H2O + O2 + K2MnO4 = KOH + KMnO4 + H2O2

H_2O water + O_2 oxygen + K_2MnO_4 potassium manganate ⟶ KOH potassium hydroxide + KMnO_4 potassium permanganate + H_2O_2 hydrogen peroxide

Balance the chemical equation algebraically: H_2O + O_2 + K_2MnO_4 ⟶ KOH + KMnO_4 + H_2O_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 O_2 + c_3 K_2MnO_4 ⟶ c_4 KOH + c_5 KMnO_4 + c_6 H_2O_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K and Mn: H: | 2 c_1 = c_4 + 2 c_6 O: | c_1 + 2 c_2 + 4 c_3 = c_4 + 4 c_5 + 2 c_6 K: | 2 c_3 = c_4 + c_5 Mn: | 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 = 2 c_2 = 1 c_4 = c_3 c_5 = c_3 c_6 = 2 - c_3/2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_3 = 2 and solve for the remaining coefficients: c_1 = 2 c_2 = 1 c_3 = 2 c_4 = 2 c_5 = 2 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + O_2 + 2 K_2MnO_4 ⟶ 2 KOH + 2 KMnO_4 + H_2O_2

+ + ⟶ + +

water + oxygen + potassium manganate ⟶ potassium hydroxide + potassium permanganate + hydrogen peroxide

Construct the equilibrium constant, K, expression for: H_2O + O_2 + K_2MnO_4 ⟶ KOH + KMnO_4 + H_2O_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: 2 H_2O + O_2 + 2 K_2MnO_4 ⟶ 2 KOH + 2 KMnO_4 + H_2O_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 H_2O | 2 | -2 O_2 | 1 | -1 K_2MnO_4 | 2 | -2 KOH | 2 | 2 KMnO_4 | 2 | 2 H_2O_2 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 2 | -2 | ([H2O])^(-2) O_2 | 1 | -1 | ([O2])^(-1) K_2MnO_4 | 2 | -2 | ([K2MnO4])^(-2) KOH | 2 | 2 | ([KOH])^2 KMnO_4 | 2 | 2 | ([KMnO4])^2 H_2O_2 | 1 | 1 | [H2O2] 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])^(-2) ([O2])^(-1) ([K2MnO4])^(-2) ([KOH])^2 ([KMnO4])^2 [H2O2] = (([KOH])^2 ([KMnO4])^2 [H2O2])/(([H2O])^2 [O2] ([K2MnO4])^2)

Construct the rate of reaction expression for: H_2O + O_2 + K_2MnO_4 ⟶ KOH + KMnO_4 + H_2O_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: 2 H_2O + O_2 + 2 K_2MnO_4 ⟶ 2 KOH + 2 KMnO_4 + H_2O_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 H_2O | 2 | -2 O_2 | 1 | -1 K_2MnO_4 | 2 | -2 KOH | 2 | 2 KMnO_4 | 2 | 2 H_2O_2 | 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_2O | 2 | -2 | -1/2 (Δ[H2O])/(Δt) O_2 | 1 | -1 | -(Δ[O2])/(Δt) K_2MnO_4 | 2 | -2 | -1/2 (Δ[K2MnO4])/(Δt) KOH | 2 | 2 | 1/2 (Δ[KOH])/(Δt) KMnO_4 | 2 | 2 | 1/2 (Δ[KMnO4])/(Δt) H_2O_2 | 1 | 1 | (Δ[H2O2])/(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 = -1/2 (Δ[H2O])/(Δt) = -(Δ[O2])/(Δt) = -1/2 (Δ[K2MnO4])/(Δt) = 1/2 (Δ[KOH])/(Δt) = 1/2 (Δ[KMnO4])/(Δt) = (Δ[H2O2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | oxygen | potassium manganate | potassium hydroxide | potassium permanganate | hydrogen peroxide formula | H_2O | O_2 | K_2MnO_4 | KOH | KMnO_4 | H_2O_2 Hill formula | H_2O | O_2 | K_2MnO_4 | HKO | KMnO_4 | H_2O_2 name | water | oxygen | potassium manganate | potassium hydroxide | potassium permanganate | hydrogen peroxide IUPAC name | water | molecular oxygen | dipotassium dioxido-dioxomanganese | potassium hydroxide | potassium permanganate | hydrogen peroxide

| water | oxygen | potassium manganate | potassium hydroxide | potassium permanganate | hydrogen peroxide molar mass | 18.015 g/mol | 31.998 g/mol | 197.13 g/mol | 56.105 g/mol | 158.03 g/mol | 34.014 g/mol phase | liquid (at STP) | gas (at STP) | solid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) melting point | 0 °C | -218 °C | 190 °C | 406 °C | 240 °C | -0.43 °C boiling point | 99.9839 °C | -183 °C | | 1327 °C | | 150.2 °C density | 1 g/cm^3 | 0.001429 g/cm^3 (at 0 °C) | | 2.044 g/cm^3 | 1 g/cm^3 | 1.44 g/cm^3 solubility in water | | | decomposes | soluble | | miscible surface tension | 0.0728 N/m | 0.01347 N/m | | | | 0.0804 N/m dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | 2.055×10^-5 Pa s (at 25 °C) | | 0.001 Pa s (at 550 °C) | | 0.001249 Pa s (at 20 °C) odor | odorless | odorless | | | odorless |