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 ,
Parts
» COMPUTATIONAL MECHANICS and APPLIED ANALYSIS
» Explicit and Implicit Functions and Total Differentiation
» Exact (Perfect) and Inexact (Imperfect) Differentials
» Intermolecular Forces and Potential Energy
» Internal Energy, Temperature, Collision Number and Mean Free Path
» Vector or Cross Product r The area A due to a vector product
» First Law for a Closed System
» First Law For an Open System
» STATEMENTS OF THE SECOND LAW
» Cyclical Integral for a Reversible Heat Engine
» Irreversibility and Entropy of an Isolated System
» Degradation and Quality of Energy
» SINGLE–COMPONENT INCOMPRESSIBLE FLUIDS
» Evaluation of Entropy for a Control Volume
» Internally Reversible Work for an Open System
» MAXIMUM ENTROPY AND MINIMUM ENERGY
» Generalized Derivation for a Single Phase
» LaGrange Multiplier Method for Equilibrium
» Absolute and Relative Availability Under Interactions with Ambient
» Irreversibility or Lost Work
» Applications of the Availability Balance Equation
» Closed System (Non–Flow Systems)
» Heat Pumps and Refrigerators
» Work Producing and Consumption Devices
» Graphical Illustration of Lost, Isentropic, and Optimum Work
» Flow Processes or Heat Exchangers
» CLASSICAL RATIONALE FOR POSTULATORY APPROACH
» Generalized Legendre Transform
» Van der Waals (VW) Equation of State
» Other Two–Parameter Equations of State
» Compressibility Charts (Principle of Corresponding States)
» Boyle Temperature and Boyle Curves
» Three Parameter Equations of State
» Empirical Equations Of State
» State Equations for Liquids/Solids
» Internal Energy (du) Relation
» EXPERIMENTS TO MEASURE (u O – u)
» Vapor Pressure and the Clapeyron Equation
» Saturation Relations with Surface Tension Effects
» Temperature Change During Throttling
» Throttling in Closed Systems
» Procedure for Determining Thermodynamic Properties
» Euler and Gibbs–Duhem Equations
» Relationship Between Molal and Pure Properties
» Relations between Partial Molal and Pure Properties
» Mixing Rules for Equations of State
» Partial Molal Properties Using Mixture State Equations
» Ideal Solution and Raoult’s Law
» Completely Miscible Mixtures
» DEVIATIONS FROM RAOULT’S LAW
» Mathematical Criterion for Stability
» APPLICATION TO BOILING AND CONDENSATION
» Physical Processes and Stability
» Constant Temperature and Volume
» Equivalence Ratio, Stoichiometric Ratio
» Entropy, Gibbs Function, and Gibbs Function of Formation
» Entropy Generated During an Adiabatic Chemical Reaction
» MASS CONSERVATION AND MOLE BALANCE EQUATIONS
» Evaluation of Properties During an Irreversible Chemical Reaction
» Criteria in Terms of Chemical Force Potential
» Generalized Relation for the Chemical Potential
» Nonideal Mixtures and Solutions
» Gas, Liquid and Solid Mixtures
» Availability Balance Equation
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