HNO3 + MnSO4 + PbO2 = H2O + Pb(NO3)2 + PbSO4 + HMnO4

HNO_3 (nitric acid) + MnSO_4 (manganese(II) sulfate) + PbO_2 (lead dioxide) ⟶ H_2O (water) + Pb(NO_3)_2 (lead(II) nitrate) + PbSO_4 (lead(II) sulfate) + HMnO4

Balance the chemical equation algebraically: HNO_3 + MnSO_4 + PbO_2 ⟶ H_2O + Pb(NO_3)_2 + PbSO_4 + HMnO4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 MnSO_4 + c_3 PbO_2 ⟶ c_4 H_2O + c_5 Pb(NO_3)_2 + c_6 PbSO_4 + c_7 HMnO4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, Mn, S and Pb: H: | c_1 = 2 c_4 + c_7 N: | c_1 = 2 c_5 O: | 3 c_1 + 4 c_2 + 2 c_3 = c_4 + 6 c_5 + 4 c_6 + 4 c_7 Mn: | c_2 = c_7 S: | c_2 = c_6 Pb: | c_3 = c_5 + 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 3 c_2 = 1 c_3 = 5/2 c_4 = 1 c_5 = 3/2 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 = 2 c_5 = 3 c_6 = 2 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 HNO_3 + 2 MnSO_4 + 5 PbO_2 ⟶ 2 H_2O + 3 Pb(NO_3)_2 + 2 PbSO_4 + 2 HMnO4

+ + ⟶ + + + HMnO4

nitric acid + manganese(II) sulfate + lead dioxide ⟶ water + lead(II) nitrate + lead(II) sulfate + HMnO4

Construct the equilibrium constant, K, expression for: HNO_3 + MnSO_4 + PbO_2 ⟶ H_2O + Pb(NO_3)_2 + PbSO_4 + HMnO4 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 MnSO_4 + 5 PbO_2 ⟶ 2 H_2O + 3 Pb(NO_3)_2 + 2 PbSO_4 + 2 HMnO4 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 MnSO_4 | 2 | -2 PbO_2 | 5 | -5 H_2O | 2 | 2 Pb(NO_3)_2 | 3 | 3 PbSO_4 | 2 | 2 HMnO4 | 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) MnSO_4 | 2 | -2 | ([MnSO4])^(-2) PbO_2 | 5 | -5 | ([PbO2])^(-5) H_2O | 2 | 2 | ([H2O])^2 Pb(NO_3)_2 | 3 | 3 | ([Pb(NO3)2])^3 PbSO_4 | 2 | 2 | ([PbSO4])^2 HMnO4 | 2 | 2 | ([HMnO4])^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) ([MnSO4])^(-2) ([PbO2])^(-5) ([H2O])^2 ([Pb(NO3)2])^3 ([PbSO4])^2 ([HMnO4])^2 = (([H2O])^2 ([Pb(NO3)2])^3 ([PbSO4])^2 ([HMnO4])^2)/(([HNO3])^6 ([MnSO4])^2 ([PbO2])^5)

Construct the rate of reaction expression for: HNO_3 + MnSO_4 + PbO_2 ⟶ H_2O + Pb(NO_3)_2 + PbSO_4 + HMnO4 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 MnSO_4 + 5 PbO_2 ⟶ 2 H_2O + 3 Pb(NO_3)_2 + 2 PbSO_4 + 2 HMnO4 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 MnSO_4 | 2 | -2 PbO_2 | 5 | -5 H_2O | 2 | 2 Pb(NO_3)_2 | 3 | 3 PbSO_4 | 2 | 2 HMnO4 | 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) MnSO_4 | 2 | -2 | -1/2 (Δ[MnSO4])/(Δt) PbO_2 | 5 | -5 | -1/5 (Δ[PbO2])/(Δt) H_2O | 2 | 2 | 1/2 (Δ[H2O])/(Δt) Pb(NO_3)_2 | 3 | 3 | 1/3 (Δ[Pb(NO3)2])/(Δt) PbSO_4 | 2 | 2 | 1/2 (Δ[PbSO4])/(Δt) HMnO4 | 2 | 2 | 1/2 (Δ[HMnO4])/(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[MnSO4])/(Δt) = -1/5 (Δ[PbO2])/(Δt) = 1/2 (Δ[H2O])/(Δt) = 1/3 (Δ[Pb(NO3)2])/(Δt) = 1/2 (Δ[PbSO4])/(Δt) = 1/2 (Δ[HMnO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | manganese(II) sulfate | lead dioxide | water | lead(II) nitrate | lead(II) sulfate | HMnO4 formula | HNO_3 | MnSO_4 | PbO_2 | H_2O | Pb(NO_3)_2 | PbSO_4 | HMnO4 Hill formula | HNO_3 | MnSO_4 | O_2Pb | H_2O | N_2O_6Pb | O_4PbS | HMnO4 name | nitric acid | manganese(II) sulfate | lead dioxide | water | lead(II) nitrate | lead(II) sulfate | IUPAC name | nitric acid | manganese(+2) cation sulfate | | water | plumbous dinitrate | |