H2SO4 + H2S + KMn4 = H2O + K2SO4 + S + MnSO4

H_2SO_4 sulfuric acid + H_2S hydrogen sulfide + KMn4 ⟶ H_2O water + K_2SO_4 potassium sulfate + S mixed sulfur + MnSO_4 manganese(II) sulfate

Balance the chemical equation algebraically: H_2SO_4 + H_2S + KMn4 ⟶ 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 H_2S + c_3 KMn4 ⟶ 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_2 = 2 c_4 O: | 4 c_1 = c_4 + 4 c_5 + 4 c_7 S: | c_1 + c_2 = c_5 + c_6 + c_7 K: | c_3 = 2 c_5 Mn: | 4 c_3 = 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_2 = 1 and solve the system of equations for the remaining coefficients: c_2 = 1 c_3 = c_1/6 - 1/18 c_4 = c_1 + 1 c_5 = c_1/12 - 1/36 c_6 = c_1/4 + 5/4 c_7 = (2 c_1)/3 - 2/9 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_2 = 3 c_3 = c_1/6 - 1/6 c_4 = c_1 + 3 c_5 = c_1/12 - 1/12 c_6 = c_1/4 + 15/4 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 = 49 and solve for the remaining coefficients: c_1 = 49 c_2 = 3 c_3 = 8 c_4 = 52 c_5 = 4 c_6 = 16 c_7 = 32 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 49 H_2SO_4 + 3 H_2S + 8 KMn4 ⟶ 52 H_2O + 4 K_2SO_4 + 16 S + 32 MnSO_4

+ + KMn4 ⟶ + + +

sulfuric acid + hydrogen sulfide + KMn4 ⟶ water + potassium sulfate + mixed sulfur + manganese(II) sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + H_2S + KMn4 ⟶ 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: 49 H_2SO_4 + 3 H_2S + 8 KMn4 ⟶ 52 H_2O + 4 K_2SO_4 + 16 S + 32 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 | 49 | -49 H_2S | 3 | -3 KMn4 | 8 | -8 H_2O | 52 | 52 K_2SO_4 | 4 | 4 S | 16 | 16 MnSO_4 | 32 | 32 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 49 | -49 | ([H2SO4])^(-49) H_2S | 3 | -3 | ([H2S])^(-3) KMn4 | 8 | -8 | ([KMn4])^(-8) H_2O | 52 | 52 | ([H2O])^52 K_2SO_4 | 4 | 4 | ([K2SO4])^4 S | 16 | 16 | ([S])^16 MnSO_4 | 32 | 32 | ([MnSO4])^32 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])^(-49) ([H2S])^(-3) ([KMn4])^(-8) ([H2O])^52 ([K2SO4])^4 ([S])^16 ([MnSO4])^32 = (([H2O])^52 ([K2SO4])^4 ([S])^16 ([MnSO4])^32)/(([H2SO4])^49 ([H2S])^3 ([KMn4])^8)

Construct the rate of reaction expression for: H_2SO_4 + H_2S + KMn4 ⟶ 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: 49 H_2SO_4 + 3 H_2S + 8 KMn4 ⟶ 52 H_2O + 4 K_2SO_4 + 16 S + 32 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 | 49 | -49 H_2S | 3 | -3 KMn4 | 8 | -8 H_2O | 52 | 52 K_2SO_4 | 4 | 4 S | 16 | 16 MnSO_4 | 32 | 32 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 | 49 | -49 | -1/49 (Δ[H2SO4])/(Δt) H_2S | 3 | -3 | -1/3 (Δ[H2S])/(Δt) KMn4 | 8 | -8 | -1/8 (Δ[KMn4])/(Δt) H_2O | 52 | 52 | 1/52 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) S | 16 | 16 | 1/16 (Δ[S])/(Δt) MnSO_4 | 32 | 32 | 1/32 (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 = -1/49 (Δ[H2SO4])/(Δt) = -1/3 (Δ[H2S])/(Δt) = -1/8 (Δ[KMn4])/(Δt) = 1/52 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/16 (Δ[S])/(Δt) = 1/32 (Δ[MnSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | hydrogen sulfide | KMn4 | water | potassium sulfate | mixed sulfur | manganese(II) sulfate formula | H_2SO_4 | H_2S | KMn4 | H_2O | K_2SO_4 | S | MnSO_4 Hill formula | H_2O_4S | H_2S | KMn4 | H_2O | K_2O_4S | S | MnSO_4 name | sulfuric acid | hydrogen sulfide | | water | potassium sulfate | mixed sulfur | manganese(II) sulfate IUPAC name | sulfuric acid | hydrogen sulfide | | water | dipotassium sulfate | sulfur | manganese(+2) cation sulfate

| sulfuric acid | hydrogen sulfide | KMn4 | water | potassium sulfate | mixed sulfur | manganese(II) sulfate molar mass | 98.07 g/mol | 34.08 g/mol | 258.8505 g/mol | 18.015 g/mol | 174.25 g/mol | 32.06 g/mol | 150.99 g/mol phase | liquid (at STP) | gas (at STP) | | liquid (at STP) | | solid (at STP) | solid (at STP) melting point | 10.371 °C | -85 °C | | 0 °C | | 112.8 °C | 710 °C boiling point | 279.6 °C | -60 °C | | 99.9839 °C | | 444.7 °C | density | 1.8305 g/cm^3 | 0.001393 g/cm^3 (at 25 °C) | | 1 g/cm^3 | | 2.07 g/cm^3 | 3.25 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 | | |