H2SO4 + KMnO4 + H2O2 = H2O + O2 + MnSO4 + K2S4

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + H_2O_2 hydrogen peroxide ⟶ H_2O water + O_2 oxygen + MnSO_4 manganese(II) sulfate + K2S4

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + H_2O_2 ⟶ H_2O + O_2 + MnSO_4 + K2S4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 H_2O_2 ⟶ c_4 H_2O + c_5 O_2 + c_6 MnSO_4 + c_7 K2S4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, K and Mn: H: | 2 c_1 + 2 c_3 = 2 c_4 O: | 4 c_1 + 4 c_2 + 2 c_3 = c_4 + 2 c_5 + 4 c_6 S: | c_1 = c_6 + 4 c_7 K: | c_2 = 2 c_7 Mn: | c_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_7 = 1 and solve the system of equations for the remaining coefficients: c_1 = 6 c_2 = 2 c_4 = c_3 + 6 c_5 = c_3/2 + 9 c_6 = 2 c_7 = 1 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_3 = 2 and solve for the remaining coefficients: c_1 = 6 c_2 = 2 c_3 = 2 c_4 = 8 c_5 = 10 c_6 = 2 c_7 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 H_2SO_4 + 2 KMnO_4 + 2 H_2O_2 ⟶ 8 H_2O + 10 O_2 + 2 MnSO_4 + K2S4

+ + ⟶ + + + K2S4

sulfuric acid + potassium permanganate + hydrogen peroxide ⟶ water + oxygen + manganese(II) sulfate + K2S4

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + H_2O_2 ⟶ H_2O + O_2 + MnSO_4 + K2S4 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 H_2SO_4 + 2 KMnO_4 + 2 H_2O_2 ⟶ 8 H_2O + 10 O_2 + 2 MnSO_4 + K2S4 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_2SO_4 | 6 | -6 KMnO_4 | 2 | -2 H_2O_2 | 2 | -2 H_2O | 8 | 8 O_2 | 10 | 10 MnSO_4 | 2 | 2 K2S4 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 6 | -6 | ([H2SO4])^(-6) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) H_2O_2 | 2 | -2 | ([H2O2])^(-2) H_2O | 8 | 8 | ([H2O])^8 O_2 | 10 | 10 | ([O2])^10 MnSO_4 | 2 | 2 | ([MnSO4])^2 K2S4 | 1 | 1 | [K2S4] 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 = ([H2SO4])^(-6) ([KMnO4])^(-2) ([H2O2])^(-2) ([H2O])^8 ([O2])^10 ([MnSO4])^2 [K2S4] = (([H2O])^8 ([O2])^10 ([MnSO4])^2 [K2S4])/(([H2SO4])^6 ([KMnO4])^2 ([H2O2])^2)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + H_2O_2 ⟶ H_2O + O_2 + MnSO_4 + K2S4 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 H_2SO_4 + 2 KMnO_4 + 2 H_2O_2 ⟶ 8 H_2O + 10 O_2 + 2 MnSO_4 + K2S4 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_2SO_4 | 6 | -6 KMnO_4 | 2 | -2 H_2O_2 | 2 | -2 H_2O | 8 | 8 O_2 | 10 | 10 MnSO_4 | 2 | 2 K2S4 | 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_2SO_4 | 6 | -6 | -1/6 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) H_2O_2 | 2 | -2 | -1/2 (Δ[H2O2])/(Δt) H_2O | 8 | 8 | 1/8 (Δ[H2O])/(Δt) O_2 | 10 | 10 | 1/10 (Δ[O2])/(Δt) MnSO_4 | 2 | 2 | 1/2 (Δ[MnSO4])/(Δt) K2S4 | 1 | 1 | (Δ[K2S4])/(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[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/2 (Δ[H2O2])/(Δt) = 1/8 (Δ[H2O])/(Δt) = 1/10 (Δ[O2])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) = (Δ[K2S4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | hydrogen peroxide | water | oxygen | manganese(II) sulfate | K2S4 formula | H_2SO_4 | KMnO_4 | H_2O_2 | H_2O | O_2 | MnSO_4 | K2S4 Hill formula | H_2O_4S | KMnO_4 | H_2O_2 | H_2O | O_2 | MnSO_4 | K2S4 name | sulfuric acid | potassium permanganate | hydrogen peroxide | water | oxygen | manganese(II) sulfate | IUPAC name | sulfuric acid | potassium permanganate | hydrogen peroxide | water | molecular oxygen | manganese(+2) cation sulfate |

| sulfuric acid | potassium permanganate | hydrogen peroxide | water | oxygen | manganese(II) sulfate | K2S4 molar mass | 98.07 g/mol | 158.03 g/mol | 34.014 g/mol | 18.015 g/mol | 31.998 g/mol | 150.99 g/mol | 206.4 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | liquid (at STP) | gas (at STP) | solid (at STP) | melting point | 10.371 °C | 240 °C | -0.43 °C | 0 °C | -218 °C | 710 °C | boiling point | 279.6 °C | | 150.2 °C | 99.9839 °C | -183 °C | | density | 1.8305 g/cm^3 | 1 g/cm^3 | 1.44 g/cm^3 | 1 g/cm^3 | 0.001429 g/cm^3 (at 0 °C) | 3.25 g/cm^3 | solubility in water | very soluble | | miscible | | | soluble | surface tension | 0.0735 N/m | | 0.0804 N/m | 0.0728 N/m | 0.01347 N/m | | dynamic viscosity | 0.021 Pa s (at 25 °C) | | 0.001249 Pa s (at 20 °C) | 8.9×10^-4 Pa s (at 25 °C) | 2.055×10^-5 Pa s (at 25 °C) | | odor | odorless | odorless | | odorless | odorless | |