HNO3 + NO2 + K2Cr2O7 = H2O + KNO3 + Cr(NO3)3

HNO_3 nitric acid + NO_2 nitrogen dioxide + K_2Cr_2O_7 potassium dichromate ⟶ H_2O water + KNO_3 potassium nitrate + CrN_3O_9 chromium nitrate

Balance the chemical equation algebraically: HNO_3 + NO_2 + K_2Cr_2O_7 ⟶ H_2O + KNO_3 + CrN_3O_9 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 NO_2 + c_3 K_2Cr_2O_7 ⟶ c_4 H_2O + c_5 KNO_3 + c_6 CrN_3O_9 Set the number of atoms in the reactants equal to the number of atoms in the products for H, N, O, Cr and K: H: | c_1 = 2 c_4 N: | c_1 + c_2 = c_5 + 3 c_6 O: | 3 c_1 + 2 c_2 + 7 c_3 = c_4 + 3 c_5 + 9 c_6 Cr: | 2 c_3 = c_6 K: | 2 c_3 = 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_3 = 1 and solve the system of equations for the remaining coefficients: c_1 = 2 c_2 = 6 c_3 = 1 c_4 = 1 c_5 = 2 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 HNO_3 + 6 NO_2 + K_2Cr_2O_7 ⟶ H_2O + 2 KNO_3 + 2 CrN_3O_9

+ + ⟶ + +

nitric acid + nitrogen dioxide + potassium dichromate ⟶ water + potassium nitrate + chromium nitrate

Construct the equilibrium constant, K, expression for: HNO_3 + NO_2 + K_2Cr_2O_7 ⟶ H_2O + KNO_3 + CrN_3O_9 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: 2 HNO_3 + 6 NO_2 + K_2Cr_2O_7 ⟶ H_2O + 2 KNO_3 + 2 CrN_3O_9 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 HNO_3 | 2 | -2 NO_2 | 6 | -6 K_2Cr_2O_7 | 1 | -1 H_2O | 1 | 1 KNO_3 | 2 | 2 CrN_3O_9 | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression HNO_3 | 2 | -2 | ([HNO3])^(-2) NO_2 | 6 | -6 | ([NO2])^(-6) K_2Cr_2O_7 | 1 | -1 | ([K2Cr2O7])^(-1) H_2O | 1 | 1 | [H2O] KNO_3 | 2 | 2 | ([KNO3])^2 CrN_3O_9 | 2 | 2 | ([CrN3O9])^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 = ([HNO3])^(-2) ([NO2])^(-6) ([K2Cr2O7])^(-1) [H2O] ([KNO3])^2 ([CrN3O9])^2 = ([H2O] ([KNO3])^2 ([CrN3O9])^2)/(([HNO3])^2 ([NO2])^6 [K2Cr2O7])

Construct the rate of reaction expression for: HNO_3 + NO_2 + K_2Cr_2O_7 ⟶ H_2O + KNO_3 + CrN_3O_9 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: 2 HNO_3 + 6 NO_2 + K_2Cr_2O_7 ⟶ H_2O + 2 KNO_3 + 2 CrN_3O_9 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 HNO_3 | 2 | -2 NO_2 | 6 | -6 K_2Cr_2O_7 | 1 | -1 H_2O | 1 | 1 KNO_3 | 2 | 2 CrN_3O_9 | 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 HNO_3 | 2 | -2 | -1/2 (Δ[HNO3])/(Δt) NO_2 | 6 | -6 | -1/6 (Δ[NO2])/(Δt) K_2Cr_2O_7 | 1 | -1 | -(Δ[K2Cr2O7])/(Δt) H_2O | 1 | 1 | (Δ[H2O])/(Δt) KNO_3 | 2 | 2 | 1/2 (Δ[KNO3])/(Δt) CrN_3O_9 | 2 | 2 | 1/2 (Δ[CrN3O9])/(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 = -1/2 (Δ[HNO3])/(Δt) = -1/6 (Δ[NO2])/(Δt) = -(Δ[K2Cr2O7])/(Δt) = (Δ[H2O])/(Δt) = 1/2 (Δ[KNO3])/(Δt) = 1/2 (Δ[CrN3O9])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | nitrogen dioxide | potassium dichromate | water | potassium nitrate | chromium nitrate formula | HNO_3 | NO_2 | K_2Cr_2O_7 | H_2O | KNO_3 | CrN_3O_9 Hill formula | HNO_3 | NO_2 | Cr_2K_2O_7 | H_2O | KNO_3 | CrN_3O_9 name | nitric acid | nitrogen dioxide | potassium dichromate | water | potassium nitrate | chromium nitrate IUPAC name | nitric acid | Nitrogen dioxide | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | water | potassium nitrate | chromium(+3) cation trinitrate