H2SO4 + KMnO4 + H2S = H2O + K2SO4 + MnSO4 + S8

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + H_2S hydrogen sulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + S_8 rhombic sulfur

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + H_2S ⟶ H_2O + K_2SO_4 + MnSO_4 + S_8 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 H_2S ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 MnSO_4 + c_7 S_8 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_3 = 2 c_4 O: | 4 c_1 + 4 c_2 = c_4 + 4 c_5 + 4 c_6 S: | c_1 + c_3 = c_5 + c_6 + 8 c_7 K: | c_2 = 2 c_5 Mn: | c_2 = c_6 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_7 = 1 and solve the system of equations for the remaining coefficients: c_2 = (8 c_1)/7 - 16/7 c_3 = (5 c_1)/7 + 32/7 c_4 = (12 c_1)/7 + 32/7 c_5 = (4 c_1)/7 - 8/7 c_6 = (8 c_1)/7 - 16/7 c_7 = 1 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 9 and solve for the remaining coefficients: c_1 = 9 c_2 = 8 c_3 = 11 c_4 = 20 c_5 = 4 c_6 = 8 c_7 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 9 H_2SO_4 + 8 KMnO_4 + 11 H_2S ⟶ 20 H_2O + 4 K_2SO_4 + 8 MnSO_4 + S_8

+ + ⟶ + + +

sulfuric acid + potassium permanganate + hydrogen sulfide ⟶ water + potassium sulfate + manganese(II) sulfate + rhombic sulfur

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + H_2S ⟶ H_2O + K_2SO_4 + MnSO_4 + S_8 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: 9 H_2SO_4 + 8 KMnO_4 + 11 H_2S ⟶ 20 H_2O + 4 K_2SO_4 + 8 MnSO_4 + S_8 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 | 9 | -9 KMnO_4 | 8 | -8 H_2S | 11 | -11 H_2O | 20 | 20 K_2SO_4 | 4 | 4 MnSO_4 | 8 | 8 S_8 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 9 | -9 | ([H2SO4])^(-9) KMnO_4 | 8 | -8 | ([KMnO4])^(-8) H_2S | 11 | -11 | ([H2S])^(-11) H_2O | 20 | 20 | ([H2O])^20 K_2SO_4 | 4 | 4 | ([K2SO4])^4 MnSO_4 | 8 | 8 | ([MnSO4])^8 S_8 | 1 | 1 | [S8] 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])^(-9) ([KMnO4])^(-8) ([H2S])^(-11) ([H2O])^20 ([K2SO4])^4 ([MnSO4])^8 [S8] = (([H2O])^20 ([K2SO4])^4 ([MnSO4])^8 [S8])/(([H2SO4])^9 ([KMnO4])^8 ([H2S])^11)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + H_2S ⟶ H_2O + K_2SO_4 + MnSO_4 + S_8 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: 9 H_2SO_4 + 8 KMnO_4 + 11 H_2S ⟶ 20 H_2O + 4 K_2SO_4 + 8 MnSO_4 + S_8 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 | 9 | -9 KMnO_4 | 8 | -8 H_2S | 11 | -11 H_2O | 20 | 20 K_2SO_4 | 4 | 4 MnSO_4 | 8 | 8 S_8 | 1 | 1 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 | 9 | -9 | -1/9 (Δ[H2SO4])/(Δt) KMnO_4 | 8 | -8 | -1/8 (Δ[KMnO4])/(Δt) H_2S | 11 | -11 | -1/11 (Δ[H2S])/(Δt) H_2O | 20 | 20 | 1/20 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) MnSO_4 | 8 | 8 | 1/8 (Δ[MnSO4])/(Δt) S_8 | 1 | 1 | (Δ[S8])/(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 = -1/9 (Δ[H2SO4])/(Δt) = -1/8 (Δ[KMnO4])/(Δt) = -1/11 (Δ[H2S])/(Δt) = 1/20 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/8 (Δ[MnSO4])/(Δt) = (Δ[S8])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | hydrogen sulfide | water | potassium sulfate | manganese(II) sulfate | rhombic sulfur formula | H_2SO_4 | KMnO_4 | H_2S | H_2O | K_2SO_4 | MnSO_4 | S_8 Hill formula | H_2O_4S | KMnO_4 | H_2S | H_2O | K_2O_4S | MnSO_4 | S_8 name | sulfuric acid | potassium permanganate | hydrogen sulfide | water | potassium sulfate | manganese(II) sulfate | rhombic sulfur IUPAC name | sulfuric acid | potassium permanganate | hydrogen sulfide | water | dipotassium sulfate | manganese(+2) cation sulfate | octathiocane

| sulfuric acid | potassium permanganate | hydrogen sulfide | water | potassium sulfate | manganese(II) sulfate | rhombic sulfur molar mass | 98.07 g/mol | 158.03 g/mol | 34.08 g/mol | 18.015 g/mol | 174.25 g/mol | 150.99 g/mol | 256.5 g/mol phase | liquid (at STP) | solid (at STP) | gas (at STP) | liquid (at STP) | | solid (at STP) | solid (at STP) melting point | 10.371 °C | 240 °C | -85 °C | 0 °C | | 710 °C | boiling point | 279.6 °C | | -60 °C | 99.9839 °C | | | density | 1.8305 g/cm^3 | 1 g/cm^3 | 0.001393 g/cm^3 (at 25 °C) | 1 g/cm^3 | | 3.25 g/cm^3 | 2.07 g/cm^3 solubility in water | very soluble | | | | soluble | soluble | surface tension | 0.0735 N/m | | | 0.0728 N/m | | | dynamic viscosity | 0.021 Pa s (at 25 °C) | | 1.239×10^-5 Pa s (at 25 °C) | 8.9×10^-4 Pa s (at 25 °C) | | | odor | odorless | odorless | | odorless | | |