ChemE 123 LE1 Conceptual

Properties

Quantities that are characteristic of the system

T, P, xi, yi, density, H, U, V

Examples of properties

Intensive property

Independent of quantity of material

Extensive property

Dependent on quantity of material

State

Condition in which system is found at any given time; established by fixed values of intensive properties of a substance

Phase

Homogeneous region of matter

Phase diagram

Can only know definite phase if in equilibrium

Phase rule

State of equilibrium is defined by a set of intensive properties

Degrees of Freedom

No. of independent (intensive) variables that must be arbitrarily fixed to establish state of system in equilibrium

Gibb’s Phase Rule

F = 2 - pi + N (pi phases, N components); F = No. of independent intensive variables - No. of independent equations

Gibb’s Phase Rule

No. of variables = 2 + (N-1)(pi); No. of equations = (pi-1)(N)

Effect of liquid miscibility on pi

Miscible = 1, Immiscible = 1+

Pi of solid components

1 different solid, pi = 1

Three (3)

Max. no. of variables that can be fixed to a binary system [P, T, and 1 mole/mass fraction]

Effect of pi on Degrees of Freedom

↑ pi ↓ F

Duhem’s Rule

“For any closed system, formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined when any 2 independent variables are fixed.”

Duhem’s Rule

Applied to closed systems in equilibrium, in which intensive and extensive states are kept as constant.

Duhem’s Rule

2 variables can be intensive or extensive. When F = 1, at least one of the two variables must be extensive, and when F = 0, both must be extensive

Txy Diagram

Lower line = bubble line, Upper line = dew line

Pxy Diagram

Lower line = dew line, Upper line = bubble line

Pxy/Txy Diagrams

Lower BP = more volatile, Higher P = more volatile (harder to compress)

Bubble point

• First bubble of vapor from a liquid mixture appears at a given composition

• Last bubble of a vapor mixture disappears at a given composition

Dew point

• First drop of liquid from a vapor mixture appears at a given composition

• Last liquid drop of a liquid mixture disappears at a given composition

Azeotrope

Point at which the equilibrium liquid and vapor compositions are equal at an intermediate mixture composition

Azeotrope

• constant-boiling mixtures

• separation by distillation not possible

• crossed 45-deg line (xy-diagram)

Types of azeotropes

1. Maximum-pressure, Minimum-boiling

2. Minimum-pressure, Maximum-boiling

Raoult’s law

yiP = xiPisat

Raoult’s Law

• Assumptions

  • Vapor phase is an ideal gas (negligible intermolecular interactions)

    • Low to moderate pressures

Raoult’s Law

• Assumptions

  • Liquid phase is an ideal solution (similar structures)

    • mixture of isomers

    • mixture of adjacent members of a homologous series

Bubble P calc

1. ΣxiPisat = P

2. yi = (xiPisat)/P

Dew P calc

1. 1/(Σyi/Pisat) = P

2. xi = (yiP)/(Pisat)

Bubble T calc

1. ΣxiTisat = T0

2. ΣxiPisat = P where Pisat is in exponential form

3. Shift-solve with T = T0

4. yi = (xiPisat)/P

Dew T calc

1. ΣxiTisat = T0

2. 1/(Σyi/Pisat) = P where Pisat is in exponential form

3. Shift-solve with T = T0

4. xi = (yiP)/(Pisat)

Henry’s Law

yiP = xiHi

Henry’s Law

• Assumptions

  • vapor phase is an ideal gas

    • low to moderate pressures

  • species i is a very dilute solute in the liquid phase

Modified Raoult’s Law

yiP = γixiPisat

Modified Raoult’s Law

• Assumptions

  • Vapor phase is an ideal gas

    • low to moderate pressures

Activity coefficient (γ)

Takes account deviation of liquid phase from ideality; function of composition and/or temperature

Modified Bubble P calc

1. Solve: P1sat(T), P2sat(T), A, & 𝛾𝑖(x,T)

2. Σ𝛾𝑖xiPisat = P

3. yi = 𝛾𝑖xiPisat/P

Modified Dew P calc

1. Solve: P1sat(T), P2sat(T), A

2. Table: γ1, γ2, y1/γ1P1sat, y2/γ2P2sat, Pressure, x1, x2

3. Iteration 0: γ1 = γ2 = 1 → y1/γ1P1sat, y2/γ2P2sat → P = 1/(y1/γ1P1sat + y2/γ2P2sat) → xi0 = (yi/γiPisat)*P

4. Iteration 1: γi = f(xi0, T) → y1/γ1P1sat, y2/γ2P2sat → P = 1/(y1/γ1P1sat + y2/γ2P2sat) → xi = (yi/γiPisat)*P

5. Repeat iterations

Modified Bubble T calc

1. Solve: T1sat(P), T2sat(P), T0 = x1T1sat + x2T2sat

2. Table: Temp, A, γ1, γ2, α11, α21, x1γ1α11, x2γ2α21, P1sat, P2sat, y1, y2

3. Iteration 0: Temp = T0 → A0 = f(T0) → γi0 = f(x,T0) → αii = 1, αik = exp(Ai - (Bi/(Ci+T0)) - Ak + (Bk/(Ck+T0))) → x1γ1α11, x2γ2α21 → P1sat = P/(x1γ1α11 + x2γ2α21)

