Thermodynamics IIIIII

Department of Chemical Engineering

Prof. Kim, Jong Hak Solution Thermodynamics : theory

■ Objective : lay the theoretical foundation for applications of thermodynamics to gas mixture and liquid solution

○ Most of chemical process undergo composition changes by mixing separation. => compositions become essential variable along with T and P.

○ Fundamental property relation become more comprehensive than eqn. (6.10) dG = VdP – SdT. 11.1 Fundamental Property Relation

○ Eqn. (6.6) d(nG) = (nV)dP – (nS)dT for closed system of single phase ∂nG( ) ∂nG( ) [ ] = nV and [ ] = nS- ∂P n,T ∂P n,P

○ For a single-phase, open system

nG = f ( P, T, n 1, n 2, ….) ∂nG( ) ∂nG( ) ∂nG( ) nG(d ) = [ ] dP +[ ] dT + ∑[ ] dn n,T n,P n,T,P j i ∂P ∂T i ∂ni

By definition chemical potential of species i in the mixture ∂ nG( ) µ = ∑[ ] dn (11.1) i T,P n, j i i ∂n i

= + (11.2) ∴ nG(d ) nV( dP) nS(- dT) ∑µidn i i For special case of one mole of solution n=1, ni = x i

= + (11.3) dG VdP -SdT ∑µidx i i = ∴∴∴G G x,P,T( 1 x, 2 ,....., xi ,....)

From (11.3) ∂G ∂G V = ( ) S = ( ) ∂P x,T ∂T x,P ∂G H = G + TS = G - (T ) ∂T x,P

※ Gibbs energy plays a role of a generating function , providing the means for calculation of all other thermodynamic properties by simple mathematical operations. 11.2 The Chemical Potential and Phase Equilibria

■ Closed systems consisting of two phase in equilibrium each individual phase is open to the other => mass transfer occur b/w phases

Eqn.(11.2) applies to each phase

α = α α = α α nG(d ) nV( ) dP nS(- ) dT ∑µi dn i β = β β = β β nG(d ) nV( ) dP nS(- ) dT ∑µi dn i

Because two phases are in equilibrium -> T and P is uniform Change of total Gibbs energy = sum of two equation nG(d )α + nG(d )β

= + α α + β β nG(d ) nV( dP) - (nS)dT ∑µi dn i ∑µi dn i nM( = nM( )α + nM( )β )

at equilibrium α α + β β = ∑µi dn i ∑µi dn i 0 α + β = by mass conservation, dn i dn i 0 α β α = ∑(µi -µi dn) i 0 ①①① α = β At equilibrium μof each phase is same (µi µi )

Thus, multiple phase at same T&P are in equilibrium when the chemical potential of each species is the same in the all phase

α = β = … = π = … ∴µi µi µi i( 3,2,1 , N)

(11-6) 11.3 Partial Properties

■ Partial property is defined by Mi ∂nM( ) M ≡[ ] (11-7) i n,T,P j ∂ni Chemical potential is partial molar property of Gibbs Energy ∂nG( ) µ = ( ) = G i n,T,P j i ∂ni

※ Equations relating Molar and Partial molar properties

○ From the knowledge of the partial properties, we can calculate solution properties or we can do reversely.

○ Total thermo properties of homogeneous phase are functions of T, P and the numbers of moles of the individual species which comprise the phase = nM µ n,P,T( 1 n, 2 ,....., ni ) The total differential of nM is

∂nM( ) ∂nM( ) ∂nM( ) nM(d ) = [ ] dP +[ ] dT + ∑[ ] dn n,T n,P n,T,P j i ∂P ∂T i ∂ni

∂ ∂ = nM( ) + M + nM(d ) (n ) x,T dP (n ) x,P dT ∑Midn i (11-9) ∂P ∂T i = = + ni xin → dn i xidn ndx i nM(d ) ≡ndM + Mdn

∂M ∂M ∴ndM + Mdn = (n ) dP + (n ) dT + ∑M x( dn + ndx ) ∂P x,T ∂T x,P i i i The terms containing “n” are collected separated from those containing dn to yield ∂ ∂ M M + = dM[ (- ) x,T dP (- ) x,P dT - ∑Midx i n] [M - ∑xiMi dn] 0 ∂P ∂T i ∂ ∂ = M + M + ∴dM ( ) x,T dP ( ) x,P dT ∑Midx i (11-10) ∂P ∂T i = M ∑xi Mi (11-11) i

Multiply Egn.(11.11) by n yield

= nM ∑niMi (11-12) i

(11.10) -> special case of Eqn (11.9) by setting n=1 (11.11), (11.12) -> summability relations => allow calculation of mixture from partial property From Egn (11.11) = M ∑xi Mi

= + dM ∑xidMi ∑Midx i Compare this with (11.10) i i ∂M ∂M + = (11-13) ( ) x,T dP ( ) x,P dT - ∑xidM i 0 ∂P ∂T i => Gibbs/Duhem equation

at const T, P = (11-14) ∑xidMi 0 i ■ Rationale for partial property

= M ∑xidMi => Solution property is sum of its partial properties

= = lim M lim Mi Mi (in the limit as a solution become pure in species i) xi →1 xi →1 = ∞ By definition lim Mi Mi xi →0

■ Summary of partial property

∂ nM( ) 1. Definition M = [ ] => yield partial properties from total property i n,T,P j ∂n i 2. summability = => yield total properties from partial properties M ∑xi Mi i ∂M ∂M 3. Gibbs/Duhem = + ∑xidM i ( ) x,T dP ( ) x,P dT i ∂P ∂T => partial properties of species in solution => dependent one another ■ Partial properties in binary solutions

= For binary solution (system), From summability relations M ∑xidMi

= + ⒜⒜⒜=>=>=> dM = x dM + M dx + x dM + M dx ⒝⒝⒝ M x1 M1 x 2 M2 1 1 1 1 2 2 2 2

at constant T, P by Gibbs/Duhem Egn.

