ChE 335 · Lecture 2 · Supplementary

Vapor–Liquid Equilibrium, from a drum at rest to a live column

A working recap of the VLE thermodynamics the rest of the course runs on: bubble and dew points, K values and relative volatility, the flash drum, activity coefficients, equations of state, and the one property-method choice that decides what an ethanol–water column can ship.

Foundations · 2.1 – 2.2

Equilibrium, and the T–xy map

Both a sealed drum and a fed column read d/dt = 0. Watch the difference: one stops exchanging, the other never does.

Press play — same flat reading, opposite stories

2.1

t = 0.0 s

Closed drum · equilibrium

net transfer

–

Continuous flash · steady state

net transfer

–

Left: molecules keep hopping between liquid and vapor, but evaporation and condensation come into balance — the net falls to zero. Right: feed streams through forever; every molecule is still moving, the gauge holds steady above zero. Both read d/dt = 0. Equilibrium sets how far a separation can go; steady state sets how fast.

Every separation runs on an equilibrium

Separation	Phase a	Phase b	Equilibrium	Where in ChE 335
Distillation	liquid	vapor	VLE	Lectures 3–6, this page
Absorption / stripping	liquid	gas	GLE	Lecture 8
Liquid–liquid extraction	liquid	liquid	LLE	Lecture 7
Leaching	solid	liquid	SLE	Leaching deck

Master the vapor–liquid row and the same instinct carries into every other separation.

Drag the temperature: bubble, dew, tie line, lever rule

2.2

Boiling points ~30 °C apart, a fat lens, a modest column does the job. In Non-ideality we squeeze this lens shut and the same split turns nearly impossible.

The ideal rulebook · 2.3 – 2.4

Raoult, Dalton, K and α, in motion

Watch each species escape the liquid at a rate set by its share × its pure pushiness. Those escapes stack into the total pressure — and when the stack reaches the outside pressure, it boils.

Raoult & Dalton, as escaping molecules

2.3

Load

x₁ benzene0.30

Temp °C60.0

P total mmHg–

y₁ vapor–

K₁–

K₂–

α₁₂–

–

Each benzene or toluene molecule leaves the surface at a rate ∝ x_i × P_i^sat(T) (Raoult). The escapes stack into the total pressure P = p₁ + p₂ (Dalton). The vapor's benzene share is y₁ = p₁/P, giving K = y/x = P^sat/P and α = K₁/K₂.

Raoult

p_i = x_i P_i^sat

→

Dalton

P = Σp_i

→

K and α

K_i = P_i^satP , α = K₁K₂

α alone sets the difficulty

2.4

Propylene–propane (α ≈ 1.1) needs 150–200 trays in a tower over 80 m tall — one of the tallest in any petrochemical plant. Benzene–toluene (α ≈ 2.4) needs a fraction of that. The xy curve is the T–xy of 2.2 read off tie line by tie line.

One stage in action · 2.5

The flash drum, flowing

One drum, two equations. Watch the feed stream split: light molecules ride the vapor up, heavy ones drain as liquid. On the xy plot that split is one straight operating line crossing the curve at a single point.

Feed → V + L, live

Example 2.3

watch the streams split by volatility

ψ = V/F0.50

feed z₁0.50

α2.50

x liquid–

y vapor–

V/F–

L/F–

check–

Mass balance: F = V + L and Fz = Vy + Lx. Divide by F and you get one straight operating line z = ψy + (1−ψ)x pivoting on the feed point (z, z). Where it meets the equilibrium curve is the only (x, y) that obeys both the balance and equilibrium at once.

Example 2.3: 1 kmol/h at z₁ = 0.50, half vaporised, α = 2.5 → x₁ = 0.387, y₁ = 0.613. In industry this is the pre-flash before a crude column: the cheapest separation there is — one stage, no reflux. A column is many of these stacked, which is where Lecture 3 begins.

When Raoult breaks · 2.6 – 2.7

Deviations, the azeotrope, and the γ engine

Real molecules notice their neighbours. Slide the affinity and watch the bubble line bend, the azeotrope appear, and γ — the one correction Aspen fits — take shape.

