H2O + SO2 + HMnO4 = H2SO4 + MnSO4

H_2O water + SO_2 sulfur dioxide + HMnO4 ⟶ H_2SO_4 sulfuric acid + MnSO_4 manganese(II) sulfate

Balance the chemical equation algebraically: H_2O + SO_2 + HMnO4 ⟶ H_2SO_4 + MnSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 SO_2 + c_3 HMnO4 ⟶ c_4 H_2SO_4 + c_5 MnSO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S and Mn: H: | 2 c_1 + c_3 = 2 c_4 O: | c_1 + 2 c_2 + 4 c_3 = 4 c_4 + 4 c_5 S: | c_2 = c_4 + c_5 Mn: | c_3 = c_5 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 = 5/2 c_3 = 1 c_4 = 3/2 c_5 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 2 c_2 = 5 c_3 = 2 c_4 = 3 c_5 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + 5 SO_2 + 2 HMnO4 ⟶ 3 H_2SO_4 + 2 MnSO_4

+ + HMnO4 ⟶ +

water + sulfur dioxide + HMnO4 ⟶ sulfuric acid + manganese(II) sulfate

Construct the equilibrium constant, K, expression for: H_2O + SO_2 + HMnO4 ⟶ H_2SO_4 + 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: 2 H_2O + 5 SO_2 + 2 HMnO4 ⟶ 3 H_2SO_4 + 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_2O | 2 | -2 SO_2 | 5 | -5 HMnO4 | 2 | -2 H_2SO_4 | 3 | 3 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_2O | 2 | -2 | ([H2O])^(-2) SO_2 | 5 | -5 | ([SO2])^(-5) HMnO4 | 2 | -2 | ([HMnO4])^(-2) H_2SO_4 | 3 | 3 | ([H2SO4])^3 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 = ([H2O])^(-2) ([SO2])^(-5) ([HMnO4])^(-2) ([H2SO4])^3 ([MnSO4])^2 = (([H2SO4])^3 ([MnSO4])^2)/(([H2O])^2 ([SO2])^5 ([HMnO4])^2)

Construct the rate of reaction expression for: H_2O + SO_2 + HMnO4 ⟶ H_2SO_4 + 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: 2 H_2O + 5 SO_2 + 2 HMnO4 ⟶ 3 H_2SO_4 + 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_2O | 2 | -2 SO_2 | 5 | -5 HMnO4 | 2 | -2 H_2SO_4 | 3 | 3 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_2O | 2 | -2 | -1/2 (Δ[H2O])/(Δt) SO_2 | 5 | -5 | -1/5 (Δ[SO2])/(Δt) HMnO4 | 2 | -2 | -1/2 (Δ[HMnO4])/(Δt) H_2SO_4 | 3 | 3 | 1/3 (Δ[H2SO4])/(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 = -1/2 (Δ[H2O])/(Δt) = -1/5 (Δ[SO2])/(Δt) = -1/2 (Δ[HMnO4])/(Δt) = 1/3 (Δ[H2SO4])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | sulfur dioxide | HMnO4 | sulfuric acid | manganese(II) sulfate formula | H_2O | SO_2 | HMnO4 | H_2SO_4 | MnSO_4 Hill formula | H_2O | O_2S | HMnO4 | H_2O_4S | MnSO_4 name | water | sulfur dioxide | | sulfuric acid | manganese(II) sulfate IUPAC name | water | sulfur dioxide | | sulfuric acid | manganese(+2) cation sulfate

| water | sulfur dioxide | HMnO4 | sulfuric acid | manganese(II) sulfate molar mass | 18.015 g/mol | 64.06 g/mol | 119.94 g/mol | 98.07 g/mol | 150.99 g/mol phase | liquid (at STP) | gas (at STP) | | liquid (at STP) | solid (at STP) melting point | 0 °C | -73 °C | | 10.371 °C | 710 °C boiling point | 99.9839 °C | -10 °C | | 279.6 °C | density | 1 g/cm^3 | 0.002619 g/cm^3 (at 25 °C) | | 1.8305 g/cm^3 | 3.25 g/cm^3 solubility in water | | | | very soluble | soluble surface tension | 0.0728 N/m | 0.02859 N/m | | 0.0735 N/m | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | 1.282×10^-5 Pa s (at 25 °C) | | 0.021 Pa s (at 25 °C) | odor | odorless | | | odorless |