For any real process, the entropy change of the universe (system plus surroundings) is

never negative, and zero only for a reversible process

equal to the first law's energy change

Two phases of a multicomponent system are in equilibrium precisely when, for every species present,

the chemical potential of that species is equal in both phases

the temperatures of the two phases are equal only

the chemical potential of that species is equal in both phases

its mole fraction is equal in both phases

its partial pressure is zero in both phases

A residual property M^R of a real fluid at a given T and P is defined as

M(real fluid) minus M(ideal gas), both at the same T and P

M(real fluid) at T, P minus M(real fluid) at the critical point

the partial molar value of M for the mixture

M(ideal solution) minus M(real mixture)

A cubic equation of state such as Peng-Robinson can, at conditions below the critical point, return

up to three real roots, the smallest being liquid volume and the largest vapour volume

exactly one real root, always the vapour volume

up to three real roots, the smallest being liquid volume and the largest vapour volume

only complex roots, meaning the equation has failed

a single root equal to the ideal-gas volume

Which of the following are true about fugacity coefficient phi_i and activity coefficient gamma_i?

phi_i equals 1 when the phase behaves as an ideal gas.

gamma_i equals 1 when the liquid mixture is an ideal (Raoult's-law) solution.

gamma_i is generally a function of liquid composition and temperature.

phi_i equals 1 when the phase behaves as an ideal gas.

gamma_i equals 1 when the liquid mixture is an ideal (Raoult's-law) solution.

phi_i and gamma_i correct against the same reference state.

gamma_i is generally a function of liquid composition and temperature.

Excess Gibbs energy G^E is defined against which reference, and what does it primarily feed into?

The ideal solution; activity-coefficient models correlate it and derive gamma_i from it

The ideal gas; it feeds directly into the equilibrium constant K

The ideal solution; activity-coefficient models correlate it and derive gamma_i from it

The critical point; it feeds into the reduced-property correlations

Absolute zero; it feeds into the third law

Modified Raoult's law, y_i*P = x_i*gamma_i*P_i-sat, is a simplification of the rigorous gamma-phi VLE equations obtained by

dropping the vapour-phase fugacity-coefficient correction, valid at low to moderate pressure

dropping the liquid-phase activity coefficient entirely

dropping the vapour-phase fugacity-coefficient correction, valid at low to moderate pressure

assuming the liquid is an ideal solution as well as the vapour an ideal gas

replacing the equilibrium constant with the phase rule

The equilibrium constant K of a reaction, exp(-Delta-G-rxn-standard / RT), depends on

temperature and total pressure equally

Adding an inert diluent at constant total pressure to a gas-phase reaction that increases the number of moles will, according to Le Chatelier's principle,

shift the equilibrium composition toward more product, without changing K

change the value of K at that temperature

shift the equilibrium composition toward more product, without changing K

have no effect on composition or K

shift the equilibrium composition toward more reactant, without changing K

Properties, Mixtures and Phase and Reaction Equilibria

Section 3 of the GATE Chemical Engineering (CH) paper. Thermodynamics here is not the two-law abstraction of a physics course — it is the set of tools that answer three practical questions a plant asks every day: how much energy does this stream actually carry, will this mixture split into two phases at these conditions, and how far will this reaction actually go. Start with the laws themselves and the state functions entropy and chemical potential they define; then equations of state and residual properties, which get you from an ideal-gas answer to a real one; then partial molar properties, fugacity, excess properties and activity coefficients, which do the same correction for mixtures; then VLE prediction and chemical reaction equilibrium, where all of the above gets used to answer the two questions that actually matter to a process.

1. The laws, entropy and chemical potential

The first law is an energy balance in disguise: for a closed system, dU = dQ - dW, and for an open system in steady flow it becomes the familiar dH = dQ - dW_s plus kinetic and potential energy terms. The second law introduces entropy S as a state function whose change for the universe (system plus surroundings) is never negative for any real process, and is exactly zero only for a reversible one — this single inequality is what tells you which direction a real process can run, something the first law alone cannot say.

🎯 Chemical potential is the driving force for mass, the way temperature is for heat

Chemical potential mu_i is defined as the partial molar Gibbs energy of species i, and it plays exactly the role for mass transfer that temperature plays for heat transfer and pressure for momentum: species i moves spontaneously from a phase of higher mu_i to one of lower mu_i, and phase equilibrium between two phases is defined precisely as mu_i being equal in both, for every species present.

2. Equations of state and residual properties for pure substances

An equation of state relates P, V and T for a real fluid beyond the ideal-gas law: the virial equation truncated after the second coefficient corrects for moderate non-ideality, while cubic equations of state (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson) are fitted to also reproduce vapour-liquid behaviour and the critical point, at the cost of being cubic in volume — which is exactly why they can return three real roots, the smallest being the liquid volume and the largest the vapour volume.

