H2SO4 + KMnO4 + CH3COCH3 = H2O + CO2 + K2SO4 + MnSO4 + CH3OOH

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + CH_3COCH_3 acetone ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + CH_4O_2 methyl hydroperoxide

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + CH_3COCH_3 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_4O_2 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 CH_3COCH_3 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 MnSO_4 + c_8 CH_4O_2 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 + 6 c_3 = 2 c_4 + 4 c_8 O: | 4 c_1 + 4 c_2 + c_3 = 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: | 3 c_3 = c_5 + 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_4 = 1 and solve the system of equations for the remaining coefficients: c_2 = (2 c_1)/3 c_3 = (8 c_1)/15 - 1/5 c_4 = 1 c_5 = (3 c_1)/10 + 1/5 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = (13 c_1)/10 - 4/5 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 6 and solve for the remaining coefficients: c_1 = 6 c_2 = 4 c_3 = 3 c_4 = 1 c_5 = 2 c_6 = 2 c_7 = 4 c_8 = 7 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 6 H_2SO_4 + 4 KMnO_4 + 3 CH_3COCH_3 ⟶ H_2O + 2 CO_2 + 2 K_2SO_4 + 4 MnSO_4 + 7 CH_4O_2

+ + ⟶ + + + +

sulfuric acid + potassium permanganate + acetone ⟶ water + carbon dioxide + potassium sulfate + manganese(II) sulfate + methyl hydroperoxide

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + CH_3COCH_3 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_4O_2 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: 6 H_2SO_4 + 4 KMnO_4 + 3 CH_3COCH_3 ⟶ H_2O + 2 CO_2 + 2 K_2SO_4 + 4 MnSO_4 + 7 CH_4O_2 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 | 6 | -6 KMnO_4 | 4 | -4 CH_3COCH_3 | 3 | -3 H_2O | 1 | 1 CO_2 | 2 | 2 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 CH_4O_2 | 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 | 6 | -6 | ([H2SO4])^(-6) KMnO_4 | 4 | -4 | ([KMnO4])^(-4) CH_3COCH_3 | 3 | -3 | ([CH3COCH3])^(-3) H_2O | 1 | 1 | [H2O] CO_2 | 2 | 2 | ([CO2])^2 K_2SO_4 | 2 | 2 | ([K2SO4])^2 MnSO_4 | 4 | 4 | ([MnSO4])^4 CH_4O_2 | 7 | 7 | ([CH4O2])^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])^(-6) ([KMnO4])^(-4) ([CH3COCH3])^(-3) [H2O] ([CO2])^2 ([K2SO4])^2 ([MnSO4])^4 ([CH4O2])^7 = ([H2O] ([CO2])^2 ([K2SO4])^2 ([MnSO4])^4 ([CH4O2])^7)/(([H2SO4])^6 ([KMnO4])^4 ([CH3COCH3])^3)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + CH_3COCH_3 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_4O_2 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: 6 H_2SO_4 + 4 KMnO_4 + 3 CH_3COCH_3 ⟶ H_2O + 2 CO_2 + 2 K_2SO_4 + 4 MnSO_4 + 7 CH_4O_2 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 | 6 | -6 KMnO_4 | 4 | -4 CH_3COCH_3 | 3 | -3 H_2O | 1 | 1 CO_2 | 2 | 2 K_2SO_4 | 2 | 2 MnSO_4 | 4 | 4 CH_4O_2 | 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 | 6 | -6 | -1/6 (Δ[H2SO4])/(Δt) KMnO_4 | 4 | -4 | -1/4 (Δ[KMnO4])/(Δt) CH_3COCH_3 | 3 | -3 | -1/3 (Δ[CH3COCH3])/(Δt) H_2O | 1 | 1 | (Δ[H2O])/(Δt) CO_2 | 2 | 2 | 1/2 (Δ[CO2])/(Δt) K_2SO_4 | 2 | 2 | 1/2 (Δ[K2SO4])/(Δt) MnSO_4 | 4 | 4 | 1/4 (Δ[MnSO4])/(Δt) CH_4O_2 | 7 | 7 | 1/7 (Δ[CH4O2])/(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 = -1/6 (Δ[H2SO4])/(Δt) = -1/4 (Δ[KMnO4])/(Δt) = -1/3 (Δ[CH3COCH3])/(Δt) = (Δ[H2O])/(Δt) = 1/2 (Δ[CO2])/(Δt) = 1/2 (Δ[K2SO4])/(Δt) = 1/4 (Δ[MnSO4])/(Δt) = 1/7 (Δ[CH4O2])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | acetone | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | methyl hydroperoxide formula | H_2SO_4 | KMnO_4 | CH_3COCH_3 | H_2O | CO_2 | K_2SO_4 | MnSO_4 | CH_4O_2 Hill formula | H_2O_4S | KMnO_4 | C_3H_6O | H_2O | CO_2 | K_2O_4S | MnSO_4 | CH_4O_2 name | sulfuric acid | potassium permanganate | acetone | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | methyl hydroperoxide IUPAC name | sulfuric acid | potassium permanganate | acetone | water | carbon dioxide | dipotassium sulfate | manganese(+2) cation sulfate | hydroperoxymethane