H2O + Cr2(SO4)3 + O3 = H2SO4 + O2 + H2Cr2O7

H_2O water + Cr_2(SO_4)_3 chromium sulfate + O_3 ozone ⟶ H_2SO_4 sulfuric acid + O_2 oxygen + H_2Cr_2O_7 dichromic acid

Balance the chemical equation algebraically: H_2O + Cr_2(SO_4)_3 + O_3 ⟶ H_2SO_4 + O_2 + H_2Cr_2O_7 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 Cr_2(SO_4)_3 + c_3 O_3 ⟶ c_4 H_2SO_4 + c_5 O_2 + c_6 H_2Cr_2O_7 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, Cr and S: H: | 2 c_1 = 2 c_4 + 2 c_6 O: | c_1 + 12 c_2 + 3 c_3 = 4 c_4 + 2 c_5 + 7 c_6 Cr: | 2 c_2 = 2 c_6 S: | 3 c_2 = c_4 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 = 4 c_2 = 1 c_4 = 3 c_5 = (3 c_3)/2 - 3/2 c_6 = 1 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_3 = 3 and solve for the remaining coefficients: c_1 = 4 c_2 = 1 c_3 = 3 c_4 = 3 c_5 = 3 c_6 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 4 H_2O + Cr_2(SO_4)_3 + 3 O_3 ⟶ 3 H_2SO_4 + 3 O_2 + H_2Cr_2O_7

+ + ⟶ + +

water + chromium sulfate + ozone ⟶ sulfuric acid + oxygen + dichromic acid

Construct the equilibrium constant, K, expression for: H_2O + Cr_2(SO_4)_3 + O_3 ⟶ H_2SO_4 + O_2 + H_2Cr_2O_7 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: 4 H_2O + Cr_2(SO_4)_3 + 3 O_3 ⟶ 3 H_2SO_4 + 3 O_2 + H_2Cr_2O_7 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 | 4 | -4 Cr_2(SO_4)_3 | 1 | -1 O_3 | 3 | -3 H_2SO_4 | 3 | 3 O_2 | 3 | 3 H_2Cr_2O_7 | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 4 | -4 | ([H2O])^(-4) Cr_2(SO_4)_3 | 1 | -1 | ([Cr2(SO4)3])^(-1) O_3 | 3 | -3 | ([O3])^(-3) H_2SO_4 | 3 | 3 | ([H2SO4])^3 O_2 | 3 | 3 | ([O2])^3 H_2Cr_2O_7 | 1 | 1 | [H2Cr2O7] 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])^(-4) ([Cr2(SO4)3])^(-1) ([O3])^(-3) ([H2SO4])^3 ([O2])^3 [H2Cr2O7] = (([H2SO4])^3 ([O2])^3 [H2Cr2O7])/(([H2O])^4 [Cr2(SO4)3] ([O3])^3)

Construct the rate of reaction expression for: H_2O + Cr_2(SO_4)_3 + O_3 ⟶ H_2SO_4 + O_2 + H_2Cr_2O_7 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: 4 H_2O + Cr_2(SO_4)_3 + 3 O_3 ⟶ 3 H_2SO_4 + 3 O_2 + H_2Cr_2O_7 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 | 4 | -4 Cr_2(SO_4)_3 | 1 | -1 O_3 | 3 | -3 H_2SO_4 | 3 | 3 O_2 | 3 | 3 H_2Cr_2O_7 | 1 | 1 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 | 4 | -4 | -1/4 (Δ[H2O])/(Δt) Cr_2(SO_4)_3 | 1 | -1 | -(Δ[Cr2(SO4)3])/(Δt) O_3 | 3 | -3 | -1/3 (Δ[O3])/(Δt) H_2SO_4 | 3 | 3 | 1/3 (Δ[H2SO4])/(Δt) O_2 | 3 | 3 | 1/3 (Δ[O2])/(Δt) H_2Cr_2O_7 | 1 | 1 | (Δ[H2Cr2O7])/(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 = -1/4 (Δ[H2O])/(Δt) = -(Δ[Cr2(SO4)3])/(Δt) = -1/3 (Δ[O3])/(Δt) = 1/3 (Δ[H2SO4])/(Δt) = 1/3 (Δ[O2])/(Δt) = (Δ[H2Cr2O7])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | chromium sulfate | ozone | sulfuric acid | oxygen | dichromic acid formula | H_2O | Cr_2(SO_4)_3 | O_3 | H_2SO_4 | O_2 | H_2Cr_2O_7 Hill formula | H_2O | Cr_2O_12S_3 | O_3 | H_2O_4S | O_2 | Cr_2H_2O_7 name | water | chromium sulfate | ozone | sulfuric acid | oxygen | dichromic acid IUPAC name | water | chromium(+3) cation trisulfate | ozone | sulfuric acid | molecular oxygen | hydroxy-(hydroxy-dioxo-chromio)oxy-dioxo-chromium

| water | chromium sulfate | ozone | sulfuric acid | oxygen | dichromic acid molar mass | 18.015 g/mol | 392.2 g/mol | 47.997 g/mol | 98.07 g/mol | 31.998 g/mol | 218 g/mol phase | liquid (at STP) | liquid (at STP) | gas (at STP) | liquid (at STP) | gas (at STP) | melting point | 0 °C | | -192.2 °C | 10.371 °C | -218 °C | boiling point | 99.9839 °C | 330 °C | -111.9 °C | 279.6 °C | -183 °C | density | 1 g/cm^3 | 1.84 g/cm^3 | 0.001962 g/cm^3 (at 25 °C) | 1.8305 g/cm^3 | 0.001429 g/cm^3 (at 0 °C) | 1.66 g/cm^3 solubility in water | | | | very soluble | | surface tension | 0.0728 N/m | | | 0.0735 N/m | 0.01347 N/m | dynamic viscosity | 8.9×10^-4 Pa s (at 25 °C) | | | 0.021 Pa s (at 25 °C) | 2.055×10^-5 Pa s (at 25 °C) | odor | odorless | odorless | | odorless | odorless |