The Peng–Robinson equation of state is used to compute the extent of reaction and the correction factor due to gas-phase nonideality, as a result of the high pressure applied in the synthesis of ammonia (3H₂ + N₂ ⇌ 2NH₃). The simulation plots these two quantities versus pressure for temperatures ranging from 550 K to 850 K. The extent of reaction approaches one at high pressure and low temperature. The non-ideal gas correction factor is close to one at low pressure and considerably less than one at high pressure.
The correction factor due to gas-phase nonideality is given by \(\varphi_{\mathrm{NH_3}}\big/\big(\varphi_{\mathrm{H_2}}^{1.5}\,\varphi_{\mathrm{N_2}}^{0.5}\big)\), where \(\varphi_{\mathrm{NH_3}}\), \(\varphi_{\mathrm{N_2}}\), and \(\varphi_{\mathrm{H_2}}\) are the fugacity coefficients of the species involved; they are calculated using the Peng–Robinson equation of state.
Starting from a stoichiometric feed of 1 mol N₂ and 3 mol H₂, and letting \(\xi\) be the extent of reaction (0 to 1), the equilibrium mole fractions are
\[y_{\mathrm{H_2}}=\frac{1.5(1-\xi)}{2-\xi},\qquad y_{\mathrm{N_2}}=\frac{0.5(1-\xi)}{2-\xi},\qquad y_{\mathrm{NH_3}}=\frac{\xi}{2-\xi}\]
The vapor-phase compressibility factor \(Z\) of the mixture solves
\[Z^3-(1-B)Z^2+(A-3B^2-2B)Z-(AB-B^2-B^3)=0\]
with mixing rules and pure-component parameters
\[A=\sum_i\sum_j y_iy_jA_{ij},\quad A_{ij}=\sqrt{A_iA_j}\,(1-k_{ij}),\quad B=\sum_i y_iB_i\]
\[A_i=\frac{a_i(T)\,P}{(RT)^2},\qquad B_i=\frac{b_i P}{RT},\qquad b_i=0.0778\,R\,\frac{T_{c,i}}{P_{c,i}}\]
\[a_i(T)=0.45724\,R^2\frac{T_{c,i}^2}{P_{c,i}}\Big[1+m_i\big(1-\sqrt{T/T_{c,i}}\,\big)\Big]^2,\qquad m_i=0.37464+1.54226\,\omega_i-0.26992\,\omega_i^2\]
\[\ln\varphi_i=(Z-1)\frac{B_i}{B}-\ln(Z-B)-\frac{A}{2\sqrt2\,B}\left(\frac{2\sum_j y_jA_{ij}}{A}-\frac{B_i}{B}\right)\ln\!\left(\frac{Z+(1+\sqrt2)B}{Z+(1-\sqrt2)B}\right)\]
\[K_v=\frac{\varphi_{\mathrm{NH_3}}}{\varphi_{\mathrm{H_2}}^{1.5}\,\varphi_{\mathrm{N_2}}^{0.5}}\]
The extent of reaction is found by solving the Peng–Robinson cubic together with the equilibrium relation
\[K(T)=\frac{y_{\mathrm{NH_3}}}{y_{\mathrm{H_2}}^{1.5}\,y_{\mathrm{N_2}}^{0.5}}\cdot\frac{P^{\circ}}{P}\cdot K_v,\qquad P^{\circ}=1\text{ atm}\]
where the standard-state equilibrium constant \(K(T)\), with \(T\) in kelvin, is
\[K(T)=\frac{3.3885364\times10^{12}}{(1.8\,T)^{3.5312124}}\exp\!\left[\begin{aligned}&4641.1884\Big(\tfrac1T-\tfrac1{298.15}\Big)+0.0016755\,(T-298.15)\\[-2pt] -{}&8.219068\times10^{-9}(T^2-298.15^2)+2.7468727\times10^{-11}(T^3-298.15^3)\end{aligned}\right]\]
| \(\xi\) | extent of reaction |
| \(y_i\) | mole fraction of species \(i\) in the vapor mixture |
| \(\varphi_i\) | fugacity coefficient of species \(i\) |
| \(Z\) | compressibility factor of the vapor mixture |
| \(A,B\) | mixture Peng–Robinson parameters; \(A_i,B_i\) are the pure-component values |
| \(k_{ij}\) | binary interaction parameter (\(k_{\mathrm{H_2,N_2}}=-0.036\), \(k_{\mathrm{N_2,NH_3}}=0.222\), \(k_{\mathrm{H_2,NH_3}}=0\)) |
| \(T_{c,i},P_{c,i},\omega_i\) | critical temperature, critical pressure, and acentric factor of species \(i\) |
| \(R\) | gas constant |
| \(K(T)\) | standard-state equilibrium constant |
| \(K_v\) | non-ideal gas correction factor |
| Species | Tc (°R) | Pc (psia) | ω |
|---|---|---|---|
| H₂ | 59.7 | 190.8 | 0.000 |
| N₂ | 227.3 | 492.9 | 0.0206 |
| NH₃ | 730.2 | 1653.7 | 0.2582 |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It is a JavaScript/HTML5 implementation of a Mathematica simulation by Housam Binous. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.