H2O + MnSO4 + K2S2O8 = H2SO4 + K2SO4 + HMnO4

H_2O water + MnSO_4 manganese(II) sulfate + K_2S_2O_8 potassium persulfate ⟶ H_2SO_4 sulfuric acid + K_2SO_4 potassium sulfate + HMnO4

Balance the chemical equation algebraically: H_2O + MnSO_4 + K_2S_2O_8 ⟶ H_2SO_4 + K_2SO_4 + HMnO4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 MnSO_4 + c_3 K_2S_2O_8 ⟶ c_4 H_2SO_4 + c_5 K_2SO_4 + c_6 HMnO4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Mn, S and K: H: | 2 c_1 = 2 c_4 + c_6 O: | c_1 + 4 c_2 + 8 c_3 = 4 c_4 + 4 c_5 + 4 c_6 Mn: | c_2 = c_6 S: | c_2 + 2 c_3 = c_4 + c_5 K: | 2 c_3 = 2 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 4 c_2 = 1 c_3 = 5/2 c_4 = 7/2 c_5 = 5/2 c_6 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 8 c_2 = 2 c_3 = 5 c_4 = 7 c_5 = 5 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2O + 2 MnSO_4 + 5 K_2S_2O_8 ⟶ 7 H_2SO_4 + 5 K_2SO_4 + 2 HMnO4

+ + ⟶ + + HMnO4

water + manganese(II) sulfate + potassium persulfate ⟶ sulfuric acid + potassium sulfate + HMnO4

Construct the equilibrium constant, K, expression for: H_2O + MnSO_4 + K_2S_2O_8 ⟶ H_2SO_4 + K_2SO_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: 8 H_2O + 2 MnSO_4 + 5 K_2S_2O_8 ⟶ 7 H_2SO_4 + 5 K_2SO_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 H_2O | 8 | -8 MnSO_4 | 2 | -2 K_2S_2O_8 | 5 | -5 H_2SO_4 | 7 | 7 K_2SO_4 | 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 H_2O | 8 | -8 | ([H2O])^(-8) MnSO_4 | 2 | -2 | ([MnSO4])^(-2) K_2S_2O_8 | 5 | -5 | ([K2S2O8])^(-5) H_2SO_4 | 7 | 7 | ([H2SO4])^7 K_2SO_4 | 5 | 5 | ([K2SO4])^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 = ([H2O])^(-8) ([MnSO4])^(-2) ([K2S2O8])^(-5) ([H2SO4])^7 ([K2SO4])^5 ([HMnO4])^2 = (([H2SO4])^7 ([K2SO4])^5 ([HMnO4])^2)/(([H2O])^8 ([MnSO4])^2 ([K2S2O8])^5)

Construct the rate of reaction expression for: H_2O + MnSO_4 + K_2S_2O_8 ⟶ H_2SO_4 + K_2SO_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: 8 H_2O + 2 MnSO_4 + 5 K_2S_2O_8 ⟶ 7 H_2SO_4 + 5 K_2SO_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 H_2O | 8 | -8 MnSO_4 | 2 | -2 K_2S_2O_8 | 5 | -5 H_2SO_4 | 7 | 7 K_2SO_4 | 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 H_2O | 8 | -8 | -1/8 (Δ[H2O])/(Δt) MnSO_4 | 2 | -2 | -1/2 (Δ[MnSO4])/(Δt) K_2S_2O_8 | 5 | -5 | -1/5 (Δ[K2S2O8])/(Δt) H_2SO_4 | 7 | 7 | 1/7 (Δ[H2SO4])/(Δt) K_2SO_4 | 5 | 5 | 1/5 (Δ[K2SO4])/(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 = -1/8 (Δ[H2O])/(Δt) = -1/2 (Δ[MnSO4])/(Δt) = -1/5 (Δ[K2S2O8])/(Δt) = 1/7 (Δ[H2SO4])/(Δt) = 1/5 (Δ[K2SO4])/(Δt) = 1/2 (Δ[HMnO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | manganese(II) sulfate | potassium persulfate | sulfuric acid | potassium sulfate | HMnO4 formula | H_2O | MnSO_4 | K_2S_2O_8 | H_2SO_4 | K_2SO_4 | HMnO4 Hill formula | H_2O | MnSO_4 | K_2O_8S_2 | H_2O_4S | K_2O_4S | HMnO4 name | water | manganese(II) sulfate | potassium persulfate | sulfuric acid | potassium sulfate | IUPAC name | water | manganese(+2) cation sulfate | dipotassium sulfonatooxy sulfate | sulfuric acid | dipotassium sulfate |

| water | manganese(II) sulfate | potassium persulfate | sulfuric acid | potassium sulfate | HMnO4 molar mass | 18.015 g/mol | 150.99 g/mol | 270.31 g/mol | 98.07 g/mol | 174.25 g/mol | 119.94 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | | melting point | 0 °C | 710 °C | 100 °C | 10.371 °C | | boiling point | 99.9839 °C | | | 279.6 °C | | density | 1 g/cm^3 | 3.25 g/cm^3 | 2.477 g/cm^3 | 1.8305 g/cm^3 | | solubility in water | | soluble | soluble | very soluble | soluble | surface tension | 0.0728 N/m | | | 0.0735 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.021 Pa s (at 25 °C) | | odor | odorless | | | odorless | |