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