O2 + CH4 = H2O + C2H2

O_2 oxygen + CH_4 methane ⟶ H_2O water + C_2H_2 acetylene

Balance the chemical equation algebraically: O_2 + CH_4 ⟶ H_2O + C_2H_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 O_2 + c_2 CH_4 ⟶ c_3 H_2O + c_4 C_2H_2 Set the number of atoms in the reactants equal to the number of atoms in the products for O, C and H: O: | 2 c_1 = c_3 C: | c_2 = 2 c_4 H: | 4 c_2 = 2 c_3 + 2 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_4 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3/2 c_2 = 2 c_3 = 3 c_4 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 3 c_2 = 4 c_3 = 6 c_4 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 O_2 + 4 CH_4 ⟶ 6 H_2O + 2 C_2H_2

+ ⟶ +

oxygen + methane ⟶ water + acetylene

| oxygen | methane | water | acetylene molecular enthalpy | 0 kJ/mol | -74.6 kJ/mol | -285.8 kJ/mol | 227.4 kJ/mol total enthalpy | 0 kJ/mol | -298.4 kJ/mol | -1715 kJ/mol | 454.8 kJ/mol | H_initial = -298.4 kJ/mol | | H_final = -1260 kJ/mol | ΔH_rxn^0 | -1260 kJ/mol - -298.4 kJ/mol = -961.8 kJ/mol (exothermic) | | |

| oxygen | methane | water | acetylene molecular free energy | 231.7 kJ/mol | -51 kJ/mol | -237.1 kJ/mol | 209.9 kJ/mol total free energy | 695.1 kJ/mol | -204 kJ/mol | -1423 kJ/mol | 419.8 kJ/mol | G_initial = 491.1 kJ/mol | | G_final = -1003 kJ/mol | ΔG_rxn^0 | -1003 kJ/mol - 491.1 kJ/mol = -1494 kJ/mol (exergonic) | | |

| oxygen | methane | water | acetylene molecular entropy | 205 J/(mol K) | 186 J/(mol K) | 69.91 J/(mol K) | 201 J/(mol K) total entropy | 615 J/(mol K) | 744 J/(mol K) | 419.5 J/(mol K) | 402 J/(mol K) | S_initial = 1359 J/(mol K) | | S_final = 821.5 J/(mol K) | ΔS_rxn^0 | 821.5 J/(mol K) - 1359 J/(mol K) = -537.5 J/(mol K) (exoentropic) | | |

Construct the equilibrium constant, K, expression for: O_2 + CH_4 ⟶ H_2O + C_2H_2 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 O_2 + 4 CH_4 ⟶ 6 H_2O + 2 C_2H_2 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 O_2 | 3 | -3 CH_4 | 4 | -4 H_2O | 6 | 6 C_2H_2 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression O_2 | 3 | -3 | ([O2])^(-3) CH_4 | 4 | -4 | ([CH4])^(-4) H_2O | 6 | 6 | ([H2O])^6 C_2H_2 | 2 | 2 | ([C2H2])^2 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 = ([O2])^(-3) ([CH4])^(-4) ([H2O])^6 ([C2H2])^2 = (([H2O])^6 ([C2H2])^2)/(([O2])^3 ([CH4])^4)

Construct the rate of reaction expression for: O_2 + CH_4 ⟶ H_2O + C_2H_2 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 O_2 + 4 CH_4 ⟶ 6 H_2O + 2 C_2H_2 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 O_2 | 3 | -3 CH_4 | 4 | -4 H_2O | 6 | 6 C_2H_2 | 2 | 2 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 O_2 | 3 | -3 | -1/3 (Δ[O2])/(Δt) CH_4 | 4 | -4 | -1/4 (Δ[CH4])/(Δt) H_2O | 6 | 6 | 1/6 (Δ[H2O])/(Δt) C_2H_2 | 2 | 2 | 1/2 (Δ[C2H2])/(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[O2])/(Δt) = -1/4 (Δ[CH4])/(Δt) = 1/6 (Δ[H2O])/(Δt) = 1/2 (Δ[C2H2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| oxygen | methane | water | acetylene formula | O_2 | CH_4 | H_2O | C_2H_2 name | oxygen | methane | water | acetylene IUPAC name | molecular oxygen | methane | water | acetylene