How do I calculate activity coefficients using modified Raoult's law?

Using modified Raoult's law (y_i * P = x_i * gamma_i * P_i_sat), rearrange for each component's activity coefficient: gamma_i = (y_i * P) / (x_i * P_i_sat). You need the experimental VLE data (x_i, y_i, P) and the saturation pressures at the given temperature. Calculate gamma for each component separately, then use these values to fit activity coefficient models such as Margules or van Laar.

What is the difference between the 1-parameter Margules and the van Laar model?

The 1-parameter Margules model uses a single constant A and assumes symmetric behaviour: ln(gamma_1) = A*x_2^2 and ln(gamma_2) = A*x_1^2. The van Laar model uses two parameters (A12 and A21) and handles asymmetric systems better. For van Laar, the parameters are found from the infinite-dilution activity coefficients: A12 = ln(gamma_1_inf) and A21 = ln(gamma_2_inf). Always check which model fits your experimental gamma values more accurately.

How do I solve a bubble point temperature calculation?

Bubble point calculations require iterative solution. Start with an initial temperature guess (weighted average of pure component boiling points works well). Calculate saturation pressures using the Antoine correlation at that T. Evaluate activity coefficients at the given liquid composition. Check if the sum of y_i = sum(x_i * gamma_i * P_i_sat / P) equals 1.0. If not, adjust T and repeat. Convergence is usually reached in 5-10 iterations when the temperature step is small.

How do I relate equilibrium constant to temperature using Gibbs free energy data?

Calculate the Gibbs free energy of reaction (delta_G_rxn) at multiple temperatures using: delta_G_rxn = sum(nu_i * delta_g_f_i). Then use delta_G_rxn = -RT*ln(Ka) to get Ka at each temperature. Fit ln(Ka) vs 1/T to get a linear or polynomial expression relating Ka and T. For Problem 4 in this sheet, you have data at nine temperatures, so regression gives a reliable expression.

Chemical Engineering Thermodynamics Problem Sheet

VLE · ACTIVITY COEFFICIENTS · BUBBLE POINT · EQUILIBRIUM CONSTANT · GIBBS FREE ENERGY · EQUILIBRIUM CONVERSION

How to Approach Each Question

Five problems, two CILOs, and a mark sheet that will punish rushed working. The questions span vapour-liquid equilibrium, activity coefficient modelling, bubble point iteration, Gibbs free energy-based equilibrium constants, and equilibrium conversion at operating conditions. Here is exactly how to set up each one — step by step — before you put pen to paper.

11–14 min read Chemical Engineering / Thermodynamics EN8701T — CILO 3 & CILO 4 100 Marks / 30% Weightage

Custom University Papers — Academic Writing Team

Technical guidance for engineering and science students on thermodynamics, process calculations, and assignment structure. See also: Engineering assignment help, complex technical assignment assistance, and chemistry homework help.

This problem sheet covers two distinct competency areas: vapour-liquid equilibrium modelling (CILO 3) and chemical reaction equilibrium (CILO 4). The marks are split unevenly — Problems 4 and 5 together account for 50 marks — so if you lose the thread on the Gibbs energy section, that is half your grade. Read the setup for each problem before you write a single number. Units, sign conventions, and the correct form of each equation matter more here than in most assessments.

Modified Raoult’s Law Activity Coefficient Models Bubble Point Iteration Gibbs Free Energy & Ka Equilibrium Conversion Units & Common Errors Hand-Written Requirements

What This Guide Covers

Assignment Overview and Mark Distribution Problem 1 — VLE, Activity Coefficients, Margules Problem 2 — Van Laar Model Fitting Problem 3 — Bubble Point Temperature Problem 4 — Ka vs T from Gibbs Data Problem 5 — Equilibrium Conversion Where Students Lose Marks Hand-Written Submission Requirements Frequently Asked Questions

Assignment Overview and Mark Distribution

Before anything else, understand what each problem is actually worth and which CILO it assesses. The mark split tells you where to spend your time.

