" the="" change="" of="" gibbs="" free="" energy:="" for="" an="" isothermal="" process="" (remembering="">< s2-s1)="" (="" )="" (="" )="" )(="" )(="" 2121="" 2121="" reversibleggw="" leirreversibggw="" s="" s="" −="−" −="" −="" performs="" same="" task="" as="" f,="" but="" for="" open="" systems="" (be="" careful!)="" why="" bother?="" consider="" the="" non-flow="" ‘availability-like’="" properties="" dstdvpduda="" stvpua="" 00="" 00="" −+="−+=" sdttdsdudf="" tsuf="" −−="−=" suppose="" our="" system="" evolves="" isothermally="" at="" temperature="" t="T0" t="T0=const." dstdudf="" 0−="This" is="" what="" the="" system="" actually="" does="" dstdvpduda="" 00="" −+="This" is="" how="" much="" of="" it="" is="" transmitted="" to="" the="" surroundings="" (i.e.="" useful)="" dvpdfda="" 0+="Why" bother?="" -="" combustion="" •="" chemical="" equilibrium="" –="" f="" &="" g="" provide="" the="" ‘system’s="" view’="" of="" availability="" –="" equilibrium="" defined="" as="" df="0" (closed="" system),="" dg="0" (open="" system)="" •="" why?="" –="" because="" if="" they="" were="" non-zero,="" then="" spontaneous="" work="" transfers="" would="" arise="" rate="" kinetics="" controlled="" equilibrium="" controlled="" examples="" •="" in="" an="" isothermal="" expansion="" of="" a="" gas="" from="" v1="" to="" v2,="" what="" is="" the="" change="" in="" the="" specific="" helmholtz="" free="" energy="" f?="" •="" calculate="" the="" specific="" gibbs="" free="" energy="" for="" saturated="" liquid="" and="" saturated="" steam="" at="" 5bar="" equilibrium="" of="" phases="" remember="" ‘pure="" substances’="" from="" your="" first="" year="" anywhere="" on="" the="" surface="" is="" an="" equilibrium="" point="" dg="0" a="" point="" off="" the="" surface="" will="" spontaneously="" move=""><0) to="" the="" equilibrium="" surface.="" this="" can="" happen="" very="" quickly="" https://www.youtube.com/watch?v="Fot3m7kyLn4" https://www.youtube.com/watch?v="Fot3m7kyLn4" helmholtz="" function="" and="" non-="" newtonian="" fluid="" mechanics="" stir="" a="" beaker="" of="" (newtonian)="" water="" fluid="" forms="" a="" hollow="" whirlpool="" stir="" a="" beaker="" of="" polyisobutylene="" in="" polybutene="" fluid="" climbs="" up="" the="" spoon="" rheological="" phenomena="" in="" focus="" by="" d.v.="" boger,="" k.="" walters="" treat="" microstructure="" as="" ensemble="" of="" springs="" stress="" arises="" from="" pulling="" springs="" out="" of="" equilibrium="" configuration="" https://www.youtube.com/watch?v="nX6GxoiCneY" phenomena="" discovered="" in="" wwii,="" when="" natural="" rubbers="" dissolved="" in="" gasoline="" to="" produce="" flame="" thrower="" fuel="" https://www.youtube.com/watch?v="nX6GxoiCneY" interlude:="" calculus="" of="" small="" increments="" suppose="" you="" want="" to="" climb="" the="" hill="" by="" moving="" east="" and="" north="" only="" part="" 1="" (="" )="" de="" e="" z="" dz="" n="" ="" ="" ="" ="" ="" ="" ="" ="1" part="" 2="" (="" )="" dn="" n="" z="" dz="" e="" ="" ="" ="" ="" ="" ="" ="" ="2" z="Z(N,E):" total="" change="" of="" height="" (="" )="" (="" )="" dn="" n="" z="" de="" e="" z="" dzdzdz="" en="" ="" ="" ="" ="" ="" ="" ="" ="" +="" ="" ="" ="" ="" ="" ="" ="+=" 21="" interlude:="" calculus="" of="" small="" increments="" (="" )="" de="" e="" z="" dz="" n="" ="" ="" ="" ="" ="" ="" ="" ="1" partial="" derivative="" means="" ‘rate="" of="" change="" of="" z="" with="" respect="" to="" e;="" all="" other="" things="" being="" constant’="" why="" do="" we="" put="" the="" n="" subscript="" on="" the="" partial="" derivative?="" to="" remind="" us="" that="" this="" equation="" only="" holds="" if="" z="Z(E,N)" (as="" opposed="" to="" z="Z(SE,NE)," say)="" example="" (newton’s="" mistake)="" calculate="" the="" speed="" of="" sound="" in="" air="" at="" 300k,="" given="" s="" p="" a="" ="" ="" ="" ="" ="" ="" ="" ="" 2="" try="" using="" p="RT" rta="" p="=" ="" ="" 2="" ="" ),(="" 2="" rtaactually="" ="Went" wrong="" because="" we="" calculated="" t="" p="" ="" ="" ="" ="" ="" ="" ="" ="" ="" solution="" (try="" it="" again="" with="" p-="const." –="" why="" does="" that="" work?)