H2SO4 + KMnO4 = H2O + K2SO4 + MnO2 + O3

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate ⟶ H_2O water + K_2SO_4 potassium sulfate + MnO_2 manganese dioxide + O_3 ozone

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 ⟶ H_2O + K_2SO_4 + MnO_2 + O_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 ⟶ c_3 H_2O + c_4 K_2SO_4 + c_5 MnO_2 + c_6 O_3 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 O: | 4 c_1 + 4 c_2 = c_3 + 4 c_4 + 2 c_5 + 3 c_6 S: | c_1 = c_4 K: | c_2 = 2 c_4 Mn: | c_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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 2 c_3 = 1 c_4 = 1 c_5 = 2 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2SO_4 + 2 KMnO_4 ⟶ H_2O + K_2SO_4 + 2 MnO_2 + O_3

+ ⟶ + + +

sulfuric acid + potassium permanganate ⟶ water + potassium sulfate + manganese dioxide + ozone

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 ⟶ H_2O + K_2SO_4 + MnO_2 + O_3 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: H_2SO_4 + 2 KMnO_4 ⟶ H_2O + K_2SO_4 + 2 MnO_2 + O_3 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 | 1 | -1 KMnO_4 | 2 | -2 H_2O | 1 | 1 K_2SO_4 | 1 | 1 MnO_2 | 2 | 2 O_3 | 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 | 1 | -1 | ([H2SO4])^(-1) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) H_2O | 1 | 1 | [H2O] K_2SO_4 | 1 | 1 | [K2SO4] MnO_2 | 2 | 2 | ([MnO2])^2 O_3 | 1 | 1 | [O3] 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])^(-1) ([KMnO4])^(-2) [H2O] [K2SO4] ([MnO2])^2 [O3] = ([H2O] [K2SO4] ([MnO2])^2 [O3])/([H2SO4] ([KMnO4])^2)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 ⟶ H_2O + K_2SO_4 + MnO_2 + O_3 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: H_2SO_4 + 2 KMnO_4 ⟶ H_2O + K_2SO_4 + 2 MnO_2 + O_3 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 | 1 | -1 KMnO_4 | 2 | -2 H_2O | 1 | 1 K_2SO_4 | 1 | 1 MnO_2 | 2 | 2 O_3 | 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 | 1 | -1 | -(Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) H_2O | 1 | 1 | (Δ[H2O])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) MnO_2 | 2 | 2 | 1/2 (Δ[MnO2])/(Δt) O_3 | 1 | 1 | (Δ[O3])/(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 = -(Δ[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = (Δ[H2O])/(Δt) = (Δ[K2SO4])/(Δt) = 1/2 (Δ[MnO2])/(Δt) = (Δ[O3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | water | potassium sulfate | manganese dioxide | ozone formula | H_2SO_4 | KMnO_4 | H_2O | K_2SO_4 | MnO_2 | O_3 Hill formula | H_2O_4S | KMnO_4 | H_2O | K_2O_4S | MnO_2 | O_3 name | sulfuric acid | potassium permanganate | water | potassium sulfate | manganese dioxide | ozone IUPAC name | sulfuric acid | potassium permanganate | water | dipotassium sulfate | dioxomanganese | ozone

| sulfuric acid | potassium permanganate | water | potassium sulfate | manganese dioxide | ozone molar mass | 98.07 g/mol | 158.03 g/mol | 18.015 g/mol | 174.25 g/mol | 86.936 g/mol | 47.997 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | | solid (at STP) | gas (at STP) melting point | 10.371 °C | 240 °C | 0 °C | | 535 °C | -192.2 °C boiling point | 279.6 °C | | 99.9839 °C | | | -111.9 °C density | 1.8305 g/cm^3 | 1 g/cm^3 | 1 g/cm^3 | | 5.03 g/cm^3 | 0.001962 g/cm^3 (at 25 °C) solubility in water | very soluble | | | soluble | insoluble | surface tension | 0.0735 N/m | | 0.0728 N/m | | | dynamic viscosity | 0.021 Pa s (at 25 °C) | | 8.9×10^-4 Pa s (at 25 °C) | | | odor | odorless | odorless | odorless | | |