🧠 A residual property is exactly the ideal-gas correction

A residual property M^R is defined as M(actual, real fluid) minus M(ideal gas, same T and P) — so the real enthalpy or entropy at a given T, P is simply the ideal-gas value plus its residual, and the residual itself is computed from an equation of state alone (via a departure-function integral), without ever needing a reference state. This is the single trick behind every generalized correlation and every steam-table-style property chart.

3. Mixtures: partial molar properties, fugacity, excess properties, activity coefficients

A partial molar property M_i-bar is how much a mixture's total property M changes per mole of species i added, at constant T, P and moles of everything else — it is generally NOT the same as the pure-species property, and the difference is exactly what mixing does. Fugacity f_i extends the pure-substance idea of escaping tendency to a component in a mixture, defined so that mu_i = mu_i(reference) + RT ln(f_i/f_i-reference); it reduces to partial pressure only in the ideal-gas limit.

Two ways real-mixture behaviour is corrected
Correction factor	Applies to	Equals 1 when
Fugacity coefficient, phi_i	Vapour (or any) phase, vs ideal gas	The phase behaves as an ideal gas
Activity coefficient, gamma_i	Liquid phase, vs ideal (Raoult's-law) solution	The liquid mixture is an ideal solution

An excess property M^E is defined the same way a residual property was, but against the ideal-SOLUTION reference instead of the ideal-gas one: M^E = M(actual mixture) - M(ideal solution, same T, P, composition). Excess Gibbs energy G^E is what activity-coefficient models (Margules, van Laar, Wilson, NRTL, UNIQUAC) actually correlate, and gamma_i is obtained from it by the same partial-molar-derivative operation used everywhere else in this section.

4. Phase equilibria: predicting VLE, and chemical reaction equilibrium

Predicting VLE means solving f_i(vapour) = f_i(liquid) for every species, at the system's T and P. The gamma-phi approach keeps this exact by pairing an equation of state (for the vapour's fugacity coefficient) with an activity-coefficient model (for the liquid), while the simpler phi-phi approach uses one equation of state for both phases; the even simpler modified Raoult's law, y_iP = x_igamma_i*P_i-sat, drops the vapour-phase correction entirely and is accurate only at low to moderate pressure.

⚠️ Kp is temperature-only; the composition still has to be solved for

For a reaction at chemical equilibrium, the equilibrium constant K = exp(-Delta-G-rxn-standard / RT) depends on temperature ALONE, through the standard Gibbs energy change of reaction — never on pressure or composition directly. Pressure and inert dilution still shift the EQUILIBRIUM COMPOSITION (via Le Chatelier's principle and how K relates to mole fractions and fugacity coefficients), but they do this without changing K itself; conflating 'K is fixed at this T' with 'composition cannot shift' is the standard trap.

Key takeaways

The second law's entropy inequality, never negative for the universe, is what tells you which direction a real process can run — something an energy balance alone cannot say.
A residual property is the real-fluid value minus the ideal-gas value at the same T and P, computed straight from an equation of state — the trick behind every generalized property correlation.
Fugacity coefficient corrects a phase against ideal-gas behaviour; activity coefficient corrects a liquid against ideal-solution behaviour — different reference states for different phases.
Predicting VLE means solving vapour fugacity equals liquid fugacity for every species; the gamma-phi, phi-phi and modified-Raoult's-law routes trade rigour for simplicity in that order.
The equilibrium constant K depends on temperature alone; pressure and inert dilution shift the equilibrium COMPOSITION through Le Chatelier's principle without changing K itself at fixed T.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

