HNO3 + K2Cr2O7 + KNO2 = H2O + KNO3 + Cr(NO3)3

HNO_3 (nitric acid) + K_2Cr_2O_7 (potassium dichromate) + KNO_2 (potassium nitrite) ⟶ H_2O (water) + KNO_3 (potassium nitrate) + CrN_3O_9 (chromium nitrate)

Balance the chemical equation algebraically: HNO_3 + K_2Cr_2O_7 + KNO_2 ⟶ H_2O + KNO_3 + CrN_3O_9 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 HNO_3 + c_2 K_2Cr_2O_7 + c_3 KNO_2 ⟶ 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_3 = c_5 + 3 c_6 O: | 3 c_1 + 7 c_2 + 2 c_3 = c_4 + 3 c_5 + 9 c_6 Cr: | 2 c_2 = c_6 K: | 2 c_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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 8 c_2 = 1 c_3 = 3 c_4 = 4 c_5 = 5 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 HNO_3 + K_2Cr_2O_7 + 3 KNO_2 ⟶ 4 H_2O + 5 KNO_3 + 2 CrN_3O_9

+ + ⟶ + +

nitric acid + potassium dichromate + potassium nitrite ⟶ water + potassium nitrate + chromium nitrate

Construct the equilibrium constant, K, expression for: HNO_3 + K_2Cr_2O_7 + KNO_2 ⟶ 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: 8 HNO_3 + K_2Cr_2O_7 + 3 KNO_2 ⟶ 4 H_2O + 5 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 | 8 | -8 K_2Cr_2O_7 | 1 | -1 KNO_2 | 3 | -3 H_2O | 4 | 4 KNO_3 | 5 | 5 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 | 8 | -8 | ([HNO3])^(-8) K_2Cr_2O_7 | 1 | -1 | ([K2Cr2O7])^(-1) KNO_2 | 3 | -3 | ([KNO2])^(-3) H_2O | 4 | 4 | ([H2O])^4 KNO_3 | 5 | 5 | ([KNO3])^5 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])^(-8) ([K2Cr2O7])^(-1) ([KNO2])^(-3) ([H2O])^4 ([KNO3])^5 ([CrN3O9])^2 = (([H2O])^4 ([KNO3])^5 ([CrN3O9])^2)/(([HNO3])^8 [K2Cr2O7] ([KNO2])^3)

Construct the rate of reaction expression for: HNO_3 + K_2Cr_2O_7 + KNO_2 ⟶ 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: 8 HNO_3 + K_2Cr_2O_7 + 3 KNO_2 ⟶ 4 H_2O + 5 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 | 8 | -8 K_2Cr_2O_7 | 1 | -1 KNO_2 | 3 | -3 H_2O | 4 | 4 KNO_3 | 5 | 5 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 | 8 | -8 | -1/8 (Δ[HNO3])/(Δt) K_2Cr_2O_7 | 1 | -1 | -(Δ[K2Cr2O7])/(Δt) KNO_2 | 3 | -3 | -1/3 (Δ[KNO2])/(Δt) H_2O | 4 | 4 | 1/4 (Δ[H2O])/(Δt) KNO_3 | 5 | 5 | 1/5 (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 = -1/8 (Δ[HNO3])/(Δt) = -(Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[KNO2])/(Δt) = 1/4 (Δ[H2O])/(Δt) = 1/5 (Δ[KNO3])/(Δt) = 1/2 (Δ[CrN3O9])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| nitric acid | potassium dichromate | potassium nitrite | water | potassium nitrate | chromium nitrate formula | HNO_3 | K_2Cr_2O_7 | KNO_2 | H_2O | KNO_3 | CrN_3O_9 Hill formula | HNO_3 | Cr_2K_2O_7 | KNO_2 | H_2O | KNO_3 | CrN_3O_9 name | nitric acid | potassium dichromate | potassium nitrite | water | potassium nitrate | chromium nitrate IUPAC name | nitric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | potassium nitrite | water | potassium nitrate | chromium(+3) cation trinitrate

| nitric acid | potassium dichromate | potassium nitrite | water | potassium nitrate | chromium nitrate molar mass | 63.012 g/mol | 294.18 g/mol | 85.103 g/mol | 18.015 g/mol | 101.1 g/mol | 238.01 g/mol phase | liquid (at STP) | solid (at STP) | solid (at STP) | liquid (at STP) | solid (at STP) | solid (at STP) melting point | -41.6 °C | 398 °C | 350 °C | 0 °C | 334 °C | 66 °C boiling point | 83 °C | | | 99.9839 °C | | density | 1.5129 g/cm^3 | 2.67 g/cm^3 | 1.915 g/cm^3 | 1 g/cm^3 | | 1.8 g/cm^3 solubility in water | miscible | | | | soluble | soluble surface tension | | | | 0.0728 N/m | | dynamic viscosity | 7.6×10^-4 Pa s (at 25 °C) | | | 8.9×10^-4 Pa s (at 25 °C) | | odor | | odorless | | odorless | odorless |