100 Total Marks — 30% of Final Grade

5 Problems Across CILOs 3 and 4

50 Marks From Problems 4 & 5 Alone

10 Pages — Must Be Hand-Written

Problem	Topic	CILO	Marks	Key Equations
1	VLE — activity coefficients, g^E/RT, 1-parameter Margules	CILO 3	15 (5+5+5)	Modified Raoult’s law, Margules equation
2	VLE dataset — activity coefficients + van Laar model fitting	CILO 3	20	Modified Raoult’s law, van Laar equations
3	Bubble point temperature — methanol/water system	CILO 3	15	Antoine correlation, modified Raoult’s law, iterative method
4	Equilibrium constant vs T from Gibbs free energy data	CILO 4	25 (20+5)	ΔG°_rxn = −RT ln K_a
5	Equilibrium conversion from experimental K_a expression	CILO 4	25 (20+5)	K_a expression, stoichiometric table, fugacity/pressure terms

Hand-Written and Scanned — Not Typed, Not Printed

The assignment explicitly states it must be hand-written. Marks are deducted if it is not. Scan or photograph each page clearly, compile into a single PDF, and upload to Moodle or the instructor’s email. Make sure equations, subscripts, and Greek letters are legible in the scan — blurry photos of pencil working will cost you presentation marks even if the method is correct.

Problem 1 — VLE, Activity Coefficients, g^E/RT, and 1-Parameter Margules

You have a single data point: x₁ = 0.27, y₁ = 0.37, P = 110.1 kPa at 20°C. Saturation pressures are P₁^sat = 90.2 kPa and P₂^sat = 120.4 kPa. Three sub-questions, five marks each.

Part A — Activity Coefficients

Apply Modified Raoult’s Law to Each Component Separately

Modified Raoult’s law states: y_iP = x_iγ_iP_i^sat. Rearrange for γ_i:

γ₁ = (y₁ × P) / (x₁ × P₁^sat) and γ₂ = (y₂ × P) / (x₂ × P₂^sat)

Remember: y₂ = 1 − y₁ = 0.63 and x₂ = 1 − x₁ = 0.73. Calculate both γ values numerically and show each step. Both should be close to 1 but not exactly 1 — this is a non-ideal system. If you get exactly 1 for either component, check you used the right y and x values.

Part B — g^E/RT

Use the Excess Gibbs Energy Relationship at This Composition

The molar excess Gibbs energy is related to activity coefficients by:

g^E/RT = x₁ ln(γ₁) + x₂ ln(γ₂)

Plug in x₁, x₂, and the γ values you calculated in Part A. Take natural logs — not log₁₀. The result is dimensionless. Show the numerical substitution clearly rather than just writing the final number. Five marks here means the marker wants to follow your working.

Part C — 1-Parameter Margules Equation

Identify the Margules Constant A from One of the Activity Coefficients

The one-parameter Margules model assumes symmetric non-ideality. For a binary system:

ln(γ₁) = A·x₂² and ln(γ₂) = A·x₁²

You have only one data point, so calculate A from each equation and note whether they agree. In a strictly symmetric system they should be the same. Use the average if they differ slightly. Clearly state your value of A with units (it is dimensionless) and show how it was derived from your Part A γ values. Do not use a different value of A for each component — the whole point of the one-parameter model is a single constant.

Problem 2 — n-Heptane / Toluene — Van Laar Model Fitting

Twenty marks. You have 9 interior data points (excluding pure component endpoints) and Antoine-type saturation pressure correlations. This is the most calculation-heavy problem in the CILO 3 section. Organise your working in a table.

Convert Temperature Before Using the Antoine Correlations

The correlations are given in Kelvin but the experimental data is at 25°C. T = 25 + 273.15 = 298.15 K. Calculate P₁^sat and P₂^sat at this temperature first. Check the valid range stated in the problem: 323–523 K for both. 298.15 K is below this range — that is a given constraint of the problem and the instructor knows this, so use the correlations anyway. Note the extrapolation in your working.

Calculate Activity Coefficients at Each Interior Data Point

For each of the 9 x₁ values (0.029 through 0.934), apply modified Raoult’s law to get γ₁ and γ₂. Use the corresponding y₁, y₂ = 1−y₁, and P values from the table. Build a results table with columns: x₁, y₁, P, γ₁, γ₂. This table is your primary output and makes the van Laar fitting much easier to show systematically.

