S + KMnO4 = K2SO4 + MnO2

S mixed sulfur + KMnO_4 potassium permanganate ⟶ K_2SO_4 potassium sulfate + MnO_2 manganese dioxide

Balance the chemical equation algebraically: S + KMnO_4 ⟶ K_2SO_4 + MnO_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 S + c_2 KMnO_4 ⟶ c_3 K_2SO_4 + c_4 MnO_2 Set the number of atoms in the reactants equal to the number of atoms in the products for S, K, Mn and O: S: | c_1 = c_3 K: | c_2 = 2 c_3 Mn: | c_2 = c_4 O: | 4 c_2 = 4 c_3 + 2 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 2 c_3 = 1 c_4 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | S + 2 KMnO_4 ⟶ K_2SO_4 + 2 MnO_2

+ ⟶ +

mixed sulfur + potassium permanganate ⟶ potassium sulfate + manganese dioxide

Construct the equilibrium constant, K, expression for: S + KMnO_4 ⟶ 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: S + 2 KMnO_4 ⟶ K_2SO_4 + 2 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 S | 1 | -1 KMnO_4 | 2 | -2 K_2SO_4 | 1 | 1 MnO_2 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression S | 1 | -1 | ([S])^(-1) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) K_2SO_4 | 1 | 1 | [K2SO4] MnO_2 | 2 | 2 | ([MnO2])^2 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 = ([S])^(-1) ([KMnO4])^(-2) [K2SO4] ([MnO2])^2 = ([K2SO4] ([MnO2])^2)/([S] ([KMnO4])^2)

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

| mixed sulfur | potassium permanganate | potassium sulfate | manganese dioxide formula | S | KMnO_4 | K_2SO_4 | MnO_2 Hill formula | S | KMnO_4 | K_2O_4S | MnO_2 name | mixed sulfur | potassium permanganate | potassium sulfate | manganese dioxide IUPAC name | sulfur | potassium permanganate | dipotassium sulfate | dioxomanganese

| mixed sulfur | potassium permanganate | potassium sulfate | manganese dioxide molar mass | 32.06 g/mol | 158.03 g/mol | 174.25 g/mol | 86.936 g/mol phase | solid (at STP) | solid (at STP) | | solid (at STP) melting point | 112.8 °C | 240 °C | | 535 °C boiling point | 444.7 °C | | | density | 2.07 g/cm^3 | 1 g/cm^3 | | 5.03 g/cm^3 solubility in water | | | soluble | insoluble odor | | odorless | |