H2SO4 + K2Cr2O7 + (NH4)2S = H2O + K2SO4 + NH3 + Cr2(SO4)3

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + (NH_4)_2S diammonium sulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + NH_3 ammonia + Cr_2(SO_4)_3 chromium sulfate

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + (NH_4)_2S ⟶ H_2O + K_2SO_4 + NH_3 + Cr_2(SO_4)_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 (NH_4)_2S ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 NH_3 + 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, K and N: H: | 2 c_1 + 8 c_3 = 2 c_4 + 3 c_6 O: | 4 c_1 + 7 c_2 = c_4 + 4 c_5 + 12 c_7 S: | c_1 + c_3 = c_5 + 3 c_7 Cr: | 2 c_2 = 2 c_7 K: | 2 c_2 = 2 c_5 N: | 2 c_3 = 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 13/3 c_2 = 4/3 c_3 = 1 c_4 = 16/3 c_5 = 4/3 c_6 = 2 c_7 = 4/3 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_1 = 13 c_2 = 4 c_3 = 3 c_4 = 16 c_5 = 4 c_6 = 6 c_7 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 13 H_2SO_4 + 4 K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 16 H_2O + 4 K_2SO_4 + 6 NH_3 + 4 Cr_2(SO_4)_3

+ + ⟶ + + +

sulfuric acid + potassium dichromate + diammonium sulfide ⟶ water + potassium sulfate + ammonia + chromium sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + (NH_4)_2S ⟶ H_2O + K_2SO_4 + NH_3 + 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: 13 H_2SO_4 + 4 K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 16 H_2O + 4 K_2SO_4 + 6 NH_3 + 4 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 | 13 | -13 K_2Cr_2O_7 | 4 | -4 (NH_4)_2S | 3 | -3 H_2O | 16 | 16 K_2SO_4 | 4 | 4 NH_3 | 6 | 6 Cr_2(SO_4)_3 | 4 | 4 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 13 | -13 | ([H2SO4])^(-13) K_2Cr_2O_7 | 4 | -4 | ([K2Cr2O7])^(-4) (NH_4)_2S | 3 | -3 | ([(NH4)2S])^(-3) H_2O | 16 | 16 | ([H2O])^16 K_2SO_4 | 4 | 4 | ([K2SO4])^4 NH_3 | 6 | 6 | ([NH3])^6 Cr_2(SO_4)_3 | 4 | 4 | ([Cr2(SO4)3])^4 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])^(-13) ([K2Cr2O7])^(-4) ([(NH4)2S])^(-3) ([H2O])^16 ([K2SO4])^4 ([NH3])^6 ([Cr2(SO4)3])^4 = (([H2O])^16 ([K2SO4])^4 ([NH3])^6 ([Cr2(SO4)3])^4)/(([H2SO4])^13 ([K2Cr2O7])^4 ([(NH4)2S])^3)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + (NH_4)_2S ⟶ H_2O + K_2SO_4 + NH_3 + 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: 13 H_2SO_4 + 4 K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 16 H_2O + 4 K_2SO_4 + 6 NH_3 + 4 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 | 13 | -13 K_2Cr_2O_7 | 4 | -4 (NH_4)_2S | 3 | -3 H_2O | 16 | 16 K_2SO_4 | 4 | 4 NH_3 | 6 | 6 Cr_2(SO_4)_3 | 4 | 4 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 | 13 | -13 | -1/13 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 4 | -4 | -1/4 (Δ[K2Cr2O7])/(Δt) (NH_4)_2S | 3 | -3 | -1/3 (Δ[(NH4)2S])/(Δt) H_2O | 16 | 16 | 1/16 (Δ[H2O])/(Δt) K_2SO_4 | 4 | 4 | 1/4 (Δ[K2SO4])/(Δt) NH_3 | 6 | 6 | 1/6 (Δ[NH3])/(Δt) Cr_2(SO_4)_3 | 4 | 4 | 1/4 (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 = -1/13 (Δ[H2SO4])/(Δt) = -1/4 (Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[(NH4)2S])/(Δt) = 1/16 (Δ[H2O])/(Δt) = 1/4 (Δ[K2SO4])/(Δt) = 1/6 (Δ[NH3])/(Δt) = 1/4 (Δ[Cr2(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | diammonium sulfide | water | potassium sulfate | ammonia | chromium sulfate formula | H_2SO_4 | K_2Cr_2O_7 | (NH_4)_2S | H_2O | K_2SO_4 | NH_3 | Cr_2(SO_4)_3 Hill formula | H_2O_4S | Cr_2K_2O_7 | H_8N_2S | H_2O | K_2O_4S | H_3N | Cr_2O_12S_3 name | sulfuric acid | potassium dichromate | diammonium sulfide | water | potassium sulfate | ammonia | chromium sulfate IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | diammonium sulfide | water | dipotassium sulfate | ammonia | chromium(+3) cation trisulfate