H2SO4 + KMnO4 + Sb = H2O + K2SO4 + MnSO4 + Sb2O3

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + Sb gray antimony ⟶ H_2O water + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + Sb_2O_3 antimony trioxide

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + Sb ⟶ H_2O + K_2SO_4 + MnSO_4 + Sb_2O_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 Sb ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 Sb_2O_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K, Mn and Sb: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_6 + 3 c_7 S: | c_1 = c_5 + c_6 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 Sb: | 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_5 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 10/3 c_4 = 3 c_5 = 1 c_6 = 2 c_7 = 5/3 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 9 c_2 = 6 c_3 = 10 c_4 = 9 c_5 = 3 c_6 = 6 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 9 H_2SO_4 + 6 KMnO_4 + 10 Sb ⟶ 9 H_2O + 3 K_2SO_4 + 6 MnSO_4 + 5 Sb_2O_3

+ + ⟶ + + +

sulfuric acid + potassium permanganate + gray antimony ⟶ water + potassium sulfate + manganese(II) sulfate + antimony trioxide

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + Sb ⟶ H_2O + K_2SO_4 + MnSO_4 + Sb_2O_3 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: 9 H_2SO_4 + 6 KMnO_4 + 10 Sb ⟶ 9 H_2O + 3 K_2SO_4 + 6 MnSO_4 + 5 Sb_2O_3 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_2SO_4 | 9 | -9 KMnO_4 | 6 | -6 Sb | 10 | -10 H_2O | 9 | 9 K_2SO_4 | 3 | 3 MnSO_4 | 6 | 6 Sb_2O_3 | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 9 | -9 | ([H2SO4])^(-9) KMnO_4 | 6 | -6 | ([KMnO4])^(-6) Sb | 10 | -10 | ([Sb])^(-10) H_2O | 9 | 9 | ([H2O])^9 K_2SO_4 | 3 | 3 | ([K2SO4])^3 MnSO_4 | 6 | 6 | ([MnSO4])^6 Sb_2O_3 | 5 | 5 | ([Sb2O3])^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 = ([H2SO4])^(-9) ([KMnO4])^(-6) ([Sb])^(-10) ([H2O])^9 ([K2SO4])^3 ([MnSO4])^6 ([Sb2O3])^5 = (([H2O])^9 ([K2SO4])^3 ([MnSO4])^6 ([Sb2O3])^5)/(([H2SO4])^9 ([KMnO4])^6 ([Sb])^10)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + Sb ⟶ H_2O + K_2SO_4 + MnSO_4 + Sb_2O_3 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: 9 H_2SO_4 + 6 KMnO_4 + 10 Sb ⟶ 9 H_2O + 3 K_2SO_4 + 6 MnSO_4 + 5 Sb_2O_3 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_2SO_4 | 9 | -9 KMnO_4 | 6 | -6 Sb | 10 | -10 H_2O | 9 | 9 K_2SO_4 | 3 | 3 MnSO_4 | 6 | 6 Sb_2O_3 | 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 H_2SO_4 | 9 | -9 | -1/9 (Δ[H2SO4])/(Δt) KMnO_4 | 6 | -6 | -1/6 (Δ[KMnO4])/(Δt) Sb | 10 | -10 | -1/10 (Δ[Sb])/(Δt) H_2O | 9 | 9 | 1/9 (Δ[H2O])/(Δt) K_2SO_4 | 3 | 3 | 1/3 (Δ[K2SO4])/(Δt) MnSO_4 | 6 | 6 | 1/6 (Δ[MnSO4])/(Δt) Sb_2O_3 | 5 | 5 | 1/5 (Δ[Sb2O3])/(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 = -1/9 (Δ[H2SO4])/(Δt) = -1/6 (Δ[KMnO4])/(Δt) = -1/10 (Δ[Sb])/(Δt) = 1/9 (Δ[H2O])/(Δt) = 1/3 (Δ[K2SO4])/(Δt) = 1/6 (Δ[MnSO4])/(Δt) = 1/5 (Δ[Sb2O3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | gray antimony | water | potassium sulfate | manganese(II) sulfate | antimony trioxide formula | H_2SO_4 | KMnO_4 | Sb | H_2O | K_2SO_4 | MnSO_4 | Sb_2O_3 Hill formula | H_2O_4S | KMnO_4 | Sb | H_2O | K_2O_4S | MnSO_4 | O_3Sb_2 name | sulfuric acid | potassium permanganate | gray antimony | water | potassium sulfate | manganese(II) sulfate | antimony trioxide IUPAC name | sulfuric acid | potassium permanganate | antimony | water | dipotassium sulfate | manganese(+2) cation sulfate | oxo-oxostibanyloxystibane