Domain: Liquid-phase equilibrium

Isothermal liquid \[(T=350.00, \mathrm{K})\], \[(P=1.000, \mathrm{bar})\]. Reaction: 

\[ (\mathrm{A+B\rightleftharpoons C+D}) \].

Feed (total \[(=1.0000\ \mathrm{mol})\]): \[(n_A^0=n_B^0=0.5000,\ n_C^0=n_D^0=0)\]. Extent: 

\[(n_A=0.5-\xi,\ n_B=0.5-\xi,\ n_C=\xi,\ n_D=\xi)\]; mole fractions \[(x_i=n_i)\].

Standard state: pure liquid at \[(T,P)\]. Activities: \[(a_i=\gamma_i x_i)\].

\[ (\Delta_r G^\circ(350,\mathrm{K})=-1.850\ \mathrm{kJ,mol^{-1}}\Rightarrow K^\circ=\exp(-\Delta_r G^\circ/RT)) \].

NRTL with \[(\alpha_{ij}=0.30)\] for \[(i\neq j)\]:

\[\displaystyle \ln\gamma_i=\sum_{j}\frac{G_{ji}\tau_{ji}x_j}{\sum_k G_{jk}x_k}+\sum_{j}\frac{x_j G_{ij}}{\sum_k G_{ik}x_k}\left(\tau_{ij}-\frac{\sum_m x_m G_{im}\tau_{im}}{\sum_k G_{ik}x_k}\right),\quad G_{ij}=e^{-\alpha_{ij}\tau_{ij}}.\]

Binary interaction parameters at \[(350,\mathrm{K})\] (only off-diagonals listed, order 
\[(\tau_{ij}))\]:

\[(\tau_{AB}=1.20,\ \tau_{BA}=0.10,\ \tau_{AC}=0.65,\ \tau_{CA}=0.90,\ \tau_{AD}=-0.30,\ \tau_{DA}=1.40,)\]

\[(\tau_{BC}=0.25,\ \tau_{CB}=0.55,\ \tau_{BD}=0.80,\ \tau_{DB}=0.35,\ \tau_{CD}=1.10,\ \tau_{DC}=0.20.)\]

Equilibrium condition for \[(\xi\in[0,0.5])\]:

\[\displaystyle K^\circ=\frac{\gamma_C(\xi)\gamma_D(\xi),x_C(\xi),x_D(\xi)}{\gamma_A(\xi)\gamma_B(\xi),x_A(\xi),x_B(\xi)}.\]      

Calculate \[\xi^{*}\] (4 s.f.).

Grading rubric:
criterion: The solution computes the numerical value of the standard equilibrium constant as K° = 1.88846 using ΔrG° = −1850 J mol⁻¹, R = 8.314 J mol⁻¹ K⁻¹, T = 350 K.
weight: 10
criterion: The solution constructs the extent-of-reaction mole balances from first principles for a closed, constant-mole liquid system, yielding nA = nB = 0.5 − ξ and nC = nD = ξ with no extraneous normalization
weight: 10
criterion: The solution accurately constructs the equilibrium reaction quotient strictly as Q = (γC γD ξ²)/(γA γB (0.5 − ξ)²) without alternative formulations
weight: 10
criterion: The solution correctly reports the ordered NRTL interaction factor pair (GAB, GAD) = (0.6977, 1.0942) computed from τAB = 1.20 and τAD = −0.30 with α = 0.30
weight: 10
criterion: The solution accurately reports the numerical activity coefficient value γA = 1.773 at ξ = 0.3073
weight: 20
criterion: The solution correctly reports the numerical activity coefficient value γC = 1.398 at ξ = 0.3073
weight: 20
criterion: The solution correctly identifies ξ = 0.3073 as the only physically admissible root within 0 ≤ ξ ≤ 0.5
weight: 20

Answer:
xi* = 0.3073