H2O + KMnO4 + C2H2 = KOH + MnO2 + C2H2O4

H_2O water + KMnO_4 potassium permanganate + C_2H_2 acetylene ⟶ KOH potassium hydroxide + MnO_2 manganese dioxide + HO_2CCO_2H oxalic acid

Balance the chemical equation algebraically: H_2O + KMnO_4 + C_2H_2 ⟶ KOH + MnO_2 + HO_2CCO_2H Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 C_2H_2 ⟶ c_4 KOH + c_5 MnO_2 + c_6 HO_2CCO_2H Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn and C: H: | 2 c_1 + 2 c_3 = c_4 + 2 c_6 O: | c_1 + 4 c_2 = c_4 + 2 c_5 + 4 c_6 K: | c_2 = c_4 Mn: | c_2 = c_5 C: | 2 c_3 = 2 c_6 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 4/3 c_2 = 8/3 c_3 = 1 c_4 = 8/3 c_5 = 8/3 c_6 = 1 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 4 c_2 = 8 c_3 = 3 c_4 = 8 c_5 = 8 c_6 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2O + 8 KMnO_4 + 3 C_2H_2 ⟶ 8 KOH + 8 MnO_2 + 3 HO_2CCO_2H

+ + ⟶ + +

water + potassium permanganate + acetylene ⟶ potassium hydroxide + manganese dioxide + oxalic acid

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + C_2H_2 ⟶ KOH + MnO_2 + HO_2CCO_2H 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: 4 H_2O + 8 KMnO_4 + 3 C_2H_2 ⟶ 8 KOH + 8 MnO_2 + 3 HO_2CCO_2H 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_2O | 4 | -4 KMnO_4 | 8 | -8 C_2H_2 | 3 | -3 KOH | 8 | 8 MnO_2 | 8 | 8 HO_2CCO_2H | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 4 | -4 | ([H2O])^(-4) KMnO_4 | 8 | -8 | ([KMnO4])^(-8) C_2H_2 | 3 | -3 | ([C2H2])^(-3) KOH | 8 | 8 | ([KOH])^8 MnO_2 | 8 | 8 | ([MnO2])^8 HO_2CCO_2H | 3 | 3 | ([HO2CCO2H])^3 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 = ([H2O])^(-4) ([KMnO4])^(-8) ([C2H2])^(-3) ([KOH])^8 ([MnO2])^8 ([HO2CCO2H])^3 = (([KOH])^8 ([MnO2])^8 ([HO2CCO2H])^3)/(([H2O])^4 ([KMnO4])^8 ([C2H2])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + C_2H_2 ⟶ KOH + MnO_2 + HO_2CCO_2H 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: 4 H_2O + 8 KMnO_4 + 3 C_2H_2 ⟶ 8 KOH + 8 MnO_2 + 3 HO_2CCO_2H 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_2O | 4 | -4 KMnO_4 | 8 | -8 C_2H_2 | 3 | -3 KOH | 8 | 8 MnO_2 | 8 | 8 HO_2CCO_2H | 3 | 3 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_2O | 4 | -4 | -1/4 (Δ[H2O])/(Δt) KMnO_4 | 8 | -8 | -1/8 (Δ[KMnO4])/(Δt) C_2H_2 | 3 | -3 | -1/3 (Δ[C2H2])/(Δt) KOH | 8 | 8 | 1/8 (Δ[KOH])/(Δt) MnO_2 | 8 | 8 | 1/8 (Δ[MnO2])/(Δt) HO_2CCO_2H | 3 | 3 | 1/3 (Δ[HO2CCO2H])/(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 = -1/4 (Δ[H2O])/(Δt) = -1/8 (Δ[KMnO4])/(Δt) = -1/3 (Δ[C2H2])/(Δt) = 1/8 (Δ[KOH])/(Δt) = 1/8 (Δ[MnO2])/(Δt) = 1/3 (Δ[HO2CCO2H])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | acetylene | potassium hydroxide | manganese dioxide | oxalic acid formula | H_2O | KMnO_4 | C_2H_2 | KOH | MnO_2 | HO_2CCO_2H Hill formula | H_2O | KMnO_4 | C_2H_2 | HKO | MnO_2 | C_2H_2O_4 name | water | potassium permanganate | acetylene | potassium hydroxide | manganese dioxide | oxalic acid IUPAC name | water | potassium permanganate | acetylene | potassium hydroxide | dioxomanganese | oxalic acid