H2SO4 + KMnO4 + NO = H2O + HNO3 + K2SO4 + MnSO4

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + NO nitric oxide ⟶ H_2O water + HNO_3 nitric acid + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + NO ⟶ H_2O + HNO_3 + K_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 NO ⟶ c_4 H_2O + c_5 HNO_3 + c_6 K_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 N: H: | 2 c_1 = 2 c_4 + c_5 O: | 4 c_1 + 4 c_2 + c_3 = c_4 + 3 c_5 + 4 c_6 + 4 c_7 S: | c_1 = c_6 + c_7 K: | c_2 = 2 c_6 Mn: | c_2 = c_7 N: | 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_6 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 10/3 c_4 = 4/3 c_5 = 10/3 c_6 = 1 c_7 = 2 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 9 c_2 = 6 c_3 = 10 c_4 = 4 c_5 = 10 c_6 = 3 c_7 = 6 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 9 H_2SO_4 + 6 KMnO_4 + 10 NO ⟶ 4 H_2O + 10 HNO_3 + 3 K_2SO_4 + 6 MnSO_4

+ + ⟶ + + +

sulfuric acid + potassium permanganate + nitric oxide ⟶ water + nitric acid + potassium sulfate + manganese(II) sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + NO ⟶ H_2O + HNO_3 + K_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: 9 H_2SO_4 + 6 KMnO_4 + 10 NO ⟶ 4 H_2O + 10 HNO_3 + 3 K_2SO_4 + 6 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 | 9 | -9 KMnO_4 | 6 | -6 NO | 10 | -10 H_2O | 4 | 4 HNO_3 | 10 | 10 K_2SO_4 | 3 | 3 MnSO_4 | 6 | 6 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 9 | -9 | ([H2SO4])^(-9) KMnO_4 | 6 | -6 | ([KMnO4])^(-6) NO | 10 | -10 | ([NO])^(-10) H_2O | 4 | 4 | ([H2O])^4 HNO_3 | 10 | 10 | ([HNO3])^10 K_2SO_4 | 3 | 3 | ([K2SO4])^3 MnSO_4 | 6 | 6 | ([MnSO4])^6 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])^(-9) ([KMnO4])^(-6) ([NO])^(-10) ([H2O])^4 ([HNO3])^10 ([K2SO4])^3 ([MnSO4])^6 = (([H2O])^4 ([HNO3])^10 ([K2SO4])^3 ([MnSO4])^6)/(([H2SO4])^9 ([KMnO4])^6 ([NO])^10)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + NO ⟶ H_2O + HNO_3 + K_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: 9 H_2SO_4 + 6 KMnO_4 + 10 NO ⟶ 4 H_2O + 10 HNO_3 + 3 K_2SO_4 + 6 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 | 9 | -9 KMnO_4 | 6 | -6 NO | 10 | -10 H_2O | 4 | 4 HNO_3 | 10 | 10 K_2SO_4 | 3 | 3 MnSO_4 | 6 | 6 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 | 9 | -9 | -1/9 (Δ[H2SO4])/(Δt) KMnO_4 | 6 | -6 | -1/6 (Δ[KMnO4])/(Δt) NO | 10 | -10 | -1/10 (Δ[NO])/(Δt) H_2O | 4 | 4 | 1/4 (Δ[H2O])/(Δt) HNO_3 | 10 | 10 | 1/10 (Δ[HNO3])/(Δt) K_2SO_4 | 3 | 3 | 1/3 (Δ[K2SO4])/(Δt) MnSO_4 | 6 | 6 | 1/6 (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 = -1/9 (Δ[H2SO4])/(Δt) = -1/6 (Δ[KMnO4])/(Δt) = -1/10 (Δ[NO])/(Δt) = 1/4 (Δ[H2O])/(Δt) = 1/10 (Δ[HNO3])/(Δt) = 1/3 (Δ[K2SO4])/(Δt) = 1/6 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | nitric oxide | water | nitric acid | potassium sulfate | manganese(II) sulfate formula | H_2SO_4 | KMnO_4 | NO | H_2O | HNO_3 | K_2SO_4 | MnSO_4 Hill formula | H_2O_4S | KMnO_4 | NO | H_2O | HNO_3 | K_2O_4S | MnSO_4 name | sulfuric acid | potassium permanganate | nitric oxide | water | nitric acid | potassium sulfate | manganese(II) sulfate IUPAC name | sulfuric acid | potassium permanganate | nitric oxide | water | nitric acid | dipotassium sulfate | manganese(+2) cation sulfate