ChemE 123 LE1 ProbSol

Txy Diagram

Lower line = bubble line, Upper line = dew line

Pxy Diagram

Lower line = dew line, Upper line = bubble line

Raoult’s law

yiP = xiPisat

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

Modified Raoult’s Law

yiP = γixiPisat

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

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

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

Pure-species component property

when xi = 1, Mi = Mi_bar = M

Infinite dilution partial property

when xi = 0, Mi_bar = Mi_bar,∞

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