Acrylonitrile–Water Two-Phase System (Margules): 23-Chem-A1 Process Balances and Chemical Thermodynamics, December 2014, Q6

Question 6: Acrylonitrile–Water Two-Phase System (Margules) (Part C, 30 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Liquid–liquid equilibrium at 70 °C: $x_{1A}=0.968$ ($x_{2A}=0.032$), $x_{1W}=0.073$ ($x_{2W}=0.927$). Two-parameter Margules $\ln\gamma_1 = x_2^2[A_{12}+2(A_{21}-A_{12})x_1]$, $\ln\gamma_2 = x_1^2[A_{21}+2(A_{12}-A_{21})x_2]$; $P_1^{\text{sat}}=0.791$, $P_2^{\text{sat}}=0.312$ bar.

Find. (a) the four activity coefficients (each species in each phase); (b) the three-phase bubble pressure and vapour composition.

Figure 5 — Two liquid layers (acrylonitrile-rich and water-rich) in equilibrium with a common vapour; each species has equal activity in both liquids.

Approach. Liquid–liquid equilibrium equates each species’ activity in the two phases, giving two equations for the two Margules constants; the fitted model then delivers all $\gamma$’s and, through modified Raoult’s law, the bubble pressure.

Equal-activity (LLE) conditions. At equilibrium $x_{iA}\gamma_{iA} = x_{iW}\gamma_{iW}$ for each component: $$0.968\,\gamma_{1A} = 0.073\,\gamma_{1W},\qquad 0.032\,\gamma_{2A} = 0.927\,\gamma_{2W}.$$
Solve for the Margules constants. Substituting the Margules expressions and solving the $2\times2$ system (Newton) gives $$A_{12} = 2.90,\qquad A_{21} = 3.65.$$
Activity coefficients (a). Evaluating the model in each phase: $$\boxed{\gamma_{1A}=1.00,\ \gamma_{2A}=29.3\ \text{(AN-rich)};\quad \gamma_{1W}=13.3,\ \gamma_{2W}=1.01\ \text{(water-rich)}.}$$ Each species is nearly ideal ($\gamma\to1$) in the phase it dominates and strongly non-ideal (large $\gamma$) where it is dilute — the signature of a nearly-immiscible pair.
Partial pressures (modified Raoult). Using either phase (activities are equal), $p_i = x_{i}\gamma_{i}P_i^{\text{sat}}$: $$p_1 = 0.968(1.00)(0.791)=0.769\ \text{bar},\qquad p_2 = 0.032(29.3)(0.312)=0.293\ \text{bar}.$$
Bubble pressure and vapour (b). $$P = p_1+p_2 = \boxed{1.06\ \text{bar}},\qquad y_1 = \frac{p_1}{P} = \boxed{0.724},\ y_2 = 0.276.$$

Quantity	Result
Margules constants	$A_{12}=2.90$, $A_{21}=3.65$
(a) γ (AN-rich / water-rich)	γ₁=1.00, γ₂=29.3 / γ₁=13.3, γ₂=1.01
(b) Bubble-point pressure	1.06 bar
(b) Vapour composition	y₁=0.724 (AN), y₂=0.276 (H₂O)

Concept & Theory

Topic. Liquid–liquid equilibrium fitted to a Margules model; three-phase (VLLE) bubble pressure by modified Raoult’s law.

Key relations. LLE: $x_{iA}\gamma_{iA}=x_{iW}\gamma_{iW}$; modified Raoult $p_i = x_i\gamma_i P_i^{\text{sat}}$; two-parameter Margules for $\ln\gamma_1,\ln\gamma_2$.

Why this works. Equality of each species’ fugacity (hence activity $x_i\gamma_i$) across the two liquids is the LLE criterion; two such equations exactly determine the two Margules constants. Because the two liquids share the same vapour, either phase reproduces the same partial pressures, so the bubble pressure is unambiguous.

Common pitfall. Picking the wrong root of the Margules fit (choose the branch with both $A_{ij}>0$, giving $\gamma>1$ everywhere for this positively-deviating, partially-miscible pair), or evaluating $p_i$ in only one phase and doubting the other — the check $x_{1A}\gamma_{1A}P_1^{\text{sat}}=x_{1W}\gamma_{1W}P_1^{\text{sat}}$ confirms consistency.

Source. Smith, Van Ness, Abbott & Swihart, ICET §12.1–12.7 and §14 (excess Gibbs models, VLE and LLE; Margules equation).

Question 6 of 7: Acrylonitrile–Water Two-Phase System (Margules)

Question 6: Acrylonitrile–Water Two-Phase System (Margules) (Part C, 30 marks)