4. Iteration 1: Temp = T(P1sat) → A = f(T) → γi = f(x,T) → αii = 1, αik = exp(Ai - (Bi/(Ci+T)) - Ak + (Bk/(Ck+T))) → x1γ1α11, x2γ2α21 → P1sat = P/(x1γ1α11 + x2γ2α21) → P2sat = f(T)

5. Repeat iterations until convergence

6. yi = 𝛾𝑖xiPisat/P

Modified Dew T calc

1. Solve: T1sat(P), T2sat(P), T0 = y1T1sat + y2T2sat

2. Table: α11, α21, x1, x2, A, γ1, γ2, y1/γ1α11, y2/γ2α21, P1sat, Temp, P2sat

3. Iteration 0: γi0 = 1 → Temp = T0 → Pisat0 = f(T0)

4. Iteration 1: αii = 1, αik = Pisat0/Pksat0 → xi = yiP/γi0Pisat0 → A = f(T0) → γi = f(x,T) → y1/γ1α11, y2/γ2α21, P1sat = P*(y1/γ1α11 + y2/γ2α21), Temp = f(P1sat), P2sat = f(T)

5. Repeat iterations until convergence

K-value

Partition ratio; preference to vapor over liquid

K-value (formula)

Ki = yi/xi

K-value (Raoult’s Law)

K = Pisat/P

K-value (Modified Raoult’s Law)

K = γiPisat/P

Bubble point (K-values)

ΣKixi = 1

Dew point (K-values)

Σyi/Ki = 1

Flashing

Partial evaporation; liquid to liquid and vapor phase as a result of pressure reduction

Gibbs Free Energy

Easy to measure because of T&P experimental values

Fundamental Property Relation

d(nG) = (nV)dP - (nS)dT + Σµidni (single-phase fluid system; variable mass and composition)

Chemical potential

µi = [∂(nG)/∂ni]_P,T,nj

Chemical potential

Partial molar Gibbs energy; measures response of total Gibbs energy of a solution to addition of a differential amount of species i at constant T&P

Gibbs energy as generating function

Provides the means for calculation of all other thermodynamic properties by simple mathematical operations and implicitly represents complete property information

Functions generated by Gibbs free energy

1. V = [∂G/∂P]_T,x

2. S = -[∂G/∂T]_P,x

3. H = G - T[∂G/∂T]_P,x

4. U = H - P[∂G/∂P]_T,x

5. A = T[∂G/∂T]_P,x - P[∂G/∂P]_T,x

Phase equilibria

µi,alpha = µi,beta = … = µi,pi (for closed system, multiple pi phases with N number of species at same T and P in equilibrium)

Phase equilibria

Multiple phases at same T&P are in equilibrium when the chemical potential of each species is the same in all phases

Solution properties

Properties of WHOLE solution (usually per mole of solution) [ex. V, U, H, S, G]

Partial properties

Properties of species of solution when it’s MIXED with other species [ex. Vi_bar, Ui_bar, Hi_bar, Si_bar, Gi_bar]

Pure species properties

Properties of species when it’s ALONE in solution [ex. Vi, Ui, Hi, Si, Gi]

Pure-species component property

when xi = 1, Mi = Mi_bar = M

Infinite dilution partial property

when xi = 0, Mi_bar = Mi_bar,∞

Partial molar property

Mi_bar = [∂(nM)/∂ni]_P,T,nj

Partial molar property

Measures response of solution property nM to addition at constant T&P of differential amount of species i to a finite amount of solution

Summability relation

M = ΣxiMi_bar; nM = ΣniMi_bar

Gibbs/Duhem equation

(∂M/∂P)_T,x dP + (∂M/∂T)_P,x dT = Σxid(Mi_bar); at constant T&P, Σxid(Mi_bar) = 0

Relations in binary solution

• M1_bar = M + x2(dM/dx1)

• M2_bar = M - x1(dM/dx1)

• x1(dM1_bar/dx1) + x2(dM2_bar/dx1) = 0

Equation analog

• Solution: nH = nU + P(nV); Partial: Hi_bar = Ui_bar + PVi_bar

• Solution: nA = nU - T(nS); Partial: Ai_bar = Ui_bar - TSi_bar

• Solution: nG = nH - T(nS); Partial: Gi_bar = Hi_bar - TSi_bar