H2O + KMnO4 + MnSO4 = H2SO4 + KOH + MnO2

H_2O water + KMnO_4 potassium permanganate + MnSO_4 manganese(II) sulfate ⟶ H_2SO_4 sulfuric acid + KOH potassium hydroxide + MnO_2 manganese dioxide

Balance the chemical equation algebraically: H_2O + KMnO_4 + MnSO_4 ⟶ H_2SO_4 + KOH + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 MnSO_4 ⟶ c_4 H_2SO_4 + c_5 KOH + c_6 MnO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn and S: H: | 2 c_1 = 2 c_4 + c_5 O: | c_1 + 4 c_2 + 4 c_3 = 4 c_4 + c_5 + 2 c_6 K: | c_2 = c_5 Mn: | c_2 + c_3 = c_6 S: | c_3 = c_4 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 = 3/2 c_4 = 3/2 c_5 = 1 c_6 = 5/2 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_1 = 4 c_2 = 2 c_3 = 3 c_4 = 3 c_5 = 2 c_6 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2O + 2 KMnO_4 + 3 MnSO_4 ⟶ 3 H_2SO_4 + 2 KOH + 5 MnO_2

+ + ⟶ + +

water + potassium permanganate + manganese(II) sulfate ⟶ sulfuric acid + potassium hydroxide + manganese dioxide

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + MnSO_4 ⟶ H_2SO_4 + KOH + MnO_2 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: 4 H_2O + 2 KMnO_4 + 3 MnSO_4 ⟶ 3 H_2SO_4 + 2 KOH + 5 MnO_2 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 | 4 | -4 KMnO_4 | 2 | -2 MnSO_4 | 3 | -3 H_2SO_4 | 3 | 3 KOH | 2 | 2 MnO_2 | 5 | 5 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 4 | -4 | ([H2O])^(-4) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) MnSO_4 | 3 | -3 | ([MnSO4])^(-3) H_2SO_4 | 3 | 3 | ([H2SO4])^3 KOH | 2 | 2 | ([KOH])^2 MnO_2 | 5 | 5 | ([MnO2])^5 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])^(-4) ([KMnO4])^(-2) ([MnSO4])^(-3) ([H2SO4])^3 ([KOH])^2 ([MnO2])^5 = (([H2SO4])^3 ([KOH])^2 ([MnO2])^5)/(([H2O])^4 ([KMnO4])^2 ([MnSO4])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + MnSO_4 ⟶ H_2SO_4 + KOH + MnO_2 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: 4 H_2O + 2 KMnO_4 + 3 MnSO_4 ⟶ 3 H_2SO_4 + 2 KOH + 5 MnO_2 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 | 4 | -4 KMnO_4 | 2 | -2 MnSO_4 | 3 | -3 H_2SO_4 | 3 | 3 KOH | 2 | 2 MnO_2 | 5 | 5 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 | 4 | -4 | -1/4 (Δ[H2O])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) MnSO_4 | 3 | -3 | -1/3 (Δ[MnSO4])/(Δt) H_2SO_4 | 3 | 3 | 1/3 (Δ[H2SO4])/(Δt) KOH | 2 | 2 | 1/2 (Δ[KOH])/(Δt) MnO_2 | 5 | 5 | 1/5 (Δ[MnO2])/(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 = -1/4 (Δ[H2O])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/3 (Δ[MnSO4])/(Δt) = 1/3 (Δ[H2SO4])/(Δt) = 1/2 (Δ[KOH])/(Δt) = 1/5 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | manganese(II) sulfate | sulfuric acid | potassium hydroxide | manganese dioxide formula | H_2O | KMnO_4 | MnSO_4 | H_2SO_4 | KOH | MnO_2 Hill formula | H_2O | KMnO_4 | MnSO_4 | H_2O_4S | HKO | MnO_2 name | water | potassium permanganate | manganese(II) sulfate | sulfuric acid | potassium hydroxide | manganese dioxide IUPAC name | water | potassium permanganate | manganese(+2) cation sulfate | sulfuric acid | potassium hydroxide | dioxomanganese

| water | potassium permanganate | manganese(II) sulfate | sulfuric acid | potassium hydroxide | manganese dioxide molar mass | 18.015 g/mol | 158.03 g/mol | 150.99 g/mol | 98.07 g/mol | 56.105 g/mol | 86.936 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 | 240 °C | 710 °C | 10.371 °C | 406 °C | 535 °C boiling point | 99.9839 °C | | | 279.6 °C | 1327 °C | density | 1 g/cm^3 | 1 g/cm^3 | 3.25 g/cm^3 | 1.8305 g/cm^3 | 2.044 g/cm^3 | 5.03 g/cm^3 solubility in water | | | soluble | very soluble | 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) | 0.001 Pa s (at 550 °C) | odor | odorless | odorless | | odorless | |