H2SO4 + KMnO4 + K2S = H2O + K2SO4 + S + MnSO4

H_2SO_4 (sulfuric acid) + KMnO_4 (potassium permanganate) + K2S ⟶ H_2O (water) + K_2SO_4 (potassium sulfate) + S (mixed sulfur) + MnSO_4 (manganese(II) sulfate)

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + K2S ⟶ H_2O + K_2SO_4 + S + MnSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 K2S ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 S + c_7 MnSO_4 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_4 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_7 S: | c_1 + c_3 = c_5 + c_6 + c_7 K: | c_2 + 2 c_3 = 2 c_5 Mn: | c_2 = c_7 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_6 = 1 and solve the system of equations for the remaining coefficients: c_2 = (2 c_1)/3 - 2/3 c_3 = (5 c_1)/12 + 1/3 c_4 = c_1 c_5 = (3 c_1)/4 c_6 = 1 c_7 = (2 c_1)/3 - 2/3 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 4 and solve for the remaining coefficients: c_1 = 4 c_2 = 2 c_3 = 2 c_4 = 4 c_5 = 3 c_6 = 1 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2SO_4 + 2 KMnO_4 + 2 K2S ⟶ 4 H_2O + 3 K_2SO_4 + S + 2 MnSO_4

+ + K2S ⟶ + + +

sulfuric acid + potassium permanganate + K2S ⟶ water + potassium sulfate + mixed sulfur + manganese(II) sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + K2S ⟶ H_2O + K_2SO_4 + S + MnSO_4 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_2SO_4 + 2 KMnO_4 + 2 K2S ⟶ 4 H_2O + 3 K_2SO_4 + S + 2 MnSO_4 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 | 4 | -4 KMnO_4 | 2 | -2 K2S | 2 | -2 H_2O | 4 | 4 K_2SO_4 | 3 | 3 S | 1 | 1 MnSO_4 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 4 | -4 | ([H2SO4])^(-4) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) K2S | 2 | -2 | ([K2S])^(-2) H_2O | 4 | 4 | ([H2O])^4 K_2SO_4 | 3 | 3 | ([K2SO4])^3 S | 1 | 1 | [S] MnSO_4 | 2 | 2 | ([MnSO4])^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 = ([H2SO4])^(-4) ([KMnO4])^(-2) ([K2S])^(-2) ([H2O])^4 ([K2SO4])^3 [S] ([MnSO4])^2 = (([H2O])^4 ([K2SO4])^3 [S] ([MnSO4])^2)/(([H2SO4])^4 ([KMnO4])^2 ([K2S])^2)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + K2S ⟶ H_2O + K_2SO_4 + S + MnSO_4 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_2SO_4 + 2 KMnO_4 + 2 K2S ⟶ 4 H_2O + 3 K_2SO_4 + S + 2 MnSO_4 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 | 4 | -4 KMnO_4 | 2 | -2 K2S | 2 | -2 H_2O | 4 | 4 K_2SO_4 | 3 | 3 S | 1 | 1 MnSO_4 | 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 H_2SO_4 | 4 | -4 | -1/4 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) K2S | 2 | -2 | -1/2 (Δ[K2S])/(Δt) H_2O | 4 | 4 | 1/4 (Δ[H2O])/(Δt) K_2SO_4 | 3 | 3 | 1/3 (Δ[K2SO4])/(Δt) S | 1 | 1 | (Δ[S])/(Δt) MnSO_4 | 2 | 2 | 1/2 (Δ[MnSO4])/(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[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -1/2 (Δ[K2S])/(Δt) = 1/4 (Δ[H2O])/(Δt) = 1/3 (Δ[K2SO4])/(Δt) = (Δ[S])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | K2S | water | potassium sulfate | mixed sulfur | manganese(II) sulfate formula | H_2SO_4 | KMnO_4 | K2S | H_2O | K_2SO_4 | S | MnSO_4 Hill formula | H_2O_4S | KMnO_4 | K2S | H_2O | K_2O_4S | S | MnSO_4 name | sulfuric acid | potassium permanganate | | water | potassium sulfate | mixed sulfur | manganese(II) sulfate IUPAC name | sulfuric acid | potassium permanganate | | water | dipotassium sulfate | sulfur | manganese(+2) cation sulfate