Thermodynamics Kinetics - PDF document

1 MIT Open Co urseWare http://oc w .mit.e
MIT Open Co urseWare http://oc w .mit.edu 5.60 Thermodynamics & Kinetics Spring 2008 For info rmati on abo ut citin g these mate rials or o u r Te rms of Use, visit: http://ocw.mit.edu/term s . 5. 60 S p ri ng 2 0 0 8 Lect ure # 2 0 page 1 Two-Component Phase Equilibria Go al : Un d e rs ta nd & pred i c t eff e cts of m i xin g of su bstan c es on va por pre s sure , bo iling poin t, fr eezin g p o int, etc . Bin a ry l i quid -g as mixt ur es (non -r eact in g): Tot a l # of var i ables: 4 ( T , p , x A , y A ) Constraints due to coexistence: 2 A ( ) = A (g ) B ( ) = B (g ) # in

2 depe nden t variab le s F = 4 – 2
depe nden t variab le s F = 4 – 2 = 2 Onl y 2! e . g. k n o w ing (T,p ) un ique ly determines the compositions in the liquid & gas ph ases Ge ner a l i zat i on: Gibbs phase rule g i ve s # in depen d e n t variables neede d t o de scr i be a m u l t i-co mponent system where different phases coexist in equilibrium F = C – P +2 F # degrees of freedom (independent variables)  C # co mpone n ts P # p h ases Ho w do we ge t t his?  Suppo se a system ha s C co mpone n t s an d P p h a s e s.  Wh at ar e all the var i abl es? First, T and p.  5. 60 S p r

3 i ng 2 0 0 8 Lect ure # 2 0 page 2
i ng 2 0 0 8 Lect ure # 2 0 page 2 Then in each phase “ ”, each component is spec ifie d by it s mole C () fractio n , with the constraint th at x i 1 . i 1 So the composition of each phase is specified by (C – 1) variables.  With P phas es , we have P(C – 1) variables.  Including T and p, the to tal # va ri ab les is P( C – 1) + 2. Now a dd cons traints du e to phase equilibria:  C h emi c al poten t ial of ea ch com p onen t is the same in all the p h ase s.  ( ) e.g. for component “i”, i (1) i ( 2) i P P – 1 con s t r a

4 int s For C co mpone n t s , it’ s
int s For C co mpone n t s , it’ s C(P – 1) constraints altogeth er So total # independent variable s is F = P(C – 1) + 2 - C( P – 1) F = C – P + 2 Gibbs phas e rule F o r 1 - com p on en t sys t em : F = 3 – P P = 1 F = 2 Can vary freely in (T,p) plane P = 2 F = 1 Can vary alon g coexistence curve T(p) P = 3 F = 0 No free variables at triple point (T t ,p t ) P = 4 I m possibl e! Ca n’ t hav e 4 phas es i n eq ui lib r iu m Ra oult’s Law and Ideal S o lutions “A” is a volatile so lve nt (e.g. water)  “B”

5 is a nonvolatile so lu te ( e .g. a n t
is a nonvolatile so lu te ( e .g. a n tif r eez e)  p A * vap o r pre ssure o f p u re A at temperature T 5. 60 S p ri ng 2 0 0 8 Lect ure # 2 0 page 4 The gas phase is described by y A or y B . If T and x A are given, then y A an d y B are fixed (by Gibbs phas e ru le). That is, if T and the com p osi t i o n of t h e l i qu id phas e a r e k n ow n , then t h e com p osi t i o n of the gas phase is determined. So how d o we g e t y A ? p A = y A p ( D a l ton’s Law ) p A = x A p A * and p B = x B p B * = (1 –x A )p B * (Raoult’s Law) p x p A p A A A y A

6 p p p x p 1 x p A B A A A B
p p p x p 1 x p A B A A A B xp AA y A p p A p x A B B y p In ve rt in g this e x pre s sio n x AB A p p p y A B A A xp A Co mb in in g the s e t w o r e sul t s p p A A y A y A p p AB or p p p p y A B A A This is summarized in the following diagram: 5. 60 S p ri ng 2 0 0 8 Lect ure # 2 0 page 5 Co mbin in g both d i agr a m s into one p l ot: This allows us to see the composit ions of both liquid and gas phas es If we know the composition of one phase at a given T, we can determined the compo s ition o f th e other phase from the diag