H2SO4 + KMnO4 + C6H10 = H2O + CO2 + K2SO4 + MnSO4 + C2H4O2 + C3H6O2

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + C_6H_10 cyclohexene ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + CH_3CO_2H acetic acid + CH_3CH_2COOH propionic acid

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + C_6H_10 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_3CO_2H + CH_3CH_2COOH Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 C_6H_10 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 MnSO_4 + c_8 CH_3CO_2H + c_9 CH_3CH_2COOH 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 + 4 c_8 + 6 c_9 O: | 4 c_1 + 4 c_2 = c_4 + 2 c_5 + 4 c_6 + 4 c_7 + 2 c_8 + 2 c_9 S: | c_1 = c_6 + c_7 K: | c_2 = 2 c_6 Mn: | c_2 = c_7 C: | 6 c_3 = c_5 + 2 c_8 + 3 c_9 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_5 = 1 and solve the system of equations for the remaining coefficients: c_2 = (2 c_1)/3 c_4 = 1 + c_1 - c_3 c_5 = 1 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = -7/2 + (5 c_1)/2 - (9 c_3)/2 c_9 = 2 - (5 c_1)/3 + 5 c_3 The resulting system of equations is still underdetermined, so additional coefficients must be set arbitrarily. Set c_1 = 51 and c_3 = 20 and solve for the remaining coefficients: c_1 = 51 c_2 = 34 c_3 = 20 c_4 = 32 c_5 = 1 c_6 = 17 c_7 = 34 c_8 = 34 c_9 = 17 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 51 H_2SO_4 + 34 KMnO_4 + 20 C_6H_10 ⟶ 32 H_2O + CO_2 + 17 K_2SO_4 + 34 MnSO_4 + 34 CH_3CO_2H + 17 CH_3CH_2COOH

+ + ⟶ + + + + +

sulfuric acid + potassium permanganate + cyclohexene ⟶ water + carbon dioxide + potassium sulfate + manganese(II) sulfate + acetic acid + propionic acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + C_6H_10 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_3CO_2H + CH_3CH_2COOH 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: 51 H_2SO_4 + 34 KMnO_4 + 20 C_6H_10 ⟶ 32 H_2O + CO_2 + 17 K_2SO_4 + 34 MnSO_4 + 34 CH_3CO_2H + 17 CH_3CH_2COOH 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 | 51 | -51 KMnO_4 | 34 | -34 C_6H_10 | 20 | -20 H_2O | 32 | 32 CO_2 | 1 | 1 K_2SO_4 | 17 | 17 MnSO_4 | 34 | 34 CH_3CO_2H | 34 | 34 CH_3CH_2COOH | 17 | 17 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 51 | -51 | ([H2SO4])^(-51) KMnO_4 | 34 | -34 | ([KMnO4])^(-34) C_6H_10 | 20 | -20 | ([C6H10])^(-20) H_2O | 32 | 32 | ([H2O])^32 CO_2 | 1 | 1 | [CO2] K_2SO_4 | 17 | 17 | ([K2SO4])^17 MnSO_4 | 34 | 34 | ([MnSO4])^34 CH_3CO_2H | 34 | 34 | ([CH3CO2H])^34 CH_3CH_2COOH | 17 | 17 | ([CH3CH2COOH])^17 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])^(-51) ([KMnO4])^(-34) ([C6H10])^(-20) ([H2O])^32 [CO2] ([K2SO4])^17 ([MnSO4])^34 ([CH3CO2H])^34 ([CH3CH2COOH])^17 = (([H2O])^32 [CO2] ([K2SO4])^17 ([MnSO4])^34 ([CH3CO2H])^34 ([CH3CH2COOH])^17)/(([H2SO4])^51 ([KMnO4])^34 ([C6H10])^20)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + C_6H_10 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_3CO_2H + CH_3CH_2COOH 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: 51 H_2SO_4 + 34 KMnO_4 + 20 C_6H_10 ⟶ 32 H_2O + CO_2 + 17 K_2SO_4 + 34 MnSO_4 + 34 CH_3CO_2H + 17 CH_3CH_2COOH 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 | 51 | -51 KMnO_4 | 34 | -34 C_6H_10 | 20 | -20 H_2O | 32 | 32 CO_2 | 1 | 1 K_2SO_4 | 17 | 17 MnSO_4 | 34 | 34 CH_3CO_2H | 34 | 34 CH_3CH_2COOH | 17 | 17 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 | 51 | -51 | -1/51 (Δ[H2SO4])/(Δt) KMnO_4 | 34 | -34 | -1/34 (Δ[KMnO4])/(Δt) C_6H_10 | 20 | -20 | -1/20 (Δ[C6H10])/(Δt) H_2O | 32 | 32 | 1/32 (Δ[H2O])/(Δt) CO_2 | 1 | 1 | (Δ[CO2])/(Δt) K_2SO_4 | 17 | 17 | 1/17 (Δ[K2SO4])/(Δt) MnSO_4 | 34 | 34 | 1/34 (Δ[MnSO4])/(Δt) CH_3CO_2H | 34 | 34 | 1/34 (Δ[CH3CO2H])/(Δt) CH_3CH_2COOH | 17 | 17 | 1/17 (Δ[CH3CH2COOH])/(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 = -1/51 (Δ[H2SO4])/(Δt) = -1/34 (Δ[KMnO4])/(Δt) = -1/20 (Δ[C6H10])/(Δt) = 1/32 (Δ[H2O])/(Δt) = (Δ[CO2])/(Δt) = 1/17 (Δ[K2SO4])/(Δt) = 1/34 (Δ[MnSO4])/(Δt) = 1/34 (Δ[CH3CO2H])/(Δt) = 1/17 (Δ[CH3CH2COOH])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | cyclohexene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | acetic acid | propionic acid formula | H_2SO_4 | KMnO_4 | C_6H_10 | H_2O | CO_2 | K_2SO_4 | MnSO_4 | CH_3CO_2H | CH_3CH_2COOH Hill formula | H_2O_4S | KMnO_4 | C_6H_10 | H_2O | CO_2 | K_2O_4S | MnSO_4 | C_2H_4O_2 | C_3H_6O_2 name | sulfuric acid | potassium permanganate | cyclohexene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | acetic acid | propionic acid IUPAC name | sulfuric acid | potassium permanganate | cyclohexene | water | carbon dioxide | dipotassium sulfate | manganese(+2) cation sulfate | acetic acid | propionic acid