H2O + HNO3 + K2SO4 + MnSO4 = H2SO4 + KMnO4 + HNO2

H_2O water + HNO_3 nitric acid + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate ⟶ H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + HNO_2 nitrous acid

Balance the chemical equation algebraically: H_2O + HNO_3 + K_2SO_4 + MnSO_4 ⟶ H_2SO_4 + KMnO_4 + HNO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 HNO_3 + c_3 K_2SO_4 + c_4 MnSO_4 ⟶ c_5 H_2SO_4 + c_6 KMnO_4 + c_7 HNO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, N, K, S and Mn: H: | 2 c_1 + c_2 = 2 c_5 + c_7 O: | c_1 + 3 c_2 + 4 c_3 + 4 c_4 = 4 c_5 + 4 c_6 + 2 c_7 N: | c_2 = c_7 K: | 2 c_3 = c_6 S: | c_3 + c_4 = c_5 Mn: | 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 5 c_3 = 1 c_4 = 2 c_5 = 3 c_6 = 2 c_7 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2O + 5 HNO_3 + K_2SO_4 + 2 MnSO_4 ⟶ 3 H_2SO_4 + 2 KMnO_4 + 5 HNO_2

+ + + ⟶ + +

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

Construct the equilibrium constant, K, expression for: H_2O + HNO_3 + K_2SO_4 + MnSO_4 ⟶ H_2SO_4 + KMnO_4 + HNO_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: 3 H_2O + 5 HNO_3 + K_2SO_4 + 2 MnSO_4 ⟶ 3 H_2SO_4 + 2 KMnO_4 + 5 HNO_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 H_2O | 3 | -3 HNO_3 | 5 | -5 K_2SO_4 | 1 | -1 MnSO_4 | 2 | -2 H_2SO_4 | 3 | 3 KMnO_4 | 2 | 2 HNO_2 | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 3 | -3 | ([H2O])^(-3) HNO_3 | 5 | -5 | ([HNO3])^(-5) K_2SO_4 | 1 | -1 | ([K2SO4])^(-1) MnSO_4 | 2 | -2 | ([MnSO4])^(-2) H_2SO_4 | 3 | 3 | ([H2SO4])^3 KMnO_4 | 2 | 2 | ([KMnO4])^2 HNO_2 | 5 | 5 | ([HNO2])^5 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])^(-3) ([HNO3])^(-5) ([K2SO4])^(-1) ([MnSO4])^(-2) ([H2SO4])^3 ([KMnO4])^2 ([HNO2])^5 = (([H2SO4])^3 ([KMnO4])^2 ([HNO2])^5)/(([H2O])^3 ([HNO3])^5 [K2SO4] ([MnSO4])^2)

Construct the rate of reaction expression for: H_2O + HNO_3 + K_2SO_4 + MnSO_4 ⟶ H_2SO_4 + KMnO_4 + HNO_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: 3 H_2O + 5 HNO_3 + K_2SO_4 + 2 MnSO_4 ⟶ 3 H_2SO_4 + 2 KMnO_4 + 5 HNO_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 H_2O | 3 | -3 HNO_3 | 5 | -5 K_2SO_4 | 1 | -1 MnSO_4 | 2 | -2 H_2SO_4 | 3 | 3 KMnO_4 | 2 | 2 HNO_2 | 5 | 5 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 | 3 | -3 | -1/3 (Δ[H2O])/(Δt) HNO_3 | 5 | -5 | -1/5 (Δ[HNO3])/(Δt) K_2SO_4 | 1 | -1 | -(Δ[K2SO4])/(Δt) MnSO_4 | 2 | -2 | -1/2 (Δ[MnSO4])/(Δt) H_2SO_4 | 3 | 3 | 1/3 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | 2 | 1/2 (Δ[KMnO4])/(Δt) HNO_2 | 5 | 5 | 1/5 (Δ[HNO2])/(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[H2O])/(Δt) = -1/5 (Δ[HNO3])/(Δt) = -(Δ[K2SO4])/(Δt) = -1/2 (Δ[MnSO4])/(Δt) = 1/3 (Δ[H2SO4])/(Δt) = 1/2 (Δ[KMnO4])/(Δt) = 1/5 (Δ[HNO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | nitric acid | potassium sulfate | manganese(II) sulfate | sulfuric acid | potassium permanganate | nitrous acid formula | H_2O | HNO_3 | K_2SO_4 | MnSO_4 | H_2SO_4 | KMnO_4 | HNO_2 Hill formula | H_2O | HNO_3 | K_2O_4S | MnSO_4 | H_2O_4S | KMnO_4 | HNO_2 name | water | nitric acid | potassium sulfate | manganese(II) sulfate | sulfuric acid | potassium permanganate | nitrous acid IUPAC name | water | nitric acid | dipotassium sulfate | manganese(+2) cation sulfate | sulfuric acid | potassium permanganate | nitrous acid