H2O + KMnO4 + H3PO3 = KOH + MnO2 + H3PO4

H_2O water + KMnO_4 potassium permanganate + HP(O)(OH)_2 phosphorous acid ⟶ KOH potassium hydroxide + MnO_2 manganese dioxide + H_3PO_4 phosphoric acid

Balance the chemical equation algebraically: H_2O + KMnO_4 + HP(O)(OH)_2 ⟶ KOH + MnO_2 + H_3PO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 HP(O)(OH)_2 ⟶ c_4 KOH + c_5 MnO_2 + c_6 H_3PO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn and P: H: | 2 c_1 + 3 c_3 = c_4 + 3 c_6 O: | c_1 + 4 c_2 + 3 c_3 = c_4 + 2 c_5 + 4 c_6 K: | c_2 = c_4 Mn: | c_2 = c_5 P: | c_3 = 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 2 c_3 = 3 c_4 = 2 c_5 = 2 c_6 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + 2 KMnO_4 + 3 HP(O)(OH)_2 ⟶ 2 KOH + 2 MnO_2 + 3 H_3PO_4

+ + ⟶ + +

water + potassium permanganate + phosphorous acid ⟶ potassium hydroxide + manganese dioxide + phosphoric acid

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + HP(O)(OH)_2 ⟶ KOH + MnO_2 + H_3PO_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: H_2O + 2 KMnO_4 + 3 HP(O)(OH)_2 ⟶ 2 KOH + 2 MnO_2 + 3 H_3PO_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 H_2O | 1 | -1 KMnO_4 | 2 | -2 HP(O)(OH)_2 | 3 | -3 KOH | 2 | 2 MnO_2 | 2 | 2 H_3PO_4 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 1 | -1 | ([H2O])^(-1) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) HP(O)(OH)_2 | 3 | -3 | ([HP(O)(OH)2])^(-3) KOH | 2 | 2 | ([KOH])^2 MnO_2 | 2 | 2 | ([MnO2])^2 H_3PO_4 | 3 | 3 | ([H3PO4])^3 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])^(-1) ([KMnO4])^(-2) ([HP(O)(OH)2])^(-3) ([KOH])^2 ([MnO2])^2 ([H3PO4])^3 = (([KOH])^2 ([MnO2])^2 ([H3PO4])^3)/([H2O] ([KMnO4])^2 ([HP(O)(OH)2])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + HP(O)(OH)_2 ⟶ KOH + MnO_2 + H_3PO_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: H_2O + 2 KMnO_4 + 3 HP(O)(OH)_2 ⟶ 2 KOH + 2 MnO_2 + 3 H_3PO_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 H_2O | 1 | -1 KMnO_4 | 2 | -2 HP(O)(OH)_2 | 3 | -3 KOH | 2 | 2 MnO_2 | 2 | 2 H_3PO_4 | 3 | 3 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 | 1 | -1 | -(Δ[H2O])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) HP(O)(OH)_2 | 3 | -3 | -1/3 (Δ[HP(O)(OH)2])/(Δt) KOH | 2 | 2 | 1/2 (Δ[KOH])/(Δt) MnO_2 | 2 | 2 | 1/2 (Δ[MnO2])/(Δt) H_3PO_4 | 3 | 3 | 1/3 (Δ[H3PO4])/(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 = -(Δ[H2O])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/3 (Δ[HP(O)(OH)2])/(Δt) = 1/2 (Δ[KOH])/(Δt) = 1/2 (Δ[MnO2])/(Δt) = 1/3 (Δ[H3PO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | phosphorous acid | potassium hydroxide | manganese dioxide | phosphoric acid formula | H_2O | KMnO_4 | HP(O)(OH)_2 | KOH | MnO_2 | H_3PO_4 Hill formula | H_2O | KMnO_4 | H_3O_3P | HKO | MnO_2 | H_3O_4P name | water | potassium permanganate | phosphorous acid | potassium hydroxide | manganese dioxide | phosphoric acid IUPAC name | water | potassium permanganate | phosphorous acid | potassium hydroxide | dioxomanganese | phosphoric acid

| water | potassium permanganate | phosphorous acid | potassium hydroxide | manganese dioxide | phosphoric acid molar mass | 18.015 g/mol | 158.03 g/mol | 81.995 g/mol | 56.105 g/mol | 86.936 g/mol | 97.994 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) melting point | 0 °C | 240 °C | 74 °C | 406 °C | 535 °C | 42.4 °C boiling point | 99.9839 °C | | | 1327 °C | | 158 °C density | 1 g/cm^3 | 1 g/cm^3 | 1.597 g/cm^3 | 2.044 g/cm^3 | 5.03 g/cm^3 | 1.685 g/cm^3 solubility in water | | | | soluble | insoluble | very soluble surface tension | 0.0728 N/m | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.001 Pa s (at 550 °C) | | odor | odorless | odorless | | | | odorless