H2O + MnSO4 + (NH4)2S2O8 = H2SO4 + MnO2 + (NH4)2SO4

H_2O water + MnSO_4 manganese(II) sulfate + (NH_4)_2S_2O_8 ammonium persulfate ⟶ H_2SO_4 sulfuric acid + MnO_2 manganese dioxide + (NH_4)_2SO_4 ammonium sulfate

Balance the chemical equation algebraically: H_2O + MnSO_4 + (NH_4)_2S_2O_8 ⟶ H_2SO_4 + MnO_2 + (NH_4)_2SO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 MnSO_4 + c_3 (NH_4)_2S_2O_8 ⟶ c_4 H_2SO_4 + c_5 MnO_2 + c_6 (NH_4)_2SO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Mn, S and N: H: | 2 c_1 + 8 c_3 = 2 c_4 + 8 c_6 O: | c_1 + 4 c_2 + 8 c_3 = 4 c_4 + 2 c_5 + 4 c_6 Mn: | c_2 = c_5 S: | c_2 + 2 c_3 = c_4 + c_6 N: | 2 c_3 = 2 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 = 2 c_2 = 1 c_3 = 1 c_4 = 2 c_5 = 1 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + MnSO_4 + (NH_4)_2S_2O_8 ⟶ 2 H_2SO_4 + MnO_2 + (NH_4)_2SO_4

+ + ⟶ + +

water + manganese(II) sulfate + ammonium persulfate ⟶ sulfuric acid + manganese dioxide + ammonium sulfate

Construct the equilibrium constant, K, expression for: H_2O + MnSO_4 + (NH_4)_2S_2O_8 ⟶ H_2SO_4 + MnO_2 + (NH_4)_2SO_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 + MnSO_4 + (NH_4)_2S_2O_8 ⟶ 2 H_2SO_4 + MnO_2 + (NH_4)_2SO_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 MnSO_4 | 1 | -1 (NH_4)_2S_2O_8 | 1 | -1 H_2SO_4 | 2 | 2 MnO_2 | 1 | 1 (NH_4)_2SO_4 | 1 | 1 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) MnSO_4 | 1 | -1 | ([MnSO4])^(-1) (NH_4)_2S_2O_8 | 1 | -1 | ([(NH4)2S2O8])^(-1) H_2SO_4 | 2 | 2 | ([H2SO4])^2 MnO_2 | 1 | 1 | [MnO2] (NH_4)_2SO_4 | 1 | 1 | [(NH4)2SO4] 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) ([MnSO4])^(-1) ([(NH4)2S2O8])^(-1) ([H2SO4])^2 [MnO2] [(NH4)2SO4] = (([H2SO4])^2 [MnO2] [(NH4)2SO4])/(([H2O])^2 [MnSO4] [(NH4)2S2O8])

Construct the rate of reaction expression for: H_2O + MnSO_4 + (NH_4)_2S_2O_8 ⟶ H_2SO_4 + MnO_2 + (NH_4)_2SO_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 + MnSO_4 + (NH_4)_2S_2O_8 ⟶ 2 H_2SO_4 + MnO_2 + (NH_4)_2SO_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 MnSO_4 | 1 | -1 (NH_4)_2S_2O_8 | 1 | -1 H_2SO_4 | 2 | 2 MnO_2 | 1 | 1 (NH_4)_2SO_4 | 1 | 1 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) MnSO_4 | 1 | -1 | -(Δ[MnSO4])/(Δt) (NH_4)_2S_2O_8 | 1 | -1 | -(Δ[(NH4)2S2O8])/(Δt) H_2SO_4 | 2 | 2 | 1/2 (Δ[H2SO4])/(Δt) MnO_2 | 1 | 1 | (Δ[MnO2])/(Δt) (NH_4)_2SO_4 | 1 | 1 | (Δ[(NH4)2SO4])/(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) = -(Δ[MnSO4])/(Δt) = -(Δ[(NH4)2S2O8])/(Δt) = 1/2 (Δ[H2SO4])/(Δt) = (Δ[MnO2])/(Δt) = (Δ[(NH4)2SO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | manganese(II) sulfate | ammonium persulfate | sulfuric acid | manganese dioxide | ammonium sulfate formula | H_2O | MnSO_4 | (NH_4)_2S_2O_8 | H_2SO_4 | MnO_2 | (NH_4)_2SO_4 Hill formula | H_2O | MnSO_4 | H_8N_2O_8S_2 | H_2O_4S | MnO_2 | H_8N_2O_4S name | water | manganese(II) sulfate | ammonium persulfate | sulfuric acid | manganese dioxide | ammonium sulfate IUPAC name | water | manganese(+2) cation sulfate | diammonium sulfonatooxy sulfate | sulfuric acid | dioxomanganese |

| water | manganese(II) sulfate | ammonium persulfate | sulfuric acid | manganese dioxide | ammonium sulfate molar mass | 18.015 g/mol | 150.99 g/mol | 228.2 g/mol | 98.07 g/mol | 86.936 g/mol | 132.1 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) melting point | 0 °C | 710 °C | 120 °C | 10.371 °C | 535 °C | 280 °C boiling point | 99.9839 °C | | | 279.6 °C | | density | 1 g/cm^3 | 3.25 g/cm^3 | 1.98 g/cm^3 | 1.8305 g/cm^3 | 5.03 g/cm^3 | 1.77 g/cm^3 solubility in water | | soluble | | very soluble | insoluble | 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 | odorless | | odorless