H2SO4 + KMnO4 + K4(Fe(CN)6) = H2O + K2SO4 + MnSO4 + K3(Fe(CN)6)

H_2SO_4 (sulfuric acid) + KMnO_4 (potassium permanganate) + K4Fe(CN)6 ⟶ H_2O (water) + K_2SO_4 (potassium sulfate) + MnSO_4 (manganese(II) sulfate) + 1 km^3 of Fe(CN)6

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + K4Fe(CN)6 ⟶ H_2O + K_2SO_4 + MnSO_4 + Fe(CN)6 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 K4Fe(CN)6 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 Fe(CN)6 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K, Mn, Fe, C and N: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_6 S: | c_1 = c_5 + c_6 K: | c_2 + 4 c_3 = 2 c_5 Mn: | c_2 = c_6 Fe: | c_3 = c_7 C: | 6 c_3 = 6 c_7 N: | 6 c_3 = 6 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 = 4 c_2 = 1 c_3 = 5/4 c_4 = 4 c_5 = 3 c_6 = 1 c_7 = 5/4 Multiply by the least common denominator, 4, to eliminate fractional coefficients: c_1 = 16 c_2 = 4 c_3 = 5 c_4 = 16 c_5 = 12 c_6 = 4 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 16 H_2SO_4 + 4 KMnO_4 + 5 K4Fe(CN)6 ⟶ 16 H_2O + 12 K_2SO_4 + 4 MnSO_4 + 5 Fe(CN)6

reagent | equivalents H_2SO_4 | KMnO_4 | K4Fe(CN)6 | H_2O | K_2SO_4 | MnSO_4 | Fe(CN)6 |

+ + K4Fe(CN)6 ⟶ + + + Fe(CN)6

sulfuric acid + potassium permanganate + K4Fe(CN)6 ⟶ water + potassium sulfate + manganese(II) sulfate + Fe(CN)6

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + K4Fe(CN)6 ⟶ H_2O + K_2SO_4 + MnSO_4 + Fe(CN)6 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: 16 H_2SO_4 + 4 KMnO_4 + 5 K4Fe(CN)6 ⟶ 16 H_2O + 12 K_2SO_4 + 4 MnSO_4 + 5 Fe(CN)6 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 | 16 | -16 KMnO_4 | 4 | -4 K4Fe(CN)6 | 5 | -5 H_2O | 16 | 16 K_2SO_4 | 12 | 12 MnSO_4 | 4 | 4 Fe(CN)6 | 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 | 16 | -16 | ([H2SO4])^(-16) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) K4Fe(CN)6 | 5 | -5 | ([K4Fe(CN)6])^(-5) H_2O | 16 | 16 | ([H2O])^16 K_2SO_4 | 12 | 12 | ([K2SO4])^12 MnSO_4 | 4 | 4 | ([MnSO4])^4 Fe(CN)6 | 5 | 5 | ([Fe(CN)6])^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])^(-16) ([KMnO4])^(-4) ([K4Fe(CN)6])^(-5) ([H2O])^16 ([K2SO4])^12 ([MnSO4])^4 ([Fe(CN)6])^5 = (([H2O])^16 ([K2SO4])^12 ([MnSO4])^4 ([Fe(CN)6])^5)/(([H2SO4])^16 ([KMnO4])^4 ([K4Fe(CN)6])^5)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + K4Fe(CN)6 ⟶ H_2O + K_2SO_4 + MnSO_4 + Fe(CN)6 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: 16 H_2SO_4 + 4 KMnO_4 + 5 K4Fe(CN)6 ⟶ 16 H_2O + 12 K_2SO_4 + 4 MnSO_4 + 5 Fe(CN)6 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 | 16 | -16 KMnO_4 | 4 | -4 K4Fe(CN)6 | 5 | -5 H_2O | 16 | 16 K_2SO_4 | 12 | 12 MnSO_4 | 4 | 4 Fe(CN)6 | 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 | 16 | -16 | -1/16 (Δ[H2SO4])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) K4Fe(CN)6 | 5 | -5 | -1/5 (Δ[K4Fe(CN)6])/(Δt) H_2O | 16 | 16 | 1/16 (Δ[H2O])/(Δt) K_2SO_4 | 12 | 12 | 1/12 (Δ[K2SO4])/(Δt) MnSO_4 | 4 | 4 | 1/4 (Δ[MnSO4])/(Δt) Fe(CN)6 | 5 | 5 | 1/5 (Δ[Fe(CN)6])/(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 = -1/16 (Δ[H2SO4])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/5 (Δ[K4Fe(CN)6])/(Δt) = 1/16 (Δ[H2O])/(Δt) = 1/12 (Δ[K2SO4])/(Δt) = 1/4 (Δ[MnSO4])/(Δt) = 1/5 (Δ[Fe(CN)6])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | K4Fe(CN)6 | water | potassium sulfate | manganese(II) sulfate | Fe(CN)6 formula | H_2SO_4 | KMnO_4 | K4Fe(CN)6 | H_2O | K_2SO_4 | MnSO_4 | Fe(CN)6 Hill formula | H_2O_4S | KMnO_4 | C6FeK4N6 | H_2O | K_2O_4S | MnSO_4 | C6FeN6 name | sulfuric acid | potassium permanganate | | water | potassium sulfate | manganese(II) sulfate | IUPAC name | sulfuric acid | potassium permanganate | | water | dipotassium sulfate | manganese(+2) cation sulfate |