Session 3 · optional reference
30 September 2026
Return to the Week 3 independent lesson or start page.
Use these optional details when a question from your investigation needs them. They are outside the required 180-minute holiday lesson. There is no compulsory additional homework.
Possible extensions include NRTL tie-line fitting in Lab 05, paired bootstrap from independent replicates, or feasible three-phase amounts in Lab 06. Complete the core Lab 03 and Lab 04 tasks first.
The Lab 03 definitions state the equations, thresholds, endpoint assumptions and implementation limits.
Primary references: NIST area and Herington, point test, modeling test.
In Lab 04, select Margules · 1 parameter, A = 3. Choose Binodal & spinodal map in the View menu.
The binodal marks coexisting compositions. The spinodal marks zero local curvature. Between a binodal and a spinodal, a homogeneous liquid can have positive curvature while a finite split lowers its Gibbs energy.
These equilibrium constructions do not predict nucleation barriers or separation times.
For a trial composition w relative to reference z, TPD(w;z)=g(w)-g(z)-g'(z)(w-z)
A negative value identifies a Gibbs-energy-lowering perturbation. Local curvature examines only the neighborhood of z.
Lab 04 evaluates the binary model; it is not a general multicomponent TPD package.
Positive activity deviations alone do not guarantee LLE. The mixing Gibbs-energy shape must support a split.
The labs include Margules and NRTL options with different shapes. Standard Wilson behavior is discussed in the reference deck, not implemented as an LLE solver here.
Tie-line data constrain the coexistence compositions. Fitting must reproduce distinct phases and a supported common tangent.
A small tie-line error at one T does not establish a unique parameter set or a temperature law.
Lab 05 retains the input observations and checks stability of the fitted result.
\hat f_i^V=\hat f_i^{L\alpha}=\hat f_i^{L\beta}
For a binary nonreacting system with three phases, the phase rule gives one intensive degree of freedom before imposing T or P.
Phase compositions and phase amounts are different unknowns. Specifying T and P does not always uniquely determine all amounts.
Given three fixed binary phase compositions, f_\alpha+f_\beta+f_V=1 z_1=f_\alpha x_1^\alpha+f_\beta x_1^\beta+f_Vy_1
There are two independent constraints on three fractions. Nonnegativity bounds the feasible family. Another independent extensive constraint may select a member.
Compare the supported liquid and vapor states. Move within the admissible phase-fraction family and verify the overall balance.
Identify a state where the vapor fraction is constrained by nonnegativity. Explain why equal fugacities alone cannot determine every fraction.
A fitted liquid model needs a matching component order, parameter convention and temperature before reuse.
VLE also requires pure-component vapor pressures and a vapor model. LLE experimental pressure is not a vapor-pressure correlation.
A new phase calculation brings new assumptions and validation needs.
Module reference deck · Lab assumptions and sources
The worked examples use synthetic inputs. Each lab states its supported models and validity limits.
For a binary molar property m(x), at fixed T and P: \bar m_1=m+(1-x)m',\qquad\bar m_2=m-xm'
The tangent intercepts at x=1 and x=0 give the two partial molar properties.
Paper exercise: use m(x)=x(1−x), in an arbitrary consistent molar unit, at x=0.25. Compute both intercepts and check x\bar m_1+(1-x)\bar m_2=m.
Reading locator: supplied textbook contents, 11.1–11.2 and 14.11.