="" changes="" in="" the="" free="" energies="" the="" two="" property="" rule="" (recall="" first="" year)="" means="" we="" can="" write="" u="U(S,V)" (or="" indeed,="" any="" two="" thermodynamic="" properties)="" a="" change="" in="" u="U(S,V)" can="" therefore="" be="" written="" as="" dv="" v="" u="" ds="" s="" u="" du="" sv="" ="" ="" ="" ="" ="" ="" ="" ="" +="" ="" ="" ="" ="" ="" ="" ="Compare" this="" with="" z="Z(N,E)" from="" the="" previous="" slide="" pdvtds="" wqdu="" r="" −="−=" ="" notice="" as="" well,="" from="" your="" earlier="" years="" in="" thermodynamics...="" can="" only="" be="" true="" if="" s="" v="" v="" u="" p="" s="" u="" t="" ="" ="" ="" ="" ="" ="" ="" ="" −="" ="" ="" ="" ="" ="" ="" ="These" are="" examples="" of="" the="" so-called="" maxwell="" relations="" changes="" in="" the="" free="" energies="" the="" two="" property="" rule="" only="" applies="" to="" chemically="" unchanging="" systems="" suppose="" the="" system="" comprises="" i="" species="" mk,="" k="1,2,...,i" generalising="" the="" differential="" to="" include="" multiple="" chemical="" species,="" u="U(S,V,mk)" and="" +−="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ++="" ="" ="" ="" ="" ="" ="" ="" +−="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ++="" ="" ="" ="" ="" ="" ="" ="" +="" ="" ="" ="" ="" ="" ="" ="" +="" ="" ="" ="" ="" ="" ="" ="k" kk="" k="" mvskmvs="" k="" mvskmvsmsmv="" dmpdvtds="" dm="" m="" u="" dm="" m="" u="" pdvtds="" dm="" m="" u="" dm="" m="" u="" dv="" v="" u="" ds="" s="" u="" du="" jj="" jjkk="" ="" ,,="" 1="" ,,1="" ,,="" 1="" ,,1,,="" ...="" ...="" jmvs="" k="" k="" m="" u="" ,,="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" the="" change="" internal="" energy="" in="" a="" mixture="" associated="" with="" species="" k="" alone="" chemical="" potential="" changes="" in="" the="" free="" energies="" we="" are="" not="" just="" limited="" to="" expressing="" the="" chemical="" potential="" in="" terms="" of="" internal="" energy="" suppose="" the="" system="" comprises="" i="" species="" mk,="" k="1,2,...,i" generalising="" the="" differential,="" g="G(p,T,mk)" and="" k="" k="" k="" k="" mtpkmtpmpmt="" dmsdtvdp="" dm="" m="" g="" dm="" m="" g="" dt="" t="" g="" dp="" p="" g="" dg="" jjkk="" +−="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ++="" ="" ="" ="" ="" ="" ="" ="" +="" ="" ="" ="" ="" ="" ="" ="" +="" ="" ="" ="" ="" ="" ="" ="" ,,="" 1="" ,,1,,="" ...="" sdtvdpdg="" −="jmTp" k="" k="" m="" g="" ,,="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" the="" gibbs="" function="" in="" a="" mixture="" associated="" with="" species="" k="" alone="" chemical="" potential="" changes="" in="" the="" free="" energies="" for="" fluids="" whose="" chemical="" composition="" changes,="" it="" follows="" that="" ="" ="" ="" ="" ++−="+−−=" ++="+−=" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" dmvdpsdtdg="" dmpdvsdtdf="" dmvdptdsdh="" dmpdvtdsdu="" which="" variable="" to="" use="" depends="" on="" the="" problem...="" for="" phase="" changes="" and="" chemical="" equilibrium,="" use="" the="" free="" energies="" f="" is="" typically="" used="" for="" closed="" systems="" g="" is="" typically="" used="" for="" open="" systems="" changes="" in="" the="" free="" energies="" (molar)="" for="" fluids="" whose="" chemical="" composition="" changes,="" it="" follows="" that="" ="" ="" ="" ="" ++−="+−−=" ++="+−=" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" ="" dnvdpsdtdg="" dnpdvsdtdf="" dnvdptdsdh="" dnpdvtdsdu="" which="" variable="" to="" use="" depends="" on="" the="" problem...="" for="" phase="" changes="" and="" chemical="" equilibrium,="" use="" the="" free="" energies="" f="" is="" typically="" used="" for="" closed="" systems="" g="" is="" typically="" used="" for="" open="" systems="" using="" the="" free="" energies="" for="" the="" flame="" shown,="" consider="" the="" equilibrium="" reaction="" ohcohco="" 222="" ++="" at="" equilibrium....