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Qwestrum Engineering360 · Chemical Engineering · Chemical Engineering Thermodynamics

Equation of State

Equations of state relate P, V and T for real fluids; cubic forms like van der Waals and Peng-Robinson add a volume-correction b and an attraction term a fitted to critical properties, giving Z and fugacity where the ideal-gas law fails.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

EOS gives Z and fugacity for non-ideal vapours
Critical properties link a, b in cubic EOS
Mixing rules (e.g. k_ij) needed for mixtures

Topic details

Introduction

This topic supplies the P-V-T model used in flash and compressor calculations. You learn the virial expansion for modest pressures and cubic equations of state for higher pressures and for the two-phase region. The parameters a and b are obtained from critical temperature and pressure by imposing the critical-point inflection conditions, and mixtures are handled through mixing rules with binary interaction parameters.

Key relations & formulas

P = R \frac{T}{(V - b)} - a / (V(V + b))

(van der Waals-type / cubic)

Z = 1 + B \frac{P}{(R T)} + ...

(virial expansion)

P = R \frac{T}{(V - b_{m})} - a_{m} \alpha(T) / (V(V + b_{m}) + b_{m}(V - b_{m}))

(Peng-Robinson)

Notation and sign conventions

Relation 1 —

P = R T /

P = R \frac{T}{(V - b)} - a / (V(V + b))

(van der Waals-type / cubic)

Write this relation with symbols exactly as in Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

Z = 1 + B P /

Z = 1 + B \frac{P}{(R T)} + ...

(virial expansion)

Relation 3 —

P = R T /

P = R \frac{T}{(V - b_{m})} - a_{m} \alpha(T) / (V(V + b_{m}) + b_{m}(V - b_{m}))

(Peng-Robinson)

Concept in depth

The ideal-gas law ignores molecular volume and attraction; cubic equations of state restore both. The b term shrinks the free volume (molecules are not points) and the a term lowers pressure (attractions pull molecules together). Because a cubic in V has up to three real roots in the two-phase region, the largest root is the vapour volume and the smallest the liquid, which is why one equation can describe both phases and predict phase envelopes. The virial equation, by contrast, is a rigorous power series in density best suited to gases at low to moderate pressure.

Assumptions and validity limits

State assumptions explicitly before using any relation for equation of state — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Chemical Engineering Thermodynamics viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Chemical Engineering Thermodynamics papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to equation of state.
4. Use equation 1:

P = R T /

.
5. Use equation 2:

Z = 1 + B P /

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Equation of State appears in separation and reaction design. In Indian chemical curricula this topic is tested because it connects theory to phase equilibria and property models.

GATE and semester exams often combine equation of state with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use equation of state?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Frequent mistakes are using critical pressure in gauge units, forgetting the α(T) temperature function in Peng-Robinson, and picking the wrong root of the cubic (vapour versus liquid). Students also apply the virial truncation well outside its valid pressure range.

Quick revision checklist

Before attempting equation of state problems, confirm you can:

1. EOS gives Z and fugacity for non-ideal vapours
2. Critical properties link a, b in cubic EOS
3. Mixing rules (e.g. k_ij) needed for mixtures

Revise the solved examples in Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

van der Waals constant b

Problem

For a gas with T_c = 304 K and P_c = 73.8 bar, estimate the van der Waals constant b = RT_c/(8P_c).

Solution

b = (8.314 × 304)/(8 × 73.8 × 10⁵) = 2528/(5.904×10⁷) = 4.28 × 10⁻⁵ m³/mol. This co-volume sets the minimum molar volume the fluid can reach.

Conceptual check — Equation of State

Problem

In a Chemical Engineering Thermodynamics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of equation of state." What should a complete answer include?

Exams & GATE

Know when ideal gas fails — near critical point or high pressure.

📖 Standard books (India)

Introduction to Chemical Engineering Thermodynamics — Smith, Van Ness & Abbott
Read: Syllabus unit
Phase equilibria and chemical thermo

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📚Key formulas & points