H2SO4 + K2Cr2O7 + C8H8 = H2O + CO2 + K2SO4 + Cr2(SO4)3 + C7H6O2

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + C_6H_5CH=CH_2 styrene ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + Cr_2(SO_4)_3 chromium sulfate + C_6H_5COOH benzoic acid

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + C_6H_5CH=CH_2 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + C_6H_5COOH Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 C_6H_5CH=CH_2 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 Cr_2(SO_4)_3 + c_8 C_6H_5COOH Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, S, Cr, K and C: H: | 2 c_1 + 8 c_3 = 2 c_4 + 6 c_8 O: | 4 c_1 + 7 c_2 = c_4 + 2 c_5 + 4 c_6 + 12 c_7 + 2 c_8 S: | c_1 = c_6 + 3 c_7 Cr: | 2 c_2 = 2 c_7 K: | 2 c_2 = 2 c_6 C: | 8 c_3 = c_5 + 7 c_8 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_8 = 1 and solve the system of equations for the remaining coefficients: c_2 = c_1/4 c_3 = (3 c_1)/80 + 3/4 c_4 = (23 c_1)/20 c_5 = (3 c_1)/10 - 1 c_6 = c_1/4 c_7 = c_1/4 c_8 = 1 Multiply by the least common denominator, 2, to eliminate fractional coefficients: c_2 = c_1/4 c_3 = (3 c_1)/80 + 3/2 c_4 = (23 c_1)/20 c_5 = (3 c_1)/10 - 2 c_6 = c_1/4 c_7 = c_1/4 c_8 = 2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 40 and solve for the remaining coefficients: c_1 = 40 c_2 = 10 c_3 = 3 c_4 = 46 c_5 = 10 c_6 = 10 c_7 = 10 c_8 = 2 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 40 H_2SO_4 + 10 K_2Cr_2O_7 + 3 C_6H_5CH=CH_2 ⟶ 46 H_2O + 10 CO_2 + 10 K_2SO_4 + 10 Cr_2(SO_4)_3 + 2 C_6H_5COOH

+ + ⟶ + + + +

sulfuric acid + potassium dichromate + styrene ⟶ water + carbon dioxide + potassium sulfate + chromium sulfate + benzoic acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + C_6H_5CH=CH_2 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + C_6H_5COOH 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: 40 H_2SO_4 + 10 K_2Cr_2O_7 + 3 C_6H_5CH=CH_2 ⟶ 46 H_2O + 10 CO_2 + 10 K_2SO_4 + 10 Cr_2(SO_4)_3 + 2 C_6H_5COOH 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_2SO_4 | 40 | -40 K_2Cr_2O_7 | 10 | -10 C_6H_5CH=CH_2 | 3 | -3 H_2O | 46 | 46 CO_2 | 10 | 10 K_2SO_4 | 10 | 10 Cr_2(SO_4)_3 | 10 | 10 C_6H_5COOH | 2 | 2 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 40 | -40 | ([H2SO4])^(-40) K_2Cr_2O_7 | 10 | -10 | ([K2Cr2O7])^(-10) C_6H_5CH=CH_2 | 3 | -3 | ([C6H5CH=CH2])^(-3) H_2O | 46 | 46 | ([H2O])^46 CO_2 | 10 | 10 | ([CO2])^10 K_2SO_4 | 10 | 10 | ([K2SO4])^10 Cr_2(SO_4)_3 | 10 | 10 | ([Cr2(SO4)3])^10 C_6H_5COOH | 2 | 2 | ([C6H5COOH])^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 = ([H2SO4])^(-40) ([K2Cr2O7])^(-10) ([C6H5CH=CH2])^(-3) ([H2O])^46 ([CO2])^10 ([K2SO4])^10 ([Cr2(SO4)3])^10 ([C6H5COOH])^2 = (([H2O])^46 ([CO2])^10 ([K2SO4])^10 ([Cr2(SO4)3])^10 ([C6H5COOH])^2)/(([H2SO4])^40 ([K2Cr2O7])^10 ([C6H5CH=CH2])^3)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + C_6H_5CH=CH_2 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + C_6H_5COOH 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: 40 H_2SO_4 + 10 K_2Cr_2O_7 + 3 C_6H_5CH=CH_2 ⟶ 46 H_2O + 10 CO_2 + 10 K_2SO_4 + 10 Cr_2(SO_4)_3 + 2 C_6H_5COOH 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_2SO_4 | 40 | -40 K_2Cr_2O_7 | 10 | -10 C_6H_5CH=CH_2 | 3 | -3 H_2O | 46 | 46 CO_2 | 10 | 10 K_2SO_4 | 10 | 10 Cr_2(SO_4)_3 | 10 | 10 C_6H_5COOH | 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_2SO_4 | 40 | -40 | -1/40 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 10 | -10 | -1/10 (Δ[K2Cr2O7])/(Δt) C_6H_5CH=CH_2 | 3 | -3 | -1/3 (Δ[C6H5CH=CH2])/(Δt) H_2O | 46 | 46 | 1/46 (Δ[H2O])/(Δt) CO_2 | 10 | 10 | 1/10 (Δ[CO2])/(Δt) K_2SO_4 | 10 | 10 | 1/10 (Δ[K2SO4])/(Δt) Cr_2(SO_4)_3 | 10 | 10 | 1/10 (Δ[Cr2(SO4)3])/(Δt) C_6H_5COOH | 2 | 2 | 1/2 (Δ[C6H5COOH])/(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 = -1/40 (Δ[H2SO4])/(Δt) = -1/10 (Δ[K2Cr2O7])/(Δt) = -1/3 (Δ[C6H5CH=CH2])/(Δt) = 1/46 (Δ[H2O])/(Δt) = 1/10 (Δ[CO2])/(Δt) = 1/10 (Δ[K2SO4])/(Δt) = 1/10 (Δ[Cr2(SO4)3])/(Δt) = 1/2 (Δ[C6H5COOH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | styrene | water | carbon dioxide | potassium sulfate | chromium sulfate | benzoic acid formula | H_2SO_4 | K_2Cr_2O_7 | C_6H_5CH=CH_2 | H_2O | CO_2 | K_2SO_4 | Cr_2(SO_4)_3 | C_6H_5COOH Hill formula | H_2O_4S | Cr_2K_2O_7 | C_8H_8 | H_2O | CO_2 | K_2O_4S | Cr_2O_12S_3 | C_7H_6O_2 name | sulfuric acid | potassium dichromate | styrene | water | carbon dioxide | potassium sulfate | chromium sulfate | benzoic acid IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | vinylbenzene | water | carbon dioxide | dipotassium sulfate | chromium(+3) cation trisulfate | benzoic acid