KOH + KMnO4 + MnSO4 = H2O + K2SO4 + MnO2

KOH potassium hydroxide + KMnO_4 potassium permanganate + MnSO_4 manganese(II) sulfate ⟶ H_2O water + K_2SO_4 potassium sulfate + MnO_2 manganese dioxide

Balance the chemical equation algebraically: KOH + KMnO_4 + MnSO_4 ⟶ H_2O + K_2SO_4 + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KOH + c_2 KMnO_4 + c_3 MnSO_4 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for H, K, O, Mn and S: H: | c_1 = 2 c_4 K: | c_1 + c_2 = 2 c_5 O: | c_1 + 4 c_2 + 4 c_3 = c_4 + 4 c_5 + 2 c_6 Mn: | c_2 + c_3 = c_6 S: | c_3 = 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 = 2 c_2 = 1 c_3 = 3/2 c_4 = 1 c_5 = 3/2 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 = 2 c_5 = 3 c_6 = 5 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 KOH + 2 KMnO_4 + 3 MnSO_4 ⟶ 2 H_2O + 3 K_2SO_4 + 5 MnO_2

+ + ⟶ + +

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

Construct the equilibrium constant, K, expression for: KOH + KMnO_4 + MnSO_4 ⟶ H_2O + K_2SO_4 + 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 KOH + 2 KMnO_4 + 3 MnSO_4 ⟶ 2 H_2O + 3 K_2SO_4 + 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 KOH | 4 | -4 KMnO_4 | 2 | -2 MnSO_4 | 3 | -3 H_2O | 2 | 2 K_2SO_4 | 3 | 3 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 KOH | 4 | -4 | ([KOH])^(-4) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) MnSO_4 | 3 | -3 | ([MnSO4])^(-3) H_2O | 2 | 2 | ([H2O])^2 K_2SO_4 | 3 | 3 | ([K2SO4])^3 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 = ([KOH])^(-4) ([KMnO4])^(-2) ([MnSO4])^(-3) ([H2O])^2 ([K2SO4])^3 ([MnO2])^5 = (([H2O])^2 ([K2SO4])^3 ([MnO2])^5)/(([KOH])^4 ([KMnO4])^2 ([MnSO4])^3)

Construct the rate of reaction expression for: KOH + KMnO_4 + MnSO_4 ⟶ H_2O + K_2SO_4 + 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 KOH + 2 KMnO_4 + 3 MnSO_4 ⟶ 2 H_2O + 3 K_2SO_4 + 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 KOH | 4 | -4 KMnO_4 | 2 | -2 MnSO_4 | 3 | -3 H_2O | 2 | 2 K_2SO_4 | 3 | 3 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 KOH | 4 | -4 | -1/4 (Δ[KOH])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) MnSO_4 | 3 | -3 | -1/3 (Δ[MnSO4])/(Δt) H_2O | 2 | 2 | 1/2 (Δ[H2O])/(Δt) K_2SO_4 | 3 | 3 | 1/3 (Δ[K2SO4])/(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[KOH])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/3 (Δ[MnSO4])/(Δt) = 1/2 (Δ[H2O])/(Δt) = 1/3 (Δ[K2SO4])/(Δt) = 1/5 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium hydroxide | potassium permanganate | manganese(II) sulfate | water | potassium sulfate | manganese dioxide formula | KOH | KMnO_4 | MnSO_4 | H_2O | K_2SO_4 | MnO_2 Hill formula | HKO | KMnO_4 | MnSO_4 | H_2O | K_2O_4S | MnO_2 name | potassium hydroxide | potassium permanganate | manganese(II) sulfate | water | potassium sulfate | manganese dioxide IUPAC name | potassium hydroxide | potassium permanganate | manganese(+2) cation sulfate | water | dipotassium sulfate | dioxomanganese

| potassium hydroxide | potassium permanganate | manganese(II) sulfate | water | potassium sulfate | manganese dioxide molar mass | 56.105 g/mol | 158.03 g/mol | 150.99 g/mol | 18.015 g/mol | 174.25 g/mol | 86.936 g/mol phase | solid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | | solid (at STP) melting point | 406 °C | 240 °C | 710 °C | 0 °C | | 535 °C boiling point | 1327 °C | | | 99.9839 °C | | density | 2.044 g/cm^3 | 1 g/cm^3 | 3.25 g/cm^3 | 1 g/cm^3 | | 5.03 g/cm^3 solubility in water | soluble | | soluble | | soluble | insoluble surface tension | | | | 0.0728 N/m | | dynamic viscosity | 0.001 Pa s (at 550 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | | odor | | odorless | | odorless | |