H2O + KMnO4 + H2O2 = O2 + KOH + MnO2

H_2O water + KMnO_4 potassium permanganate + H_2O_2 hydrogen peroxide ⟶ O_2 oxygen + KOH potassium hydroxide + MnO_2 manganese dioxide

Balance the chemical equation algebraically: H_2O + KMnO_4 + H_2O_2 ⟶ O_2 + KOH + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 H_2O_2 ⟶ c_4 O_2 + c_5 KOH + c_6 MnO_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 + 2 c_3 = c_5 O: | c_1 + 4 c_2 + 2 c_3 = 2 c_4 + c_5 + 2 c_6 K: | c_2 = c_5 Mn: | c_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_2 = 2 c_1 + 2 c_3 = 1 c_4 = (3 c_1)/2 + 2 c_5 = 2 c_1 + 2 c_6 = 2 c_1 + 2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 2 and solve for the remaining coefficients: c_1 = 2 c_2 = 6 c_3 = 1 c_4 = 5 c_5 = 6 c_6 = 6 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + 6 KMnO_4 + H_2O_2 ⟶ 5 O_2 + 6 KOH + 6 MnO_2

+ + ⟶ + +

water + potassium permanganate + hydrogen peroxide ⟶ oxygen + potassium hydroxide + manganese dioxide

| water | potassium permanganate | hydrogen peroxide | oxygen | potassium hydroxide | manganese dioxide molecular free energy | -237.1 kJ/mol | -737.6 kJ/mol | -120.4 kJ/mol | 231.7 kJ/mol | -379.4 kJ/mol | -465.1 kJ/mol total free energy | -474.2 kJ/mol | -4426 kJ/mol | -120.4 kJ/mol | 1159 kJ/mol | -2276 kJ/mol | -2791 kJ/mol | G_initial = -5020 kJ/mol | | | G_final = -3909 kJ/mol | | ΔG_rxn^0 | -3909 kJ/mol - -5020 kJ/mol = 1112 kJ/mol (endergonic) | | | | |

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + H_2O_2 ⟶ O_2 + KOH + MnO_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 + 6 KMnO_4 + H_2O_2 ⟶ 5 O_2 + 6 KOH + 6 MnO_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 KMnO_4 | 6 | -6 H_2O_2 | 1 | -1 O_2 | 5 | 5 KOH | 6 | 6 MnO_2 | 6 | 6 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) KMnO_4 | 6 | -6 | ([KMnO4])^(-6) H_2O_2 | 1 | -1 | ([H2O2])^(-1) O_2 | 5 | 5 | ([O2])^5 KOH | 6 | 6 | ([KOH])^6 MnO_2 | 6 | 6 | ([MnO2])^6 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) ([KMnO4])^(-6) ([H2O2])^(-1) ([O2])^5 ([KOH])^6 ([MnO2])^6 = (([O2])^5 ([KOH])^6 ([MnO2])^6)/(([H2O])^2 ([KMnO4])^6 [H2O2])

Construct the rate of reaction expression for: H_2O + KMnO_4 + H_2O_2 ⟶ O_2 + KOH + MnO_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 + 6 KMnO_4 + H_2O_2 ⟶ 5 O_2 + 6 KOH + 6 MnO_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 KMnO_4 | 6 | -6 H_2O_2 | 1 | -1 O_2 | 5 | 5 KOH | 6 | 6 MnO_2 | 6 | 6 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) KMnO_4 | 6 | -6 | -1/6 (Δ[KMnO4])/(Δt) H_2O_2 | 1 | -1 | -(Δ[H2O2])/(Δt) O_2 | 5 | 5 | 1/5 (Δ[O2])/(Δt) KOH | 6 | 6 | 1/6 (Δ[KOH])/(Δt) MnO_2 | 6 | 6 | 1/6 (Δ[MnO2])/(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) = -1/6 (Δ[KMnO4])/(Δt) = -(Δ[H2O2])/(Δt) = 1/5 (Δ[O2])/(Δt) = 1/6 (Δ[KOH])/(Δt) = 1/6 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | hydrogen peroxide | oxygen | potassium hydroxide | manganese dioxide formula | H_2O | KMnO_4 | H_2O_2 | O_2 | KOH | MnO_2 Hill formula | H_2O | KMnO_4 | H_2O_2 | O_2 | HKO | MnO_2 name | water | potassium permanganate | hydrogen peroxide | oxygen | potassium hydroxide | manganese dioxide IUPAC name | water | potassium permanganate | hydrogen peroxide | molecular oxygen | potassium hydroxide | dioxomanganese

| water | potassium permanganate | hydrogen peroxide | oxygen | potassium hydroxide | manganese dioxide molar mass | 18.015 g/mol | 158.03 g/mol | 34.014 g/mol | 31.998 g/mol | 56.105 g/mol | 86.936 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | gas (at STP) | solid (at STP) | solid (at STP) melting point | 0 °C | 240 °C | -0.43 °C | -218 °C | 406 °C | 535 °C boiling point | 99.9839 °C | | 150.2 °C | -183 °C | 1327 °C | density | 1 g/cm^3 | 1 g/cm^3 | 1.44 g/cm^3 | 0.001429 g/cm^3 (at 0 °C) | 2.044 g/cm^3 | 5.03 g/cm^3 solubility in water | | | miscible | | soluble | insoluble surface tension | 0.0728 N/m | | 0.0804 N/m | 0.01347 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | 0.001249 Pa s (at 20 °C) | 2.055×10^-5 Pa s (at 25 °C) | 0.001 Pa s (at 550 °C) | odor | odorless | odorless | | odorless | |