EXCESS FUNCTIONS MATHEMATICAL EXPLANATION

Transcription

1 EXCESS FUNCTIONS Excess functons... 1 Mathematcal exlanaton... 1 Physcal examle: Molar volume of a ure substance... 2 Chemcal otental for a ure substance... 3 Actvty, actvty coeffcent and fugacty (ure substances)... 3 Excess functons for other varables (ure substances)... 4 Excess functons n a mxture... 5 Ionc solutons... 7 Actvty quotent... 7 Equlbrum constant... 8 EXCESS FUNCTIONS Excess functons are ntroduced n Thermodynamcs to modfy smle models of substance behavour (deal gas model, deal mxture model, deal soluton model), to account for devatons shown n real substances. They were frst ntroduced by the Amercan Glbert Lews n 1907 as a dfference between deal artal ressures and real 'actvtes' or 'fugactes', as he named them. MATHEMATICAL EXPLANATION Some functons of nterest, say y(x), may vary slowly and t s then advantageous to ut t as a smle functon (e.g. a straght lne) lus a devaton. The most common case s the local exanson of the functon as the tangent lne at a gven ont of nterest, x0, lus ts devaton (Fg. 1): y(x) = y(x0) + y'(x0)(x-x0) + η(x) (1) functon = ref.value + local sloe + real dfference In many thermodynamc functons, however, the nterest s not n a local exanson but n an asymtotc exanson (Fg. 1), because the smle models only aroach real behavour n the lmt (e.g. when the ressure s very low, 0, or when the concentraton s very low, x 0). A smlar exanson to (1), y(x)=y(x )+y'(x )(x-x )+η(x) s no longer vald snce y(x ) s ll-defned. Sometmes even y'(x ) s lldefned, but ths may be crcumvented by changng the varables (e.g. from to ln, as for the molar volume versus ressure, v(), that v=0=, dv/d=0=, but dlnv/dln=0= 1). The roblem of the asymtotc-base-ont beng naccessble s solved by choosng an deal-base-ont not belongng to the functon but to the asymtotc model near the range of ractcal nterest,.e. ont y (x0) n Fg. 1, and, nstead of y(x)=y(x )+y'(x )(x-x )+η(x) use: y(x) = y (x0) + y' (x0)(x-x0) + η(x) (2) functon = deal ref.value + deal sloe + real dfference ( excess).e. excess functon real functon - deal functon. Excess functons 1

2 Fg. 1. Illustraton of a local and an asymtotc exanson of a functon. Notce the dfference between y (x0) and y(x0) n the asymtotc exanson. Physcal examle: Molar volume of a ure substance The molar volume of a ure substance, v V/n, related to the densty by v=m/ρ, s a functon of T and for a gven substance. There s no asymtotc behavour wth temerature of nterest n ths case (but the lmt T 0 s mortant n Thrd Law related matters), but there s ndeed an nterest to exand the varaton wth ressure relatve to the deal gas model v=rt/. Introducng Z v/(rt) and changng varables from v() to lnv(ln), the exanson s then lnv=ln(rt)- ln+lnz, or, to be more hyscal, ln(v/v0)=ln(t/t0)-ln(/0)+ln(z/z0), where T0, 0 and v0 are freely chosen to make the functon nondmensonal (although one usually chooses v0=rt0/0) and Z0 s measured. In concluson: ln(v/v0) = ln(t/t0) + ( 1)(ln(/0)-ln(0/0)) + lnz(t/t0,/0) (3) functon = deal ref.value + deal sloe + real dfference ( excess) For the case of ure water, f one chooses T0=288 K, 0=0.1 MPa and v0=rt0/0=0.024 m 3 /mol, the exanson s lotted n Fg. 2, where t s seen that the molar volume jums to m 3 /mol after reachng the vaour/lqud hase-change (at MPa for 288 K, ln(0.017)= 4, and at 0.1 MPa for 373 K,ln(1)=0). Fg. 2. Exanson of molar volume, v, vs. ressure,, for a ure substance at constant temerature: the case of water at 25 ºC and 100 ºC. Excess functons 2