="" 0="" 222222="+++=" ohohcocohhcoco="" dndndndndg="" ="" 0="++−=" ="" ="" ="" dnvdpsdtdg="" now,="" if="" system="" is="" at="" constant="" temperature="" and="" pressure="" (an="" open="" system),="" dt="0," dp="0" what="" does="" that="" mean="" for="" a="" flame?="" (there="" are="" many,="" many="" such="" reactions="" occurring="" simultaneously)="" using="" the="" free="" energies="" but,="" 1kmol="" of="" co2="" and="" 1kmol="" of="" h2="" is="" consumed="" for="" every="" 1kmol="" of="" co="" and="" 1kmol="" of="" h2o="" produced,="" so="" the="" changes="" in="" the="" numbers="" of="" moles="" is...="" coohcohcoco="" dndndndndndn="−=−=" 222="" ,,="" (="" )="" 0="" 0="" 222="" 222222="++−−" =="+++" coohcohco="" ohohcocohhcoco="" dn="" dgdndndndn="" ="" ="" ohcohco="" 222="" ++="" using="" the="" free="" energies="" the="" equilibrium="" condition="" is="" thus="" given="" by="" what="" we="" need="" is="" an="" expression="" for="" ="" for="" each="" species:="" (experimentally="" determined)="" (p0="" is="" a="" reference="" pressure)="" 0="" 222="++−−" ohcohco="" ="" as="" temperature="" (and="" partial="" pressures)="" change,="" the="" chemical="" potential="" changes="" (="" )="" (="" )="" (="" )0="" 0="" 0,="" ln,="" pptrttp="" ="" ="" +="(p1,T1)" (p2,t2)="" (xco2)2,="" (xh2)2,="" (xco)2,="" (xh20)2="" (xco2)1,="" (xh2)1,="" (xco)1,="" (xh20)1="" the="" relative="" proportions="" of="" co2,="" h2,="" co="" and="" h2o="" change="" with="" the="" respective="" ="" using="" the="" free="" energies="" using="" this="" relation,="" the="" equilibrium="" condition="" gives="" us="" (="" )tk="" trpp="" pp="" ohcohco="" hco="" ohco="" ="" ="" ="" ="" ="" ="" ="" −−+="0" 0,0,0,0,="" 222="" 22="" 2="" exp="" ="" 0ln="" 2="" 22="" 222="" 00="" 00="" 0="" 0,0,0,0,="" ="" ="" ="" ="" ="" ="" ="" −++−−="" ohco="" hco="" ohcohco="" p="" p="" p="" p="" p="" p="" p="" p="" tr="" 0="" 222="++−−" ohcohco="" ="" (="" )="" (="" )="" (="" )0="" 0="" 0,="" ln,="" pptrttp="" ="" ="" +="Look" at="" page="" 21="" of="" steam="" tables;="" ln(k)="" is="" tabulated="" using="" the="" free="" energies="" -="" equilibrium="" •="" example="" –="" calculate="" the="" degree="" of="" dissociation="" ()="" for="" the="" equilibrium="" reaction="" •="" at="" 1="" bar,="" 1200k="" •="" at="" 1="" bar,="" 1800k="" •="" at="" 5="" bar,="" 1200k="" •="" at="" 5="" bar,="" 1800k="" (="" )="" 222="" 2="" 1="" ocococo="" ="" ="" ++−="" using="" the="" free="" energies="" -="" equilibrium="" 222222283="" 21="" 795="" khfcoeodnobhaconohc="" +++++→="" ="" ="" ="" ="" ="" ++="" ="" suppose="" we="" had="" the="" reaction="" (with="" ="" known)="" earlier="" in="" the="" year,="" we="" assumed="" certain="" behaviours="" for="" different="" ranges="" of="" ="" now,="" we="" can="" use="" additional="" insights="" from="" (known)="" equilibria="" to="" help="" with="" this="" problem="" ohcohco="" 222="" ++="" suppose="" principal="" equilibrium="" step="" in="" action="" is="" the="" water-gas="" shift="" reaction,="" and="" that="" ="">1 22222283 21 795 kHfCOdNObHaCONOHC ++++→ ++ Using the free energies - equilibrium Use K(T) to help close our equations; suppose K(T) is known dNfbaO kbHfaC 2 21 795 2:2 5 2: 228:,3: =++= +=+= OHCOHCO 222 ++ 22222283 21 795 kHfCOdNObHaCONOHC ++++→ ++ 5 unknowns in 4 equations ak fb n n nn nn p p xx xx pp pp Tk HCO OHCO HCO OHCO HCO OHCO ==== 2 2 2 2 22 2 22 2 22 2)( The 5th equation Using the free energies • Estimate the adiabatic flame temperature of propane assuming an equivalence ratio of 1.1. Assume that the equilibrium conditions are governed by the water gas shift reaction • Assume that the reactants are initially at 300K, and ( ) 5.3 22 2 222 == ++ HCO COOH nn nn TK OHCOHCO Species hf(kJ/kmol) Cp(kJ/kgK) C3H8 -103900 1.679 O2 0 1.090 N2 0 1