H2SO4 + H2S + K2CrO4 = H2O + K2SO4 + S + Cr2(SO4)3

H_2SO_4 sulfuric acid + H_2S hydrogen sulfide + K_2CrO_4 potassium chromate ⟶ H_2O water + K_2SO_4 potassium sulfate + S mixed sulfur + Cr_2(SO_4)_3 chromium sulfate

Balance the chemical equation algebraically: H_2SO_4 + H_2S + K_2CrO_4 ⟶ H_2O + K_2SO_4 + S + Cr_2(SO_4)_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 H_2S + c_3 K_2CrO_4 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 S + c_7 Cr_2(SO_4)_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Cr and K: H: | 2 c_1 + 2 c_2 = 2 c_4 O: | 4 c_1 + 4 c_3 = c_4 + 4 c_5 + 12 c_7 S: | c_1 + c_2 = c_5 + c_6 + 3 c_7 Cr: | c_3 = 2 c_7 K: | 2 c_3 = 2 c_5 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 = (3 c_1)/17 + 12/17 c_3 = (8 c_1)/17 - 2/17 c_4 = (20 c_1)/17 + 12/17 c_5 = (8 c_1)/17 - 2/17 c_6 = 1 c_7 = (4 c_1)/17 - 1/17 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_2 = (3 c_1)/17 + 24/17 c_3 = (8 c_1)/17 - 4/17 c_4 = (20 c_1)/17 + 24/17 c_5 = (8 c_1)/17 - 4/17 c_6 = 2 c_7 = (4 c_1)/17 - 2/17 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 = 3 c_3 = 4 c_4 = 12 c_5 = 4 c_6 = 2 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 9 H_2SO_4 + 3 H_2S + 4 K_2CrO_4 ⟶ 12 H_2O + 4 K_2SO_4 + 2 S + 2 Cr_2(SO_4)_3

+ + ⟶ + + +

sulfuric acid + hydrogen sulfide + potassium chromate ⟶ water + potassium sulfate + mixed sulfur + chromium sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + H_2S + K_2CrO_4 ⟶ H_2O + K_2SO_4 + S + Cr_2(SO_4)_3 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 + 3 H_2S + 4 K_2CrO_4 ⟶ 12 H_2O + 4 K_2SO_4 + 2 S + 2 Cr_2(SO_4)_3 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 H_2S | 3 | -3 K_2CrO_4 | 4 | -4 H_2O | 12 | 12 K_2SO_4 | 4 | 4 S | 2 | 2 Cr_2(SO_4)_3 | 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 | 9 | -9 | ([H2SO4])^(-9) H_2S | 3 | -3 | ([H2S])^(-3) K_2CrO_4 | 4 | -4 | ([K2CrO4])^(-4) H_2O | 12 | 12 | ([H2O])^12 K_2SO_4 | 4 | 4 | ([K2SO4])^4 S | 2 | 2 | ([S])^2 Cr_2(SO_4)_3 | 2 | 2 | ([Cr2(SO4)3])^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])^(-9) ([H2S])^(-3) ([K2CrO4])^(-4) ([H2O])^12 ([K2SO4])^4 ([S])^2 ([Cr2(SO4)3])^2 = (([H2O])^12 ([K2SO4])^4 ([S])^2 ([Cr2(SO4)3])^2)/(([H2SO4])^9 ([H2S])^3 ([K2CrO4])^4)

Construct the rate of reaction expression for: H_2SO_4 + H_2S + K_2CrO_4 ⟶ H_2O + K_2SO_4 + S + Cr_2(SO_4)_3 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 + 3 H_2S + 4 K_2CrO_4 ⟶ 12 H_2O + 4 K_2SO_4 + 2 S + 2 Cr_2(SO_4)_3 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 H_2S | 3 | -3 K_2CrO_4 | 4 | -4 H_2O | 12 | 12 K_2SO_4 | 4 | 4 S | 2 | 2 Cr_2(SO_4)_3 | 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 | 9 | -9 | -1/9 (Δ[H2SO4])/(Δt) H_2S | 3 | -3 | -1/3 (Δ[H2S])/(Δt) K_2CrO_4 | 4 | -4 | -1/4 (Δ[K2CrO4])/(Δt) H_2O | 12 | 12 | 1/12 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) S | 2 | 2 | 1/2 (Δ[S])/(Δt) Cr_2(SO_4)_3 | 2 | 2 | 1/2 (Δ[Cr2(SO4)3])/(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) = -1/3 (Δ[H2S])/(Δt) = -1/4 (Δ[K2CrO4])/(Δt) = 1/12 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/2 (Δ[S])/(Δt) = 1/2 (Δ[Cr2(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | hydrogen sulfide | potassium chromate | water | potassium sulfate | mixed sulfur | chromium sulfate formula | H_2SO_4 | H_2S | K_2CrO_4 | H_2O | K_2SO_4 | S | Cr_2(SO_4)_3 Hill formula | H_2O_4S | H_2S | CrK_2O_4 | H_2O | K_2O_4S | S | Cr_2O_12S_3 name | sulfuric acid | hydrogen sulfide | potassium chromate | water | potassium sulfate | mixed sulfur | chromium sulfate IUPAC name | sulfuric acid | hydrogen sulfide | dipotassium dioxido-dioxochromium | water | dipotassium sulfate | sulfur | chromium(+3) cation trisulfate