Fugacity Coefficients of HCl and DCM: 23-Chem-A1 Process Balances and Chemical Thermodynamics, May 2018, Q4

Question 4: Fugacity Coefficients of HCl and DCM (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. $T=500\,{}^{\circ}\text{R}$, $P=200$ psia; $y_{\text{HCl}}=0.80$, $y_{\text{DCM}}=0.20$. Critical data: HCl $T_c=584\,{}^{\circ}\text{R}$, $P_c=1209.6$ psia; DCM $T_c=933\,{}^{\circ}\text{R}$, $P_c=893$ psia. Only $T_c$ and $P_c$ are supplied (no acentric factors, no binary interaction data).

Species	$T_r=T/T_c$	$P_r=P/P_c$
HCl	$500/584=0.856$	$200/1209.6=0.165$
DCM	$500/933=0.536$	$200/893=0.224$

Find. The pure-component fugacity coefficients $\varphi_{\text{HCl}}$ and $\varphi_{\text{DCM}}$ at $T,P$ (and, by Lewis–Randall, the mixture fugacities).

Approach. With only $T_c,P_c$ available, apply the Pitzer generalized second-virial correlation $\ln\varphi=(P_r/T_r)\,B^0$, using $B^0(T_r)=0.083-0.422/T_r^{1.6}$ for each species (acentric term unavailable, so the $\omega B^1$ contribution is dropped).

Generalized virial fugacity coefficient. For a pure gas the truncated-virial result is $$\ln\varphi=\frac{P_r}{T_r}\,B^0,\qquad B^0=0.083-\frac{0.422}{T_r^{1.6}}.$$ This is the two-parameter ($\omega=0$) form; it needs only reduced $T$ and $P$.
HCl. With $T_r=0.856,\ P_r=0.165$: $$B^0=0.083-\frac{0.422}{0.856^{1.6}}=-0.458,\quad \ln\varphi=\frac{0.165}{0.856}(-0.458)=-0.0884,$$ $$\boxed{\varphi_{\text{HCl}}=e^{-0.0884}=0.915.}$$
DCM. With $T_r=0.536,\ P_r=0.224$: $$B^0=0.083-\frac{0.422}{0.536^{1.6}}=-1.062,\quad \ln\varphi=\frac{0.224}{0.536}(-1.062)=-0.444,$$ $$\boxed{\varphi_{\text{DCM}}=e^{-0.444}=0.642.}$$
Mixture fugacities (Lewis–Randall). Treating each component’s fugacity coefficient in the mixture as its pure value, $\hat f_i=y_i\varphi_iP$: $$\hat f_{\text{HCl}}=0.80(0.915)(200)=146\ \text{psia},\qquad \hat f_{\text{DCM}}=0.20(0.642)(200)=25.7\ \text{psia}.$$

Check — method dictated by the data given

Only $T_c$ and $P_c$ are provided, so the two-parameter virial correlation is the intended route; acentric factors ($\omega\,B^1$) and cross-coefficients ($B_{12}$) are unavailable. DCM’s $T_r=0.54$ is at the low-temperature edge of the correlation’s reliable range, so $\varphi_{\text{DCM}}$ is the best estimate consistent with the supplied data (an EOS with $\omega$ would refine it).

Quantity	Result
$\varphi_{\text{HCl}}$	0.915
$\varphi_{\text{DCM}}$	0.642
$\hat f_{\text{HCl}},\ \hat f_{\text{DCM}}$ (Lewis–Randall)	146 psia, 25.7 psia

Concept & Theory

Topic. Fugacity coefficients from the Pitzer generalized (corresponding-states) second-virial correlation.

Key relations. $\ln\varphi=(P_r/T_r)(B^0+\omega B^1)$ with $B^0=0.083-0.422/T_r^{1.6}$, $B^1=0.139-0.172/T_r^{4.2}$; here $\omega$ is not given, so only $B^0$ is used; Lewis–Randall $\hat f_i=y_i\varphi_iP$.

Why this works. The truncated virial EOS $Z=1+BP/RT$ integrates to a closed form for $\ln\varphi$ that depends only on reduced properties, so $T_c$ and $P_c$ suffice. A fugacity coefficient below 1 signals attraction-dominated non-ideality; DCM (larger molecule, closer to its critical temperature at 500 °R) departs from ideality far more than HCl.

Common pitfall. Forgetting to convert to reduced properties, or applying the correlation at conditions ($T_r<0.7$, high $P_r$) where the second-virial truncation is invalid without noting the limitation. Using $\omega$-dependent $B^1$ when no acentric factor is provided is also inconsistent with the data.

Source. Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics §3.6 (generalized correlations) and §11.6 (fugacity in mixtures, Lewis–Randall rule).

Question 4 of 6: Fugacity Coefficients of HCl and DCM

Question 4: Fugacity Coefficients of HCl and DCM (Part B — equal value)