H2O + KMnO4 + Sb2(SO4)3 = H2SO4 + K2SO4 + MnSO4 + H3SbO4

H_2O water + KMnO_4 potassium permanganate + O_12S_3Sb_2 antimony(III) sulfate ⟶ H_2SO_4 sulfuric acid + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + H3SbO4

Balance the chemical equation algebraically: H_2O + KMnO_4 + O_12S_3Sb_2 ⟶ H_2SO_4 + K_2SO_4 + MnSO_4 + H3SbO4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 O_12S_3Sb_2 ⟶ c_4 H_2SO_4 + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 H3SbO4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn, S and Sb: H: | 2 c_1 = 2 c_4 + 3 c_7 O: | c_1 + 4 c_2 + 12 c_3 = 4 c_4 + 4 c_5 + 4 c_6 + 4 c_7 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 S: | 3 c_3 = c_4 + c_5 + c_6 Sb: | 2 c_3 = 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 = 12 c_2 = 2 c_3 = 5/2 c_4 = 9/2 c_5 = 1 c_6 = 2 c_7 = 5 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 24 c_2 = 4 c_3 = 5 c_4 = 9 c_5 = 2 c_6 = 4 c_7 = 10 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 24 H_2O + 4 KMnO_4 + 5 O_12S_3Sb_2 ⟶ 9 H_2SO_4 + 2 K_2SO_4 + 4 MnSO_4 + 10 H3SbO4

+ + ⟶ + + + H3SbO4

water + potassium permanganate + antimony(III) sulfate ⟶ sulfuric acid + potassium sulfate + manganese(II) sulfate + H3SbO4

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + O_12S_3Sb_2 ⟶ H_2SO_4 + K_2SO_4 + MnSO_4 + H3SbO4 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: 24 H_2O + 4 KMnO_4 + 5 O_12S_3Sb_2 ⟶ 9 H_2SO_4 + 2 K_2SO_4 + 4 MnSO_4 + 10 H3SbO4 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 | 24 | -24 KMnO_4 | 4 | -4 O_12S_3Sb_2 | 5 | -5 H_2SO_4 | 9 | 9 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 H3SbO4 | 10 | 10 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 24 | -24 | ([H2O])^(-24) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) O_12S_3Sb_2 | 5 | -5 | ([O12S3Sb2])^(-5) H_2SO_4 | 9 | 9 | ([H2SO4])^9 K_2SO_4 | 2 | 2 | ([K2SO4])^2 MnSO_4 | 4 | 4 | ([MnSO4])^4 H3SbO4 | 10 | 10 | ([H3SbO4])^10 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])^(-24) ([KMnO4])^(-4) ([O12S3Sb2])^(-5) ([H2SO4])^9 ([K2SO4])^2 ([MnSO4])^4 ([H3SbO4])^10 = (([H2SO4])^9 ([K2SO4])^2 ([MnSO4])^4 ([H3SbO4])^10)/(([H2O])^24 ([KMnO4])^4 ([O12S3Sb2])^5)

Construct the rate of reaction expression for: H_2O + KMnO_4 + O_12S_3Sb_2 ⟶ H_2SO_4 + K_2SO_4 + MnSO_4 + H3SbO4 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: 24 H_2O + 4 KMnO_4 + 5 O_12S_3Sb_2 ⟶ 9 H_2SO_4 + 2 K_2SO_4 + 4 MnSO_4 + 10 H3SbO4 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 | 24 | -24 KMnO_4 | 4 | -4 O_12S_3Sb_2 | 5 | -5 H_2SO_4 | 9 | 9 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 H3SbO4 | 10 | 10 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 | 24 | -24 | -1/24 (Δ[H2O])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) O_12S_3Sb_2 | 5 | -5 | -1/5 (Δ[O12S3Sb2])/(Δt) H_2SO_4 | 9 | 9 | 1/9 (Δ[H2SO4])/(Δt) K_2SO_4 | 2 | 2 | 1/2 (Δ[K2SO4])/(Δt) MnSO_4 | 4 | 4 | 1/4 (Δ[MnSO4])/(Δt) H3SbO4 | 10 | 10 | 1/10 (Δ[H3SbO4])/(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 = -1/24 (Δ[H2O])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/5 (Δ[O12S3Sb2])/(Δt) = 1/9 (Δ[H2SO4])/(Δt) = 1/2 (Δ[K2SO4])/(Δt) = 1/4 (Δ[MnSO4])/(Δt) = 1/10 (Δ[H3SbO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | antimony(III) sulfate | sulfuric acid | potassium sulfate | manganese(II) sulfate | H3SbO4 formula | H_2O | KMnO_4 | O_12S_3Sb_2 | H_2SO_4 | K_2SO_4 | MnSO_4 | H3SbO4 Hill formula | H_2O | KMnO_4 | O_12S_3Sb_2 | H_2O_4S | K_2O_4S | MnSO_4 | H3O4Sb name | water | potassium permanganate | antimony(III) sulfate | sulfuric acid | potassium sulfate | manganese(II) sulfate | IUPAC name | water | potassium permanganate | antimony(+3) cation trisulfate | sulfuric acid | dipotassium sulfate | manganese(+2) cation sulfate |