H2O + K2Cr2O7 + H2O2 = O2 + KOH + Cr(OH)3

H_2O water + K_2Cr_2O_7 potassium dichromate + H_2O_2 hydrogen peroxide ⟶ O_2 oxygen + KOH potassium hydroxide + Cr(OH)3

Balance the chemical equation algebraically: H_2O + K_2Cr_2O_7 + H_2O_2 ⟶ O_2 + KOH + Cr(OH)3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 K_2Cr_2O_7 + c_3 H_2O_2 ⟶ c_4 O_2 + c_5 KOH + c_6 Cr(OH)3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Cr and K: H: | 2 c_1 + 2 c_3 = c_5 + 3 c_6 O: | c_1 + 7 c_2 + 2 c_3 = 2 c_4 + c_5 + 3 c_6 Cr: | 2 c_2 = c_6 K: | 2 c_2 = 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_2 = 1 c_3 = 4 - c_1 c_4 = 7/2 - c_1/2 c_5 = 2 c_6 = 2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 3 and solve for the remaining coefficients: c_1 = 3 c_2 = 1 c_3 = 1 c_4 = 2 c_5 = 2 c_6 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2O + K_2Cr_2O_7 + H_2O_2 ⟶ 2 O_2 + 2 KOH + 2 Cr(OH)3

+ + ⟶ + + Cr(OH)3

water + potassium dichromate + hydrogen peroxide ⟶ oxygen + potassium hydroxide + Cr(OH)3

Construct the equilibrium constant, K, expression for: H_2O + K_2Cr_2O_7 + H_2O_2 ⟶ O_2 + KOH + Cr(OH)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: 3 H_2O + K_2Cr_2O_7 + H_2O_2 ⟶ 2 O_2 + 2 KOH + 2 Cr(OH)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 | 3 | -3 K_2Cr_2O_7 | 1 | -1 H_2O_2 | 1 | -1 O_2 | 2 | 2 KOH | 2 | 2 Cr(OH)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 | 3 | -3 | ([H2O])^(-3) K_2Cr_2O_7 | 1 | -1 | ([K2Cr2O7])^(-1) H_2O_2 | 1 | -1 | ([H2O2])^(-1) O_2 | 2 | 2 | ([O2])^2 KOH | 2 | 2 | ([KOH])^2 Cr(OH)3 | 2 | 2 | ([Cr(OH)3])^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])^(-3) ([K2Cr2O7])^(-1) ([H2O2])^(-1) ([O2])^2 ([KOH])^2 ([Cr(OH)3])^2 = (([O2])^2 ([KOH])^2 ([Cr(OH)3])^2)/(([H2O])^3 [K2Cr2O7] [H2O2])

Construct the rate of reaction expression for: H_2O + K_2Cr_2O_7 + H_2O_2 ⟶ O_2 + KOH + Cr(OH)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: 3 H_2O + K_2Cr_2O_7 + H_2O_2 ⟶ 2 O_2 + 2 KOH + 2 Cr(OH)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 | 3 | -3 K_2Cr_2O_7 | 1 | -1 H_2O_2 | 1 | -1 O_2 | 2 | 2 KOH | 2 | 2 Cr(OH)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 | 3 | -3 | -1/3 (Δ[H2O])/(Δt) K_2Cr_2O_7 | 1 | -1 | -(Δ[K2Cr2O7])/(Δt) H_2O_2 | 1 | -1 | -(Δ[H2O2])/(Δt) O_2 | 2 | 2 | 1/2 (Δ[O2])/(Δt) KOH | 2 | 2 | 1/2 (Δ[KOH])/(Δt) Cr(OH)3 | 2 | 2 | 1/2 (Δ[Cr(OH)3])/(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 = -1/3 (Δ[H2O])/(Δt) = -(Δ[K2Cr2O7])/(Δt) = -(Δ[H2O2])/(Δt) = 1/2 (Δ[O2])/(Δt) = 1/2 (Δ[KOH])/(Δt) = 1/2 (Δ[Cr(OH)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium dichromate | hydrogen peroxide | oxygen | potassium hydroxide | Cr(OH)3 formula | H_2O | K_2Cr_2O_7 | H_2O_2 | O_2 | KOH | Cr(OH)3 Hill formula | H_2O | Cr_2K_2O_7 | H_2O_2 | O_2 | HKO | H3CrO3 name | water | potassium dichromate | hydrogen peroxide | oxygen | potassium hydroxide | IUPAC name | water | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | hydrogen peroxide | molecular oxygen | potassium hydroxide |

| water | potassium dichromate | hydrogen peroxide | oxygen | potassium hydroxide | Cr(OH)3 molar mass | 18.015 g/mol | 294.18 g/mol | 34.014 g/mol | 31.998 g/mol | 56.105 g/mol | 103.02 g/mol phase | liquid (at STP) | solid (at STP) | liquid (at STP) | gas (at STP) | solid (at STP) | melting point | 0 °C | 398 °C | -0.43 °C | -218 °C | 406 °C | boiling point | 99.9839 °C | | 150.2 °C | -183 °C | 1327 °C | density | 1 g/cm^3 | 2.67 g/cm^3 | 1.44 g/cm^3 | 0.001429 g/cm^3 (at 0 °C) | 2.044 g/cm^3 | solubility in water | | | miscible | | soluble | surface tension | 0.0728 N/m | | 0.0804 N/m | 0.01347 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | 0.001249 Pa s (at 20 °C) | 2.055×10^-5 Pa s (at 25 °C) | 0.001 Pa s (at 550 °C) | odor | odorless | odorless | | odorless | |