#### Chemical Equilibrium for Multiple Reactions: Screencasts

Demonstrates how to calculate the change in Gibbs free energy for a reaction at elevated temperature when the heat of reaction and heat capacities are functions of temperature. The same procedure is used for ΔG^{o}_{f,T} and ΔG^{o}_{rxn,T}.

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Demonstrates how to use an Excel spreadsheet to calculate the Gibbs free energy as a function of temperature when heat capacities and heat of reaction are functions of temperature. Spreadsheet: Gibbs Free Energy as a Function of Temperature

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Explains how to calculate chemical equilibrium using Gibbs minimization instead of equilibrium constants and extents of reaction.

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##### Important Equations:

**Gibbs free energies **

\[\overline{G}_i = G_i + RTlny_i\] \(G_i = \Delta G_{f,i}^o +RTln \left( \frac{P}{P^{\circ}} \right)\)

where \(\overline{G}_i\) = partial molar Gibbs free energy for component \(i\)

\(G_i\) = Gibbs free energy of pure component \(i\)

\(\Delta G_{f,i}^{\circ}\) = Gibbs free energy of formation of component \(i\) at standard conditions

\(y_i\) = mole fraction of component \(i\)

\(T\) = temperature (K)

\(R\) = ideal gas constant

\(P\) = pressure

\(P^{\circ}\) = standard state pressure (1 bar) \[nG = \sum n_i\Delta G_{f,T}^{\circ} + \sum n_iRTlnP + \sum n_iRTlny_i\]

\(n\) = total number of moles

\(G\) = Gibbs free energy per mole of mixture

\(n_i\) = number of moles of component \(i\)

**Atomic balances on C, O, H for this example **\[n_{c,in} = n_{c,out} \,\,\,\,n_{O,in}=n_{O,out}\,\,\,\,n_{H,in}=n_{H,out}\]

\(n_{i,in}\)= moles of \(i\) atoms in molecules fed to reactor

\(n_{i,out}\)= moles of \(i\) atoms in molecules leaving reactor

**Heat of formation as a function of temperature** \[\Delta H_T^o = \Delta T_R^o = \Delta A(T-T_R)+\frac{\Delta B}{2}(T^2-T_R^2)+\frac{\Delta C}{3}(T^3-T_R^3)+\frac{\Delta D}{4}(T^4-T_R^4)\] where \(\Delta H_R^o\) = heat of reaction at temperature \(T\) at standard conditions

\(\Delta H_R^o\) = heat of reaction at reference temperature \(T_R\) at standard conditions

\(T_R\) = reference temperature (298K) \[C_{pi}=A_i+B_iT=C_iT^2+D_iT^3\] where \(C_{pi}\)= heat capacity of component \(i\)

\(A_i, B_i, C_i, D_i\) are constants for component \(i\) \[\Delta C_p = \sum \nu _iC_{pi}\] where \(\nu _i\)= stoichiometric coefficient (positive for products, negative for reactants) \[\Delta A = \sum \nu _iA_i\] (likewise for \(\Delta B, \Delta C, \Delta D\)) \[J = \Delta H_R^o – \Delta AT_R – \frac{\Delta B}{2}T_R^2-\frac{\Delta C}{3}T_R^3-\frac{\Delta D}{4}T_R^4\] \[\Delta H_T^o = J + \Delta AT + \frac{\Delta B}{2}T^2+\frac{\Delta C}{3}T^3+\frac{\Delta D}{4}T^4\] \(J\) = constant for a given reaction

**van’t Hoff equation**\[\frac{\partial (G_T^o)/RT)}{\partial T} = -\frac{\Delta H_T^o}{RT^2}\]

Gibbs free energy of formation as a function of temperature \[\frac{\Delta G_T^{\circ}}{RT}=\frac{\Delta G_R^{\circ}}{RT_R}-\frac{1}{R}\left[-\frac{J}{T}+\frac{J}{T_R}+\Delta A(lnT -lnT_R)+\frac{\Delta B}{2}(T-T_R)+\frac{\Delta C}{6}(T^2-T_R^2)+\frac{\Delta D}{12}(T^3-T_R^3)\right]\] where \(\Delta G_R^{\circ}\) = change in Gibbs free energy for a reaction at reference temperature, \(T_R\), which is 298 K, at standard conditions, and \(J\) is defined above. \[I=\frac{1}{R}\left[-\frac{J}{T_R}+\Delta AlnT_R+\frac{\Delta B}{2}T_R+\frac{\Delta C}{6}T_R^2+\frac{\Delta D}{12}T_R^3\right]\] where \(I\) = constant for a given reaction \[\frac{\Delta G_T^{\circ}}{RT}=\frac{\Delta G_R^{\circ}}{RT_R}+I+\frac{1}{R}\left[\frac{J}{T}-\Delta AlnT-\frac{\Delta B}{2}T-\frac{\Delta C}{6}T^2-\frac{\Delta D}{12}T^3\right]\]