H2SO4 + SO2 + K2Cr2O7 = H2O + K2SO4 + S + Cr2(SO4)3

H_2SO_4 sulfuric acid + SO_2 sulfur dioxide + K_2Cr_2O_7 potassium dichromate ⟶ 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 + SO_2 + K_2Cr_2O_7 ⟶ 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 SO_2 + c_3 K_2Cr_2O_7 ⟶ 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_4 O: | 4 c_1 + 2 c_2 + 7 c_3 = c_4 + 4 c_5 + 12 c_7 S: | c_1 + c_2 = c_5 + c_6 + 3 c_7 Cr: | 2 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_3 = (2 c_2)/9 + 1/3 c_4 = 1 c_5 = (2 c_2)/9 + 1/3 c_6 = c_2/9 - 1/3 c_7 = (2 c_2)/9 + 1/3 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_2 = 12 and solve for the remaining coefficients: c_1 = 1 c_2 = 12 c_3 = 3 c_4 = 1 c_5 = 3 c_6 = 1 c_7 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2SO_4 + 12 SO_2 + 3 K_2Cr_2O_7 ⟶ H_2O + 3 K_2SO_4 + S + 3 Cr_2(SO_4)_3

+ + ⟶ + + +

sulfuric acid + sulfur dioxide + potassium dichromate ⟶ water + potassium sulfate + mixed sulfur + chromium sulfate

Construct the equilibrium constant, K, expression for: H_2SO_4 + SO_2 + K_2Cr_2O_7 ⟶ 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: H_2SO_4 + 12 SO_2 + 3 K_2Cr_2O_7 ⟶ H_2O + 3 K_2SO_4 + S + 3 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 | 1 | -1 SO_2 | 12 | -12 K_2Cr_2O_7 | 3 | -3 H_2O | 1 | 1 K_2SO_4 | 3 | 3 S | 1 | 1 Cr_2(SO_4)_3 | 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 | 1 | -1 | ([H2SO4])^(-1) SO_2 | 12 | -12 | ([SO2])^(-12) K_2Cr_2O_7 | 3 | -3 | ([K2Cr2O7])^(-3) H_2O | 1 | 1 | [H2O] K_2SO_4 | 3 | 3 | ([K2SO4])^3 S | 1 | 1 | [S] Cr_2(SO_4)_3 | 3 | 3 | ([Cr2(SO4)3])^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])^(-1) ([SO2])^(-12) ([K2Cr2O7])^(-3) [H2O] ([K2SO4])^3 [S] ([Cr2(SO4)3])^3 = ([H2O] ([K2SO4])^3 [S] ([Cr2(SO4)3])^3)/([H2SO4] ([SO2])^12 ([K2Cr2O7])^3)

Construct the rate of reaction expression for: H_2SO_4 + SO_2 + K_2Cr_2O_7 ⟶ 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: H_2SO_4 + 12 SO_2 + 3 K_2Cr_2O_7 ⟶ H_2O + 3 K_2SO_4 + S + 3 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 | 1 | -1 SO_2 | 12 | -12 K_2Cr_2O_7 | 3 | -3 H_2O | 1 | 1 K_2SO_4 | 3 | 3 S | 1 | 1 Cr_2(SO_4)_3 | 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 | 1 | -1 | -(Δ[H2SO4])/(Δt) SO_2 | 12 | -12 | -1/12 (Δ[SO2])/(Δt) K_2Cr_2O_7 | 3 | -3 | -1/3 (Δ[K2Cr2O7])/(Δt) H_2O | 1 | 1 | (Δ[H2O])/(Δt) K_2SO_4 | 3 | 3 | 1/3 (Δ[K2SO4])/(Δt) S | 1 | 1 | (Δ[S])/(Δt) Cr_2(SO_4)_3 | 3 | 3 | 1/3 (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 = -(Δ[H2SO4])/(Δt) = -1/12 (Δ[SO2])/(Δt) = -1/3 (Δ[K2Cr2O7])/(Δt) = (Δ[H2O])/(Δt) = 1/3 (Δ[K2SO4])/(Δt) = (Δ[S])/(Δt) = 1/3 (Δ[Cr2(SO4)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | sulfur dioxide | potassium dichromate | water | potassium sulfate | mixed sulfur | chromium sulfate formula | H_2SO_4 | SO_2 | K_2Cr_2O_7 | H_2O | K_2SO_4 | S | Cr_2(SO_4)_3 Hill formula | H_2O_4S | O_2S | Cr_2K_2O_7 | H_2O | K_2O_4S | S | Cr_2O_12S_3 name | sulfuric acid | sulfur dioxide | potassium dichromate | water | potassium sulfate | mixed sulfur | chromium sulfate IUPAC name | sulfuric acid | sulfur dioxide | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | water | dipotassium sulfate | sulfur | chromium(+3) cation trisulfate