H2 + KMnO4 = H2O + KOH + MnO2

H_2 hydrogen + KMnO_4 potassium permanganate ⟶ H_2O water + KOH potassium hydroxide + MnO_2 manganese dioxide

Balance the chemical equation algebraically: H_2 + KMnO_4 ⟶ H_2O + KOH + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2 + c_2 KMnO_4 ⟶ c_3 H_2O + c_4 KOH + c_5 MnO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, K, Mn and O: H: | 2 c_1 = 2 c_3 + c_4 K: | c_2 = c_4 Mn: | c_2 = c_5 O: | 4 c_2 = c_3 + c_4 + 2 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 = 3/2 c_2 = 1 c_3 = 1 c_4 = 1 c_5 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 3 c_2 = 2 c_3 = 2 c_4 = 2 c_5 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2 + 2 KMnO_4 ⟶ 2 H_2O + 2 KOH + 2 MnO_2

+ ⟶ + +

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

| hydrogen | potassium permanganate | water | potassium hydroxide | manganese dioxide molecular free energy | 0 kJ/mol | -737.6 kJ/mol | -237.1 kJ/mol | -379.4 kJ/mol | -465.1 kJ/mol total free energy | 0 kJ/mol | -1475 kJ/mol | -474.2 kJ/mol | -758.8 kJ/mol | -930.2 kJ/mol | G_initial = -1475 kJ/mol | | G_final = -2163 kJ/mol | | ΔG_rxn^0 | -2163 kJ/mol - -1475 kJ/mol = -688 kJ/mol (exergonic) | | | |

Construct the equilibrium constant, K, expression for: H_2 + KMnO_4 ⟶ H_2O + 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: 3 H_2 + 2 KMnO_4 ⟶ 2 H_2O + 2 KOH + 2 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_2 | 3 | -3 KMnO_4 | 2 | -2 H_2O | 2 | 2 KOH | 2 | 2 MnO_2 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2 | 3 | -3 | ([H2])^(-3) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) H_2O | 2 | 2 | ([H2O])^2 KOH | 2 | 2 | ([KOH])^2 MnO_2 | 2 | 2 | ([MnO2])^2 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 = ([H2])^(-3) ([KMnO4])^(-2) ([H2O])^2 ([KOH])^2 ([MnO2])^2 = (([H2O])^2 ([KOH])^2 ([MnO2])^2)/(([H2])^3 ([KMnO4])^2)

Construct the rate of reaction expression for: H_2 + KMnO_4 ⟶ H_2O + 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: 3 H_2 + 2 KMnO_4 ⟶ 2 H_2O + 2 KOH + 2 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_2 | 3 | -3 KMnO_4 | 2 | -2 H_2O | 2 | 2 KOH | 2 | 2 MnO_2 | 2 | 2 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_2 | 3 | -3 | -1/3 (Δ[H2])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) H_2O | 2 | 2 | 1/2 (Δ[H2O])/(Δt) KOH | 2 | 2 | 1/2 (Δ[KOH])/(Δt) MnO_2 | 2 | 2 | 1/2 (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 = -1/3 (Δ[H2])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = 1/2 (Δ[H2O])/(Δt) = 1/2 (Δ[KOH])/(Δt) = 1/2 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| hydrogen | potassium permanganate | water | potassium hydroxide | manganese dioxide formula | H_2 | KMnO_4 | H_2O | KOH | MnO_2 Hill formula | H_2 | KMnO_4 | H_2O | HKO | MnO_2 name | hydrogen | potassium permanganate | water | potassium hydroxide | manganese dioxide IUPAC name | molecular hydrogen | potassium permanganate | water | potassium hydroxide | dioxomanganese

| hydrogen | potassium permanganate | water | potassium hydroxide | manganese dioxide molar mass | 2.016 g/mol | 158.03 g/mol | 18.015 g/mol | 56.105 g/mol | 86.936 g/mol phase | gas (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) melting point | -259.2 °C | 240 °C | 0 °C | 406 °C | 535 °C boiling point | -252.8 °C | | 99.9839 °C | 1327 °C | density | 8.99×10^-5 g/cm^3 (at 0 °C) | 1 g/cm^3 | 1 g/cm^3 | 2.044 g/cm^3 | 5.03 g/cm^3 solubility in water | | | | soluble | insoluble surface tension | | | 0.0728 N/m | | dynamic viscosity | 8.9×10^-6 Pa s (at 25 °C) | | 8.9×10^-4 Pa s (at 25 °C) | 0.001 Pa s (at 550 °C) | odor | odorless | odorless | odorless | |