H2SO4 + KMnO4 + Na2S = H2O + K2SO4 + Na2SO4 + MnSO4

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + Na_2S sodium sulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + Na_2SO_4 sodium sulfate + MnSO_4 manganese(II) sulfate

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + Na_2S ⟶ H_2O + K_2SO_4 + Na_2SO_4 + MnSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 Na_2S ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 Na_2SO_4 + c_7 MnSO_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 Na: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_6 + 4 c_7 S: | c_1 + c_3 = c_5 + c_6 + c_7 K: | c_2 = 2 c_5 Mn: | c_2 = c_7 Na: | 2 c_3 = 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_1 = 3 c_2 = 2 c_3 = 5/4 c_4 = 3 c_5 = 1 c_6 = 5/4 c_7 = 2 Multiply by the least common denominator, 4, to eliminate fractional coefficients: c_1 = 12 c_2 = 8 c_3 = 5 c_4 = 12 c_5 = 4 c_6 = 5 c_7 = 8 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 12 H_2SO_4 + 8 KMnO_4 + 5 Na_2S ⟶ 12 H_2O + 4 K_2SO_4 + 5 Na_2SO_4 + 8 MnSO_4

+ + ⟶ + + +

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

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + Na_2S ⟶ H_2O + K_2SO_4 + Na_2SO_4 + MnSO_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: 12 H_2SO_4 + 8 KMnO_4 + 5 Na_2S ⟶ 12 H_2O + 4 K_2SO_4 + 5 Na_2SO_4 + 8 MnSO_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 | 12 | -12 KMnO_4 | 8 | -8 Na_2S | 5 | -5 H_2O | 12 | 12 K_2SO_4 | 4 | 4 Na_2SO_4 | 5 | 5 MnSO_4 | 8 | 8 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 12 | -12 | ([H2SO4])^(-12) KMnO_4 | 8 | -8 | ([KMnO4])^(-8) Na_2S | 5 | -5 | ([Na2S])^(-5) H_2O | 12 | 12 | ([H2O])^12 K_2SO_4 | 4 | 4 | ([K2SO4])^4 Na_2SO_4 | 5 | 5 | ([Na2SO4])^5 MnSO_4 | 8 | 8 | ([MnSO4])^8 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])^(-12) ([KMnO4])^(-8) ([Na2S])^(-5) ([H2O])^12 ([K2SO4])^4 ([Na2SO4])^5 ([MnSO4])^8 = (([H2O])^12 ([K2SO4])^4 ([Na2SO4])^5 ([MnSO4])^8)/(([H2SO4])^12 ([KMnO4])^8 ([Na2S])^5)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + Na_2S ⟶ H_2O + K_2SO_4 + Na_2SO_4 + MnSO_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: 12 H_2SO_4 + 8 KMnO_4 + 5 Na_2S ⟶ 12 H_2O + 4 K_2SO_4 + 5 Na_2SO_4 + 8 MnSO_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 | 12 | -12 KMnO_4 | 8 | -8 Na_2S | 5 | -5 H_2O | 12 | 12 K_2SO_4 | 4 | 4 Na_2SO_4 | 5 | 5 MnSO_4 | 8 | 8 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 | 12 | -12 | -1/12 (Δ[H2SO4])/(Δt) KMnO_4 | 8 | -8 | -1/8 (Δ[KMnO4])/(Δt) Na_2S | 5 | -5 | -1/5 (Δ[Na2S])/(Δt) H_2O | 12 | 12 | 1/12 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) Na_2SO_4 | 5 | 5 | 1/5 (Δ[Na2SO4])/(Δt) MnSO_4 | 8 | 8 | 1/8 (Δ[MnSO4])/(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 = -1/12 (Δ[H2SO4])/(Δt) = -1/8 (Δ[KMnO4])/(Δt) = -1/5 (Δ[Na2S])/(Δt) = 1/12 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/5 (Δ[Na2SO4])/(Δt) = 1/8 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | sodium sulfide | water | potassium sulfate | sodium sulfate | manganese(II) sulfate formula | H_2SO_4 | KMnO_4 | Na_2S | H_2O | K_2SO_4 | Na_2SO_4 | MnSO_4 Hill formula | H_2O_4S | KMnO_4 | Na_2S_1 | H_2O | K_2O_4S | Na_2O_4S | MnSO_4 name | sulfuric acid | potassium permanganate | sodium sulfide | water | potassium sulfate | sodium sulfate | manganese(II) sulfate IUPAC name | sulfuric acid | potassium permanganate | | water | dipotassium sulfate | disodium sulfate | manganese(+2) cation sulfate