HNO3 + MnSO4 + PbO2 = H2O + H2SO4 + Pb(NO3)2 + HMnO4

HNO_3 nitric acid + MnSO_4 manganese(II) sulfate + PbO_2 lead dioxide ⟶ H_2O water + H_2SO_4 sulfuric acid + Pb(NO_3)_2 lead(II) nitrate + HMnO4

Balance the chemical equation algebraically: HNO_3 + MnSO_4 + PbO_2 ⟶ H_2O + H_2SO_4 + Pb(NO_3)_2 + 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 H_2SO_4 + c_6 Pb(NO_3)_2 + 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 + 2 c_5 + c_7 N: | c_1 = 2 c_6 O: | 3 c_1 + 4 c_2 + 2 c_3 = c_4 + 4 c_5 + 6 c_6 + 4 c_7 Mn: | c_2 = c_7 S: | c_2 = c_5 Pb: | 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 5 c_2 = 1 c_3 = 5/2 c_4 = 1 c_5 = 1 c_6 = 5/2 c_7 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 10 c_2 = 2 c_3 = 5 c_4 = 2 c_5 = 2 c_6 = 5 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 10 HNO_3 + 2 MnSO_4 + 5 PbO_2 ⟶ 2 H_2O + 2 H_2SO_4 + 5 Pb(NO_3)_2 + 2 HMnO4

+ + ⟶ + + + HMnO4

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

Construct the equilibrium constant, K, expression for: HNO_3 + MnSO_4 + PbO_2 ⟶ H_2O + H_2SO_4 + Pb(NO_3)_2 + 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: 10 HNO_3 + 2 MnSO_4 + 5 PbO_2 ⟶ 2 H_2O + 2 H_2SO_4 + 5 Pb(NO_3)_2 + 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 | 10 | -10 MnSO_4 | 2 | -2 PbO_2 | 5 | -5 H_2O | 2 | 2 H_2SO_4 | 2 | 2 Pb(NO_3)_2 | 5 | 5 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 | 10 | -10 | ([HNO3])^(-10) MnSO_4 | 2 | -2 | ([MnSO4])^(-2) PbO_2 | 5 | -5 | ([PbO2])^(-5) H_2O | 2 | 2 | ([H2O])^2 H_2SO_4 | 2 | 2 | ([H2SO4])^2 Pb(NO_3)_2 | 5 | 5 | ([Pb(NO3)2])^5 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])^(-10) ([MnSO4])^(-2) ([PbO2])^(-5) ([H2O])^2 ([H2SO4])^2 ([Pb(NO3)2])^5 ([HMnO4])^2 = (([H2O])^2 ([H2SO4])^2 ([Pb(NO3)2])^5 ([HMnO4])^2)/(([HNO3])^10 ([MnSO4])^2 ([PbO2])^5)

Construct the rate of reaction expression for: HNO_3 + MnSO_4 + PbO_2 ⟶ H_2O + H_2SO_4 + Pb(NO_3)_2 + 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: 10 HNO_3 + 2 MnSO_4 + 5 PbO_2 ⟶ 2 H_2O + 2 H_2SO_4 + 5 Pb(NO_3)_2 + 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 | 10 | -10 MnSO_4 | 2 | -2 PbO_2 | 5 | -5 H_2O | 2 | 2 H_2SO_4 | 2 | 2 Pb(NO_3)_2 | 5 | 5 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 | 10 | -10 | -1/10 (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) H_2SO_4 | 2 | 2 | 1/2 (Δ[H2SO4])/(Δt) Pb(NO_3)_2 | 5 | 5 | 1/5 (Δ[Pb(NO3)2])/(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 = -1/10 (Δ[HNO3])/(Δt) = -1/2 (Δ[MnSO4])/(Δt) = -1/5 (Δ[PbO2])/(Δt) = 1/2 (Δ[H2O])/(Δt) = 1/2 (Δ[H2SO4])/(Δt) = 1/5 (Δ[Pb(NO3)2])/(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 | sulfuric acid | lead(II) nitrate | HMnO4 formula | HNO_3 | MnSO_4 | PbO_2 | H_2O | H_2SO_4 | Pb(NO_3)_2 | HMnO4 Hill formula | HNO_3 | MnSO_4 | O_2Pb | H_2O | H_2O_4S | N_2O_6Pb | HMnO4 name | nitric acid | manganese(II) sulfate | lead dioxide | water | sulfuric acid | lead(II) nitrate | IUPAC name | nitric acid | manganese(+2) cation sulfate | | water | sulfuric acid | plumbous dinitrate |

| nitric acid | manganese(II) sulfate | lead dioxide | water | sulfuric acid | lead(II) nitrate | HMnO4 molar mass | 63.012 g/mol | 150.99 g/mol | 239.2 g/mol | 18.015 g/mol | 98.07 g/mol | 331.2 g/mol | 119.94 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | liquid (at STP) | solid (at STP) | melting point | -41.6 °C | 710 °C | 290 °C | 0 °C | 10.371 °C | 470 °C | boiling point | 83 °C | | | 99.9839 °C | 279.6 °C | | density | 1.5129 g/cm^3 | 3.25 g/cm^3 | 9.58 g/cm^3 | 1 g/cm^3 | 1.8305 g/cm^3 | | solubility in water | miscible | soluble | insoluble | | very soluble | | surface tension | | | | 0.0728 N/m | 0.0735 N/m | | dynamic viscosity | 7.6×10^-4 Pa s (at 25 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | 0.021 Pa s (at 25 °C) | | odor | | | | odorless | odorless | odorless |