KOH + Mn3O4 + KCl3 = H2O + KCl + K2MnO4

KOH potassium hydroxide + Mn_3O_4 manganese(II, III) oxide + KCl3 ⟶ H_2O water + KCl potassium chloride + K_2MnO_4 potassium manganate

Balance the chemical equation algebraically: KOH + Mn_3O_4 + KCl3 ⟶ H_2O + KCl + K_2MnO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KOH + c_2 Mn_3O_4 + c_3 KCl3 ⟶ c_4 H_2O + c_5 KCl + c_6 K_2MnO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, K, O, Mn and Cl: H: | c_1 + c_2 = 2 c_4 K: | c_1 + c_3 = c_5 + 2 c_6 O: | c_1 + 2 c_2 = c_4 + 4 c_6 Mn: | 2 c_2 = c_6 Cl: | 3 c_3 = c_5 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 = 13 c_2 = 1 c_3 = 9/2 c_4 = 7 c_5 = 27/2 c_6 = 2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 26 c_2 = 2 c_3 = 9 c_4 = 14 c_5 = 27 c_6 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 26 KOH + 2 Mn_3O_4 + 9 KCl3 ⟶ 14 H_2O + 27 KCl + 4 K_2MnO_4

+ + KCl3 ⟶ + +

potassium hydroxide + manganese(II, III) oxide + KCl3 ⟶ water + potassium chloride + potassium manganate

Construct the equilibrium constant, K, expression for: KOH + Mn_3O_4 + KCl3 ⟶ H_2O + KCl + K_2MnO_4 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: 26 KOH + 2 Mn_3O_4 + 9 KCl3 ⟶ 14 H_2O + 27 KCl + 4 K_2MnO_4 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 KOH | 26 | -26 Mn_3O_4 | 2 | -2 KCl3 | 9 | -9 H_2O | 14 | 14 KCl | 27 | 27 K_2MnO_4 | 4 | 4 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KOH | 26 | -26 | ([KOH])^(-26) Mn_3O_4 | 2 | -2 | ([Mn3O4])^(-2) KCl3 | 9 | -9 | ([KCl3])^(-9) H_2O | 14 | 14 | ([H2O])^14 KCl | 27 | 27 | ([KCl])^27 K_2MnO_4 | 4 | 4 | ([K2MnO4])^4 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 = ([KOH])^(-26) ([Mn3O4])^(-2) ([KCl3])^(-9) ([H2O])^14 ([KCl])^27 ([K2MnO4])^4 = (([H2O])^14 ([KCl])^27 ([K2MnO4])^4)/(([KOH])^26 ([Mn3O4])^2 ([KCl3])^9)

Construct the rate of reaction expression for: KOH + Mn_3O_4 + KCl3 ⟶ H_2O + KCl + K_2MnO_4 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: 26 KOH + 2 Mn_3O_4 + 9 KCl3 ⟶ 14 H_2O + 27 KCl + 4 K_2MnO_4 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 KOH | 26 | -26 Mn_3O_4 | 2 | -2 KCl3 | 9 | -9 H_2O | 14 | 14 KCl | 27 | 27 K_2MnO_4 | 4 | 4 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 KOH | 26 | -26 | -1/26 (Δ[KOH])/(Δt) Mn_3O_4 | 2 | -2 | -1/2 (Δ[Mn3O4])/(Δt) KCl3 | 9 | -9 | -1/9 (Δ[KCl3])/(Δt) H_2O | 14 | 14 | 1/14 (Δ[H2O])/(Δt) KCl | 27 | 27 | 1/27 (Δ[KCl])/(Δt) K_2MnO_4 | 4 | 4 | 1/4 (Δ[K2MnO4])/(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 = -1/26 (Δ[KOH])/(Δt) = -1/2 (Δ[Mn3O4])/(Δt) = -1/9 (Δ[KCl3])/(Δt) = 1/14 (Δ[H2O])/(Δt) = 1/27 (Δ[KCl])/(Δt) = 1/4 (Δ[K2MnO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium hydroxide | manganese(II, III) oxide | KCl3 | water | potassium chloride | potassium manganate formula | KOH | Mn_3O_4 | KCl3 | H_2O | KCl | K_2MnO_4 Hill formula | HKO | Mn_3O_4 | Cl3K | H_2O | ClK | K_2MnO_4 name | potassium hydroxide | manganese(II, III) oxide | | water | potassium chloride | potassium manganate IUPAC name | potassium hydroxide | | | water | potassium chloride | dipotassium dioxido-dioxomanganese

| potassium hydroxide | manganese(II, III) oxide | KCl3 | water | potassium chloride | potassium manganate molar mass | 56.105 g/mol | 142.882 g/mol | 145.4 g/mol | 18.015 g/mol | 74.55 g/mol | 197.13 g/mol phase | solid (at STP) | | | liquid (at STP) | solid (at STP) | solid (at STP) melting point | 406 °C | | | 0 °C | 770 °C | 190 °C boiling point | 1327 °C | | | 99.9839 °C | 1420 °C | density | 2.044 g/cm^3 | 4.8 g/cm^3 | | 1 g/cm^3 | 1.98 g/cm^3 | solubility in water | soluble | | | | soluble | decomposes surface tension | | | | 0.0728 N/m | | dynamic viscosity | 0.001 Pa s (at 550 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | | odor | | | | odorless | odorless |