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

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + (NH_4)_2S diammonium sulfide ⟶ H_2O water + K_2SO_4 potassium sulfate + S mixed sulfur + Cr_2(SO_4)_3 chromium sulfate + (NH_4)_2SO_4 ammonium sulfate

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + (NH_4)_2S ⟶ H_2O + K_2SO_4 + S + Cr_2(SO_4)_3 + (NH_4)_2SO_4 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 S + c_7 Cr_2(SO_4)_3 + c_8 (NH_4)_2SO_4 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 + 8 c_8 O: | 4 c_1 + 7 c_2 = c_4 + 4 c_5 + 12 c_7 + 4 c_8 S: | c_1 + c_3 = c_5 + c_6 + 3 c_7 + c_8 Cr: | 2 c_2 = 2 c_7 K: | 2 c_2 = 2 c_5 N: | 2 c_3 = 2 c_8 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 = (3 c_1)/4 - 9/4 c_4 = c_1 c_5 = 1 c_6 = c_1 - 4 c_7 = 1 c_8 = (3 c_1)/4 - 9/4 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 7 and solve for the remaining coefficients: c_1 = 7 c_2 = 1 c_3 = 3 c_4 = 7 c_5 = 1 c_6 = 3 c_7 = 1 c_8 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 7 H_2SO_4 + K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 7 H_2O + K_2SO_4 + 3 S + Cr_2(SO_4)_3 + 3 (NH_4)_2SO_4

sulfuric acid + potassium dichromate + diammonium sulfide ⟶ water + potassium sulfate + mixed sulfur + chromium sulfate + ammonium sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + (NH_4)_2S ⟶ H_2O + K_2SO_4 + S + Cr_2(SO_4)_3 + (NH_4)_2SO_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: 7 H_2SO_4 + K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 7 H_2O + K_2SO_4 + 3 S + Cr_2(SO_4)_3 + 3 (NH_4)_2SO_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 | 7 | -7 K_2Cr_2O_7 | 1 | -1 (NH_4)_2S | 3 | -3 H_2O | 7 | 7 K_2SO_4 | 1 | 1 S | 3 | 3 Cr_2(SO_4)_3 | 1 | 1 (NH_4)_2SO_4 | 3 | 3 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 7 | -7 | ([H2SO4])^(-7) K_2Cr_2O_7 | 1 | -1 | ([K2Cr2O7])^(-1) (NH_4)_2S | 3 | -3 | ([(NH4)2S])^(-3) H_2O | 7 | 7 | ([H2O])^7 K_2SO_4 | 1 | 1 | [K2SO4] S | 3 | 3 | ([S])^3 Cr_2(SO_4)_3 | 1 | 1 | [Cr2(SO4)3] (NH_4)_2SO_4 | 3 | 3 | ([(NH4)2SO4])^3 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])^(-7) ([K2Cr2O7])^(-1) ([(NH4)2S])^(-3) ([H2O])^7 [K2SO4] ([S])^3 [Cr2(SO4)3] ([(NH4)2SO4])^3 = (([H2O])^7 [K2SO4] ([S])^3 [Cr2(SO4)3] ([(NH4)2SO4])^3)/(([H2SO4])^7 [K2Cr2O7] ([(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 + S + Cr_2(SO_4)_3 + (NH_4)_2SO_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: 7 H_2SO_4 + K_2Cr_2O_7 + 3 (NH_4)_2S ⟶ 7 H_2O + K_2SO_4 + 3 S + Cr_2(SO_4)_3 + 3 (NH_4)_2SO_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 | 7 | -7 K_2Cr_2O_7 | 1 | -1 (NH_4)_2S | 3 | -3 H_2O | 7 | 7 K_2SO_4 | 1 | 1 S | 3 | 3 Cr_2(SO_4)_3 | 1 | 1 (NH_4)_2SO_4 | 3 | 3 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 | 7 | -7 | -1/7 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 1 | -1 | -(Δ[K2Cr2O7])/(Δt) (NH_4)_2S | 3 | -3 | -1/3 (Δ[(NH4)2S])/(Δt) H_2O | 7 | 7 | 1/7 (Δ[H2O])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) S | 3 | 3 | 1/3 (Δ[S])/(Δt) Cr_2(SO_4)_3 | 1 | 1 | (Δ[Cr2(SO4)3])/(Δt) (NH_4)_2SO_4 | 3 | 3 | 1/3 (Δ[(NH4)2SO4])/(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 = -1/7 (Δ[H2SO4])/(Δt) = -(Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[(NH4)2S])/(Δt) = 1/7 (Δ[H2O])/(Δt) = (Δ[K2SO4])/(Δt) = 1/3 (Δ[S])/(Δt) = (Δ[Cr2(SO4)3])/(Δt) = 1/3 (Δ[(NH4)2SO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | diammonium sulfide | water | potassium sulfate | mixed sulfur | chromium sulfate | ammonium sulfate formula | H_2SO_4 | K_2Cr_2O_7 | (NH_4)_2S | H_2O | K_2SO_4 | S | Cr_2(SO_4)_3 | (NH_4)_2SO_4 Hill formula | H_2O_4S | Cr_2K_2O_7 | H_8N_2S | H_2O | K_2O_4S | S | Cr_2O_12S_3 | H_8N_2O_4S name | sulfuric acid | potassium dichromate | diammonium sulfide | water | potassium sulfate | mixed sulfur | chromium sulfate | ammonium sulfate IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | diammonium sulfide | water | dipotassium sulfate | sulfur | chromium(+3) cation trisulfate |