HNO3 + KMnO4 + H2S = H2O + S + KNO3 + Mn(NO3)2

HNO_3 nitric acid + KMnO_4 potassium permanganate + H_2S hydrogen sulfide ⟶ H_2O water + S mixed sulfur + KNO_3 potassium nitrate + Mn(NO_3)_2 manganese(II) nitrate

Balance the chemical equation algebraically: HNO_3 + KMnO_4 + H_2S ⟶ H_2O + S + KNO_3 + Mn(NO_3)_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 KMnO_4 + c_3 H_2S ⟶ c_4 H_2O + c_5 S + c_6 KNO_3 + c_7 Mn(NO_3)_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, K, Mn and S: H: | c_1 + 2 c_3 = 2 c_4 N: | c_1 = c_6 + 2 c_7 O: | 3 c_1 + 4 c_2 = c_4 + 3 c_6 + 6 c_7 K: | c_2 = c_6 Mn: | c_2 = c_7 S: | c_3 = 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 = 3 c_2 = 1 c_3 = 5/2 c_4 = 4 c_5 = 5/2 c_6 = 1 c_7 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 6 c_2 = 2 c_3 = 5 c_4 = 8 c_5 = 5 c_6 = 2 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 HNO_3 + 2 KMnO_4 + 5 H_2S ⟶ 8 H_2O + 5 S + 2 KNO_3 + 2 Mn(NO_3)_2

+ + ⟶ + + +

nitric acid + potassium permanganate + hydrogen sulfide ⟶ water + mixed sulfur + potassium nitrate + manganese(II) nitrate

Construct the equilibrium constant, K, expression for: HNO_3 + KMnO_4 + H_2S ⟶ H_2O + S + KNO_3 + Mn(NO_3)_2 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: 6 HNO_3 + 2 KMnO_4 + 5 H_2S ⟶ 8 H_2O + 5 S + 2 KNO_3 + 2 Mn(NO_3)_2 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 HNO_3 | 6 | -6 KMnO_4 | 2 | -2 H_2S | 5 | -5 H_2O | 8 | 8 S | 5 | 5 KNO_3 | 2 | 2 Mn(NO_3)_2 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HNO_3 | 6 | -6 | ([HNO3])^(-6) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) H_2S | 5 | -5 | ([H2S])^(-5) H_2O | 8 | 8 | ([H2O])^8 S | 5 | 5 | ([S])^5 KNO_3 | 2 | 2 | ([KNO3])^2 Mn(NO_3)_2 | 2 | 2 | ([Mn(NO3)2])^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 = ([HNO3])^(-6) ([KMnO4])^(-2) ([H2S])^(-5) ([H2O])^8 ([S])^5 ([KNO3])^2 ([Mn(NO3)2])^2 = (([H2O])^8 ([S])^5 ([KNO3])^2 ([Mn(NO3)2])^2)/(([HNO3])^6 ([KMnO4])^2 ([H2S])^5)

Construct the rate of reaction expression for: HNO_3 + KMnO_4 + H_2S ⟶ H_2O + S + KNO_3 + Mn(NO_3)_2 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: 6 HNO_3 + 2 KMnO_4 + 5 H_2S ⟶ 8 H_2O + 5 S + 2 KNO_3 + 2 Mn(NO_3)_2 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 HNO_3 | 6 | -6 KMnO_4 | 2 | -2 H_2S | 5 | -5 H_2O | 8 | 8 S | 5 | 5 KNO_3 | 2 | 2 Mn(NO_3)_2 | 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 HNO_3 | 6 | -6 | -1/6 (Δ[HNO3])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) H_2S | 5 | -5 | -1/5 (Δ[H2S])/(Δt) H_2O | 8 | 8 | 1/8 (Δ[H2O])/(Δt) S | 5 | 5 | 1/5 (Δ[S])/(Δt) KNO_3 | 2 | 2 | 1/2 (Δ[KNO3])/(Δt) Mn(NO_3)_2 | 2 | 2 | 1/2 (Δ[Mn(NO3)2])/(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 = -1/6 (Δ[HNO3])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/5 (Δ[H2S])/(Δt) = 1/8 (Δ[H2O])/(Δt) = 1/5 (Δ[S])/(Δt) = 1/2 (Δ[KNO3])/(Δt) = 1/2 (Δ[Mn(NO3)2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | potassium permanganate | hydrogen sulfide | water | mixed sulfur | potassium nitrate | manganese(II) nitrate formula | HNO_3 | KMnO_4 | H_2S | H_2O | S | KNO_3 | Mn(NO_3)_2 Hill formula | HNO_3 | KMnO_4 | H_2S | H_2O | S | KNO_3 | MnN_2O_6 name | nitric acid | potassium permanganate | hydrogen sulfide | water | mixed sulfur | potassium nitrate | manganese(II) nitrate IUPAC name | nitric acid | potassium permanganate | hydrogen sulfide | water | sulfur | potassium nitrate | manganese(2+) dinitrate