H2O + KMnO4 + CH3COCH3 = CO2 + KOH + MnO2 + CH3COOH

H_2O water + KMnO_4 potassium permanganate + CH_3COCH_3 acetone ⟶ CO_2 carbon dioxide + KOH potassium hydroxide + MnO_2 manganese dioxide + CH_3CO_2H acetic acid

Balance the chemical equation algebraically: H_2O + KMnO_4 + CH_3COCH_3 ⟶ CO_2 + KOH + MnO_2 + CH_3CO_2H Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 KMnO_4 + c_3 CH_3COCH_3 ⟶ c_4 CO_2 + c_5 KOH + c_6 MnO_2 + c_7 CH_3CO_2H Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, K, Mn and C: H: | 2 c_1 + 6 c_3 = c_5 + 4 c_7 O: | c_1 + 4 c_2 + c_3 = 2 c_4 + c_5 + 2 c_6 + 2 c_7 K: | c_2 = c_5 Mn: | c_2 = c_6 C: | 3 c_3 = c_4 + 2 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_1 = 1 and solve the system of equations for the remaining coefficients: c_1 = 1 c_3 = c_2/4 + 1 c_4 = c_2/2 - 1 c_5 = c_2 c_6 = c_2 c_7 = c_2/8 + 2 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_2 = 16 and solve for the remaining coefficients: c_1 = 1 c_2 = 16 c_3 = 5 c_4 = 7 c_5 = 16 c_6 = 16 c_7 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | H_2O + 16 KMnO_4 + 5 CH_3COCH_3 ⟶ 7 CO_2 + 16 KOH + 16 MnO_2 + 4 CH_3CO_2H

+ + ⟶ + + +

water + potassium permanganate + acetone ⟶ carbon dioxide + potassium hydroxide + manganese dioxide + acetic acid

Construct the equilibrium constant, K, expression for: H_2O + KMnO_4 + CH_3COCH_3 ⟶ CO_2 + KOH + MnO_2 + CH_3CO_2H 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: H_2O + 16 KMnO_4 + 5 CH_3COCH_3 ⟶ 7 CO_2 + 16 KOH + 16 MnO_2 + 4 CH_3CO_2H 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_2O | 1 | -1 KMnO_4 | 16 | -16 CH_3COCH_3 | 5 | -5 CO_2 | 7 | 7 KOH | 16 | 16 MnO_2 | 16 | 16 CH_3CO_2H | 4 | 4 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2O | 1 | -1 | ([H2O])^(-1) KMnO_4 | 16 | -16 | ([KMnO4])^(-16) CH_3COCH_3 | 5 | -5 | ([CH3COCH3])^(-5) CO_2 | 7 | 7 | ([CO2])^7 KOH | 16 | 16 | ([KOH])^16 MnO_2 | 16 | 16 | ([MnO2])^16 CH_3CO_2H | 4 | 4 | ([CH3CO2H])^4 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 = ([H2O])^(-1) ([KMnO4])^(-16) ([CH3COCH3])^(-5) ([CO2])^7 ([KOH])^16 ([MnO2])^16 ([CH3CO2H])^4 = (([CO2])^7 ([KOH])^16 ([MnO2])^16 ([CH3CO2H])^4)/([H2O] ([KMnO4])^16 ([CH3COCH3])^5)

Construct the rate of reaction expression for: H_2O + KMnO_4 + CH_3COCH_3 ⟶ CO_2 + KOH + MnO_2 + CH_3CO_2H 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: H_2O + 16 KMnO_4 + 5 CH_3COCH_3 ⟶ 7 CO_2 + 16 KOH + 16 MnO_2 + 4 CH_3CO_2H 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_2O | 1 | -1 KMnO_4 | 16 | -16 CH_3COCH_3 | 5 | -5 CO_2 | 7 | 7 KOH | 16 | 16 MnO_2 | 16 | 16 CH_3CO_2H | 4 | 4 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_2O | 1 | -1 | -(Δ[H2O])/(Δt) KMnO_4 | 16 | -16 | -1/16 (Δ[KMnO4])/(Δt) CH_3COCH_3 | 5 | -5 | -1/5 (Δ[CH3COCH3])/(Δt) CO_2 | 7 | 7 | 1/7 (Δ[CO2])/(Δt) KOH | 16 | 16 | 1/16 (Δ[KOH])/(Δt) MnO_2 | 16 | 16 | 1/16 (Δ[MnO2])/(Δt) CH_3CO_2H | 4 | 4 | 1/4 (Δ[CH3CO2H])/(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 = -(Δ[H2O])/(Δt) = -1/16 (Δ[KMnO4])/(Δt) = -1/5 (Δ[CH3COCH3])/(Δt) = 1/7 (Δ[CO2])/(Δt) = 1/16 (Δ[KOH])/(Δt) = 1/16 (Δ[MnO2])/(Δt) = 1/4 (Δ[CH3CO2H])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | potassium permanganate | acetone | carbon dioxide | potassium hydroxide | manganese dioxide | acetic acid formula | H_2O | KMnO_4 | CH_3COCH_3 | CO_2 | KOH | MnO_2 | CH_3CO_2H Hill formula | H_2O | KMnO_4 | C_3H_6O | CO_2 | HKO | MnO_2 | C_2H_4O_2 name | water | potassium permanganate | acetone | carbon dioxide | potassium hydroxide | manganese dioxide | acetic acid IUPAC name | water | potassium permanganate | acetone | carbon dioxide | potassium hydroxide | dioxomanganese | acetic acid