Thermodynamcs II Department of Chemcal Engneerng Prof. Km, Jong Hak
Soluton Thermodynamcs : theory Obectve : lay the theoretcal foundaton for applcatons of thermodynamcs to gas mxture and lqud soluton Most of chemcal process undergo composton changes by mxng separaton. > compostons become essental varable along wth T and P. Fundamental property relaton become more comprehensve than eqn. (6.0) dg VdP SdT.
. Fundamental Property Relaton Eqn. (6.6) d(ng) (nv)dp (ns)dt for closed system of sngle phase (ng) ] T, nv and [ ] P, n -ns [ n (ng) For a sngle-phase, open system ng f ( P, T, n, n,.) d(ng) (ng) [ ] T,n (ng) dp + [ ] T P,n dt + (ng) [ ] n P,T,n dn By defnton chemcal potental of speces n the mxture (ng ) (.) dn µ [ ] P,T,n n d(ng) (nv)dp - (ns)dt + (.) µ dn
For specal case of one mole of soluton n, n x G dg VdP -SdT + (.3) G(T, P, x, x,..., x µ dx,...) From (.3) G V ( ) T,x H G + TS G G - T( T G S ( T ) P,x ) P, x Gbbs energy plays a role of a generatng functon, provdng the means for calculaton of all other thermodynamc propertes by smple mathematcal operatons.
. The Chemcal Potental and Phase Equlbra Closed systems consstng of two phase n equlbrum each ndvdual phase s open to the other > mass transfer occur b/w phases Eqn.(.) apples to each phase d(ng) d(ng) (nv) (nv) (nm (nm) dp - (ns) dp - (ns) + dt dt (nm) ) Because two phases are n equlbrum -> T and P s unform Change of total Gbbs energy sum of two equaton d (ng) (nv)dp - (ns)dt + µ dn + µ µ dn dn d (ng) + µ dn d(ng) at equlbrum µ dn + µ dn 0 by mass conservaton, dn + dn 0 (µ -µ )dn 0
At equlbrum μof each phase s same (µ µ ) Thus, multple phase at same T&P are n equlbrum when the chemcal potental of each speces s the same n the all phase µ µ π µ (,,3, N) (-6)
.3 Partal Propertes Partal property M s defned by M (nm) [ ] P,T,n n (-7) Chemcal potental s partal molar property of Gbbs Energy (ng) µ ( ) P,T,n n G Equatons relatng Molar and Partal molar propertes From the knowledge of the partal propertes, we can calculate soluton propertes or we can do reversely. Total thermo propertes of homogeneous phase are functons of T, P and the numbers of moles of the ndvdual speces whch comprse the phase nm µ(t, P,n,n,...,n )
The total dfferental of nm s (nm) (nm) (nm) d(nm) [ ] T,n dp + [ ] P,n dt + [ ] P,T,n T n dn (nm) M d (nm) n( ) T,x dp + n( ) P,x dt + T M dn (-9) n x n dn x dn d(nm) ndm + Mdn + ndx ndm + Mdn M n( ) M dp + n( ) T dt + M (x dn ndx ) T,x P,x +
The terms contanng n are collected separated from those contanng dn to yeld M M [ dm - ( ) T,x dp - ( ) P,x dt - Mdx ]n + [M - xm ]dn 0 T M M dm ( ) T,x dp + ( ) P,x dt + T M Multply Egn.(.) by n yeld x M M dx (-0) (-) nm n M (-) (.0) -> specal case of Eqn (.9) by settng n (.), (.) -> summablty relatons > allow calculaton of mxture from partal property
From Egn (.) M x M dm x dm + M dx M M ( ) dp + ( T,x ) P,x dt - xdm 0 T > Gbbs/Duhem equaton Compare ths wth (.0) (-3) at const T, P x (-4) dm 0
Ratonale for partal property M x dm > Soluton property s sum of ts partal propertes lm M x lm M M (n the lmt as a soluton become pure n speces ) x By defnton lm M x 0 M Summary of partal property. Defnton M (nm). summablty M [ ] P,T,n n x M 3. Gbbs/Duhem ) dt > yeld partal propertes from total property > yeld total propertes from partal propertes M M xdm ( ) T,x dp + ( T > partal propertes of speces n soluton > dependent one another P, x