Fit the Van Laar Model Parameters A₁₂ and A₂₁

The van Laar equations are: ln(γ₁) = A₁₂ / (1 + A₁₂x₁/A₂₁x₂)² and the symmetric form for γ₂. To find A₁₂ and A₂₁, linearise the model using the Carlson-Colburn method: plot 1/√(ln γ₁) vs x₁/x₂ and 1/√(ln γ₂) vs x₂/x₁. The slopes and intercepts give A₁₂ and A₂₁ directly. Alternatively, use data at the infinite-dilution limits (x₁ → 0 and x₁ → 1) to extract the parameters from your calculated γ values at the lowest and highest compositions.

Infinite-Dilution Shortcut for Van Laar Parameters

As x₁ → 0, ln(γ₁^∞) = A₁₂. As x₂ → 0 (x₁ → 1), ln(γ₂^∞) = A₂₁. Your data at x₁ = 0.029 is close to the first limit and x₁ = 0.934 is close to the second. Use these points to get initial estimates of A₁₂ and A₂₁, then verify they give a reasonable fit across the full composition range. This is faster than regression and acceptable at this level.

Problem 3 — Bubble Point Temperature for Methanol / Water at 101.3 kPa

Fifteen marks. Bubble point calculation at fixed pressure. You are given both the Antoine correlations and the activity coefficient expressions — so the γ_i are not constant here, they depend on composition and temperature. This makes the iteration slightly more involved than a simple Raoult’s law bubble point.

How to Set Up the Iteration

The Bubble Point Condition Is: Σ(x_i γ_i P_i^sat/P) = 1

The feed is 20 mol% methanol (x₁ = 0.2) and 80 mol% water (x₂ = 0.8) at P = 101.3 kPa. You need to find T such that this sum equals exactly 1.0. The γ values are given as functions of x₁ and x₂ only — composition is fixed — so the only variable you iterate on is T through the Antoine equations.

Iteration procedure:
1. Guess an initial T. A reasonable start: the boiling point of water at 101.3 kPa is ~373 K. Methanol boils lower (~338 K). For a water-rich mixture try 360–365 K as a first guess.
2. Evaluate P₁^sat(T) and P₂^sat(T) using the Antoine correlations. Note these use T in Kelvin directly in the denominator expressions as written.
3. Evaluate γ₁ and γ₂ using the given expressions at x₁ = 0.2, x₂ = 0.8.
4. Compute Σ = x₁γ₁P₁^sat/P + x₂γ₂P₂^sat/P.
5. If Σ > 1, T is too high (saturation pressures are too large). If Σ < 1, T is too low. Adjust and repeat. 5–8 iterations typically converge to ±0.1 K.

Check Your Antoine Equation Format Carefully

The correlations given are written as ln(P^sat) = A − B/(C + T). The constants A, B, C are specific to each component and differ from the standard Antoinecorrection for log₁₀. Do not mix up the two forms. Also check whether the denominator term contains +T or −T (the forms in the problem have −T inside the denominator bracket). Substitute values mechanically and double-check your P^sat results against any known boiling point reference before proceeding with the iteration.

Once you have converged on T, calculate the vapour compositions: y_i = x_iγ_iP_i^sat/P. Confirm they sum to 1.0 as a check. Report both T (in K and °C) and the y values.

Problem 4 — Equilibrium Constant vs Temperature from Gibbs Free Energy Data

Twenty-five marks, split 20+5. The reaction is methylcyclohexane ⇌ toluene + 3H₂. You have tabulated Gibbs free energies of formation for the reactant and products at nine temperatures from 298 K to 1000 K.

Calculate ΔG°_rxn at Each Temperature

Apply ΔG°_rxn = Σ ν_i ΔG°_f,i. Stoichiometry: ν_A = −1 (reactant), ν_B = +1 (toluene), ν_H₂ = +3. You are only given data for A and B — note that ΔG°_f for H₂ is zero by convention (it is an element in its standard state). So ΔG°_rxn(T) = ΔG°_f,B(T) − ΔG°_f,A(T). Calculate this at all nine temperatures and set up a table.

