H2SO4 + KMnO4 + C6H5C2H3 = H2O + CO2 + K2SO4 + MnSO4 + C6H5COOH

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + C_6H_5CH=CH_2 styrene ⟶ 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_5CH=CH_2 ⟶ 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_5CH=CH_2 ⟶ 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 + 8 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_3 = 1 and solve the system of equations for the remaining coefficients: c_2 = (2 c_1)/3 c_3 = 1 c_4 = (4 c_1)/3 c_5 = (7 c_1)/9 - 4/3 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = 4/3 - c_1/9 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 3 and solve for the remaining coefficients: c_1 = 3 c_2 = 2 c_3 = 1 c_4 = 4 c_5 = 1 c_6 = 1 c_7 = 2 c_8 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 3 H_2SO_4 + 2 KMnO_4 + C_6H_5CH=CH_2 ⟶ 4 H_2O + CO_2 + K_2SO_4 + 2 MnSO_4 + C_6H_5COOH

+ + ⟶ + + + +

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

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + C_6H_5CH=CH_2 ⟶ 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: 3 H_2SO_4 + 2 KMnO_4 + C_6H_5CH=CH_2 ⟶ 4 H_2O + CO_2 + K_2SO_4 + 2 MnSO_4 + 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 | 3 | -3 KMnO_4 | 2 | -2 C_6H_5CH=CH_2 | 1 | -1 H_2O | 4 | 4 CO_2 | 1 | 1 K_2SO_4 | 1 | 1 MnSO_4 | 2 | 2 C_6H_5COOH | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 3 | -3 | ([H2SO4])^(-3) KMnO_4 | 2 | -2 | ([KMnO4])^(-2) C_6H_5CH=CH_2 | 1 | -1 | ([C6H5CH=CH2])^(-1) H_2O | 4 | 4 | ([H2O])^4 CO_2 | 1 | 1 | [CO2] K_2SO_4 | 1 | 1 | [K2SO4] MnSO_4 | 2 | 2 | ([MnSO4])^2 C_6H_5COOH | 1 | 1 | [C6H5COOH] 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])^(-3) ([KMnO4])^(-2) ([C6H5CH=CH2])^(-1) ([H2O])^4 [CO2] [K2SO4] ([MnSO4])^2 [C6H5COOH] = (([H2O])^4 [CO2] [K2SO4] ([MnSO4])^2 [C6H5COOH])/(([H2SO4])^3 ([KMnO4])^2 [C6H5CH=CH2])

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + C_6H_5CH=CH_2 ⟶ 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: 3 H_2SO_4 + 2 KMnO_4 + C_6H_5CH=CH_2 ⟶ 4 H_2O + CO_2 + K_2SO_4 + 2 MnSO_4 + 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 | 3 | -3 KMnO_4 | 2 | -2 C_6H_5CH=CH_2 | 1 | -1 H_2O | 4 | 4 CO_2 | 1 | 1 K_2SO_4 | 1 | 1 MnSO_4 | 2 | 2 C_6H_5COOH | 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_2SO_4 | 3 | -3 | -1/3 (Δ[H2SO4])/(Δt) KMnO_4 | 2 | -2 | -1/2 (Δ[KMnO4])/(Δt) C_6H_5CH=CH_2 | 1 | -1 | -(Δ[C6H5CH=CH2])/(Δt) H_2O | 4 | 4 | 1/4 (Δ[H2O])/(Δt) CO_2 | 1 | 1 | (Δ[CO2])/(Δt) K_2SO_4 | 1 | 1 | (Δ[K2SO4])/(Δt) MnSO_4 | 2 | 2 | 1/2 (Δ[MnSO4])/(Δt) C_6H_5COOH | 1 | 1 | (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 = -1/3 (Δ[H2SO4])/(Δt) = -1/2 (Δ[KMnO4])/(Δt) = -(Δ[C6H5CH=CH2])/(Δt) = 1/4 (Δ[H2O])/(Δt) = (Δ[CO2])/(Δt) = (Δ[K2SO4])/(Δt) = 1/2 (Δ[MnSO4])/(Δt) = (Δ[C6H5COOH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | styrene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | benzoic acid formula | H_2SO_4 | KMnO_4 | C_6H_5CH=CH_2 | H_2O | CO_2 | K_2SO_4 | MnSO_4 | C_6H_5COOH Hill formula | H_2O_4S | KMnO_4 | C_8H_8 | H_2O | CO_2 | K_2O_4S | MnSO_4 | C_7H_6O_2 name | sulfuric acid | potassium permanganate | styrene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | benzoic acid IUPAC name | sulfuric acid | potassium permanganate | vinylbenzene | water | carbon dioxide | dipotassium sulfate | manganese(+2) cation sulfate | benzoic acid