#### 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 ΔGof,T and ΔGorxn,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 $\frac{\Delta G_T^{\circ}}{RT}=\frac{\Delta G_R^{\circ}}{RT_R}+I+\frac{1}{R}\left[\frac{J}{T}-\Delta AlnT-\frac{\Delta B}{2}T-\frac{\Delta C}{6}T^2-\frac{\Delta D}{12}T^3\right]$