H2SO4 + K2Cr2O7 + FeCO3 = H2O + CO2 + K2SO4 + Cr2(SO4)3 + Fe2(SO4)3

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + FeCO_3 iron(II) carbonate ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + Cr_2(SO_4)_3 chromium sulfate + Fe_2(SO_4)_3·xH_2O iron(III) sulfate hydrate

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + FeCO_3 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + Fe_2(SO_4)_3·xH_2O Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 FeCO_3 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 Cr_2(SO_4)_3 + c_8 Fe_2(SO_4)_3·xH_2O Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Cr, K, C and Fe: H: | 2 c_1 = 2 c_4 O: | 4 c_1 + 7 c_2 + 3 c_3 = c_4 + 2 c_5 + 4 c_6 + 12 c_7 + 12 c_8 S: | c_1 = c_6 + 3 c_7 + 3 c_8 Cr: | 2 c_2 = 2 c_7 K: | 2 c_2 = 2 c_6 C: | c_3 = c_5 Fe: | 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_1 = 13 c_2 = 1 c_3 = 6 c_4 = 13 c_5 = 6 c_6 = 1 c_7 = 1 c_8 = 3 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 13 H_2SO_4 + K_2Cr_2O_7 + 6 FeCO_3 ⟶ 13 H_2O + 6 CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + 3 Fe_2(SO_4)_3·xH_2O

+ + ⟶ + + + +

sulfuric acid + potassium dichromate + iron(II) carbonate ⟶ water + carbon dioxide + potassium sulfate + chromium sulfate + iron(III) sulfate hydrate

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + FeCO_3 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + Fe_2(SO_4)_3·xH_2O 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 + K_2Cr_2O_7 + 6 FeCO_3 ⟶ 13 H_2O + 6 CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + 3 Fe_2(SO_4)_3·xH_2O 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 | 1 | -1 FeCO_3 | 6 | -6 H_2O | 13 | 13 CO_2 | 6 | 6 K_2SO_4 | 1 | 1 Cr_2(SO_4)_3 | 1 | 1 Fe_2(SO_4)_3·xH_2O | 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 | 13 | -13 | ([H2SO4])^(-13) K_2Cr_2O_7 | 1 | -1 | ([K2Cr2O7])^(-1) FeCO_3 | 6 | -6 | ([FeCO3])^(-6) H_2O | 13 | 13 | ([H2O])^13 CO_2 | 6 | 6 | ([CO2])^6 K_2SO_4 | 1 | 1 | [K2SO4] Cr_2(SO_4)_3 | 1 | 1 | [Cr2(SO4)3] Fe_2(SO_4)_3·xH_2O | 3 | 3 | ([Fe2(SO4)3·xH2O])^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])^(-13) ([K2Cr2O7])^(-1) ([FeCO3])^(-6) ([H2O])^13 ([CO2])^6 [K2SO4] [Cr2(SO4)3] ([Fe2(SO4)3·xH2O])^3 = (([H2O])^13 ([CO2])^6 [K2SO4] [Cr2(SO4)3] ([Fe2(SO4)3·xH2O])^3)/(([H2SO4])^13 [K2Cr2O7] ([FeCO3])^6)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + FeCO_3 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + Fe_2(SO_4)_3·xH_2O 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 + K_2Cr_2O_7 + 6 FeCO_3 ⟶ 13 H_2O + 6 CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + 3 Fe_2(SO_4)_3·xH_2O 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 | 1 | -1 FeCO_3 | 6 | -6 H_2O | 13 | 13 CO_2 | 6 | 6 K_2SO_4 | 1 | 1 Cr_2(SO_4)_3 | 1 | 1 Fe_2(SO_4)_3·xH_2O | 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 | 13 | -13 | -1/13 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 1 | -1 | -(Δ[K2Cr2O7])/(Δt) FeCO_3 | 6 | -6 | -1/6 (Δ[FeCO3])/(Δt) H_2O | 13 | 13 | 1/13 (Δ[H2O])/(Δt) CO_2 | 6 | 6 | 1/6 (Δ[CO2])/(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) (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) = -(Δ[K2Cr2O7])/(Δt) = -1/6 (Δ[FeCO3])/(Δt) = 1/13 (Δ[H2O])/(Δt) = 1/6 (Δ[CO2])/(Δt) = (Δ[K2SO4])/(Δt) = (Δ[Cr2(SO4)3])/(Δt) = 1/3 (Δ[Fe2(SO4)3·xH2O])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | iron(II) carbonate | water | carbon dioxide | potassium sulfate | chromium sulfate | iron(III) sulfate hydrate formula | H_2SO_4 | K_2Cr_2O_7 | FeCO_3 | H_2O | CO_2 | K_2SO_4 | Cr_2(SO_4)_3 | Fe_2(SO_4)_3·xH_2O Hill formula | H_2O_4S | Cr_2K_2O_7 | CFeO_3 | H_2O | CO_2 | K_2O_4S | Cr_2O_12S_3 | Fe_2O_12S_3 name | sulfuric acid | potassium dichromate | iron(II) carbonate | water | carbon dioxide | potassium sulfate | chromium sulfate | iron(III) sulfate hydrate IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | ferrous carbonate | water | carbon dioxide | dipotassium sulfate | chromium(+3) cation trisulfate | diferric trisulfate