Thermodynamics II. Department of Chemical Engineering. Prof. Kim, Jong Hak

Transcription

1 Thermodynamcs II Department o Chemca ngneerng ro. Km, Jong Hak

2 .5 Fugacty & Fugacty Coecent : ure Speces µ > provdes undamenta crteron or phase equbrum not easy to appy to sove probem Lmtaton o gn (.9 ( H - TS -> or y -> > µ - (whch s not true or dea gas gn (.7 > vad ony or pure speces n the dea gas For rea gas, ugacty s ntroduced nstead o Γ (T + n g Γ (T + n g - n nφ ( : ugacty coecent nφ (.33 φ

3 For an dea gas, g, & φ In chapter 6, set J as zero, ( J, m( mnφ mnφ J + (z J d - m n( whch means that mφ m From egn (6.49 φ > : obtaned rom VT data or rom compressbty actor nφ d (z -,nφ (z B B d d d - B z - B nφ B

4 vauaton o ugacty coe. rom Cubc OS nφ (.33 Z -- n(z-β - qi (6.66 nφ Z -- n(z-β - q I (.37 where β b a (T q Ztδβ b I n( δ -ε Z tεβ (3.5 (3.5 (6.65b Z ρ (.37 I, δ ε I ρb + εβ b Z β + ε β

5 Vapor-Lqud equbrum or pure speces Consder vapor qud equbrum For speces as a urated vapor, usng eqn(.3 Γ (T + n v v (.38a For speces as a urated qud Γ (T + n (.38b d v v - n v at equbrum ( : ether urated qud or vapor Aternatve ormuaton φ p φ φ > Crteron o VL or pure speces, μ,, Φ v φ

6 Fugacty o a pure qud - Fugacty o pure speces as a compressed Lq. > cacuated rom the product o easy evauated rato ( ato A > vapor phase ugacty coe., φ at VL nφ v v ( A (z v v ( ( B d - ( ( at const T ato B > snce at VL ato C > the eect o pressure on the ugacty o pure qud ; C d - Vd p p (6. at const T Vd

7 rom (-3 Γ (T + n n p p - V d n ato C, snce ( V ( ( exp( A B C V d φ p exp( V d Because s a very weak uncton o > assumed constant φ p V ( - exp( oyntng actor φ V : can be cacuated rom z v : vaue or urated qud : Antone eqn., Z - B φ B X(

8 .6 Fugacty & Fugacty Coecent : Speces n Souton For speces n a mxture o rea gas or n a souton o qud µ Γ (T + n ˆ (.46 ˆ : Fugacty o speces n souton (repacng parta pressure at equbrum, the ugacty o each component s the same n a phases ˆ ˆ α β ˆ π -> Mutcomponent VL, ˆ ˆ v

9 esdua property g M M - M Mutpy n and derentate wth respect to n at constant T,, n (nm (nm [ ],T,n [ ] n n M M,T,n - M (nm -[ n g g ],T, n Wrtten or, - g From egn(.9 µ g µ Γ (T + n ˆ ˆ µ -µ n, µ -µ g Γ (T + n(y g g y nφˆ (φˆ (.46 ˆ y (snce µ φ : ugacty coe. o speces n souton For dea gas : g g & φˆ ˆ y

10 The undamenta resdua-property reaton Aternatve orm d (n (nvd - (nsdt + n ( n n ( d(n - d n d( nv nh d - dt µ dn dt + dn ( H - TS d Vd -SdT (.54 Functon o a o canonca varabe (T,, n > Aow evauaton o a other thermodynamc propertes compare (6.37 d( V d - H o dt Speca case o mo o a constant-composton > Snce we cannot evauate absoute vaue o thermodynamc propertyes > use resdua property

11 For dea gas g g g n nv nh d ( d - + Subtractng ths eqn rom (.54 g dn n nv nh d ( d - dt + dn > The undamenta resdua-property reaton by ntroducng ugacty coe. ( nφˆ n nv nh d ( d - dt + nφˆ dn V ( / [ ] T,x, H ( / -T[ ] T,x > Use eqn (6.46, (6.48, (6.49 or the cacuaton (n / n φˆ [ ] nφˆ,t,n n : parta property o /

12 For n mo o constant composton mxture n (nz - n d Derentaton wth respect to n at constant T,, n nφˆ (nz - n [ ] n,t,n d snce, (nz n Z, n n nφˆ (Z d - (.6

13 Fugacty coecents rom the vra equaton o state z + B B y y B φˆ vouton o vaue o rom OS Smpest orm o vra eqn For bnary mxture (3.38 (.6 nφ d (z - B y y B + y y B + y y B + y y B (.6 where B(T composton second vra coe. where B : characterze bmoecuar nteracton b/w and (B B y B + y y B + y B (.6 (B, B : vra coe. o pure speces (B : cross coe (mxture property For n mo o bnary gas mxture eqn(3.38 becomes nz n + nb

14 Derentaton wth respect to n From eqn (.6 (nz z + nφˆ [ ],T,n [ ] T, n n Second vra coe. can be wrtten (nb n (nb (nb [ ] T,n d [ ] T, n n n B y ( - y B + y y B + y ( - y B y B - y y B + y y B + y B y y B y B + y B + y y δ, δ B B B nb n + n n n B + n B δ snce y n / n, mutpyng by n by derentaton (nb n [ ] T,n + n nφˆ (B + yδ nφˆ (B + yδ by smar method B ( - n δ B + (- yyδ B + yδ n n