One slider: like/unlike affinity bends the bubble line

2.6

Constant α was a convenience — real data collapses it

γ: measured, and its infinite-dilution limit

2.7

a lone ethanol molecule in water is the most non-ideal state it can occupy

γ ethanol–

γ water–

Measured from one experiment

γ_i = y_iPx_iP_i^sat

Infinite dilutionAs x → 0, γ freezes at γ^∞ — ethanol ≈ 4.5, water ≈ 2.6. That single number is what a γ model is fitted to, and Henry's law is the same limit from the gas side: H = γ^∞P^sat.

Henry's law: a soda bottle with two settings

The four γ models — four ways to picture a liquid

A γ model turns a few fitted numbers into γ at every composition. Each panel below animates the molecular picture behind it — the names on Aspen's menu.

WilsonWILSON

Biased neighbourhoods. A molecule pulls the neighbours it likes closest, so its local shell differs from the bulk. 2 parameters. Polar VLE, fully miscible — but it cannot split a liquid in two.

NRTLNRTL

Clusters. A third dial, the non-randomness α, sets how hard like molecules huddle. 3 parameters. Owns azeotropes and liquid–liquid extraction (Lecture 7).

UNIQUACUNIQUAC

Sizes and surfaces. Each molecule gets a volume r and surface q; mixing costs packing entropy plus contact energy. Great for very different molecule sizes; covers VLE and LLE.

UNIFACUNIFAC

Bags of groups. CH₃, CH₂, OH, H₂O from a published table — no data from you. The fallback for brand-new systems; approximate, never beats a fitted model.

Beyond the slidesThe vapor can be non-ideal too. Two acetic-acid molecules hydrogen-bond into a dimer ring, so the vapor acts heavier than its formula; Aspen repairs it with a Hayden–O'Connell term (NRTL-HOC). HF associates all the way to hexamers.

Engines & the ladder · 2.8 – 2.9

The vapor engine, and one ladder

For light gases and high pressure the vapor itself turns non-ideal, and an equation of state computes both phases. Then every K on this page turns out to be one rung of a single ladder.

Equations of state: the vapor engine

2.8

1 · Fugacity — the pressure molecules act like they have

f = φ P ; φ = f/P

2 · The equilibrium criterion — escaping tendencies match (Lewis 1901)

f̂_i^V = f̂_i^L

In 2.3 it hid inside p_i = x_iP_i^sat; in 2.6 inside γ; here it appears undisguised.

3 · A cubic EoS supplies the P–V–T behaviour ⇒ Z ⇒ φ

cubic EoS ⇒ Z = PVRT ⇒ φ

Equation	Year	What it added	In Aspen	Reach for it
van der Waals	1873	first attraction term a and size b	(teaching)	the idea, not the numbers
Redlich–Kwong	1949	attraction weakening with T	inside hybrids	gas-phase φ at moderate P
Soave–RK	1972	a(T) tuned per substance via ω	RK-SOAVE	gas processing, light-HC VLE
Peng–Robinson	1976	better liquid density, near-critical	PENG-ROB	refinery, cryogenic, LNG

At 1 bar all four read Z near 1; by 60 bar they disagree by several percent (RK 0.680, SRK 0.687, PR 0.661), and percent-level Z errors are real money in compressor duty and column loading.

Describe both phases with the same EoS — the φ–φ method, no γ, no P^sat

K_i = y_ix_i = φ̂_i^Lφ̂_i^V

Why the N₂–CH₄ loops never reach the edgePure nitrogen's critical temperature is 126 K. On any warmer isotherm no pure liquid nitrogen exists, so the bubble and dew curves cannot reach the x = 1 edge; they bend together and merge at a mixture critical point inside the diagram. Cryogenic air separation and LNG nitrogen-rejection columns work right in this near-critical territory — precisely why they are modelled with an equation of state and not activity coefficients.

One criterion, one ladder

2.9

Every K on this page is one rung of a single ladder. At the top sit the two rigorous engines of 2.7 and 2.8. Let the corrections relax to 1 and you slide down to the cheap approximations of the early sections.

