H2 + CO2 = H2O + CH3OH

H_2 hydrogen + CO_2 carbon dioxide ⟶ H_2O water + CH_3OH methanol

Balance the chemical equation algebraically: H_2 + CO_2 ⟶ H_2O + CH_3OH Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2 + c_2 CO_2 ⟶ c_3 H_2O + c_4 CH_3OH Set the number of atoms in the reactants equal to the number of atoms in the products for H, C and O: H: | 2 c_1 = 2 c_3 + 4 c_4 C: | c_2 = c_4 O: | 2 c_2 = c_3 + c_4 Since the coefficients are relative quantities and underdetermined, choose a coefficient to set arbitrarily. To keep the coefficients small, the arbitrary value is ordinarily one. For instance, set c_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 1 c_3 = 1 c_4 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2 + CO_2 ⟶ H_2O + CH_3OH

+ ⟶ +

hydrogen + carbon dioxide ⟶ water + methanol

| hydrogen | carbon dioxide | water | methanol molecular enthalpy | 0 kJ/mol | -393.5 kJ/mol | -285.8 kJ/mol | -238.7 kJ/mol total enthalpy | 0 kJ/mol | -393.5 kJ/mol | -285.8 kJ/mol | -238.7 kJ/mol | H_initial = -393.5 kJ/mol | | H_final = -524.5 kJ/mol | ΔH_rxn^0 | -524.5 kJ/mol - -393.5 kJ/mol = -131 kJ/mol (exothermic) | | |

| hydrogen | carbon dioxide | water | methanol molecular free energy | 0 kJ/mol | -394.4 kJ/mol | -237.1 kJ/mol | -166.3 kJ/mol total free energy | 0 kJ/mol | -394.4 kJ/mol | -237.1 kJ/mol | -166.3 kJ/mol | G_initial = -394.4 kJ/mol | | G_final = -403.4 kJ/mol | ΔG_rxn^0 | -403.4 kJ/mol - -394.4 kJ/mol = -8.97 kJ/mol (exergonic) | | |

| hydrogen | carbon dioxide | water | methanol molecular entropy | 115 J/(mol K) | 214 J/(mol K) | 69.91 J/(mol K) | 126.8 J/(mol K) total entropy | 345 J/(mol K) | 214 J/(mol K) | 69.91 J/(mol K) | 126.8 J/(mol K) | S_initial = 559 J/(mol K) | | S_final = 196.7 J/(mol K) | ΔS_rxn^0 | 196.7 J/(mol K) - 559 J/(mol K) = -362.3 J/(mol K) (exoentropic) | | |

Construct the equilibrium constant, K, expression for: H_2 + CO_2 ⟶ H_2O + CH_3OH Plan: • Balance the chemical equation. • Determine the stoichiometric numbers. • Assemble the activity expression for each chemical species. • Use the activity expressions to build the equilibrium constant expression. Write the balanced chemical equation: 3 H_2 + CO_2 ⟶ H_2O + CH_3OH Assign stoichiometric numbers, ν_i, using the stoichiometric coefficients, c_i, from the balanced chemical equation in the following manner: ν_i = -c_i for reactants and ν_i = c_i for products: chemical species | c_i | ν_i H_2 | 3 | -3 CO_2 | 1 | -1 H_2O | 1 | 1 CH_3OH | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2 | 3 | -3 | ([H2])^(-3) CO_2 | 1 | -1 | ([CO2])^(-1) H_2O | 1 | 1 | [H2O] CH_3OH | 1 | 1 | [CH3OH] The equilibrium constant symbol in the concentration basis is: K_c Mulitply the activity expressions to arrive at the K_c expression: Answer: | | K_c = ([H2])^(-3) ([CO2])^(-1) [H2O] [CH3OH] = ([H2O] [CH3OH])/(([H2])^3 [CO2])

Construct the rate of reaction expression for: H_2 + CO_2 ⟶ H_2O + CH_3OH Plan: • Balance the chemical equation. • Determine the stoichiometric numbers. • Assemble the rate term for each chemical species. • Write the rate of reaction expression. Write the balanced chemical equation: 3 H_2 + CO_2 ⟶ H_2O + CH_3OH Assign stoichiometric numbers, ν_i, using the stoichiometric coefficients, c_i, from the balanced chemical equation in the following manner: ν_i = -c_i for reactants and ν_i = c_i for products: chemical species | c_i | ν_i H_2 | 3 | -3 CO_2 | 1 | -1 H_2O | 1 | 1 CH_3OH | 1 | 1 The rate term for each chemical species, B_i, is 1/ν_i(Δ[B_i])/(Δt) where [B_i] is the amount concentration and t is time: chemical species | c_i | ν_i | rate term H_2 | 3 | -3 | -1/3 (Δ[H2])/(Δt) CO_2 | 1 | -1 | -(Δ[CO2])/(Δt) H_2O | 1 | 1 | (Δ[H2O])/(Δt) CH_3OH | 1 | 1 | (Δ[CH3OH])/(Δt) (for infinitesimal rate of change, replace Δ with d) Set the rate terms equal to each other to arrive at the rate expression: Answer: | | rate = -1/3 (Δ[H2])/(Δt) = -(Δ[CO2])/(Δt) = (Δ[H2O])/(Δt) = (Δ[CH3OH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| hydrogen | carbon dioxide | water | methanol formula | H_2 | CO_2 | H_2O | CH_3OH Hill formula | H_2 | CO_2 | H_2O | CH_4O name | hydrogen | carbon dioxide | water | methanol IUPAC name | molecular hydrogen | carbon dioxide | water | methanol