15 For mutcomponent gas mxture, the genera eqns. nφˆ [B k kk + I J y y (δ k -δ ] δ δ k B B k - B - B - B - B kk δ δ kk δ k δ k

16 .7 enerazed Correatons or the Fugacty Coecent To cacuateφ, use enerazed methods or compressbty actor Z z z + nφ, d d c r ωz Correaton or (3.57 ω -. - og( r where Tr. 7 qn (.35 (z - c d d nφ + r nφ r d r r r r (z - ω z r r (const T transormed nto generaze orm (z d - usng (.65 Usng eqn (3.57 r r nφ nφ + ωnφ Three parameter generazed correaton or Φ φ (φ (φ ω use tabe 3 ~ 4

17 Usng tzer correatons or the second vra coe. where Z + Bˆ Z - T r T r r r (B + ωb Bˆ B + ωb r r nφ (B + ωb φ exp[ (B + ωb ] Tr Tr , B T T B.6 r (3.6 (3.63 > Insert nto eqn(.5 and ntegrate r enerazed correatons or ugacty coecent n gas mxture nφˆ k [B kk + More genera orm or coe. B C B (.69 C (δ k Bˆ + -δ ] T r T / T C B ωb > B, B -> uncton o T r >use rausntz s combnng rue or the cacuaton o ω,t C, C

18 ω c ω z c + ω V c c T c (T z c T c / + z (- k (.7 (.7 c c (.7 zc (.73 Where k emprca nteracton parameter and or chemcay smar speces k -> a eqns reduce to vaue or pure speces rocedure to obtan φˆ Fnd vaue o B rom (.69 nsert B nto eqn(.6 B 3 usng eqn(.4 φ obtan vaue o nφˆ [B k kk + nφˆ vaue o the pure-speces vra coe. B kk, B or eqn y y B y y (δ k -δ ] >nd B B C Bˆ + C B ωb

19 -8. The dea souton mode Denton : a moecues are o the same sze a orces b/w moecues (ke and unke are equa The chemca potenta rom dea-gas mxture mode g g µ (T, + n y (.4 g > Ony appcabe or dea gas g repace (T, (T, (bbs energy o pure n ts rea physca state o gas qud, sod chemca potenta o an dea souton From eqn(.8 d d µ (T, + n x > appcabe gas, qud, sod d d V ( T,x ( T d ( T,x V V d (.75 V V (.76

20 From eqn(.9 (,x d d S -(,x -( p - nx T T -S T S - nx (.77 H d, S V V d d d d d d H + TS For, usng (.75 (.77 d by summabty reaton, x x x x S V V d H + n x + TS + TS H M x M + x n x - x n x d, H - nx x H

21 The Lews / anda ue From eqn (.46 ( > µ Γ (T + n ˆ (.3 µ d µ + ˆ n( d µ + n x + ˆ n( For the speca case o an dea souton I compared wth eqn(.75 d d Γ (T + n ˆ x d (.83 > Show the composton dependence o the ugacty n an dea souton (Lews/anda rue Lews/anda ue : ugacty o each speces moe racton proportona constant ugacty o pure speces Aternatve orm o Lews/anda ue : (dvde.83 wth x d φˆ φ (-8 Fugacty coe. o speces n an dea souton ugacty coe. o pure speces

22 eca the reaton b/w (z nφ.9. xcess ropertes d -,φ,φˆ (6.49 and VT data d d (z - (.35 nφˆ (z - (.6 -> obtan thermodynamc property usng esdua propertes In the case o qud, measure the departure rom deaty not rom dea gas, but rom dea souton > excess property Mathematca ormuaton M M - M d (Derence b/w actua property and dea souton For exampe, consderng M - - M d, H d H - H, S d S-S, g M M - M ure speces M -(M d - M (dea gas mxture > dea souton o dea gas g H g g x + - TS x n x

23 M g Compare and d M H g g d H yh g g d S y S - y n y S x S - g g d y - y n y x - x g x M -x M x d g M - M d g M - M > M - M M M M - xcess roperty : apped to ony mxture esdua property : apped to both pure speces & mxture x H x n x x n x M For parta excess property : M M - M d Fundamenta excess-property reaton d( n nv nh d - dt + dn > Smar to undamenta resdua-property reaton

24 The xcess bbs nergy and the Actvty Coecent From eqn (.46 From Lews/anda rue, or an dea souton Derence µ Γ (T + n ˆ d d Γ (T + n ˆ Γ (T + n x d ˆ - n x ˆ (γ x actvty coecent (.9 Γ (T + n ˆ nγ nφˆ, or dea souton, γ To devce chemca potenta o mxture (.9 d d - nγ, + n x (.75 + nγ x (.9 Comparson o three equatons denng chemca potenta µ µ µ g d g + n y (.4 + n x + nγ x (.75 (.9 st qn : dea gas mxture mode nd qn : dea souton mode

25 xcess-property eaton In undamenta excess property reaton d( n nv nh d - dt + dn V ( / H ( / [ ] T,x (.9, -T[ ], x nv nh d - dt + nγ dn ( nγ T (.9 > ect o T, on the n γ (n / [ ],T,n n (.96 > nγ : parta propertes o n > Smar to eqn or esdua property Derence : n the case o reaton to -> we can use expermenta VT data ~ OS to cacuate esdua property V, H, γ γ V, H > Can be obtaned by experment : rom VL data : rom mxng experment

26 γ V nγ H nγ ( (.97, T,x - (, x n : parta property o, nγ T (.98 > ect o T, D on the γ From the summabty eqn x nγ M (.99 x M From bbs/duhem eqn (at const T, x d nγ x dm