KMnO4 = O2 + MnO2 + K2MnO2

KMnO_4 potassium permanganate ⟶ O_2 oxygen + MnO_2 manganese dioxide + K2MnO2

Balance the chemical equation algebraically: KMnO_4 ⟶ O_2 + MnO_2 + K2MnO2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KMnO_4 ⟶ c_2 O_2 + c_3 MnO_2 + c_4 K2MnO2 Set the number of atoms in the reactants equal to the number of atoms in the products for K, Mn and O: K: | c_1 = 2 c_4 Mn: | c_1 = c_3 + c_4 O: | 4 c_1 = 2 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 2 c_2 = 2 c_3 = 1 c_4 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 KMnO_4 ⟶ 2 O_2 + MnO_2 + K2MnO2

⟶ + + K2MnO2

potassium permanganate ⟶ oxygen + manganese dioxide + K2MnO2

Construct the equilibrium constant, K, expression for: KMnO_4 ⟶ O_2 + MnO_2 + K2MnO2 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 KMnO_4 ⟶ 2 O_2 + MnO_2 + K2MnO2 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 KMnO_4 | 2 | -2 O_2 | 2 | 2 MnO_2 | 1 | 1 K2MnO2 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KMnO_4 | 2 | -2 | ([KMnO4])^(-2) O_2 | 2 | 2 | ([O2])^2 MnO_2 | 1 | 1 | [MnO2] K2MnO2 | 1 | 1 | [K2MnO2] 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 = ([KMnO4])^(-2) ([O2])^2 [MnO2] [K2MnO2] = (([O2])^2 [MnO2] [K2MnO2])/([KMnO4])^2

Construct the rate of reaction expression for: KMnO_4 ⟶ O_2 + MnO_2 + K2MnO2 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 KMnO_4 ⟶ 2 O_2 + MnO_2 + K2MnO2 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 KMnO_4 | 2 | -2 O_2 | 2 | 2 MnO_2 | 1 | 1 K2MnO2 | 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 KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) O_2 | 2 | 2 | 1/2 (Δ[O2])/(Δt) MnO_2 | 1 | 1 | (Δ[MnO2])/(Δt) K2MnO2 | 1 | 1 | (Δ[K2MnO2])/(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[KMnO4])/(Δt) = 1/2 (Δ[O2])/(Δt) = (Δ[MnO2])/(Δt) = (Δ[K2MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium permanganate | oxygen | manganese dioxide | K2MnO2 formula | KMnO_4 | O_2 | MnO_2 | K2MnO2 name | potassium permanganate | oxygen | manganese dioxide | IUPAC name | potassium permanganate | molecular oxygen | dioxomanganese |

| potassium permanganate | oxygen | manganese dioxide | K2MnO2 molar mass | 158.03 g/mol | 31.998 g/mol | 86.936 g/mol | 165.133 g/mol phase | solid (at STP) | gas (at STP) | solid (at STP) | melting point | 240 °C | -218 °C | 535 °C | boiling point | | -183 °C | | density | 1 g/cm^3 | 0.001429 g/cm^3 (at 0 °C) | 5.03 g/cm^3 | solubility in water | | | insoluble | surface tension | | 0.01347 N/m | | dynamic viscosity | | 2.055×10^-5 Pa s (at 25 °C) | | odor | odorless | odorless | |