H2SO4 + KMnO4 + N2O = H2O + K2SO4 + NO + MnSO4

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + N_2O nitrous oxide ⟶ H_2O water + K_2SO_4 potassium sulfate + NO nitric oxide + MnSO_4 manganese(II) sulfate

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + N_2O ⟶ H_2O + K_2SO_4 + NO + MnSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 N_2O ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 NO + 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 O: | 4 c_1 + 4 c_2 + c_3 = c_4 + 4 c_5 + c_6 + 4 c_7 S: | c_1 = c_5 + c_7 K: | c_2 = 2 c_5 Mn: | c_2 = c_7 N: | 2 c_3 = 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 c_4 = 3 c_5 = 1 c_6 = 10 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2SO_4 + 2 KMnO_4 + 5 N_2O ⟶ 3 H_2O + K_2SO_4 + 10 NO + 2 MnSO_4

+ + ⟶ + + +

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

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + N_2O ⟶ H_2O + K_2SO_4 + NO + 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: 3 H_2SO_4 + 2 KMnO_4 + 5 N_2O ⟶ 3 H_2O + K_2SO_4 + 10 NO + 2 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 | 3 | -3 KMnO_4 | 2 | -2 N_2O | 5 | -5 H_2O | 3 | 3 K_2SO_4 | 1 | 1 NO | 10 | 10 MnSO_4 | 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 | 3 | -3 | ([H2SO4])^(-3) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) N_2O | 5 | -5 | ([N2O])^(-5) H_2O | 3 | 3 | ([H2O])^3 K_2SO_4 | 1 | 1 | [K2SO4] NO | 10 | 10 | ([NO])^10 MnSO_4 | 2 | 2 | ([MnSO4])^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])^(-3) ([KMnO4])^(-2) ([N2O])^(-5) ([H2O])^3 [K2SO4] ([NO])^10 ([MnSO4])^2 = (([H2O])^3 [K2SO4] ([NO])^10 ([MnSO4])^2)/(([H2SO4])^3 ([KMnO4])^2 ([N2O])^5)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + N_2O ⟶ H_2O + K_2SO_4 + NO + 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: 3 H_2SO_4 + 2 KMnO_4 + 5 N_2O ⟶ 3 H_2O + K_2SO_4 + 10 NO + 2 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 | 3 | -3 KMnO_4 | 2 | -2 N_2O | 5 | -5 H_2O | 3 | 3 K_2SO_4 | 1 | 1 NO | 10 | 10 MnSO_4 | 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 | 3 | -3 | -1/3 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) N_2O | 5 | -5 | -1/5 (Δ[N2O])/(Δt) H_2O | 3 | 3 | 1/3 (Δ[H2O])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) NO | 10 | 10 | 1/10 (Δ[NO])/(Δt) MnSO_4 | 2 | 2 | 1/2 (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 = -1/3 (Δ[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/5 (Δ[N2O])/(Δt) = 1/3 (Δ[H2O])/(Δt) = (Δ[K2SO4])/(Δt) = 1/10 (Δ[NO])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | nitrous oxide | water | potassium sulfate | nitric oxide | manganese(II) sulfate formula | H_2SO_4 | KMnO_4 | N_2O | H_2O | K_2SO_4 | NO | MnSO_4 Hill formula | H_2O_4S | KMnO_4 | N_2O | H_2O | K_2O_4S | NO | MnSO_4 name | sulfuric acid | potassium permanganate | nitrous oxide | water | potassium sulfate | nitric oxide | manganese(II) sulfate IUPAC name | sulfuric acid | potassium permanganate | nitrous oxide | water | dipotassium sulfate | nitric oxide | manganese(+2) cation sulfate