Convert ΔG°_rxn to K_a at Each Temperature

The relationship is ΔG°_rxn = −RT ln(K_a), so K_a = exp(−ΔG°_rxn/RT). Use R = 8.314 J/(mol·K). Critical: make sure ΔG°_rxn is in J/mol when multiplied with R in J/(mol·K) — or convert both to kJ consistently. A single unit inconsistency here will give K_a values that are off by factors of thousands.

Develop the K_a vs T Expression

Plot ln(K_a) vs 1/T (K⁻¹). For many reactions this is approximately linear — the slope gives −ΔH°/R and the intercept gives ΔS°/R via the van’t Hoff equation. Fit a straight line through your nine data points (linear regression by hand: use the slope formula Σ(xy − x̄ȳ)/Σ(x² − x̄²) where x = 1/T and y = ln K_a). Write the final expression as ln(K_a) = a/T + b, then exponentiate to give K_a as a function of T.

Evaluate K_a at 650 K Using Your Expression

The 5-mark sub-question simply asks you to substitute T = 650 K into the expression you derived. Show the substitution clearly. Also calculate K_a(650 K) directly from the ΔG° approach to cross-check — they should be close but not identical since your expression is a linear regression fit. If the two values differ significantly, your regression has an error.

Key Relationship for CILO 4

ΔG°_rxn = −RT ln(K_a) — Sign Convention Matters

A negative ΔG°_rxn means spontaneous reaction under standard conditions → K_a > 1. A positive ΔG°_rxn means non-spontaneous → K_a < 1. The dehydrogenation of methylcyclohexane is endothermic and becomes more favourable at higher temperatures — so K_a should increase with T. If your K_a values decrease as T increases, you have a sign error somewhere in ΔG°_rxn. The property data from NIST Webbook (webbook.nist.gov) can be used to sanity-check your ΔG°_f values at 298 K against published standards.

Problem 5 — Equilibrium Conversion and Composition at 5 bar, 360°C

Twenty-five marks (20+5). The experimental K_a expression is given as K_a = 3600 exp[−217650/R × (1/T − 1/650)], with R in kJ/(kmol·K). Pure methylcyclohexane feed.

Step 1 — Evaluate K_a at Operating Conditions

Substitute T = 360°C = 633.15 K, P = 5 bar Into the Given Expression

Convert temperature to Kelvin first: T = 360 + 273.15 = 633.15 K. Then calculate K_a:

K_a = 3600 × exp[−217650/R × (1/633.15 − 1/650)]

Use R = 8314 kJ/(kmol·K) as stated. The exponent will be negative and small in magnitude. Work through it carefully — the difference (1/633.15 − 1/650) is a small negative number, so the overall exponent is positive, meaning K_a > 3600. If your K_a comes out less than 3600, recheck the sign of (1/T − 1/650) at T < 650 K.

Step 2 — Set Up the Stoichiometric Table

Define Equilibrium Conversion ξ and Write Mole Fractions

Basis: 1 mole of pure methylcyclohexane (A). Let ξ = equilibrium conversion (fraction of A that reacts). The reaction is A ⇌ B + 3H₂.

Moles at equilibrium:
A: 1 − ξ B: ξ H₂: 3ξ Total: 1 + 3ξ

Mole fractions:
y_A = (1 − ξ)/(1 + 3ξ) y_B = ξ/(1 + 3ξ) y_H₂ = 3ξ/(1 + 3ξ)

Step 3 — Write K_a in Terms of Mole Fractions and Pressure

Account for the Pressure Dependence — This Reaction Produces More Moles

For an ideal gas mixture, K_a = K_y × (P/P°)^Δν where Δν = 1 + 3 − 1 = 3 (net moles of gas produced). Take P° = 1 bar as the standard state pressure.

K_y = y_B × y_H₂³ / y_A = [ξ/(1+3ξ)] × [3ξ/(1+3ξ)]³ / [(1−ξ)/(1+3ξ)]

K_a = K_y × (P/P°)³ = K_y × 5³ = K_y × 125

Rearrange to: K_y = K_a/125. Substitute the mole fraction expressions in terms of ξ and solve for ξ. This is a nonlinear equation — solve numerically by trial and error or by rearranging and iterating. A reasonable first guess is ξ = 0.7–0.9 for a high-temperature dehydrogenation.

