H2O + CrNO3 = O2 + H2 + HNO3 + Cr + Cr(OH)3

H_2O water + CrNO3 ⟶ O_2 oxygen + H_2 hydrogen + HNO_3 nitric acid + Cr chromium + Cr(OH)3

Balance the chemical equation algebraically: H_2O + CrNO3 ⟶ O_2 + H_2 + HNO_3 + Cr + Cr(OH)3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 CrNO3 ⟶ c_3 O_2 + c_4 H_2 + c_5 HNO_3 + c_6 Cr + c_7 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 N: H: | 2 c_1 = 2 c_4 + c_5 + 3 c_7 O: | c_1 + 3 c_2 = 2 c_3 + 3 c_5 + 3 c_7 Cr: | c_2 = c_6 + c_7 N: | 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_3 = 1 and solve the system of equations for the remaining coefficients: c_3 = 1 c_4 = 1 + c_1/2 - c_2/2 c_5 = c_2 c_6 = 2/3 - c_1/3 + c_2 c_7 = c_1/3 - 2/3 The resulting system of equations is still underdetermined, so additional coefficients must be set arbitrarily. Set c_1 = 8 and c_2 = 6 and solve for the remaining coefficients: c_1 = 8 c_2 = 6 c_3 = 1 c_4 = 2 c_5 = 6 c_6 = 4 c_7 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 8 H_2O + 6 CrNO3 ⟶ O_2 + 2 H_2 + 6 HNO_3 + 4 Cr + 2 Cr(OH)3

+ CrNO3 ⟶ + + + + Cr(OH)3

water + CrNO3 ⟶ oxygen + hydrogen + nitric acid + chromium + Cr(OH)3

Construct the equilibrium constant, K, expression for: H_2O + CrNO3 ⟶ O_2 + H_2 + HNO_3 + Cr + 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: 8 H_2O + 6 CrNO3 ⟶ O_2 + 2 H_2 + 6 HNO_3 + 4 Cr + 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 | 8 | -8 CrNO3 | 6 | -6 O_2 | 1 | 1 H_2 | 2 | 2 HNO_3 | 6 | 6 Cr | 4 | 4 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 | 8 | -8 | ([H2O])^(-8) CrNO3 | 6 | -6 | ([CrNO3])^(-6) O_2 | 1 | 1 | [O2] H_2 | 2 | 2 | ([H2])^2 HNO_3 | 6 | 6 | ([HNO3])^6 Cr | 4 | 4 | ([Cr])^4 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])^(-8) ([CrNO3])^(-6) [O2] ([H2])^2 ([HNO3])^6 ([Cr])^4 ([Cr(OH)3])^2 = ([O2] ([H2])^2 ([HNO3])^6 ([Cr])^4 ([Cr(OH)3])^2)/(([H2O])^8 ([CrNO3])^6)

Construct the rate of reaction expression for: H_2O + CrNO3 ⟶ O_2 + H_2 + HNO_3 + Cr + 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: 8 H_2O + 6 CrNO3 ⟶ O_2 + 2 H_2 + 6 HNO_3 + 4 Cr + 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 | 8 | -8 CrNO3 | 6 | -6 O_2 | 1 | 1 H_2 | 2 | 2 HNO_3 | 6 | 6 Cr | 4 | 4 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 | 8 | -8 | -1/8 (Δ[H2O])/(Δt) CrNO3 | 6 | -6 | -1/6 (Δ[CrNO3])/(Δt) O_2 | 1 | 1 | (Δ[O2])/(Δt) H_2 | 2 | 2 | 1/2 (Δ[H2])/(Δt) HNO_3 | 6 | 6 | 1/6 (Δ[HNO3])/(Δt) Cr | 4 | 4 | 1/4 (Δ[Cr])/(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 = -1/8 (Δ[H2O])/(Δt) = -1/6 (Δ[CrNO3])/(Δt) = (Δ[O2])/(Δt) = 1/2 (Δ[H2])/(Δt) = 1/6 (Δ[HNO3])/(Δt) = 1/4 (Δ[Cr])/(Δt) = 1/2 (Δ[Cr(OH)3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | CrNO3 | oxygen | hydrogen | nitric acid | chromium | Cr(OH)3 formula | H_2O | CrNO3 | O_2 | H_2 | HNO_3 | Cr | Cr(OH)3 Hill formula | H_2O | CrNO3 | O_2 | H_2 | HNO_3 | Cr | H3CrO3 name | water | | oxygen | hydrogen | nitric acid | chromium | IUPAC name | water | | molecular oxygen | molecular hydrogen | nitric acid | chromium |

| water | CrNO3 | oxygen | hydrogen | nitric acid | chromium | Cr(OH)3 molar mass | 18.015 g/mol | 114 g/mol | 31.998 g/mol | 2.016 g/mol | 63.012 g/mol | 51.9961 g/mol | 103.02 g/mol phase | liquid (at STP) | | gas (at STP) | gas (at STP) | liquid (at STP) | solid (at STP) | melting point | 0 °C | | -218 °C | -259.2 °C | -41.6 °C | 1857 °C | boiling point | 99.9839 °C | | -183 °C | -252.8 °C | 83 °C | 2672 °C | density | 1 g/cm^3 | | 0.001429 g/cm^3 (at 0 °C) | 8.99×10^-5 g/cm^3 (at 0 °C) | 1.5129 g/cm^3 | 7.14 g/cm^3 | solubility in water | | | | | miscible | insoluble | surface tension | 0.0728 N/m | | 0.01347 N/m | | | | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | 2.055×10^-5 Pa s (at 25 °C) | 8.9×10^-6 Pa s (at 25 °C) | 7.6×10^-4 Pa s (at 25 °C) | | odor | odorless | | odorless | odorless | | odorless |