The master formula — set γ = 1 and φ̂ = 1 and it collapses to Raoult

K_i = y_ix_i = γ_i f_i°φ̂_i P

Equation of stateK_i = φ̂_i^L / φ̂_i^V

§ 2.8 — both phases from one EoS. Hydrocarbons and light gases, cryogenic to critical. PENG-ROB, RK-SOAVE.

Activity coeff.K_i = γ_iφ_i^L / φ̂_i^V

§ 2.7 — real liquid, real vapor. All mixtures, ambient to near-critical. NRTL, UNIQUAC, Wilson, UNIFAC.

Modified RaoultK_i = γ_i P_i^sat / P

§ 2.6 — vapor allowed ideal, liquid still non-ideal. Non-ideal solutions near ambient pressure.

Raoult (ideal)K_i = P_i^sat / P

§ 2.3–2.5 — both phases ideal. Near-twins at low pressure: benzene–toluene.

HenryK_i = H_i / P

§ 2.7 — dilute supercritical gas in a liquid. Absorption and stripping.

After Seader, Henley & Roper, Table 2.3. The activity-coefficient rung is what Aspen evaluates when you choose NRTL or UNIQUAC; the top rung is what it evaluates for PENG-ROB.

The decision, the way Aspen frames it

Q1 · Are the components polar or hydrogen-bonding (water, alcohols, acids, amines)?

Yes, polar — is the pressure high or near-critical?

Low to moderate P → activity-coefficient method

NRTL, UNIQUAC or Wilson for γ, with an ideal or HOC vapor term. Ethanol–water, the column in 2.10.

High P → activity model with EoS vapor, or an advanced EoS

PSRK or a γ–φ method with Hayden–O'Connell for associating vapors. Acetic-acid recovery.

No, all non-polar (hydrocarbons, light gases)

Equation-of-state method

PENG-ROB or RK-SOAVE for both phases. Cryogenic N₂–CH₄, natural-gas and olefin trains.

Refinery / heavy hydrocarbons → hybrid

Chao–Seader or Grayson–Streed. Crude and light-ends columns.

Where every idea is put to work · 2.10

One ethanol–water column

The property model and the operating pressure decide what the column can ship. Flip the thermo toggle and read the same hardware give two completely different answers — that toggle is the entire argument for this lecture.

Thermodynamics by zone

2.10

Pressure

Pressure torr760

feed z EtOH0.40

distillate x_D EtOH–

bottoms x_B EtOH0.02

condenser T–

reboiler T–

No reflux ratio or tray count appears here: the wall is pure thermodynamics, and Lecture 4 starts stepping trays against it. Temperatures are bubble points from this page's Antoine constants; the xy curve is drawn at the set pressure, where at 760 torr the Wilson crossing computes to 0.884 against the measured 0.894.

Beyond the slidesThe toggle is the entire argument for this lecture. Same feed, same hardware, two property methods, two completely different answers about what the plant can ship. The ideal model promises near-pure ethanol from one column and is simply wrong; the γ model tells the truth and explains the dehydration unit bolted onto every real fuel-ethanol train. Choosing the property method in Aspen — the small menu of section 2.9 — is this decision. Get it wrong and you design a column that cannot exist.

Three things to try

Drop to 100 torr and the wall rises to about 0.98 while the reboiler cools toward 47 °C.
Go below 70 torr and the azeotrope disappears — the basis of vacuum and pressure-swing dehydration.
Move the feed and notice the ceiling does not care: the wall is set by thermodynamics, not by where you inject.

Exam prep · Self-check

Ten quick questions

These target the points students most often mix up across VLE. Answer each one, read the explanation, and re-take until it feels easy.

Bubble vs Dew

liquid known / vapor known

α sets difficulty

gap from the diagonal

γ bends Raoult

the azeotrope appears

One K ladder

rungs relaxed to 1

Answered 0 of 10

Exam prep · Reference

Formula sheet

Every relation from Lecture 2 in reading order, with the units and the one-line reason each is used.