The 5-Mark Sub-Question: Equilibrium Composition

Once you have ξ, calculate the equilibrium mole fractions y_A, y_B, and y_H₂ directly from the expressions in your stoichiometric table. Report them as percentages or fractions — both are acceptable but be consistent. Check they sum to 1.00. If they do not, recalculate. Five clean marks for correct substitution of ξ into three expressions.

Where Students Lose Marks

Using log₁₀ Instead of ln in the Gibbs/Ka Calculation

ΔG° = −RT ln(K_a) uses natural logarithm. The Antoine correlations in this problem also use ln (not log₁₀). Mixing the two kills your K_a values and the downstream conversion calculation.

Write “ln” Explicitly at Every Step

In hand-written work, write ln(…) in full every time it appears. Do not abbreviate to log without specifying the base. Markers deduct marks for ambiguous notation.

Forgetting the (P/P°)^Δν Term in Problem 5

Students frequently write K_a = K_y without the pressure correction. For a reaction where Δν ≠ 0, this is wrong and gives the incorrect ξ. At 5 bar and Δν = 3, the correction factor is 125 — not negligible.

State Your Standard Pressure and Δν Explicitly

Write out Δν = products − reactants = (1+3) − 1 = 3, state P° = 1 bar, and show K_a = K_y(P/P°)³ before substituting numbers. This earns method marks even if arithmetic slips.

Skipping Unit Conversion in the Antoine Equation

The correlations for n-heptane/toluene specify a Kelvin range. Temperature must be in Kelvin before substitution. Substituting 25 (Celsius) instead of 298.15 K produces dramatically wrong saturation pressures.

Write the Temperature Conversion As Part of Your Working

Show “T = 25 + 273.15 = 298.15 K” explicitly at the start of Problems 2 and 3 before any calculation. This is one line and secures the unit conversion step for the marker.

Inconsistent R Values Across Problems 4 and 5

Problem 4 uses R = 8.314 J/(mol·K). Problem 5 states R in kJ/(kmol·K). The numerical value is 8314 kJ/(kmol·K). If you use 8.314 in Problem 5’s K_a expression the exponent is wrong by a factor of 1000.

Re-read the Units Stated in Each Problem Before Substituting R

Problem 5 explicitly states “R is in kJ/(kmol·K).” That means R = 8314 in that expression. Note this at the top of your Problem 5 solution so the marker can see you used the correct value.

Hand-Written Submission Requirements

This is not optional formatting guidance — it affects your grade directly.

Pre-Submission Checklist

All five problems hand-written — not typed, not equation-editor output printed. The marker will know the difference.

Cover sheet completed — programme title, learner ID, learner name, date submitted, signature. Fill in the blue section before scanning.

Scan quality — all equations, subscripts, and Greek letters (γ, ξ, Δ, ν) must be legible. Photograph in good lighting on a flat surface. If using a phone, use a document scanner app rather than the standard camera.

Problem labelling — each solution clearly labelled “Problem 1,” “Problem 2,” etc. with sub-parts labelled (a), (b), (c) where applicable. Markers should not have to guess which working belongs to which question.

References cited — if you consulted any textbook or reference for equation forms (Smith, Van Ness and Abbott is the standard for this course material), cite it. The instructions say references must be acknowledged.

Submission route confirmed — upload to Moodle or send to the instructor’s email. Confirm the deadline (31 May 2026) and the late penalty policy before submitting.

Recommended Reference

Smith, J.M., Van Ness, H.C., and Abbott, M.M. — Introduction to Chemical Engineering Thermodynamics (8th ed., McGraw-Hill). Chapters 10–14 cover VLE, activity coefficients, and reaction equilibrium in the detail this problem sheet requires. The worked examples in these chapters use exactly the equation forms given in your problems.

Free External Resource

NIST WebBook (webbook.nist.gov) provides tabulated thermochemical data including Gibbs free energies of formation for methylcyclohexane and toluene. Use it to cross-check the ΔG°_f values given in Problem 4 at 298 K before you run all nine calculations — catching a data transcription error early saves a lot of rework.

