H2SO4 + KMnO4 + C6H5C2H5 = H2O + CO2 + K2SO4 + MnSO4 + C6H5COOH

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + C_6H_5C_2H_5 ethylbenzene ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + C_6H_5COOH benzoic acid

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + C_6H_5C_2H_5 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + C_6H_5COOH Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 C_6H_5C_2H_5 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 MnSO_4 + 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, K, Mn and C: H: | 2 c_1 + 10 c_3 = 2 c_4 + 6 c_8 O: | 4 c_1 + 4 c_2 = c_4 + 2 c_5 + 4 c_6 + 4 c_7 + 2 c_8 S: | c_1 = c_6 + c_7 K: | c_2 = 2 c_6 Mn: | c_2 = c_7 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 = (2 c_1)/3 c_3 = (5 c_1)/63 + 5/7 c_4 = (88 c_1)/63 + 4/7 c_5 = (40 c_1)/63 - 9/7 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = 1 Multiply by the least common denominator, 7, to eliminate fractional coefficients: c_2 = (2 c_1)/3 c_3 = (5 c_1)/63 + 5 c_4 = (88 c_1)/63 + 4 c_5 = (40 c_1)/63 - 9 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = 7 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 63 and solve for the remaining coefficients: c_1 = 63 c_2 = 42 c_3 = 10 c_4 = 92 c_5 = 31 c_6 = 21 c_7 = 42 c_8 = 7 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 63 H_2SO_4 + 42 KMnO_4 + 10 C_6H_5C_2H_5 ⟶ 92 H_2O + 31 CO_2 + 21 K_2SO_4 + 42 MnSO_4 + 7 C_6H_5COOH

+ + ⟶ + + + +

sulfuric acid + potassium permanganate + ethylbenzene ⟶ water + carbon dioxide + potassium sulfate + manganese(II) sulfate + benzoic acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + C_6H_5C_2H_5 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + 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: 63 H_2SO_4 + 42 KMnO_4 + 10 C_6H_5C_2H_5 ⟶ 92 H_2O + 31 CO_2 + 21 K_2SO_4 + 42 MnSO_4 + 7 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 | 63 | -63 KMnO_4 | 42 | -42 C_6H_5C_2H_5 | 10 | -10 H_2O | 92 | 92 CO_2 | 31 | 31 K_2SO_4 | 21 | 21 MnSO_4 | 42 | 42 C_6H_5COOH | 7 | 7 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 63 | -63 | ([H2SO4])^(-63) KMnO_4 | 42 | -42 | ([KMnO4])^(-42) C_6H_5C_2H_5 | 10 | -10 | ([C6H5C2H5])^(-10) H_2O | 92 | 92 | ([H2O])^92 CO_2 | 31 | 31 | ([CO2])^31 K_2SO_4 | 21 | 21 | ([K2SO4])^21 MnSO_4 | 42 | 42 | ([MnSO4])^42 C_6H_5COOH | 7 | 7 | ([C6H5COOH])^7 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])^(-63) ([KMnO4])^(-42) ([C6H5C2H5])^(-10) ([H2O])^92 ([CO2])^31 ([K2SO4])^21 ([MnSO4])^42 ([C6H5COOH])^7 = (([H2O])^92 ([CO2])^31 ([K2SO4])^21 ([MnSO4])^42 ([C6H5COOH])^7)/(([H2SO4])^63 ([KMnO4])^42 ([C6H5C2H5])^10)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + C_6H_5C_2H_5 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + 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: 63 H_2SO_4 + 42 KMnO_4 + 10 C_6H_5C_2H_5 ⟶ 92 H_2O + 31 CO_2 + 21 K_2SO_4 + 42 MnSO_4 + 7 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 | 63 | -63 KMnO_4 | 42 | -42 C_6H_5C_2H_5 | 10 | -10 H_2O | 92 | 92 CO_2 | 31 | 31 K_2SO_4 | 21 | 21 MnSO_4 | 42 | 42 C_6H_5COOH | 7 | 7 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 | 63 | -63 | -1/63 (Δ[H2SO4])/(Δt) KMnO_4 | 42 | -42 | -1/42 (Δ[KMnO4])/(Δt) C_6H_5C_2H_5 | 10 | -10 | -1/10 (Δ[C6H5C2H5])/(Δt) H_2O | 92 | 92 | 1/92 (Δ[H2O])/(Δt) CO_2 | 31 | 31 | 1/31 (Δ[CO2])/(Δt) K_2SO_4 | 21 | 21 | 1/21 (Δ[K2SO4])/(Δt) MnSO_4 | 42 | 42 | 1/42 (Δ[MnSO4])/(Δt) C_6H_5COOH | 7 | 7 | 1/7 (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 = -1/63 (Δ[H2SO4])/(Δt) = -1/42 (Δ[KMnO4])/(Δt) = -1/10 (Δ[C6H5C2H5])/(Δt) = 1/92 (Δ[H2O])/(Δt) = 1/31 (Δ[CO2])/(Δt) = 1/21 (Δ[K2SO4])/(Δt) = 1/42 (Δ[MnSO4])/(Δt) = 1/7 (Δ[C6H5COOH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | ethylbenzene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | benzoic acid formula | H_2SO_4 | KMnO_4 | C_6H_5C_2H_5 | H_2O | CO_2 | K_2SO_4 | MnSO_4 | C_6H_5COOH Hill formula | H_2O_4S | KMnO_4 | C_8H_10 | H_2O | CO_2 | K_2O_4S | MnSO_4 | C_7H_6O_2 name | sulfuric acid | potassium permanganate | ethylbenzene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | benzoic acid IUPAC name | sulfuric acid | potassium permanganate | ethylbenzene | water | carbon dioxide | dipotassium sulfate | manganese(+2) cation sulfate | benzoic acid