CHAPTER 8 RELATIONS

H. CHAPTER 8 RELATIONS

Mole and mass fractions Mole fraction (number fraction) X i =N i /N Mass fraction Y i =m i /m Molecular mass of mixture M m = ΣX i M i

Conversion from X i to Y i :Y i =X i M i /M m , Conversion from Y i to X i :X i =Y i M m /M i Molality, Mo = 10 –3 ×kmole of solute ÷ kg of solvent.

Generalized relations

G = G (T, P, N 1 ........N n )

Differentials dU = Tds – PdV + Σµ j dN j

dH = Tds + VdP + Σµ j dN j dA = –SdT – PdV + Σµ j dN j dG = –SdT – VdP + Σµ j dN j Thermodynamc potentials

 ∂ U   ∂ T= H = 

 ∂ S  1 n V, N ... N

 ∂ S  P, N , ... N 1 n

P= - 

,V =  ∂ 

 ∂ V  S , N ... N 1 n

 ∂ P  S N .... N 1 1 n

 ∂ N 1  S, V, N .... N 2 n  ∂ N 1  S, P, N .... N 2 n

∂ N 1  T, V, N .... N 2 n  ∂ N 1  T, P, N .... N 2 n

Partial molal property (B= U, A, H, G, etc.)  ∂ B 

= ˆ, 1 b e.g., v = ˆ 1 ,ˆ g=(G/N) 1 ∂ ∂ 1 T,P,N ,N ,...

 ∂ N 1  T,P N ,N ,.. 12 3  ∂ N 2 

T,P N ,N ,... 11 2

where ^ g 1 , partial molal Gibbs' function of species 1= µ 1 =( ∂G/∂N 1 ) T,P, N2, N3.. Mixture Property

B= ΣN k ˆb k , ˆb 1 = b -X 2 d b /dX 2

Gibbs Duhem equation

dP − N db k ˆ k = 0 dT

db db  ∂ B  ∂ dT B + dP  

k X db k = 0 or

dT +

dP

k = 1  ∂ T  PN ,

 ∂ P 

TN ,

Differentials of partial molal properties:

d ˆg k =– ˆs k dT + ˆv k dP, d ˆh k = Td ˆs k + ˆv k dP, d ˆg k =– ˆs k dT + ˆv k dP, d ˆh k = Td ˆs k + ˆv k dP,

Generalized Thermodynamic Relations

d ˆu k = ˆc v,k dT + (T( ∂P/∂T) – P) d ˆv k ,d ˆh k = ˆc p,k dT + ( ˆv k –T( ∂ ˆv k / ∂T)) dP,

d ˆs k =( ˆc pvk /T)dT + ( ∂P/∂T) ˆv k dP, d ˆs k =( ˆc p,k /T)dT – ( ∂ ˆv k / ∂T) dP.

P–V–T relations for ideal or real gas mixtures Dalton's law (LAP) P = Σp k (T, V, N k )

Amagat Leduc law (LAV) V = ΣV k (T, P, N k ) For ideal gases, volume fraction vf k /X k

Partial pressure for ideal gases p k =X k P Gibbs Dalton law U = N 1 u 1 (T,p 1 )+N 2 u 2 (T,p 2 ) +…, H = U+PV S=N 1 s 1 (T,p 1 )+N 2 s 2 (T,p 2 )+…, ˆs k = s k (T,P,X k )

= s 0 k – R ln (p k /1)= = s (T,P) – k R ln X k

Ideal solution/ideal mixture Any property other than g, a, or s

If b k =h k ,u k ,v k then ˆb k = b k (T,P)

For g k ,a k ,s k ˆb k = b k (T,p k ) for ideal mix of real gases ˆb k = b k (T,P, X k ) for ideal mix of liquids and solids

ˆb k = b k (T,P) – ln X k ,

b k = k (T,P) – ln X k ideal or real gases ˆs k (T, P, X k )– s k (T, P) = R ln X k , ˆg k id – g k (T, P) = R T ln X k . Fugacity of k

d ˆg k = ˆv k dP = R T d ln ( ˆf k (T, P, X k )) Lewis Randall rule

ˆf id

k (T, P, X k )=X k f k (T, P)

Henry’s law id ˆf

1 (HL) = X 1 (d ˆf 1 /dX 1 ) x 1 →0

Fugacity coefficient ˆφ k = ˆf k /(X k P) ( id /f )) = X = ˆ id ˆf k k k α k , i.e., id ˆf k =X k f k =X k φ k P, ideal mix of real gases

For ideal gas mixtures,

ˆf ig

=PX k =p k

Activity α ˆ k ˆ

α id

k =( ˆf k (T, P, X k )/f k (T, P)), α ˆ k =X k

ˆg k (T, P,X k )– g k (T, P) = R T ln α ˆ k = ∫( ˆv k (T, P,X k )– v k (T, P))dP ˆg k (T,P, X k )– ˆg k (T, P o )= R T ln (( α ˆ k (T,P)/ α ˆ k (T,P o ; T,P)) R T ln ( = ˆf k (T, P)/ ˆf k (P o ,T)) Activity coefficient, γ k γ k = α ˆ k / α ˆ id k ˆg id

k – ˆg k = R T ln ( γ k )= R T ln ( ˆ φ k / φ k ) Duhem–Margules relation Σ k N k d ln (X k γ k ) or Σ k X k d ln (X k γ k )=0

E Excess Property id b ˆ = b b E id

(ˆ k − ˆ k ) ,B = B–B = Σ k N k ( ˆb k – b k )

E E E E E ∂( E g /T)/ ∂(1/T) = h , ( ∂ g / ∂P) T = v , and – s =( ∂ g / ∂T) Mixing rules

1/2 2 1/3 β=Σ 3 k X k β k , β = (Σ k X k β k ) , β = (Σ k X k β k ) β = (1/4) Σ k X k β k + (3/4) (

k X k αβ k )(

Σ 2/3 k X k β k ), β=Σ k X j X k β kj , Kay’s rule T cm =X 1 T c1 +X 2 T c2 + ..., and P cm =X 1 T c1 +X 2 T c2 +….

RK and Other rules: a m 1/2 =( Σ k X k 2 a k ) , b m = Σ k X k b k

a m = Σ i Σ j X i X j a ij , b m = Σ i X i b i ,