Frequently Asked Questions

How do I handle the fact that the Antoine correlation range for Problem 2 starts at 323 K but I need P^sat at 298 K?

Use the correlation at 298 K anyway and note the extrapolation. This is a recognised limitation of the Antoine equation and most instructors expect you to acknowledge it rather than invent an alternative approach. Write something like: “Note: the Antoine correlation is extrapolated below its stated range (323 K). The result is used as the best available estimate.” One sentence is sufficient. Do not omit the calculation because of this — the instructor designed the problem this way intentionally.

For Problem 3, do I need to account for vapour-phase non-ideality as well as liquid-phase non-ideality?

No. The problem statement says “the mixture is expected to follow modified Raoult’s law” — which assumes an ideal vapour phase and an activity coefficient correction only in the liquid. This means fugacity coefficients in the vapour are assumed to be 1.0 and you only need to account for γ_i in the liquid phase. If the problem wanted vapour-phase corrections it would have said to use the Peng-Robinson EOS or provided fugacity coefficients.

In Problem 5, is the equilibrium constant K_a the thermodynamic equilibrium constant in terms of activities, or just pressure/fugacity?

It is the thermodynamic equilibrium constant in terms of fugacities relative to the standard state (1 bar for gases). For an ideal gas mixture, fugacity equals partial pressure, so K_a = K_p/P°^Δν = K_y(P/P°)^Δν. The experimental expression given is already K_a in this sense, so you do not need to apply a non-ideality correction to the gas phase — just the mole fraction/pressure relationship shown in the step-by-step guide above.

How many decimal places should I show in my final answers?

Three significant figures is standard for engineering thermodynamics calculations. K_a values may span several orders of magnitude — report them to three significant figures in scientific notation (e.g. 4.27 × 10³). Activity coefficients should be reported to four decimal places since small differences in γ matter for model fitting. Temperature should be reported to two decimal places (K) — e.g. 362.47 K. Always include units next to every numerical result.

Can I use a spreadsheet to do the iterations and then transcribe the working by hand?

The submission must be hand-written, but there is no prohibition on using a calculator or spreadsheet to verify your trial-and-error iterations. Run the iteration on paper first, check it with a spreadsheet if you want, then write the converged result and the final iteration step clearly in your hand-written submission. What matters to the marker is that the logical steps appear in your own writing — not that you did every arithmetic step manually. Showing two or three iteration cycles with the convergence check is enough to demonstrate the method.

How do I cite the textbook in the hand-written submission?

A brief in-text note is sufficient. At the start of a solution section, write: “Using Eq. (10.5) from Smith, Van Ness and Abbott (2005), Introduction to Chemical Engineering Thermodynamics, 7th ed.” If you consulted the NIST WebBook for thermochemical data, write: “ΔG°_f values cross-checked against NIST WebBook (webbook.nist.gov, accessed May 2026).” Academic referencing in an engineering problem sheet does not require a full bibliographic list unless the instructor has specified a particular citation style.

Related Services and Resources

Engineering assignment help · Mechanical engineering assignment help · Chemistry homework help · Maths assignment help · Complex technical & scientific assignment assistance · Lab report writing services · Report writing services

Before You Write a Single Number

Read through all five problems first. Identify which equations you will use and confirm the units demanded by each. Problems 1–3 are self-contained — you will not need output from one to solve another. Problems 4 and 5 are connected: the dehydrogenation reaction and its equilibrium properties run through both. But Problem 5 gives you its own K_a expression, so you do not need Problem 4’s result to solve it.

The most common point of failure in this type of assessment is not getting the concept wrong — it is getting the setup wrong and then executing perfectly on incorrect foundations. State your assumptions. Write your given data. Show your unit conversions. Those three habits protect your method marks even when a calculation goes sideways.

The assignment is hand-written and scanned. Budget time for that. A clear, well-organised submission is faster to mark and faster to give the benefit of the doubt to when a step is ambiguous.

Need Help With Your Thermodynamics Problem Sheet?

Our technical writing team supports chemical engineering students on VLE modelling, equilibrium calculations, lab reports, and postgraduate coursework across all levels.

Engineering Assignment Help Get Started