H2SO4 + K2Cr2O7 + C6H5CH2CH3 = H2O + K2SO4 + Cr2(SO4)3 + C6H5COOH

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + C_6H_5C_2H_5 ethylbenzene ⟶ H_2O water + 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_5C_2H_5 ⟶ H_2O + 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_5C_2H_5 ⟶ c_4 H_2O + c_5 K_2SO_4 + c_6 Cr_2(SO_4)_3 + c_7 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 + 10 c_3 = 2 c_4 + 6 c_7 O: | 4 c_1 + 7 c_2 = c_4 + 4 c_5 + 12 c_6 + 2 c_7 S: | c_1 = c_5 + 3 c_6 Cr: | 2 c_2 = 2 c_6 K: | 2 c_2 = 2 c_5 C: | 8 c_3 = 7 c_7 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_1 = 36/7 c_2 = 9/7 c_3 = 1 c_4 = 47/7 c_5 = 9/7 c_6 = 9/7 c_7 = 8/7 Multiply by the least common denominator, 7, to eliminate fractional coefficients: c_1 = 36 c_2 = 9 c_3 = 7 c_4 = 47 c_5 = 9 c_6 = 9 c_7 = 8 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 36 H_2SO_4 + 9 K_2Cr_2O_7 + 7 C_6H_5C_2H_5 ⟶ 47 H_2O + 9 K_2SO_4 + 9 Cr_2(SO_4)_3 + 8 C_6H_5COOH

+ + ⟶ + + +

sulfuric acid + potassium dichromate + ethylbenzene ⟶ water + potassium sulfate + chromium sulfate + benzoic acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + C_6H_5C_2H_5 ⟶ H_2O + 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: 36 H_2SO_4 + 9 K_2Cr_2O_7 + 7 C_6H_5C_2H_5 ⟶ 47 H_2O + 9 K_2SO_4 + 9 Cr_2(SO_4)_3 + 8 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 | 36 | -36 K_2Cr_2O_7 | 9 | -9 C_6H_5C_2H_5 | 7 | -7 H_2O | 47 | 47 K_2SO_4 | 9 | 9 Cr_2(SO_4)_3 | 9 | 9 C_6H_5COOH | 8 | 8 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 36 | -36 | ([H2SO4])^(-36) K_2Cr_2O_7 | 9 | -9 | ([K2Cr2O7])^(-9) C_6H_5C_2H_5 | 7 | -7 | ([C6H5C2H5])^(-7) H_2O | 47 | 47 | ([H2O])^47 K_2SO_4 | 9 | 9 | ([K2SO4])^9 Cr_2(SO_4)_3 | 9 | 9 | ([Cr2(SO4)3])^9 C_6H_5COOH | 8 | 8 | ([C6H5COOH])^8 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])^(-36) ([K2Cr2O7])^(-9) ([C6H5C2H5])^(-7) ([H2O])^47 ([K2SO4])^9 ([Cr2(SO4)3])^9 ([C6H5COOH])^8 = (([H2O])^47 ([K2SO4])^9 ([Cr2(SO4)3])^9 ([C6H5COOH])^8)/(([H2SO4])^36 ([K2Cr2O7])^9 ([C6H5C2H5])^7)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + C_6H_5C_2H_5 ⟶ H_2O + 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: 36 H_2SO_4 + 9 K_2Cr_2O_7 + 7 C_6H_5C_2H_5 ⟶ 47 H_2O + 9 K_2SO_4 + 9 Cr_2(SO_4)_3 + 8 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 | 36 | -36 K_2Cr_2O_7 | 9 | -9 C_6H_5C_2H_5 | 7 | -7 H_2O | 47 | 47 K_2SO_4 | 9 | 9 Cr_2(SO_4)_3 | 9 | 9 C_6H_5COOH | 8 | 8 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 | 36 | -36 | -1/36 (Δ[H2SO4])/(Δt) K_2Cr_2O_7 | 9 | -9 | -1/9 (Δ[K2Cr2O7])/(Δt) C_6H_5C_2H_5 | 7 | -7 | -1/7 (Δ[C6H5C2H5])/(Δt) H_2O | 47 | 47 | 1/47 (Δ[H2O])/(Δt) K_2SO_4 | 9 | 9 | 1/9 (Δ[K2SO4])/(Δt) Cr_2(SO_4)_3 | 9 | 9 | 1/9 (Δ[Cr2(SO4)3])/(Δt) C_6H_5COOH | 8 | 8 | 1/8 (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 = -1/36 (Δ[H2SO4])/(Δt) = -1/9 (Δ[K2Cr2O7])/(Δt) = -1/7 (Δ[C6H5C2H5])/(Δt) = 1/47 (Δ[H2O])/(Δt) = 1/9 (Δ[K2SO4])/(Δt) = 1/9 (Δ[Cr2(SO4)3])/(Δt) = 1/8 (Δ[C6H5COOH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | ethylbenzene | water | potassium sulfate | chromium sulfate | benzoic acid formula | H_2SO_4 | K_2Cr_2O_7 | C_6H_5C_2H_5 | H_2O | K_2SO_4 | Cr_2(SO_4)_3 | C_6H_5COOH Hill formula | H_2O_4S | Cr_2K_2O_7 | C_8H_10 | H_2O | K_2O_4S | Cr_2O_12S_3 | C_7H_6O_2 name | sulfuric acid | potassium dichromate | ethylbenzene | water | potassium sulfate | chromium sulfate | benzoic acid IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | ethylbenzene | water | dipotassium sulfate | chromium(+3) cation trisulfate | benzoic acid