23-Chem-A1 Process Balances and Chemical Thermodynamics · December 2019
Question 5 of 6: Fugacity of (Metastable) Liquid Hydrogen Chloride
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exam 16-Chem-A1, December 2019 — open-book, 3 hours. Two parts: Part A (Process Mass & Energy Balances, Q1–Q3) and Part B (Chemical Thermodynamics, Q1–Q3). Candidates answer TWO from each part; each question is of equal value. All six questions are solved in full below.
Reference texts: Felder, Rousseau & Bullard, Elementary Principles of Chemical Processes (4th ed., Wiley) — heats of reaction from combustion data, fuel/air combustion balances, and flash (equilibrium) energy balances; Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics (8th ed., McGraw-Hill) — reaction equilibrium and the van’t Hoff equation, generalized (Pitzer) fugacity coefficients and the liquid-fugacity/Poynting relation, and heat-of-mixing energy balances; critical-property and Rackett data from Poling, Prausnitz & O’Connell, The Properties of Gases and Liquids (5th ed.).
Part A — Process Mass & Energy Balances
Question B2: Fugacity of (Metastable) Liquid Hydrogen Chloride (Part B — equal value)
Given. Pure hydrogen chloride at $T=277.4$ K and $P=13.61$ atm.
Property
Value
Vapour pressure $P^{sat}$ at 277.4 K
28.81 atm
Critical temperature $T_c$
324.68 K
Critical pressure $P_c$
81.5 atm
Acentric factor / $Z_c$ (App. B, open-book)
0.12 / 0.249
Find. the fugacity of the pure liquid HCl at the stated temperature and pressure.
Figure B2 — Vapour-pressure curve of HCl from (illustrative) low pressure up to the critical point. The system state (13.61 atm) lies below the saturation pressure at 277.4 K, i.e. in the vapour field, so the liquid evaluated is metastable.
Phase note
At 277.4 K the vapour pressure (28.81 atm) exceeds the system pressure (13.61 atm), so the thermodynamically stable phase at this state is actually vapour — the “liquid” is a metastable (subcooled) reference state. Its fugacity is nonetheless well defined and is computed with the standard saturation-plus-Poynting construction; the outcome $f>P$ is precisely the signature that the liquid is not the stable phase here.
Approach. Anchor the liquid fugacity at saturation ($f^{sat}=\varphi^{sat}P^{sat}$, with $\varphi^{sat}$ from the Pitzer generalized second-virial correlation) and carry it to the system pressure with a Rackett-based Poynting correction.
Reduced conditions. $T_r=277.4/324.68=0.854$. The generalized second-virial (Pitzer) correlation applies at the saturation pressure ($P_r^{sat}=28.81/81.5=0.354$).
Saturated-vapour fugacity coefficient (Pitzer). With $B^0=0.083-0.422/T_r^{1.6}=-0.460$ and $B^1=0.139-0.172/T_r^{4.2}=-0.194$, $$\ln\varphi^{sat}=\frac{P_r^{sat}}{T_r}\,(B^0+\omega B^1)=-0.200\;\Rightarrow\;\varphi^{sat}=0.819,\quad f^{sat}=\varphi^{sat}P^{sat}=23.6\ \text{atm}.$$ At saturation the liquid and vapour fugacities are equal, so $f^{sat}$ is the liquid fugacity at $P^{sat}$.
Poynting correction to 13.61 atm. $$\exp\!\left[\frac{V^L(P-P^{sat})}{RT}\right]=\exp\!\left[\frac{36.5(13.61-28.81)}{82.06(277.4)}\right]=\exp(-0.024)=0.976.$$
Liquid fugacity. $$f^{L}=\varphi^{sat}P^{sat}\exp\!\left[\frac{V^L(P-P^{sat})}{RT}\right]=23.6(0.976)=\boxed{23.0\ \text{atm}}.$$ Because $PP=13.61$ atm confirms the metastability of the liquid state.