3 CHEMICAL POTENTIAL FOR A PURE SUBSTANCE Smlar to temerature and ressure beng the escang force for thermal energy ( TdS) and mechancal energy ( dv), resectvely, the chemcal otental, µ, s the escang force for chemcal energy ( Σµdn),.e. energy assocated to the flow of chemcal seces through a fronter. Although µ S/ n U,V,nj, they are not absolute values, as can be seen from G=Σnµ (G s an energy functon, defned wth an arbtrary constant, and for a ure substance G=nµ, thus µ s defned wth an arbtrary constant). For a ure substance G=nµ and µ=h-ts, dµ = sdt+vd, and n the deal gas lmt v=rt/, thus the deal chemcal otental s µ(t,)=µ (T, )+RTln(/ ), and one chooses the deal model as (wth ln(/ ) as the ndeendent varable, see Fg. 1): µ(t,)/(rt) (T, )/(RT) + ln(/ ) + lnγ(t,) (4) functon = deal ref.value + deal sloe + real dfference ( excess) Notce n Fg. 3 that from dg=-sdt+vd+µdn, µ/ T= s, and that, although once condensed (µ/(rt)/ <<1, t grows exonentally n the µ-ln lot. Fg. 3. Exanson of chemcal otental, µ, vs. ressure,, for a ure substance at constant temerature: the case of water at 25 ºC and 100 ºC. ACTIVITY, ACTIVITY COEFFICIENT AND FUGACITY (PURE SUBSTANCES) Absolute actvty, λ(t,), relatve actvty, a(t,), actvty coeffcent, γ(t,), and fugacty, f(t,), for ure substances (or fxed comoston mxtures) are defned as follows. In general: (, ) ln d f µ T = RT λ ( T, ) + RT ln γ ( T, ) + RT ln a ( T, ) + RT ln (5) Notce that both γ and a are nondmensonal, whereas f has unts of a ressure. For the choce of the reference state there are two ossbltes: ether choose the same reference state, µ (T, ), for gaseous states and for condensed states (see Fg. 3), what s usefull f hase changes are of nterest, or choose a Excess functons 3

4 dfferent reference state for condensed states, the real state at the reference ressure (Fg. 3), µ (T, ), what s advantageous f hase changes are not of nterest. That s: For a gas (as above): d µ ( T, ) µ ( T, ) + RTln (6).e. the reference state, µ, s chosen as a vrtual deal-gas state at =100 kpa (Fg. 3). Actvtes are comuted from measured values of the comressblty factor Z(T,) as: ln (, ) (, ) 1 ln ( a T ) = [ Z T ] d, γ = [ ] 0 ln ZT (, ) 1 dln, f γ = a (7) 0 For a condensed substance one may choose nstead: d µ ( T, ) µ ( T, ) + v( ) µ ( T, ) (8) where the reference state, µ, s a the real state at =100 kpa (Fg. 3). Actvtes are comuted from measured values of the molar volume v(t,) as: ( a T ) ln (, ) 1 v( ) = v d RT RT, lnγ ( ) 1 ( v v )( ) = v v d RT (9) RT Notce that for mult-hase equlbrum n ure substances, ts chemcal otental n the coexstng hases must concde. EXCESS FUNCTIONS FOR OTHER VARIABLES (PURE SUBSTANCES) Table 1. Exanson for ure substances n terms of the deal gas model: Z(,T) and c(t, 0). Functon = Ideal ref.value + Ideal sloe + Excess Z = Z ln(v/v0) = ln(t/t0) + ( 1)(ln(/0) + lnz u = u0+ cvdt RT 2 ZTdln RT(Z 1) h = h0+ cdt RT 2 ZTdln s = s0+ (c/t)dt + ( 1)Rln(/0) + R (Z 1+TZT)dln g( µ) = g (T,0) + RTln(/0) + RT (Z 1)dln µ (T,0) + RTln(/0) + RT (Z 1)dln α = 1/T ZT/Z κ = Z/Z c = c(t, 0) RT (2ZT+TZTT)dln ln(f/) = (Z 1)dln lna = (Z 1)dln Notce that u0 and h0 are related by h0=u0+0v0. ZT= Z/ T. All ntegrals extend from 0 to. Excess functons 4

