KMnO4 + C3H8O3 = H2O + CO2 + K2CO3 + Mn2O3

KMnO_4 potassium permanganate + HOCH_2CH(OH)CH_2OH glycerol ⟶ H_2O water + CO_2 carbon dioxide + K_2CO_3 pearl ash + Mn_2O_3 manganese(III) oxide

Balance the chemical equation algebraically: KMnO_4 + HOCH_2CH(OH)CH_2OH ⟶ H_2O + CO_2 + K_2CO_3 + Mn_2O_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KMnO_4 + c_2 HOCH_2CH(OH)CH_2OH ⟶ c_3 H_2O + c_4 CO_2 + c_5 K_2CO_3 + c_6 Mn_2O_3 Set the number of atoms in the reactants equal to the number of atoms in the products for K, Mn, O, C and H: K: | c_1 = 2 c_5 Mn: | c_1 = 2 c_6 O: | 4 c_1 + 3 c_2 = c_3 + 2 c_4 + 3 c_5 + 3 c_6 C: | 3 c_2 = c_4 + c_5 H: | 8 c_2 = 2 c_3 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_2 = 1 and solve the system of equations for the remaining coefficients: c_1 = 7/2 c_2 = 1 c_3 = 4 c_4 = 5/4 c_5 = 7/4 c_6 = 7/4 Multiply by the least common denominator, 4, to eliminate fractional coefficients: c_1 = 14 c_2 = 4 c_3 = 16 c_4 = 5 c_5 = 7 c_6 = 7 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 14 KMnO_4 + 4 HOCH_2CH(OH)CH_2OH ⟶ 16 H_2O + 5 CO_2 + 7 K_2CO_3 + 7 Mn_2O_3

+ ⟶ + + +

potassium permanganate + glycerol ⟶ water + carbon dioxide + pearl ash + manganese(III) oxide

Construct the equilibrium constant, K, expression for: KMnO_4 + HOCH_2CH(OH)CH_2OH ⟶ H_2O + CO_2 + K_2CO_3 + Mn_2O_3 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: 14 KMnO_4 + 4 HOCH_2CH(OH)CH_2OH ⟶ 16 H_2O + 5 CO_2 + 7 K_2CO_3 + 7 Mn_2O_3 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 KMnO_4 | 14 | -14 HOCH_2CH(OH)CH_2OH | 4 | -4 H_2O | 16 | 16 CO_2 | 5 | 5 K_2CO_3 | 7 | 7 Mn_2O_3 | 7 | 7 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KMnO_4 | 14 | -14 | ([KMnO4])^(-14) HOCH_2CH(OH)CH_2OH | 4 | -4 | ([HOCH2CH(OH)CH2OH])^(-4) H_2O | 16 | 16 | ([H2O])^16 CO_2 | 5 | 5 | ([CO2])^5 K_2CO_3 | 7 | 7 | ([K2CO3])^7 Mn_2O_3 | 7 | 7 | ([Mn2O3])^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 = ([KMnO4])^(-14) ([HOCH2CH(OH)CH2OH])^(-4) ([H2O])^16 ([CO2])^5 ([K2CO3])^7 ([Mn2O3])^7 = (([H2O])^16 ([CO2])^5 ([K2CO3])^7 ([Mn2O3])^7)/(([KMnO4])^14 ([HOCH2CH(OH)CH2OH])^4)

Construct the rate of reaction expression for: KMnO_4 + HOCH_2CH(OH)CH_2OH ⟶ H_2O + CO_2 + K_2CO_3 + Mn_2O_3 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: 14 KMnO_4 + 4 HOCH_2CH(OH)CH_2OH ⟶ 16 H_2O + 5 CO_2 + 7 K_2CO_3 + 7 Mn_2O_3 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 KMnO_4 | 14 | -14 HOCH_2CH(OH)CH_2OH | 4 | -4 H_2O | 16 | 16 CO_2 | 5 | 5 K_2CO_3 | 7 | 7 Mn_2O_3 | 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 KMnO_4 | 14 | -14 | -1/14 (Δ[KMnO4])/(Δt) HOCH_2CH(OH)CH_2OH | 4 | -4 | -1/4 (Δ[HOCH2CH(OH)CH2OH])/(Δt) H_2O | 16 | 16 | 1/16 (Δ[H2O])/(Δt) CO_2 | 5 | 5 | 1/5 (Δ[CO2])/(Δt) K_2CO_3 | 7 | 7 | 1/7 (Δ[K2CO3])/(Δt) Mn_2O_3 | 7 | 7 | 1/7 (Δ[Mn2O3])/(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 = -1/14 (Δ[KMnO4])/(Δt) = -1/4 (Δ[HOCH2CH(OH)CH2OH])/(Δt) = 1/16 (Δ[H2O])/(Δt) = 1/5 (Δ[CO2])/(Δt) = 1/7 (Δ[K2CO3])/(Δt) = 1/7 (Δ[Mn2O3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium permanganate | glycerol | water | carbon dioxide | pearl ash | manganese(III) oxide formula | KMnO_4 | HOCH_2CH(OH)CH_2OH | H_2O | CO_2 | K_2CO_3 | Mn_2O_3 Hill formula | KMnO_4 | C_3H_8O_3 | H_2O | CO_2 | CK_2O_3 | Mn_2O_3 name | potassium permanganate | glycerol | water | carbon dioxide | pearl ash | manganese(III) oxide IUPAC name | potassium permanganate | glycerol | water | carbon dioxide | dipotassium carbonate | oxo-(oxomanganiooxy)manganese