H2O + K2Cr2O7 + C = Cr2O3 + K2CO3 + KHCO3

H_2O water + K_2Cr_2O_7 potassium dichromate + C activated charcoal ⟶ Cr_2O_3 chromium(III) oxide + K_2CO_3 pearl ash + KHCO_3 potassium bicarbonate

Balance the chemical equation algebraically: H_2O + K_2Cr_2O_7 + C ⟶ Cr_2O_3 + K_2CO_3 + KHCO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 K_2Cr_2O_7 + c_3 C ⟶ c_4 Cr_2O_3 + c_5 K_2CO_3 + c_6 KHCO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Cr, K and C: H: | 2 c_1 = c_6 O: | c_1 + 7 c_2 = 3 c_4 + 3 c_5 + 3 c_6 Cr: | 2 c_2 = 2 c_4 K: | 2 c_2 = 2 c_5 + c_6 C: | c_3 = c_5 + 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_2 = 2 c_3 = 3 c_4 = 2 c_5 = 1 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + 2 K_2Cr_2O_7 + 3 C ⟶ 2 Cr_2O_3 + K_2CO_3 + 2 KHCO_3

+ + ⟶ + +

water + potassium dichromate + activated charcoal ⟶ chromium(III) oxide + pearl ash + potassium bicarbonate

Construct the equilibrium constant, K, expression for: H_2O + K_2Cr_2O_7 + C ⟶ Cr_2O_3 + K_2CO_3 + KHCO_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_2O + 2 K_2Cr_2O_7 + 3 C ⟶ 2 Cr_2O_3 + K_2CO_3 + 2 KHCO_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_2O | 1 | -1 K_2Cr_2O_7 | 2 | -2 C | 3 | -3 Cr_2O_3 | 2 | 2 K_2CO_3 | 1 | 1 KHCO_3 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 1 | -1 | ([H2O])^(-1) K_2Cr_2O_7 | 2 | -2 | ([K2Cr2O7])^(-2) C | 3 | -3 | ([C])^(-3) Cr_2O_3 | 2 | 2 | ([Cr2O3])^2 K_2CO_3 | 1 | 1 | [K2CO3] KHCO_3 | 2 | 2 | ([KHCO3])^2 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])^(-1) ([K2Cr2O7])^(-2) ([C])^(-3) ([Cr2O3])^2 [K2CO3] ([KHCO3])^2 = (([Cr2O3])^2 [K2CO3] ([KHCO3])^2)/([H2O] ([K2Cr2O7])^2 ([C])^3)

Construct the rate of reaction expression for: H_2O + K_2Cr_2O_7 + C ⟶ Cr_2O_3 + K_2CO_3 + KHCO_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_2O + 2 K_2Cr_2O_7 + 3 C ⟶ 2 Cr_2O_3 + K_2CO_3 + 2 KHCO_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_2O | 1 | -1 K_2Cr_2O_7 | 2 | -2 C | 3 | -3 Cr_2O_3 | 2 | 2 K_2CO_3 | 1 | 1 KHCO_3 | 2 | 2 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 | 1 | -1 | -(Δ[H2O])/(Δt) K_2Cr_2O_7 | 2 | -2 | -1/2 (Δ[K2Cr2O7])/(Δt) C | 3 | -3 | -1/3 (Δ[C])/(Δt) Cr_2O_3 | 2 | 2 | 1/2 (Δ[Cr2O3])/(Δt) K_2CO_3 | 1 | 1 | (Δ[K2CO3])/(Δt) KHCO_3 | 2 | 2 | 1/2 (Δ[KHCO3])/(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[H2O])/(Δt) = -1/2 (Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[C])/(Δt) = 1/2 (Δ[Cr2O3])/(Δt) = (Δ[K2CO3])/(Δt) = 1/2 (Δ[KHCO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium dichromate | activated charcoal | chromium(III) oxide | pearl ash | potassium bicarbonate formula | H_2O | K_2Cr_2O_7 | C | Cr_2O_3 | K_2CO_3 | KHCO_3 Hill formula | H_2O | Cr_2K_2O_7 | C | Cr_2O_3 | CK_2O_3 | CHKO_3 name | water | potassium dichromate | activated charcoal | chromium(III) oxide | pearl ash | potassium bicarbonate IUPAC name | water | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | carbon | | dipotassium carbonate | potassium hydrogen carbonate