5 All excess functons may be exressed n terms of f or a (e.g. h-h d =R ln(f/)/ (1/T), s- s d = R Tln(f/)/ (T), v-v d =RT ln(f/)/ () T. EXCESS FUNCTIONS IN A MIXTURE In general: d d a d f (,, ) ln (,, ) ln (,, ) ln µ T x = RT λ T x + RT γ T x + RT ( T,, x) + RT ln x xf (10) where the deal mxture refers to Raoult's law for mxtures of gases and for mxtures of lquds, but to Henry's law for lqud solutons, and the reference state s chosen deendng on the tye of mxture, as follows. In a mxture of gases, the reference state for a seces s the deal state of the ure comonent at the temerature of nterest and at a standard ressure (Fg. 4), and actvtes are defned by: µ ( T,, x, x ) ( T,,1,0) + RTln a ( T,, x, x ) = j j ( T,,1,0) + RT ln + RT ln x + RT ln γ( T,, x, x j) (11) and comuted from measures of the artal molar volume of seces, v= V/ n T,,nj as: ( a T x x j ) ln (,,, ) x x v( T, x,, xj) 1 = d RT, 0 f γ xf = af (12) Fg. 4. Exanson of chemcal otental, µ, vs. artal ressure, x, for a gas n a gaseous mxture at constant temerature. In a mxture of lquds, and for the solvent n a soluton, the reference state for a seces s chosen as the real state of the ure comonent at the temerature and ressure of nterest (Fg. 5), and actvtes are defned by: µ + = ( T,, x, xj) ( T,,1,0) RTln a( T,, x, xj) ( ) ( T,,1,0) + v + RT ln x + RT ln γ( T,, x, x ) (13) j Excess functons 5

6 (wth the ressure term, v ( ), usually neglected) and comuted from measures of the artal molar volume of seces n the gaseous-hase-mxture at equlbrum wth the lqud mxture (at the workng temerature and the corresondng ressure, snce the nfluence of ressure on lqud roertes was shown to be small, see Eq. (9)), lus measures for the ure substance at two-hase equlbrum. For a gven seces at a gven temerature, for lqud/vaour equlbrum n the mxture µ L, L, + RT ln al, G, G, + RT ln ag,, and for lqud/vaour equlbrum n the ure state µ + RT ln a + RT ln a, so that the actvty n the lqud mxture s: L, L, L, G, G, G, a ( T,,1, 0) x a T x x = a T x x, a ( T,,1, 0) ( T) L,, L(,,, j), G(, eq,, j), G eq f γ xf = af (14) That s, the actvty of a lqud (n a lqud mxture) s measured as the roduct of molar fracton tmes ressure, n the gas hase (also known as artal ressure of seces ), dvded by the lqud/vaour-equlbrum ressure at that temerature for that ure seces ; for deal lqud mxtures the molar fracton s recsely a,l=1. x = T (Raoult's law) and thus the actvty s G, ( )/ Fg. 5. Exanson of chemcal otental vs. molar fracton, for a lqud n a lqud mxture (e.g. ethanol n a water/ethanol soluton) at constant temerature, wth the reference state at µ. Also alcable to a solute n a solvent s, wth the reference state at µ,s (see below). Solutons. In a lqud mxture of a lqud solvent and sold or gaseous solutes, the reference state for the solvent s s as just descrbed (µ n Fg. 5), but for a seces n soluton t s the deal state of the nfntely dluted soluton of just ths seces n that solvent, at the temerature and ressure of nterest. The reference state and the corresondng actvty may be defned n molar-fracton bases (µ,s n Fg. 5) or n molar-concentraton bases (µ,s n Fg. 6) by: µ ( T,, x, x, x ) ( T,,0,1,0) + RTln a ( T,, x, x, x ) = (15) s, s j s, s, s j ( T,,1,1,0) + RT ln a ( T,,1,1,0) + RT ln x + RT lnγ (16) sx,, sx,, sx,, c sc,,( T,, c0,1,0) + RT ln asc,, sc,,( T,, c0,1,0) + RT ln + RT lnγ (17) c Excess functons 6 0

