Fugacity of Liquid Hydrogen Chloride: 23-Chem-A1 Process Balances and Chemical Thermodynamics, December 2016, Q4

Question 4: Fugacity of Liquid Hydrogen Chloride (Part B — equal value)

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. $T=277.4$ K, system $P=13.61$ atm, $P^{\text{sat}}=28.81$ atm, $T_c=324.68$ K, $P_c=81.5$ atm. Open-book look-ups (not printed): acentric factor $\omega\approx0.12$, Rackett $Z_c\approx0.249$, $R=82.06\ \text{cm}^3\text{atm}\,\text{mol}^{-1}\text{K}^{-1}$.

Check (phase note): the stated system pressure (13.61 atm) is below the vapour pressure (28.81 atm) at this temperature, so a true liquid cannot exist here — HCl would flash to vapour. The question therefore asks for the fugacity of a metastable (hypothetical) liquid, computed by continuing the saturated-liquid fugacity down to $P$ with a Poynting correction. The result exceeding $P$ is the mathematical signature of that metastability.

Find. the liquid-phase fugacity $f^{\,L}$ at the stated $T,P$.

Approach. At saturation the liquid and vapour fugacities are equal, so $f^{\text{sat}}=\phi^{\text{sat}}P^{\text{sat}}$; obtain $\phi^{\text{sat}}$ from the Pitzer generalized second-virial correlation, then correct from $P^{\text{sat}}$ to the system $P$ with the Poynting factor using a Rackett liquid molar volume.

Reduced conditions and virial coefficients. $T_r=277.4/324.68=0.854$, $P_r^{\text{sat}}=28.81/81.5=0.353$. The Pitzer functions: $$B^0=0.083-\frac{0.422}{T_r^{1.6}}=-0.460,\qquad B^1=0.139-\frac{0.172}{T_r^{4.2}}=-0.194.$$
Saturated-vapour fugacity coefficient. $$\ln\phi^{\text{sat}}=\frac{P_r^{\text{sat}}}{T_r}\left(B^0+\omega B^1\right)=0.414\big(-0.460+0.12(-0.194)\big)=-0.200,\quad \phi^{\text{sat}}=0.819.$$
Liquid molar volume (Rackett). $$V^L=\frac{RT_c}{P_c}\,Z_c^{\,[1+(1-T_r)^{2/7}]}=\frac{82.06(324.68)}{81.5}(0.249)^{1.577}=36.5\ \text{cm}^3/\text{mol}.$$
Poynting correction and fugacity. Because $P

Quantity	Result
$\phi^{\text{sat}}$	0.819
$V^L$ (Rackett)	36.5 cm³/mol
Poynting factor	0.976
Fugacity of the (metastable) liquid	≈ 23.0 atm

Concept & Theory

Topic. Liquid-phase fugacity from the saturated state and a Poynting correction.

Key relations. $f^L=\phi^{\text{sat}}P^{\text{sat}}\exp[V^L(P-P^{\text{sat}})/RT]$; Pitzer $\ln\phi=(P_r/T_r)(B^0+\omega B^1)$; Rackett $V^L=(RT_c/P_c)Z_c^{[1+(1-T_r)^{2/7}]}$.

Why this works. At the saturation pressure the two phases are in equilibrium, so the liquid fugacity equals the (easily computed) saturated-vapour fugacity. Away from $P^{\text{sat}}$ the nearly incompressible liquid’s fugacity changes only through the small Poynting exponential, which needs just a liquid molar volume. The generalized virial correlation supplies $\phi^{\text{sat}}$ from critical constants alone.

Common pitfall. Getting the sign of the Poynting exponent wrong when $PP(13.61)$ precisely because the “liquid” is metastable at this pressure.

Source. Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics §§3.6, 10.5–11.6 (generalized correlations, fugacity of pure liquids).

Question 4 of 6: Fugacity of Liquid Hydrogen Chloride

Question 4: Fugacity of Liquid Hydrogen Chloride (Part B — equal value)