H2SO4 + KMnO4 + FeSO4 = H2O + K2SO4 + MnSO4 + Fe2(SO4)3 + Mn(SO4)4

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + FeSO_4 duretter ⟶ H_2O water + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + Fe_2(SO_4)_3·xH_2O iron(III) sulfate hydrate + Mn(SO4)4

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + FeSO_4 ⟶ H_2O + K_2SO_4 + MnSO_4 + Fe_2(SO_4)_3·xH_2O + Mn(SO4)4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 FeSO_4 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 Fe_2(SO_4)_3·xH_2O + c_8 Mn(SO4)4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K, Mn and Fe: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 + 4 c_3 = c_4 + 4 c_5 + 4 c_6 + 12 c_7 + 16 c_8 S: | c_1 + c_3 = c_5 + c_6 + 3 c_7 + 4 c_8 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 + c_8 Fe: | 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_6 = 1 and solve the system of equations for the remaining coefficients: c_2 = c_1/4 c_3 = 6 - c_1/4 c_4 = c_1 c_5 = c_1/8 c_6 = 1 c_7 = 3 - c_1/8 c_8 = c_1/4 - 1 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_2 = c_1/4 c_3 = 18 - c_1/4 c_4 = c_1 c_5 = c_1/8 c_6 = 3 c_7 = 9 - c_1/8 c_8 = c_1/4 - 3 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 32 and solve for the remaining coefficients: c_1 = 32 c_2 = 8 c_3 = 10 c_4 = 32 c_5 = 4 c_6 = 3 c_7 = 5 c_8 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 32 H_2SO_4 + 8 KMnO_4 + 10 FeSO_4 ⟶ 32 H_2O + 4 K_2SO_4 + 3 MnSO_4 + 5 Fe_2(SO_4)_3·xH_2O + 5 Mn(SO4)4

+ + ⟶ + + + + Mn(SO4)4

sulfuric acid + potassium permanganate + duretter ⟶ water + potassium sulfate + manganese(II) sulfate + iron(III) sulfate hydrate + Mn(SO4)4

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + FeSO_4 ⟶ H_2O + K_2SO_4 + MnSO_4 + Fe_2(SO_4)_3·xH_2O + Mn(SO4)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: 32 H_2SO_4 + 8 KMnO_4 + 10 FeSO_4 ⟶ 32 H_2O + 4 K_2SO_4 + 3 MnSO_4 + 5 Fe_2(SO_4)_3·xH_2O + 5 Mn(SO4)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_2SO_4 | 32 | -32 KMnO_4 | 8 | -8 FeSO_4 | 10 | -10 H_2O | 32 | 32 K_2SO_4 | 4 | 4 MnSO_4 | 3 | 3 Fe_2(SO_4)_3·xH_2O | 5 | 5 Mn(SO4)4 | 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 | 32 | -32 | ([H2SO4])^(-32) KMnO_4 | 8 | -8 | ([KMnO4])^(-8) FeSO_4 | 10 | -10 | ([FeSO4])^(-10) H_2O | 32 | 32 | ([H2O])^32 K_2SO_4 | 4 | 4 | ([K2SO4])^4 MnSO_4 | 3 | 3 | ([MnSO4])^3 Fe_2(SO_4)_3·xH_2O | 5 | 5 | ([Fe2(SO4)3·xH2O])^5 Mn(SO4)4 | 5 | 5 | ([Mn(SO4)4])^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])^(-32) ([KMnO4])^(-8) ([FeSO4])^(-10) ([H2O])^32 ([K2SO4])^4 ([MnSO4])^3 ([Fe2(SO4)3·xH2O])^5 ([Mn(SO4)4])^5 = (([H2O])^32 ([K2SO4])^4 ([MnSO4])^3 ([Fe2(SO4)3·xH2O])^5 ([Mn(SO4)4])^5)/(([H2SO4])^32 ([KMnO4])^8 ([FeSO4])^10)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + FeSO_4 ⟶ H_2O + K_2SO_4 + MnSO_4 + Fe_2(SO_4)_3·xH_2O + Mn(SO4)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: 32 H_2SO_4 + 8 KMnO_4 + 10 FeSO_4 ⟶ 32 H_2O + 4 K_2SO_4 + 3 MnSO_4 + 5 Fe_2(SO_4)_3·xH_2O + 5 Mn(SO4)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_2SO_4 | 32 | -32 KMnO_4 | 8 | -8 FeSO_4 | 10 | -10 H_2O | 32 | 32 K_2SO_4 | 4 | 4 MnSO_4 | 3 | 3 Fe_2(SO_4)_3·xH_2O | 5 | 5 Mn(SO4)4 | 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 | 32 | -32 | -1/32 (Δ[H2SO4])/(Δt) KMnO_4 | 8 | -8 | -1/8 (Δ[KMnO4])/(Δt) FeSO_4 | 10 | -10 | -1/10 (Δ[FeSO4])/(Δt) H_2O | 32 | 32 | 1/32 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) MnSO_4 | 3 | 3 | 1/3 (Δ[MnSO4])/(Δt) Fe_2(SO_4)_3·xH_2O | 5 | 5 | 1/5 (Δ[Fe2(SO4)3·xH2O])/(Δt) Mn(SO4)4 | 5 | 5 | 1/5 (Δ[Mn(SO4)4])/(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 = -1/32 (Δ[H2SO4])/(Δt) = -1/8 (Δ[KMnO4])/(Δt) = -1/10 (Δ[FeSO4])/(Δt) = 1/32 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/3 (Δ[MnSO4])/(Δt) = 1/5 (Δ[Fe2(SO4)3·xH2O])/(Δt) = 1/5 (Δ[Mn(SO4)4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | duretter | water | potassium sulfate | manganese(II) sulfate | iron(III) sulfate hydrate | Mn(SO4)4 formula | H_2SO_4 | KMnO_4 | FeSO_4 | H_2O | K_2SO_4 | MnSO_4 | Fe_2(SO_4)_3·xH_2O | Mn(SO4)4 Hill formula | H_2O_4S | KMnO_4 | FeO_4S | H_2O | K_2O_4S | MnSO_4 | Fe_2O_12S_3 | MnO16S4 name | sulfuric acid | potassium permanganate | duretter | water | potassium sulfate | manganese(II) sulfate | iron(III) sulfate hydrate | IUPAC name | sulfuric acid | potassium permanganate | iron(+2) cation sulfate | water | dipotassium sulfate | manganese(+2) cation sulfate | diferric trisulfate |