7 where subndex s stands for the solvent (solutes are seces ), subndex x stands for the mole fracton unts, and subndex c stands for molar concentraton unts. Notce that the amount of seces n µ s, s ambguously reresented by the symbol 0 n (15), but once the magntude used to measure t s secfed, t takes a more exlct form, as n (16) for mole fractons and (17) for mole concentratons. Notce that sometmes c0 s not exlctly wrtten snce t s taken as unty (e.g. 1 mol/m 3, 1 mol/ltre, even 1 mol/kg) and ts logarthm 'dros' (the actvtes a are non-dmensonal, but ther actual value deends on the unt chosen, e.g. for a 10 g/kg sugar soluton n water and assumng deal behavour, n (16) we have a= /0.342= relatve to unt mole fracton, and n (17) we have a=29 relatve to 1 mol/m 3. Actvtes of non-volatle solutes are comuted from actvtes of the solvent and Gbbs-Duhen equaton, that at constant temerature and ressure reads 0=nsdµs+ndµ,s, what yelds 0=xsdlnas+xdlna,s. Notce fnally that subscrts s, x and c are usually droed n wrtng (15-17) because ambgutes are rare f the context s well understood. Fg. 6. Exanson of the chemcal otental µ,s, of a solute (e.g. sugar, oxygen) n a solvent s (e.g. water), vs. concentraton, c, at constant temerature. IONIC SOLUTIONS Some solutes (e.g. O2(g) and saccharose C12H22O11(s)) dssolve as whole molecules n water, whereas other solutes (e.g. HCl(g) and NaCl(s)) dssolve ractcally fully onsed, slttng n two ons or more (CaCl2 yelds 3 ons), so that the actvty n the dluted lmt vares wth c 2 or c n and not wth c. As ure ons cannot be solated (they always form electrcally-balanced n ars, terns or so), actvtes and actvty coeffcents cannot be measured for one on but for the set of ons formed (e.g. ahcl(aq)=mhmclγ ±, acacl2(aq)=mcam 2 Clγ ±, etc.). ACTIVITY QUOTIENT For a system subjected to a chemcal reacton 0=ΣνM, wth the extent of reacton, ξ, and the affnty, A, defned by ξ (n-n0)/ν and A Σνµ,, the Gbbs functon of reacton s gven by: Excess functons 7

8 G g = A = νµ = νµ ( T,,1, 0) + νrt lna r ξ T, (18) where the effect of the ressure change from to n the lqud hase has been neglected, and the same symbol µ stands for ure gases, ure lquds, and nfntely-dluted solutons. Equlbrum constant The so called equlbrum constant, K(T,), and actvty quotent, Q(a), are defned by: νµ ( T, ) ν ν KT (, ) λ = ex, and QT (, x,, x) (,,, ) RT a T x x (19) j j so that at any state gr= RTln(K/Q), and at equlbrum (A=0): ν deal gaseous mxtures x KT (, ) = ν er mole-fracton unt ν KT (, ) = Q= a KT (, ) = x KT (, ) = c 0 deal condensed mxtures ν er mole-concentraton unt c (20) The name actvty 'quotent' derves from the fact that some of the stochometrc coeffcents n the Q roduct are negatve (the roduct s really a quotent of roducts). Back Excess functons 8