+ = ⒞⒞⒞ x1dM1 x 2dM2 0

Because x1 + x 2 = 1, dx 1 = -dx 2 , Eliminating dx 2 in Eq ⒝⒝⒝ = + + dM x1dM1 M1dx 1 x 2dM2 - M2dx 1 by Egn ⒞⒞⒞

= dM = dM M1dx 1 - M2dx 1 ∴ M1 - M2 ⒟⒟⒟ dx 1 = + = From Egn ⒜⒜⒜ M -1( x 2 )M1 x 2 M 2 M1 - x 2 (M1 - M 2 ) = + = + M x1 M1 -1( x1)M2 M 2 x1(M1 - M 2 )

= + dM = dM insert Egn ⒟⒟⒟ M1 M x 2 M2 M - x1 dx 1 dx 1 So, from the solution properties as a function of composition (at const T, P) => Partial properties can be calculated G/D can be written in derivative form

dM1 dM2 dM x dM x + x = 0 =>=>=> 1 = 2 2 1 dx 2 dx - 1 1 dx 1 x1 dx 1

=> When M 1 & M 2 are plotted vs x1 => sign of slope is opposite

Moreover from this Equation

dM dM lim 1 = 0 lim 2 = 0 x →1 ,,, x →1 1 dx 1 2 dx 1

=> Plots of M 1 and M 2 vs x1 => horizontal as each species approach purity ■ Relations among partial properties

= + (µ = G ) nG(d ) nV( dP) nS(- dT) ∑µ idn i i i = + ⇒ nG(d ) nV( dP) - (nS)dT ∑G idn i i Apply criterion of exactness for differential expression ∂V ∂S ∂G ∂nV( ) ( ) = -( ) ( i ) = -( ) ⇒ V n,P T n, n,T n,T,P j i ∂T ∂P ∂P ∂ni ∂G ∂nS( ) ( i ) = -( ) ⇒ - S n,P n,T,P j i ∂T ∂ni ※ Property relations used in const. composition solution has their counterpart equations for partial properties

For example, H = U + PV, for n mole nH = nU + P(nV)

Differentiation with respect to ni at const T, P, nj ∂nH( ) ∂nU( ) ∂nV( ) [ ] = [ ] + [P ] =>=>=> = + n,T,P j n,T,P j n,T,P j Hi Ui PVi ∂ni ∂ni ∂ni = dGi VidP - dTS 11.4 Ideal Gas Mixtures

※ Ideal gas mixture model : Basis to build the structure of solution thermodynamics RT Molar volume of Ideal Gas : V = P => All ideal gas, whether pure or mixture have same molar volume at the same T, P

※ Partial molar volume of species i in ideal gas mixture

ig ig ∂∂∂nV( ) ∂(nRT / )P RT ∂n RT Vi = [ ] = [ ] = ( ) = n,P,T j n j ∂ni ∂ni P ∂ni P = + since n ni ∑n j

=> partial molar volume = pure species molar volume = mixture molar volume

ig RT V = Vig = Vig = i i P Partial pressure of ideal gas for n mol of ideal gas nRT P = Vt n RT = i t t for species i , Pi (since V = V in ideal Vt i P n i = i = y ∴P = yP P n i i

In ideal gas, thermodynamic properties independent of one another

※ Gibbs theorem A partial molar property ( except volume ) of a constituent in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas but at a pressure equal to its partial pressure

ig = ig (11.21) Mi )P,T( Mi P,T( i ) ※ H of Ideal Gas => independent of P = H CPdT ig = ig = ig ∴Hi )P,T( Hi P,T( i ) Hi )P,T( ig = ig -> ①①① (11.22) ∴Hi Hi

※ S of an ideal gas -> dependent on P and T dT dP dS ig = Cig - R (6.24) P T P

ig dP at const T. dS = -RdlnP = -R i P P P ig ig = = = Integration from P i to P Si -)P,T( Si p,T( i ) -Rln -Rln R ln yi pi yiP ig ig = ig Since,= ig Gibbs theorem ∴Si p,T( i ) Si -)P,T( Rlny i Si )P,T( Si P,T( i )

ig = ig -> ②②② (11.23) Si Si - Rlny i For the Gibbs Energy of ideal gas mixture

ig ig ig ig ig ig G = H - TS ⇒ G = H - TSi

ig = ig ig + Gi Hi - TS i RT ln yi (by using egn. ①①①②②② ) ig ig ≡ = ig + -> ③ (11.24) or µi Gi Gi RT ln yi

= By summability eqn (M ∑xi Mi ) ig = ig -> ④ H ∑yiHi ig = ig -> ⑤ S ∑yiSi - R∑yi ln yi ig = ig + G ∑yiGi RT ∑yi ln yi -> ⑥

• ig ig H - ∑yiHi (enthalpy charge of mixing ) = 0 => no heat transfer for ideal gas mixing 1 • ig ig = S - ∑yiSi R∑yi ln (entropy charge of mixing ) > 0 yi => agree with second law, mixing is irreversible Alternative expression for ig µi

ig = ig ig dG i Vi dP -Si dT ⇒ VdP (at const T) RT ∴dG ig = Vig dP = dP = RTd ln P i i P By integration ig = + (11.28) Gi Γi )T( RT ln P

ig = + (11.29) ∴µi Γi )T( RT ln( yi )P

Applying summability relation

ig = + ∴G ∑yiΓi )T( RT ∑yi ln( yi )P (11.30)