HCl + KMnO4 + H3PO3 = H2O + KCl + H3PO4 + MnCl2

HCl hydrogen chloride + KMnO_4 potassium permanganate + HP(O)(OH)_2 phosphorous acid ⟶ H_2O water + KCl potassium chloride + H_3PO_4 phosphoric acid + MnCl_2 manganese(II) chloride

Balance the chemical equation algebraically: HCl + KMnO_4 + HP(O)(OH)_2 ⟶ H_2O + KCl + H_3PO_4 + MnCl_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HCl + c_2 KMnO_4 + c_3 HP(O)(OH)_2 ⟶ c_4 H_2O + c_5 KCl + c_6 H_3PO_4 + c_7 MnCl_2 Set the number of atoms in the reactants equal to the number of atoms in the products for Cl, H, K, Mn, O and P: Cl: | c_1 = c_5 + 2 c_7 H: | c_1 + 3 c_3 = 2 c_4 + 3 c_6 K: | c_2 = c_5 Mn: | c_2 = c_7 O: | 4 c_2 + 3 c_3 = c_4 + 4 c_6 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 1 c_3 = 5/2 c_4 = 3/2 c_5 = 1 c_6 = 5/2 c_7 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 6 c_2 = 2 c_3 = 5 c_4 = 3 c_5 = 2 c_6 = 5 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 HCl + 2 KMnO_4 + 5 HP(O)(OH)_2 ⟶ 3 H_2O + 2 KCl + 5 H_3PO_4 + 2 MnCl_2

+ + ⟶ + + +

hydrogen chloride + potassium permanganate + phosphorous acid ⟶ water + potassium chloride + phosphoric acid + manganese(II) chloride

Construct the equilibrium constant, K, expression for: HCl + KMnO_4 + HP(O)(OH)_2 ⟶ H_2O + KCl + H_3PO_4 + MnCl_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: 6 HCl + 2 KMnO_4 + 5 HP(O)(OH)_2 ⟶ 3 H_2O + 2 KCl + 5 H_3PO_4 + 2 MnCl_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 HCl | 6 | -6 KMnO_4 | 2 | -2 HP(O)(OH)_2 | 5 | -5 H_2O | 3 | 3 KCl | 2 | 2 H_3PO_4 | 5 | 5 MnCl_2 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HCl | 6 | -6 | ([HCl])^(-6) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) HP(O)(OH)_2 | 5 | -5 | ([HP(O)(OH)2])^(-5) H_2O | 3 | 3 | ([H2O])^3 KCl | 2 | 2 | ([KCl])^2 H_3PO_4 | 5 | 5 | ([H3PO4])^5 MnCl_2 | 2 | 2 | ([MnCl2])^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 = ([HCl])^(-6) ([KMnO4])^(-2) ([HP(O)(OH)2])^(-5) ([H2O])^3 ([KCl])^2 ([H3PO4])^5 ([MnCl2])^2 = (([H2O])^3 ([KCl])^2 ([H3PO4])^5 ([MnCl2])^2)/(([HCl])^6 ([KMnO4])^2 ([HP(O)(OH)2])^5)

Construct the rate of reaction expression for: HCl + KMnO_4 + HP(O)(OH)_2 ⟶ H_2O + KCl + H_3PO_4 + MnCl_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: 6 HCl + 2 KMnO_4 + 5 HP(O)(OH)_2 ⟶ 3 H_2O + 2 KCl + 5 H_3PO_4 + 2 MnCl_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 HCl | 6 | -6 KMnO_4 | 2 | -2 HP(O)(OH)_2 | 5 | -5 H_2O | 3 | 3 KCl | 2 | 2 H_3PO_4 | 5 | 5 MnCl_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 HCl | 6 | -6 | -1/6 (Δ[HCl])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) HP(O)(OH)_2 | 5 | -5 | -1/5 (Δ[HP(O)(OH)2])/(Δt) H_2O | 3 | 3 | 1/3 (Δ[H2O])/(Δt) KCl | 2 | 2 | 1/2 (Δ[KCl])/(Δt) H_3PO_4 | 5 | 5 | 1/5 (Δ[H3PO4])/(Δt) MnCl_2 | 2 | 2 | 1/2 (Δ[MnCl2])/(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 = -1/6 (Δ[HCl])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/5 (Δ[HP(O)(OH)2])/(Δt) = 1/3 (Δ[H2O])/(Δt) = 1/2 (Δ[KCl])/(Δt) = 1/5 (Δ[H3PO4])/(Δt) = 1/2 (Δ[MnCl2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| hydrogen chloride | potassium permanganate | phosphorous acid | water | potassium chloride | phosphoric acid | manganese(II) chloride formula | HCl | KMnO_4 | HP(O)(OH)_2 | H_2O | KCl | H_3PO_4 | MnCl_2 Hill formula | ClH | KMnO_4 | H_3O_3P | H_2O | ClK | H_3O_4P | Cl_2Mn name | hydrogen chloride | potassium permanganate | phosphorous acid | water | potassium chloride | phosphoric acid | manganese(II) chloride IUPAC name | hydrogen chloride | potassium permanganate | phosphorous acid | water | potassium chloride | phosphoric acid | dichloromanganese