Fugacity of Propane over Water and Its Henry's Constant: 23-Chem-A1 Process Balances and Chemical Thermodynamics, May 2017, Q4

Question 4: Fugacity of Propane over Water and Its Henry's Constant (Part B)

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. $P = 60\ \text{bar}$, $T = 30\ ^\circ\text{C} = 303.15\ \text{K}$; $\ln\gamma_1 = A x_2^2$; propane $P^{\text{sat}} = 10.8\ \text{bar}$; liquid molar volume $V^L \approx 89\ \text{cm}^3/\text{mol}$ (open-book, $\rho\approx0.49\ \text{g/cm}^3$).

Find. (a) the fugacity of propane in each phase; (b) the Henry's-law constant $H_1(A)$.

Approach. Since $P>P^{\text{sat}}$ the propane-rich phase is a genuine compressed liquid: build its fugacity from the saturation value with a Poynting correction, then bridge the Lewis–Randall and Henry references at infinite dilution.

(a) Fugacity of the compressed liquid. Correct $P^{\text{sat}}$ by the saturation fugacity coefficient ($\phi^{\text{sat}}\approx1$ at 10.8 bar) and the Poynting factor to 60 bar: $$f_1 = \phi^{\text{sat}} P^{\text{sat}} \exp\!\left[\frac{V^L(P - P^{\text{sat}})}{RT}\right] = 10.8\,(1)\exp\!\left[\frac{89\times10^{-6}(49.2\times10^{5})}{8.314(303.15)}\right] = 10.8(1.190) = \boxed{12.8\ \text{bar}}$$ At phase equilibrium the propane fugacity is identical in the aqueous phase, so $f_1 = 12.8$ bar in either phase.
(b) Henry's constant from the infinite-dilution limit. In the aqueous phase (Lewis–Randall on pure liquid propane) $f_1 = x_1\gamma_1 f_1^{\text{pure,L}}$. As $x_1\rightarrow0$, $x_2\rightarrow1$ and $\gamma_1\rightarrow\gamma_1^\infty = e^{A}$; Henry's constant is the limiting slope $f_1/x_1$: $$\boxed{\;H_1 = \gamma_1^\infty\,f_1^{\text{pure,L}} = e^{A}\,f_1^{\text{pure,L}} \approx 12.8\,e^{A}\ \text{bar}\;}$$

Quantity	Value
Poynting factor (10.8→60 bar)	1.19
(a) Fugacity of propane, either phase	12.8 bar
(b) Henry's constant	$H_1 = 12.8\,e^{A}$ bar

Concept & Theory: Two Reference States for the Same Fugacity

Topic. Reconciling the Lewis–Randall (pure-liquid) and Henry (infinite-dilution) references for a sparingly soluble gas.

Key relations. Lewis–Randall $f_i = x_i\gamma_i f_i^{\text{pure,L}}$; Henry $f_i = H_i x_i$; the bridge $H_i = \gamma_i^\infty f_i^{\text{pure,L}}$; compressed-liquid fugacity $f = \phi^{\text{sat}}P^{\text{sat}}\exp[V^L(P-P^{\text{sat}})/RT]$.

Why this works. Fugacity is single-valued at equilibrium, so the two conventions must coincide in the dilute limit, which forces the activity coefficient at infinite dilution to be exactly the ratio of the two reference fugacities.

Common pitfall. Treating the propane as a metastable liquid (that applies only when $P<P^{\text{sat}}$); here $P>P^{\text{sat}}$ so it is a genuine liquid and the Poynting factor exceeds 1.

Source. Smith, Van Ness & Abbott, Introduction to Chemical Engineering Thermodynamics, 8th ed., §§10.4–11.8 (fugacity, activity coefficients, Henry's law, the Poynting correction).

Question 4 of 6: Fugacity of Propane over Water and Its Henry's Constant

Question 4: Fugacity of Propane over Water and Its Henry's Constant (Part B)