H2O + Mn + K2S2O8 = H2SO4 + K2SO4 + KMnO4

H_2O water + Mn manganese + K_2S_2O_8 potassium persulfate ⟶ H_2SO_4 sulfuric acid + K_2SO_4 potassium sulfate + KMnO_4 potassium permanganate

Balance the chemical equation algebraically: H_2O + Mn + K_2S_2O_8 ⟶ H_2SO_4 + K_2SO_4 + KMnO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 Mn + c_3 K_2S_2O_8 ⟶ c_4 H_2SO_4 + c_5 K_2SO_4 + c_6 KMnO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Mn, K and S: H: | 2 c_1 = 2 c_4 O: | c_1 + 8 c_3 = 4 c_4 + 4 c_5 + 4 c_6 Mn: | c_2 = c_6 K: | 2 c_3 = 2 c_5 + c_6 S: | 2 c_3 = c_4 + c_5 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 = 7/2 c_4 = 4 c_5 = 3 c_6 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 8 c_2 = 2 c_3 = 7 c_4 = 8 c_5 = 6 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2O + 2 Mn + 7 K_2S_2O_8 ⟶ 8 H_2SO_4 + 6 K_2SO_4 + 2 KMnO_4

+ + ⟶ + +

water + manganese + potassium persulfate ⟶ sulfuric acid + potassium sulfate + potassium permanganate

Construct the equilibrium constant, K, expression for: H_2O + Mn + K_2S_2O_8 ⟶ H_2SO_4 + K_2SO_4 + KMnO_4 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: 8 H_2O + 2 Mn + 7 K_2S_2O_8 ⟶ 8 H_2SO_4 + 6 K_2SO_4 + 2 KMnO_4 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 | 8 | -8 Mn | 2 | -2 K_2S_2O_8 | 7 | -7 H_2SO_4 | 8 | 8 K_2SO_4 | 6 | 6 KMnO_4 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 8 | -8 | ([H2O])^(-8) Mn | 2 | -2 | ([Mn])^(-2) K_2S_2O_8 | 7 | -7 | ([K2S2O8])^(-7) H_2SO_4 | 8 | 8 | ([H2SO4])^8 K_2SO_4 | 6 | 6 | ([K2SO4])^6 KMnO_4 | 2 | 2 | ([KMnO4])^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 = ([H2O])^(-8) ([Mn])^(-2) ([K2S2O8])^(-7) ([H2SO4])^8 ([K2SO4])^6 ([KMnO4])^2 = (([H2SO4])^8 ([K2SO4])^6 ([KMnO4])^2)/(([H2O])^8 ([Mn])^2 ([K2S2O8])^7)

Construct the rate of reaction expression for: H_2O + Mn + K_2S_2O_8 ⟶ H_2SO_4 + K_2SO_4 + KMnO_4 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: 8 H_2O + 2 Mn + 7 K_2S_2O_8 ⟶ 8 H_2SO_4 + 6 K_2SO_4 + 2 KMnO_4 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 | 8 | -8 Mn | 2 | -2 K_2S_2O_8 | 7 | -7 H_2SO_4 | 8 | 8 K_2SO_4 | 6 | 6 KMnO_4 | 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 H_2O | 8 | -8 | -1/8 (Δ[H2O])/(Δt) Mn | 2 | -2 | -1/2 (Δ[Mn])/(Δt) K_2S_2O_8 | 7 | -7 | -1/7 (Δ[K2S2O8])/(Δt) H_2SO_4 | 8 | 8 | 1/8 (Δ[H2SO4])/(Δt) K_2SO_4 | 6 | 6 | 1/6 (Δ[K2SO4])/(Δt) KMnO_4 | 2 | 2 | 1/2 (Δ[KMnO4])/(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 = -1/8 (Δ[H2O])/(Δt) = -1/2 (Δ[Mn])/(Δt) = -1/7 (Δ[K2S2O8])/(Δt) = 1/8 (Δ[H2SO4])/(Δt) = 1/6 (Δ[K2SO4])/(Δt) = 1/2 (Δ[KMnO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | manganese | potassium persulfate | sulfuric acid | potassium sulfate | potassium permanganate formula | H_2O | Mn | K_2S_2O_8 | H_2SO_4 | K_2SO_4 | KMnO_4 Hill formula | H_2O | Mn | K_2O_8S_2 | H_2O_4S | K_2O_4S | KMnO_4 name | water | manganese | potassium persulfate | sulfuric acid | potassium sulfate | potassium permanganate IUPAC name | water | manganese | dipotassium sulfonatooxy sulfate | sulfuric acid | dipotassium sulfate | potassium permanganate