H2O + K2SO4 + Cr2(SO4)3 + Fe2(SO4)3 = H2SO4 + K2Cr2O7 + FeSO4

H_2O water + K_2SO_4 potassium sulfate + Cr_2(SO_4)_3 chromium sulfate + Fe_2(SO_4)_3·xH_2O iron(III) sulfate hydrate ⟶ H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + FeSO_4 duretter

Balance the chemical equation algebraically: H_2O + K_2SO_4 + Cr_2(SO_4)_3 + Fe_2(SO_4)_3·xH_2O ⟶ H_2SO_4 + K_2Cr_2O_7 + FeSO_4 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 K_2SO_4 + c_3 Cr_2(SO_4)_3 + c_4 Fe_2(SO_4)_3·xH_2O ⟶ c_5 H_2SO_4 + c_6 K_2Cr_2O_7 + c_7 FeSO_4 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, S, Cr and Fe: H: | 2 c_1 = 2 c_5 O: | c_1 + 4 c_2 + 12 c_3 + 12 c_4 = 4 c_5 + 7 c_6 + 4 c_7 K: | 2 c_2 = 2 c_6 S: | c_2 + 3 c_3 + 3 c_4 = c_5 + c_7 Cr: | 2 c_3 = 2 c_6 Fe: | 2 c_4 = 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_1 = 7 c_2 = 1 c_3 = 1 c_4 = 3 c_5 = 7 c_6 = 1 c_7 = 6 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 7 H_2O + K_2SO_4 + Cr_2(SO_4)_3 + 3 Fe_2(SO_4)_3·xH_2O ⟶ 7 H_2SO_4 + K_2Cr_2O_7 + 6 FeSO_4

+ + + ⟶ + +

water + potassium sulfate + chromium sulfate + iron(III) sulfate hydrate ⟶ sulfuric acid + potassium dichromate + duretter

Construct the equilibrium constant, K, expression for: H_2O + K_2SO_4 + Cr_2(SO_4)_3 + Fe_2(SO_4)_3·xH_2O ⟶ H_2SO_4 + K_2Cr_2O_7 + FeSO_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_2O + K_2SO_4 + Cr_2(SO_4)_3 + 3 Fe_2(SO_4)_3·xH_2O ⟶ 7 H_2SO_4 + K_2Cr_2O_7 + 6 FeSO_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_2O | 7 | -7 K_2SO_4 | 1 | -1 Cr_2(SO_4)_3 | 1 | -1 Fe_2(SO_4)_3·xH_2O | 3 | -3 H_2SO_4 | 7 | 7 K_2Cr_2O_7 | 1 | 1 FeSO_4 | 6 | 6 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 7 | -7 | ([H2O])^(-7) K_2SO_4 | 1 | -1 | ([K2SO4])^(-1) Cr_2(SO_4)_3 | 1 | -1 | ([Cr2(SO4)3])^(-1) Fe_2(SO_4)_3·xH_2O | 3 | -3 | ([Fe2(SO4)3·xH2O])^(-3) H_2SO_4 | 7 | 7 | ([H2SO4])^7 K_2Cr_2O_7 | 1 | 1 | [K2Cr2O7] FeSO_4 | 6 | 6 | ([FeSO4])^6 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 = ([H2O])^(-7) ([K2SO4])^(-1) ([Cr2(SO4)3])^(-1) ([Fe2(SO4)3·xH2O])^(-3) ([H2SO4])^7 [K2Cr2O7] ([FeSO4])^6 = (([H2SO4])^7 [K2Cr2O7] ([FeSO4])^6)/(([H2O])^7 [K2SO4] [Cr2(SO4)3] ([Fe2(SO4)3·xH2O])^3)

Construct the rate of reaction expression for: H_2O + K_2SO_4 + Cr_2(SO_4)_3 + Fe_2(SO_4)_3·xH_2O ⟶ H_2SO_4 + K_2Cr_2O_7 + FeSO_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_2O + K_2SO_4 + Cr_2(SO_4)_3 + 3 Fe_2(SO_4)_3·xH_2O ⟶ 7 H_2SO_4 + K_2Cr_2O_7 + 6 FeSO_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_2O | 7 | -7 K_2SO_4 | 1 | -1 Cr_2(SO_4)_3 | 1 | -1 Fe_2(SO_4)_3·xH_2O | 3 | -3 H_2SO_4 | 7 | 7 K_2Cr_2O_7 | 1 | 1 FeSO_4 | 6 | 6 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_2O | 7 | -7 | -1/7 (Δ[H2O])/(Δt) K_2SO_4 | 1 | -1 | -(Δ[K2SO4])/(Δt) Cr_2(SO_4)_3 | 1 | -1 | -(Δ[Cr2(SO4)3])/(Δt) Fe_2(SO_4)_3·xH_2O | 3 | -3 | -1/3 (Δ[Fe2(SO4)3·xH2O])/(Δt) H_2SO_4 | 7 | 7 | 1/7 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 1 | 1 | (Δ[K2Cr2O7])/(Δt) FeSO_4 | 6 | 6 | 1/6 (Δ[FeSO4])/(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[H2O])/(Δt) = -(Δ[K2SO4])/(Δt) = -(Δ[Cr2(SO4)3])/(Δt) = -1/3 (Δ[Fe2(SO4)3·xH2O])/(Δt) = 1/7 (Δ[H2SO4])/(Δt) = (Δ[K2Cr2O7])/(Δt) = 1/6 (Δ[FeSO4])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium sulfate | chromium sulfate | iron(III) sulfate hydrate | sulfuric acid | potassium dichromate | duretter formula | H_2O | K_2SO_4 | Cr_2(SO_4)_3 | Fe_2(SO_4)_3·xH_2O | H_2SO_4 | K_2Cr_2O_7 | FeSO_4 Hill formula | H_2O | K_2O_4S | Cr_2O_12S_3 | Fe_2O_12S_3 | H_2O_4S | Cr_2K_2O_7 | FeO_4S name | water | potassium sulfate | chromium sulfate | iron(III) sulfate hydrate | sulfuric acid | potassium dichromate | duretter IUPAC name | water | dipotassium sulfate | chromium(+3) cation trisulfate | diferric trisulfate | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | iron(+2) cation sulfate