Favourable T–P Region for Methanol Synthesis: 23-Chem-A1 Process Balances and Chemical Thermodynamics, December 2016, Q5

Question 5: Favourable T–P Region for Methanol Synthesis (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. $\Delta G^\circ_{298}=-25.2$ kJ/mol, $\Delta H^\circ_{298}=-90.7$ kJ/mol, stoichiometric feed (CO:H₂ = 1:2), $\Delta n=-2$. Assume $\Delta H^\circ,\Delta S^\circ$ roughly constant over the range.

Find. the boundary curve $P_{\min}(T)$ above which equilibrium CO conversion $X\ge0.10$, and the favourable region on a $P$–$T$ plot.

Approach. Get $\Delta S^\circ$ from $\Delta G^\circ=\Delta H^\circ-T\Delta S^\circ$, build $K(T)$; write the equilibrium condition as a conversion group $G(X)=K(P/P^\circ)^2$, evaluate $G$ at $X=0.10$, and invert for the pressure boundary $P_{\min}(T)=\sqrt{G(0.10)/K(T)}$.

Figure 2 — Boundary $P_{\min}(T)$ for 10% equilibrium conversion. The shaded region (low temperature, high pressure) satisfies $X\ge10\%$; high temperature demands impractically high pressure.

Standard entropy and $K(T)$. $$\Delta S^\circ=\frac{\Delta H^\circ-\Delta G^\circ}{T}=\frac{(-90.7+25.2)\times10^3}{298}=-219.8\ \tfrac{\text{J}}{\text{mol K}},$$ $$\Delta G^\circ(T)=-90{,}700+219.8\,T,\qquad K(T)=\exp\!\Big[\!-\Delta G^\circ(T)/RT\Big].$$
Conversion group. For stoichiometric feed (1 mol CO, 2 mol H₂), at conversion $X$ the totals give $y_{\text{CH}_3\text{OH}}=X/(3-2X)$, etc. With $\Delta n=-2$ the equilibrium relation rearranges to $$\underbrace{\frac{X(3-2X)^2}{4(1-X)^3}}_{G(X)}=K\left(\frac{P}{P^\circ}\right)^2.$$
Evaluate at the 10% threshold. $$G(0.10)=\frac{0.10(2.8)^2}{4(0.9)^3}=0.2689.$$
Pressure boundary. Solving for the minimum pressure that reaches 10%: $$\boxed{P_{\min}(T)=P^\circ\sqrt{\frac{0.2689}{K(T)}}}.$$ Because $G$ increases with $X$, the condition $X\ge10\%$ is equivalent to $P\ge P_{\min}(T)$.
Tabulate the boundary. Using $K(T)$: $$\begin{array}{c|ccccc}T\,(\text{K}) & 400 & 450 & 500 & 550 & 600\\\hline P_{\min}\,(\text{atm}) & 0.34 & 1.55 & 5.22 & 14.1 & 32.2\end{array}$$ At 298 K, $K\approx2.6\times10^4$ so $P_{\min}\approx0.003$ atm (essentially any pressure works); by 600 K one needs $\sim$32 atm. (Strictly, $P^\circ$ is the 1-bar standard state, so these boundary values are in bar; 1 bar = 0.987 atm, a difference of about 1% that does not change the region.) The favourable region is low temperature and/or high pressure — consistent with the exothermic, mole-reducing reaction (Le Chatelier).

Quantity	Result
$\Delta S^\circ$	−219.8 J·mol⁻¹K⁻¹
10% conversion group $G(0.10)$	0.2689
Boundary	$P_{\min}(T)=\sqrt{0.2689/K(T)}$ atm
$P_{\min}$ at 400 / 500 / 600 K	0.34 / 5.22 / 32.2 atm
Favourable region	low $T$, high $P$

Concept & Theory

Topic. Gas-phase reaction equilibrium; effect of $T$ and $P$ on conversion.

Key relations. $K=\exp(-\Delta G^\circ/RT)$; van’t Hoff via $\Delta G^\circ(T)=\Delta H^\circ-T\Delta S^\circ$; $\prod y_i^{\nu_i}(P/P^\circ)^{\Delta n}=K$.

Why this works. Separating the composition dependence (the group $G(X)$) from the temperature dependence ($K$) and the pressure dependence $(P/P^\circ)^{\Delta n}$ turns the equilibrium condition into a single explicit boundary $P_{\min}(T)$. Exothermic ($\Delta H^\circ<0$) means $K$ falls with $T$; mole-reducing ($\Delta n<0$) means higher $P$ helps — both push the favourable zone to the lower-left of the $P$–$T$ plane.

Common pitfall. Dropping the $(3-2X)^2$ total-mole factor, or mishandling the $\Delta n=-2$ pressure exponent (the pressure enters squared). Assuming $\Delta H^\circ,\Delta S^\circ$ constant is acceptable for an order-of-magnitude region but should be flagged.

Source. Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics §§13.1–13.5 (single-reaction equilibrium, effect of $T$ and $P$).

Question 5 of 6: Favourable T–P Region for Methanol Synthesis

Question 5: Favourable T–P Region for Methanol Synthesis (Part B — equal value)