H2O + KMnO4 + K2S = K2SO4 + KOH + MnO2

H_2O water + KMnO_4 potassium permanganate + K2S ⟶ K_2SO_4 potassium sulfate + KOH potassium hydroxide + MnO_2 manganese dioxide

Balance the chemical equation algebraically: H_2O + KMnO_4 + K2S ⟶ K_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 K2S ⟶ c_4 K_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 = c_5 O: | c_1 + 4 c_2 = 4 c_4 + c_5 + 2 c_6 K: | c_2 + 2 c_3 = 2 c_4 + c_5 Mn: | c_2 = 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 4/3 c_2 = 8/3 c_3 = 1 c_4 = 1 c_5 = 8/3 c_6 = 8/3 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 4 c_2 = 8 c_3 = 3 c_4 = 3 c_5 = 8 c_6 = 8 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2O + 8 KMnO_4 + 3 K2S ⟶ 3 K_2SO_4 + 8 KOH + 8 MnO_2

+ + K2S ⟶ + +

water + potassium permanganate + K2S ⟶ potassium sulfate + potassium hydroxide + manganese dioxide

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + K2S ⟶ K_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 + 8 KMnO_4 + 3 K2S ⟶ 3 K_2SO_4 + 8 KOH + 8 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 | 8 | -8 K2S | 3 | -3 K_2SO_4 | 3 | 3 KOH | 8 | 8 MnO_2 | 8 | 8 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 | 8 | -8 | ([KMnO4])^(-8) K2S | 3 | -3 | ([K2S])^(-3) K_2SO_4 | 3 | 3 | ([K2SO4])^3 KOH | 8 | 8 | ([KOH])^8 MnO_2 | 8 | 8 | ([MnO2])^8 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])^(-8) ([K2S])^(-3) ([K2SO4])^3 ([KOH])^8 ([MnO2])^8 = (([K2SO4])^3 ([KOH])^8 ([MnO2])^8)/(([H2O])^4 ([KMnO4])^8 ([K2S])^3)

Construct the rate of reaction expression for: H_2O + KMnO_4 + K2S ⟶ K_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 + 8 KMnO_4 + 3 K2S ⟶ 3 K_2SO_4 + 8 KOH + 8 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 | 8 | -8 K2S | 3 | -3 K_2SO_4 | 3 | 3 KOH | 8 | 8 MnO_2 | 8 | 8 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 | 8 | -8 | -1/8 (Δ[KMnO4])/(Δt) K2S | 3 | -3 | -1/3 (Δ[K2S])/(Δt) K_2SO_4 | 3 | 3 | 1/3 (Δ[K2SO4])/(Δt) KOH | 8 | 8 | 1/8 (Δ[KOH])/(Δt) MnO_2 | 8 | 8 | 1/8 (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) = -1/8 (Δ[KMnO4])/(Δt) = -1/3 (Δ[K2S])/(Δt) = 1/3 (Δ[K2SO4])/(Δt) = 1/8 (Δ[KOH])/(Δt) = 1/8 (Δ[MnO2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | K2S | potassium sulfate | potassium hydroxide | manganese dioxide formula | H_2O | KMnO_4 | K2S | K_2SO_4 | KOH | MnO_2 Hill formula | H_2O | KMnO_4 | K2S | K_2O_4S | HKO | MnO_2 name | water | potassium permanganate | | potassium sulfate | potassium hydroxide | manganese dioxide IUPAC name | water | potassium permanganate | | dipotassium sulfate | potassium hydroxide | dioxomanganese

| water | potassium permanganate | K2S | potassium sulfate | potassium hydroxide | manganese dioxide molar mass | 18.015 g/mol | 158.03 g/mol | 110.26 g/mol | 174.25 g/mol | 56.105 g/mol | 86.936 g/mol phase | liquid (at STP) | solid (at STP) | | | solid (at STP) | solid (at STP) melting point | 0 °C | 240 °C | | | 406 °C | 535 °C boiling point | 99.9839 °C | | | | 1327 °C | density | 1 g/cm^3 | 1 g/cm^3 | | | 2.044 g/cm^3 | 5.03 g/cm^3 solubility in water | | | | soluble | soluble | insoluble surface tension | 0.0728 N/m | | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | | 0.001 Pa s (at 550 °C) | odor | odorless | odorless | | | |