H2SO4 + KMnO4 + H2S = H2O + K2SO4 + MnSO4 + H2SO3

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + H_2S hydrogen sulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + H_2SO_3 sulfurous acid

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + H_2S ⟶ H_2O + K_2SO_4 + MnSO_4 + H_2SO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 H_2S ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 H_2SO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K and Mn: H: | 2 c_1 + 2 c_3 = 2 c_4 + 2 c_7 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_6 + 3 c_7 S: | c_1 + c_3 = c_5 + c_6 + c_7 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 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_2 = 2 c_3 = c_1/3 + 2/3 c_4 = 3 c_5 = 1 c_6 = 2 c_7 = (4 c_1)/3 - 7/3 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_2 = 4 c_3 = c_1/3 + 4/3 c_4 = 6 c_5 = 2 c_6 = 4 c_7 = (4 c_1)/3 - 14/3 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 5 and solve for the remaining coefficients: c_1 = 5 c_2 = 4 c_3 = 3 c_4 = 6 c_5 = 2 c_6 = 4 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 5 H_2SO_4 + 4 KMnO_4 + 3 H_2S ⟶ 6 H_2O + 2 K_2SO_4 + 4 MnSO_4 + 2 H_2SO_3

+ + ⟶ + + +

sulfuric acid + potassium permanganate + hydrogen sulfide ⟶ water + potassium sulfate + manganese(II) sulfate + sulfurous acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + H_2S ⟶ H_2O + K_2SO_4 + MnSO_4 + H_2SO_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: 5 H_2SO_4 + 4 KMnO_4 + 3 H_2S ⟶ 6 H_2O + 2 K_2SO_4 + 4 MnSO_4 + 2 H_2SO_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 | 5 | -5 KMnO_4 | 4 | -4 H_2S | 3 | -3 H_2O | 6 | 6 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 H_2SO_3 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 5 | -5 | ([H2SO4])^(-5) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) H_2S | 3 | -3 | ([H2S])^(-3) H_2O | 6 | 6 | ([H2O])^6 K_2SO_4 | 2 | 2 | ([K2SO4])^2 MnSO_4 | 4 | 4 | ([MnSO4])^4 H_2SO_3 | 2 | 2 | ([H2SO3])^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 = ([H2SO4])^(-5) ([KMnO4])^(-4) ([H2S])^(-3) ([H2O])^6 ([K2SO4])^2 ([MnSO4])^4 ([H2SO3])^2 = (([H2O])^6 ([K2SO4])^2 ([MnSO4])^4 ([H2SO3])^2)/(([H2SO4])^5 ([KMnO4])^4 ([H2S])^3)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + H_2S ⟶ H_2O + K_2SO_4 + MnSO_4 + H_2SO_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: 5 H_2SO_4 + 4 KMnO_4 + 3 H_2S ⟶ 6 H_2O + 2 K_2SO_4 + 4 MnSO_4 + 2 H_2SO_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 | 5 | -5 KMnO_4 | 4 | -4 H_2S | 3 | -3 H_2O | 6 | 6 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 H_2SO_3 | 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_2SO_4 | 5 | -5 | -1/5 (Δ[H2SO4])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) H_2S | 3 | -3 | -1/3 (Δ[H2S])/(Δt) H_2O | 6 | 6 | 1/6 (Δ[H2O])/(Δt) K_2SO_4 | 2 | 2 | 1/2 (Δ[K2SO4])/(Δt) MnSO_4 | 4 | 4 | 1/4 (Δ[MnSO4])/(Δt) H_2SO_3 | 2 | 2 | 1/2 (Δ[H2SO3])/(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 = -1/5 (Δ[H2SO4])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/3 (Δ[H2S])/(Δt) = 1/6 (Δ[H2O])/(Δt) = 1/2 (Δ[K2SO4])/(Δt) = 1/4 (Δ[MnSO4])/(Δt) = 1/2 (Δ[H2SO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | hydrogen sulfide | water | potassium sulfate | manganese(II) sulfate | sulfurous acid formula | H_2SO_4 | KMnO_4 | H_2S | H_2O | K_2SO_4 | MnSO_4 | H_2SO_3 Hill formula | H_2O_4S | KMnO_4 | H_2S | H_2O | K_2O_4S | MnSO_4 | H_2O_3S name | sulfuric acid | potassium permanganate | hydrogen sulfide | water | potassium sulfate | manganese(II) sulfate | sulfurous acid IUPAC name | sulfuric acid | potassium permanganate | hydrogen sulfide | water | dipotassium sulfate | manganese(+2) cation sulfate | sulfurous acid