H2O + KMnO4 + NO2 = HNO3 + MnO2 + KNO3

H_2O water + KMnO_4 potassium permanganate + NO_2 nitrogen dioxide ⟶ HNO_3 nitric acid + MnO_2 manganese dioxide + KNO_3 potassium nitrate

Balance the chemical equation algebraically: H_2O + KMnO_4 + NO_2 ⟶ HNO_3 + MnO_2 + KNO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 NO_2 ⟶ c_4 HNO_3 + c_5 MnO_2 + c_6 KNO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn and N: H: | 2 c_1 = c_4 O: | c_1 + 4 c_2 + 2 c_3 = 3 c_4 + 2 c_5 + 3 c_6 K: | c_2 = c_6 Mn: | c_2 = c_5 N: | c_3 = c_4 + 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 1 c_3 = 3 c_4 = 2 c_5 = 1 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + KMnO_4 + 3 NO_2 ⟶ 2 HNO_3 + MnO_2 + KNO_3

+ + ⟶ + +

water + potassium permanganate + nitrogen dioxide ⟶ nitric acid + manganese dioxide + potassium nitrate

| water | potassium permanganate | nitrogen dioxide | nitric acid | manganese dioxide | potassium nitrate molecular free energy | -237.1 kJ/mol | -737.6 kJ/mol | 51.3 kJ/mol | -80.7 kJ/mol | -465.1 kJ/mol | -394.9 kJ/mol total free energy | -237.1 kJ/mol | -737.6 kJ/mol | 153.9 kJ/mol | -161.4 kJ/mol | -465.1 kJ/mol | -394.9 kJ/mol | G_initial = -820.8 kJ/mol | | | G_final = -1021 kJ/mol | | ΔG_rxn^0 | -1021 kJ/mol - -820.8 kJ/mol = -200.6 kJ/mol (exergonic) | | | | |

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + NO_2 ⟶ HNO_3 + MnO_2 + KNO_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: H_2O + KMnO_4 + 3 NO_2 ⟶ 2 HNO_3 + MnO_2 + KNO_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_2O | 1 | -1 KMnO_4 | 1 | -1 NO_2 | 3 | -3 HNO_3 | 2 | 2 MnO_2 | 1 | 1 KNO_3 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 1 | -1 | ([H2O])^(-1) KMnO_4 | 1 | -1 | ([KMnO4])^(-1) NO_2 | 3 | -3 | ([NO2])^(-3) HNO_3 | 2 | 2 | ([HNO3])^2 MnO_2 | 1 | 1 | [MnO2] KNO_3 | 1 | 1 | [KNO3] 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 = ([H2O])^(-1) ([KMnO4])^(-1) ([NO2])^(-3) ([HNO3])^2 [MnO2] [KNO3] = (([HNO3])^2 [MnO2] [KNO3])/([H2O] [KMnO4] ([NO2])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + NO_2 ⟶ HNO_3 + MnO_2 + KNO_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: H_2O + KMnO_4 + 3 NO_2 ⟶ 2 HNO_3 + MnO_2 + KNO_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_2O | 1 | -1 KMnO_4 | 1 | -1 NO_2 | 3 | -3 HNO_3 | 2 | 2 MnO_2 | 1 | 1 KNO_3 | 1 | 1 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_2O | 1 | -1 | -(Δ[H2O])/(Δt) KMnO_4 | 1 | -1 | -(Δ[KMnO4])/(Δt) NO_2 | 3 | -3 | -1/3 (Δ[NO2])/(Δt) HNO_3 | 2 | 2 | 1/2 (Δ[HNO3])/(Δt) MnO_2 | 1 | 1 | (Δ[MnO2])/(Δt) KNO_3 | 1 | 1 | (Δ[KNO3])/(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 = -(Δ[H2O])/(Δt) = -(Δ[KMnO4])/(Δt) = -1/3 (Δ[NO2])/(Δt) = 1/2 (Δ[HNO3])/(Δt) = (Δ[MnO2])/(Δt) = (Δ[KNO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | nitrogen dioxide | nitric acid | manganese dioxide | potassium nitrate formula | H_2O | KMnO_4 | NO_2 | HNO_3 | MnO_2 | KNO_3 name | water | potassium permanganate | nitrogen dioxide | nitric acid | manganese dioxide | potassium nitrate IUPAC name | water | potassium permanganate | Nitrogen dioxide | nitric acid | dioxomanganese | potassium nitrate