ChemE 123 LE1 Conceptual

Quantities that are characteristic of the system

Examples of properties

Independent of quantity of material

Dependent on quantity of material

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

Homogeneous region of matter

Can only know definite phase if in equilibrium

State of equilibrium is defined by a set of intensive properties

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

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

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

Miscible = 1, Immiscible = 1+

1 different solid, pi = 1

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

↑ pi ↓ F

“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.”

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

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

Lower line = bubble line, Upper line = dew line

Lower line = dew line, Upper line = bubble line

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

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

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

• 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

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

• constant-boiling mixtures

• separation by distillation not possible

• crossed 45-deg line (xy-diagram)

1. Maximum-pressure, Minimum-boiling

2. Minimum-pressure, Maximum-boiling

yiP = xiPisat

• Assumptions

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

    • Low to moderate pressures

• Assumptions

  • Liquid phase is an ideal solution (similar structures)

    • mixture of isomers

    • mixture of adjacent members of a homologous series

1. ΣxiPisat = P

2. yi = (xiPisat)/P

1. 1/(Σyi/Pisat) = P

2. xi = (yiP)/(Pisat)

1. ΣxiTisat = T0

2. ΣxiPisat = P where Pisat is in exponential form

3. Shift-solve with T = T0

4. yi = (xiPisat)/P

1. ΣxiTisat = T0

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

3. Shift-solve with T = T0

4. xi = (yiP)/(Pisat)

yiP = xiHi

• Assumptions

  • vapor phase is an ideal gas

    • low to moderate pressures

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

yiP = γixiPisat

• Assumptions

  • Vapor phase is an ideal gas

    • low to moderate pressures

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

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

2. Σ𝛾𝑖xiPisat = P

3. yi = 𝛾𝑖xiPisat/P

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

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

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

Partition ratio; preference to vapor over liquid

Ki = yi/xi

K = Pisat/P

K = γiPisat/P

ΣKixi = 1

Σyi/Ki = 1

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

Easy to measure because of T&P experimental values

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

µi = [∂(nG)/∂ni]_P,T,nj

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

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

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

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

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

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

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

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

when xi = 1, Mi = Mi_bar = M

when xi = 0, Mi_bar = Mi_bar,∞

Mi_bar = [∂(nM)/∂ni]_P,T,nj

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

M = ΣxiMi_bar; nM = ΣniMi_bar

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

• M1_bar = M + x2(dM/dx1)

• M2_bar = M - x1(dM/dx1)

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

• 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