H2O + Cr2O3 = Cr(OH)3

H_2O water + Cr_2O_3 chromium(III) oxide ⟶ Cr(OH)3

Balance the chemical equation algebraically: H_2O + Cr_2O_3 ⟶ Cr(OH)3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 Cr_2O_3 ⟶ c_3 Cr(OH)3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O and Cr: H: | 2 c_1 = 3 c_3 O: | c_1 + 3 c_2 = 3 c_3 Cr: | 2 c_2 = c_3 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 = 3 c_2 = 1 c_3 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2O + Cr_2O_3 ⟶ 2 Cr(OH)3

+ ⟶ Cr(OH)3

water + chromium(III) oxide ⟶ Cr(OH)3

Construct the equilibrium constant, K, expression for: H_2O + Cr_2O_3 ⟶ 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 + Cr_2O_3 ⟶ 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 Cr_2O_3 | 1 | -1 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) Cr_2O_3 | 1 | -1 | ([Cr2O3])^(-1) 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) ([Cr2O3])^(-1) ([Cr(OH)3])^2 = ([Cr(OH)3])^2/(([H2O])^3 [Cr2O3])

Construct the rate of reaction expression for: H_2O + Cr_2O_3 ⟶ 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 + Cr_2O_3 ⟶ 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 Cr_2O_3 | 1 | -1 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) Cr_2O_3 | 1 | -1 | -(Δ[Cr2O3])/(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[Cr2O3])/(Δt) = 1/2 (Δ[Cr(OH)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | chromium(III) oxide | Cr(OH)3 formula | H_2O | Cr_2O_3 | Cr(OH)3 Hill formula | H_2O | Cr_2O_3 | H3CrO3 name | water | chromium(III) oxide |

| water | chromium(III) oxide | Cr(OH)3 molar mass | 18.015 g/mol | 151.99 g/mol | 103.02 g/mol phase | liquid (at STP) | solid (at STP) | melting point | 0 °C | 2435 °C | boiling point | 99.9839 °C | 4000 °C | density | 1 g/cm^3 | 4.8 g/cm^3 | solubility in water | | insoluble | surface tension | 0.0728 N/m | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | odor | odorless | |