Partal propertes n bnary solutons For bnary soluton (system), From summablty relatons M x dm M + dm xdm + M dx + x dm + M dx xm x M > at constant T, P by Gbbs/Duhem Egn. > x dm + x dm 0 Because x + x, dx -dx, Elmnatng dx n Eq From Egn nsert Egn dm x + dm M dx dm + Mdx x dm - Mdx - Mdx M M + x dm dx dm M dx M M - x - M M (- x )M + x M M - x (M - M ) M x M + (- x)m M + x(m - M ) by Egn dm dx So, from the soluton propertes as a functon of composton (at const T, P) > Partal propertes can be calculated
G/D can be wrtten n dervatve form dm dm x + x dx dx 0 > dm dx - x x dm dx > When M & are plotted vs x > sgn of slope s opposte M Moreover from ths Equaton dm dm lm 0 lm 0 x dx, x dx > Plots of M and vs x > horzontal as each speces approach purty M
Relatons among partal propertes d (ng ) (nv)dp - (ns)dt + d(ng) (nv)dp - (ns)dt + V ( ) T S P,n -( ) T,n µ dn G dn G G ( µ G ) Apply crteron of exactness for dfferental expresson ( ) T,n -( P,T,n n (nv) ) (ns) ( ) P,n -( ) P,T,n T n V Property relatons used n const. composton soluton has ther counterpart equatons for partal propertes For example, H U + PV, for n mole nh nu + P(nV) - S Dfferentaton wth respect to n at const T, P, n (nh) (nu) (nv) [ ] P,T,n [ ] P,T,n + P[ ] P,T, n > n n n H U + PV dg V dp -SdT
.4 Ideal Gas Mxtures Ideal gas mxture model : Bass to buld the structure of soluton thermodynamcs Molar volume of Ideal Gas : V RT P > All deal gas, whether pure or mxture have same molar volume at the same T, P Partal molar volume of speces n deal gas mxture g g (nv ) (nrt / P) RT n V [ ] T,P,n [ ] ( ) n n n P n snce n n + n RT P > partal molar volume pure speces molar volume mxture molar volume g g g V V V RT P
Partal pressure of deal gas for n mol of deal gas for speces, P P nrt P t V n n y P nrt t V t t V V (snce n deal P In deal gas, thermodynamc propertes ndependent of one another yp Gbbs theorem A partal molar property (except volume) of a consttuent n an deal gas mxture s equal to the correspondng molar property of the speces as a pure deal gas but at a pressure equal to ts partal pressure M g (T, P) M g (T, P ) (.)
H of Ideal Gas > ndependent of P H C P dt g g H (T,P) H (T, P ) H g (T,P) g H H -> (.) g S of an deal gas -> dependent on P and T at const T. ds g (6.4) ds g Integraton from P to P g g dt dp CP - R T P dp -RdlnP -R P g g P P S (T, P) -S (T,p) -Rln -Rln R ln y p y P g S (T,p ) S (T, P) - Rlny Snce, S (T, P) S (T,P ) g g Gbbs theorem g S S - Rlny -> (.3) g
For the Gbbs Energy of deal gas mxture g g g g G H - TS G H g - TS g or g G H - TS + RT ln y g g g g µ G G + RT ln y g (by usng egn. ) -> 3 (.4) By summablty eqn H S G g g g ( M x M ) y y S H g g y G - R g + y RT ln y y ln y -> 4 -> 5 -> 6 H - y g H g (enthalpy charge of mxng) 0 > no heat transfer for deal gas mxng g g S - ys R y ln y (entropy charge of mxng) > 0 > agree wth second law, mxng s rreversble
Alternatve expresson for g µ dg By ntegraton g V g dp -S g dt VdP g g RT dg V dp dp P g G Γ (T) + RT ln P g µ Γ (T) + RT ln(y RTd ln P P) (at const T) (.8) (.9) Applyng summablty relaton G g yγ (T) + RT y ln(y P) (.30)