HCl + KMnO4 + As2O3 = H2O + KCl + MnCl2 + As2O5

HCl hydrogen chloride + KMnO_4 potassium permanganate + As_2O_3 arsenic trioxide ⟶ H_2O water + KCl potassium chloride + MnCl_2 manganese(II) chloride + As_2O_5 arsenic pentoxide

Balance the chemical equation algebraically: HCl + KMnO_4 + As_2O_3 ⟶ H_2O + KCl + MnCl_2 + As_2O_5 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HCl + c_2 KMnO_4 + c_3 As_2O_3 ⟶ c_4 H_2O + c_5 KCl + c_6 MnCl_2 + c_7 As_2O_5 Set the number of atoms in the reactants equal to the number of atoms in the products for Cl, H, K, Mn, O and As: Cl: | c_1 = c_5 + 2 c_6 H: | c_1 = 2 c_4 K: | c_2 = c_5 Mn: | c_2 = c_6 O: | 4 c_2 + 3 c_3 = c_4 + 5 c_7 As: | 2 c_3 = 2 c_7 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 = 5/4 c_4 = 3/2 c_5 = 1 c_6 = 1 c_7 = 5/4 Multiply by the least common denominator, 4, to eliminate fractional coefficients: c_1 = 12 c_2 = 4 c_3 = 5 c_4 = 6 c_5 = 4 c_6 = 4 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 12 HCl + 4 KMnO_4 + 5 As_2O_3 ⟶ 6 H_2O + 4 KCl + 4 MnCl_2 + 5 As_2O_5

+ + ⟶ + + +

hydrogen chloride + potassium permanganate + arsenic trioxide ⟶ water + potassium chloride + manganese(II) chloride + arsenic pentoxide

Construct the equilibrium constant, K, expression for: HCl + KMnO_4 + As_2O_3 ⟶ H_2O + KCl + MnCl_2 + As_2O_5 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: 12 HCl + 4 KMnO_4 + 5 As_2O_3 ⟶ 6 H_2O + 4 KCl + 4 MnCl_2 + 5 As_2O_5 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 | 12 | -12 KMnO_4 | 4 | -4 As_2O_3 | 5 | -5 H_2O | 6 | 6 KCl | 4 | 4 MnCl_2 | 4 | 4 As_2O_5 | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HCl | 12 | -12 | ([HCl])^(-12) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) As_2O_3 | 5 | -5 | ([As2O3])^(-5) H_2O | 6 | 6 | ([H2O])^6 KCl | 4 | 4 | ([KCl])^4 MnCl_2 | 4 | 4 | ([MnCl2])^4 As_2O_5 | 5 | 5 | ([As2O5])^5 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])^(-12) ([KMnO4])^(-4) ([As2O3])^(-5) ([H2O])^6 ([KCl])^4 ([MnCl2])^4 ([As2O5])^5 = (([H2O])^6 ([KCl])^4 ([MnCl2])^4 ([As2O5])^5)/(([HCl])^12 ([KMnO4])^4 ([As2O3])^5)

Construct the rate of reaction expression for: HCl + KMnO_4 + As_2O_3 ⟶ H_2O + KCl + MnCl_2 + As_2O_5 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: 12 HCl + 4 KMnO_4 + 5 As_2O_3 ⟶ 6 H_2O + 4 KCl + 4 MnCl_2 + 5 As_2O_5 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 | 12 | -12 KMnO_4 | 4 | -4 As_2O_3 | 5 | -5 H_2O | 6 | 6 KCl | 4 | 4 MnCl_2 | 4 | 4 As_2O_5 | 5 | 5 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 | 12 | -12 | -1/12 (Δ[HCl])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) As_2O_3 | 5 | -5 | -1/5 (Δ[As2O3])/(Δt) H_2O | 6 | 6 | 1/6 (Δ[H2O])/(Δt) KCl | 4 | 4 | 1/4 (Δ[KCl])/(Δt) MnCl_2 | 4 | 4 | 1/4 (Δ[MnCl2])/(Δt) As_2O_5 | 5 | 5 | 1/5 (Δ[As2O5])/(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 = -1/12 (Δ[HCl])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/5 (Δ[As2O3])/(Δt) = 1/6 (Δ[H2O])/(Δt) = 1/4 (Δ[KCl])/(Δt) = 1/4 (Δ[MnCl2])/(Δt) = 1/5 (Δ[As2O5])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| hydrogen chloride | potassium permanganate | arsenic trioxide | water | potassium chloride | manganese(II) chloride | arsenic pentoxide formula | HCl | KMnO_4 | As_2O_3 | H_2O | KCl | MnCl_2 | As_2O_5 Hill formula | ClH | KMnO_4 | As_2O_3 | H_2O | ClK | Cl_2Mn | As_2O_5 name | hydrogen chloride | potassium permanganate | arsenic trioxide | water | potassium chloride | manganese(II) chloride | arsenic pentoxide IUPAC name | hydrogen chloride | potassium permanganate | 2, 4, 5-trioxa-1, 3-diarsabicyclo[1.1.1]pentane | water | potassium chloride | dichloromanganese |