Vapor pressure & the ideal rulebook

2.3

Antoine (Basmadjian Table 6.1; mmHg, °C, log₁₀)

log₁₀ P_i^sat = A − BT + C

Raoult, Dalton, and the K value

p_i = x_i P_i^sat ; P = Σ p_i ; K_i = y_ix_i = P_i^satP

Relative volatility and the constant-α curve

α₁₂ = K₁K₂ = y(1−x)x(1−y) ⇒ y = αx1+(α−1)x

Bubble / dew / flash

2.2 & 2.5

Bubble point (liquid known) & dew point (vapor known): goal seek T

Σ K_ix_i = 1 ; Σ y_iK_i = 1

Flash: mass balance and the operating line (ψ = V/F)

F = V+L ; Fz = Vy+Lx ⇒ z = ψy+(1−ψ)x

Lever rule (vapor fraction from a tie line)

VF = z − xy − x

Non-ideality: γ, Henry, fugacity

2.6 – 2.8

Modified Raoult, measured γ, and excess Gibbs energy

y_iP = γ_ix_iP_i^sat ; γ_i = y_iPx_iP_i^sat ; G^ERT = Σ x_i ln γ_i

Henry's law (dilute limit) and its K value

H = γ^∞P^sat ; p_i = H_ix_i ; K_i = H_i/P

Fugacity, equal-fugacity criterion, and Z

f = φP ; f̂_i^V = f̂_i^L ; Z = PVRT

The K-value ladder (master formula)

2.9

Set γ = 1 and φ̂ = 1 and it collapses to Raoult

K_i = γ_i f_i°φ̂_i P → φ̂ᴱ/φ̂ᶦ · γPᵀᵃᵗ/P · Pᵀᵃᵗ/P · H/P

Antoine constants used (log₁₀, mmHg, °C)

Component	A	B	C
benzene	6.89272	1203.531	219.888
toluene	6.95805	1346.773	219.693
ethanol	8.11220	1592.864	226.184
water	7.96681	1668.210	228.000
acetone	7.11714	1210.595	229.664
chloroform	6.95465	1170.966	226.252

Common mistakes to avoid

Swapping the two summation tests. Bubble point sums K_ix_i; dew point sums y_i/K_i. If your "bubble point" lands above your dew point, this is what happened.
Assuming α is constant. Fine for benzene–toluene; dangerous for ethanol–water, where measured α runs from about 9 down to 1.0 at the azeotrope.
Reading past the azeotrope. Above x = 0.894 (ethanol–water, 1 atm) the vapor is poorer than the liquid; ordinary distillation runs backwards. No reflux or tray count passes the wall.
Choosing the wrong property method. An ideal model on ethanol–water promises near-pure ethanol from one column — and sizes a column that cannot exist. Polar → γ model; non-polar / high-P → equation of state.
Confusing equilibrium and steady state. Both read dx/dt = 0. Equilibrium has no gradients (sets how far); steady state holds gradients alive by flow (sets how fast).

Sources and data

Antoine constants: Basmadjian, Mass Transfer: Principles and Applications, CRC Press, Table 6.1 (reproduce Example 2.1 = 214.702 mmHg exactly). Raoult, Dalton, K and α: Smith, Van Ness & Abbott (SVA) Ch. 13.3 and Seader §2.2. Flash mass balance: Seader Ch. 4 (Example 2.3). Excess Gibbs energy, γ models and the γ–φ criterion: SVA §13.1–13.2. Henry's law and H = γ^∞P^sat: SVA §13.3; Henry's constants (CO₂ 1655, CH₄ 39014, O₂ 45516, N₂ 85343 atm): Sander, Atmos. Chem. Phys. 15 (2015). Equal-fugacity criterion and φ: SVA Ch. 10. Cubic EoS (van der Waals → RK → SRK → Peng–Robinson), Z and φ: SVA §3.6, 13.7 (CO₂: T_c=304.13 K, P_c=73.77 bar, ω=0.224). The K-value ladder: Seader, Henley & Roper Table 2.3. Ethanol–water VLE: Mertl, Coll. Czech. Chem. Commun. 37 (1972); Wilson parameters a₁₂=325.08, a₂₁=953.28 cal/mol (Perry 13-2). Azeotrope vs. pressure: Seader Fig. 11.22 (after Horsley). Course: Chaiwasu, J., CHE335 Mass Transfer and Equipment Design, KMUTT, 2025/2026. Every diagram is computed client-side from these constants.