For any real process, the entropy change of the universe (system plus surroundings) is
1. always exactly zero
2. never negative, and zero only for a reversible process
3. always negative
4. equal to the first law's energy change
Show answer
Answer: B — never negative, and zero only for a reversible process
The second law states the universe's entropy change is never negative for a real process, and equality (zero change) holds only in the idealised, reversible limit — this inequality is what fixes the direction a real process can run.
Two phases of a multicomponent system are in equilibrium precisely when, for every species present,
1. the temperatures of the two phases are equal only
2. the chemical potential of that species is equal in both phases
3. its mole fraction is equal in both phases
4. its partial pressure is zero in both phases
Show answer
Answer: B — the chemical potential of that species is equal in both phases
Chemical potential is the driving force for mass transfer between phases, exactly as temperature is for heat; phase equilibrium is defined as equal chemical potential (and hence equal fugacity) for each species in every phase present, not equal mole fraction, which is generally false across phases.
A residual property M^R of a real fluid at a given T and P is defined as
1. M(real fluid) minus M(ideal gas), both at the same T and P
2. M(real fluid) at T, P minus M(real fluid) at the critical point
3. the partial molar value of M for the mixture
4. M(ideal solution) minus M(real mixture)
Show answer
Answer: A — M(real fluid) minus M(ideal gas), both at the same T and P
A residual property is exactly the real-fluid value minus the ideal-gas value at the SAME temperature and pressure, computed from an equation of state via a departure-function integral — this is what lets a real enthalpy or entropy be built from an ideal-gas value plus a correction.
A cubic equation of state such as Peng-Robinson can, at conditions below the critical point, return
1. exactly one real root, always the vapour volume
2. up to three real roots, the smallest being liquid volume and the largest vapour volume
3. only complex roots, meaning the equation has failed
4. a single root equal to the ideal-gas volume
Show answer
Answer: B — up to three real roots, the smallest being liquid volume and the largest vapour volume
Being cubic in volume is exactly what lets these equations describe both liquid and vapour with one expression: in the two-phase region they return three real roots, the smallest the liquid molar volume, the largest the vapour molar volume, and a middle root with no physical meaning that is discarded.
Which of the following are true about fugacity coefficient phi_i and activity coefficient gamma_i?
1. phi_i equals 1 when the phase behaves as an ideal gas.
2. gamma_i equals 1 when the liquid mixture is an ideal (Raoult's-law) solution.
3. phi_i and gamma_i correct against the same reference state.
4. gamma_i is generally a function of liquid composition and temperature.
Show answer
Answer: A — phi_i equals 1 when the phase behaves as an ideal gas.; B — gamma_i equals 1 when the liquid mixture is an ideal (Raoult's-law) solution.; D — gamma_i is generally a function of liquid composition and temperature.
phi_i's reference is the ideal gas and gamma_i's is the ideal solution — two DIFFERENT reference states, so the third statement is false. The first, second and fourth are exactly how each coefficient is defined and how gamma_i is modelled (Margules, Wilson, NRTL and similar correlations all take composition and temperature as inputs).
Excess Gibbs energy G^E is defined against which reference, and what does it primarily feed into?
1. The ideal gas; it feeds directly into the equilibrium constant K
2. The ideal solution; activity-coefficient models correlate it and derive gamma_i from it
3. The critical point; it feeds into the reduced-property correlations
4. Absolute zero; it feeds into the third law
Show answer
Answer: B — The ideal solution; activity-coefficient models correlate it and derive gamma_i from it
An excess property is defined against the ideal-solution reference exactly as a residual property is against the ideal gas. G^E is what Margules, van Laar, Wilson, NRTL and UNIQUAC actually correlate, and gamma_i for each species is obtained from G^E by a partial-molar derivative.
Modified Raoult's law, y_iP = x_igamma_i*P_i-sat, is a simplification of the rigorous gamma-phi VLE equations obtained by
1. dropping the liquid-phase activity coefficient entirely
2. dropping the vapour-phase fugacity-coefficient correction, valid at low to moderate pressure
3. assuming the liquid is an ideal solution as well as the vapour an ideal gas
4. replacing the equilibrium constant with the phase rule
Show answer
Answer: B — dropping the vapour-phase fugacity-coefficient correction, valid at low to moderate pressure
Modified Raoult's law keeps the liquid-side activity coefficient gamma_i but drops the vapour-phase fugacity-coefficient correction (effectively setting it to 1, i.e. treating the vapour as ideal), which is a fair approximation only at low to moderate pressure where vapour non-ideality is small.
The equilibrium constant K of a reaction, exp(-Delta-G-rxn-standard / RT), depends on
1. temperature alone
2. temperature and total pressure equally
3. composition alone
4. pressure alone
Show answer
Answer: A — temperature alone
K is set entirely by the standard Gibbs energy change of reaction at the system temperature; it does not depend on pressure or composition directly, though pressure and inert dilution can shift the equilibrium composition that satisfies K at that fixed temperature.
Adding an inert diluent at constant total pressure to a gas-phase reaction that increases the number of moles will, according to Le Chatelier's principle,
1. change the value of K at that temperature
2. shift the equilibrium composition toward more product, without changing K
3. have no effect on composition or K
4. shift the equilibrium composition toward more reactant, without changing K
Show answer
Answer: B — shift the equilibrium composition toward more product, without changing K
Adding inert at constant total pressure lowers each reacting species' partial pressure (mole fraction falls at fixed total P), which for a reaction that increases mole number acts like reducing overall pressure and shifts equilibrium toward the product side to satisfy the same K — K itself, fixed by temperature, is untouched.
The virial equation of state, truncated after the second virial coefficient B, is written Z = 1 + BP/RT (Z the compressibility factor). At the ideal-gas limit, what value does Z approach?
Numerical answer — type the value.
Show answer
Answer: 1
As pressure falls toward zero (or B itself vanishes for a hypothetical ideal gas), the correction term BP/RT vanishes and Z approaches exactly 1, recovering PV = RT — the compressibility factor's whole purpose is to measure deviation from this ideal-gas value of unity.