KOH + KMnO4 + C3H6 = H2O + K2MnO4 + K2CO3 + CH3COOK

KOH potassium hydroxide + KMnO_4 potassium permanganate + C_3H_6 cyclopropane ⟶ H_2O water + K_2MnO_4 potassium manganate + K_2CO_3 pearl ash + CH_3COOK potassium acetate

Balance the chemical equation algebraically: KOH + KMnO_4 + C_3H_6 ⟶ H_2O + K_2MnO_4 + K_2CO_3 + CH_3COOK Add stoichiometric coefficients, c_i, to the reactants and products: c_1 KOH + c_2 KMnO_4 + c_3 C_3H_6 ⟶ c_4 H_2O + c_5 K_2MnO_4 + c_6 K_2CO_3 + c_7 CH_3COOK Set the number of atoms in the reactants equal to the number of atoms in the products for H, K, O, Mn and C: H: | c_1 + 6 c_3 = 2 c_4 + 3 c_7 K: | c_1 + c_2 = 2 c_5 + 2 c_6 + c_7 O: | c_1 + 4 c_2 = c_4 + 4 c_5 + 3 c_6 + 2 c_7 Mn: | c_2 = c_5 C: | 3 c_3 = c_6 + 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_3 = 1 and solve the system of equations for the remaining coefficients: c_2 = (8 c_1)/11 + 6/11 c_3 = 1 c_4 = (7 c_1)/11 - 3/11 c_5 = (8 c_1)/11 + 6/11 c_6 = (2 c_1)/11 - 15/11 c_7 = 24/11 - c_1/11 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 13 and solve for the remaining coefficients: c_1 = 13 c_2 = 10 c_3 = 1 c_4 = 8 c_5 = 10 c_6 = 1 c_7 = 1 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 13 KOH + 10 KMnO_4 + C_3H_6 ⟶ 8 H_2O + 10 K_2MnO_4 + K_2CO_3 + CH_3COOK

+ + ⟶ + + +

potassium hydroxide + potassium permanganate + cyclopropane ⟶ water + potassium manganate + pearl ash + potassium acetate

Construct the equilibrium constant, K, expression for: KOH + KMnO_4 + C_3H_6 ⟶ H_2O + K_2MnO_4 + K_2CO_3 + CH_3COOK 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: 13 KOH + 10 KMnO_4 + C_3H_6 ⟶ 8 H_2O + 10 K_2MnO_4 + K_2CO_3 + CH_3COOK 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 KOH | 13 | -13 KMnO_4 | 10 | -10 C_3H_6 | 1 | -1 H_2O | 8 | 8 K_2MnO_4 | 10 | 10 K_2CO_3 | 1 | 1 CH_3COOK | 1 | 1 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression KOH | 13 | -13 | ([KOH])^(-13) KMnO_4 | 10 | -10 | ([KMnO4])^(-10) C_3H_6 | 1 | -1 | ([C3H6])^(-1) H_2O | 8 | 8 | ([H2O])^8 K_2MnO_4 | 10 | 10 | ([K2MnO4])^10 K_2CO_3 | 1 | 1 | [K2CO3] CH_3COOK | 1 | 1 | [CH3COOK] 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 = ([KOH])^(-13) ([KMnO4])^(-10) ([C3H6])^(-1) ([H2O])^8 ([K2MnO4])^10 [K2CO3] [CH3COOK] = (([H2O])^8 ([K2MnO4])^10 [K2CO3] [CH3COOK])/(([KOH])^13 ([KMnO4])^10 [C3H6])

Construct the rate of reaction expression for: KOH + KMnO_4 + C_3H_6 ⟶ H_2O + K_2MnO_4 + K_2CO_3 + CH_3COOK 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: 13 KOH + 10 KMnO_4 + C_3H_6 ⟶ 8 H_2O + 10 K_2MnO_4 + K_2CO_3 + CH_3COOK 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 KOH | 13 | -13 KMnO_4 | 10 | -10 C_3H_6 | 1 | -1 H_2O | 8 | 8 K_2MnO_4 | 10 | 10 K_2CO_3 | 1 | 1 CH_3COOK | 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 KOH | 13 | -13 | -1/13 (Δ[KOH])/(Δt) KMnO_4 | 10 | -10 | -1/10 (Δ[KMnO4])/(Δt) C_3H_6 | 1 | -1 | -(Δ[C3H6])/(Δt) H_2O | 8 | 8 | 1/8 (Δ[H2O])/(Δt) K_2MnO_4 | 10 | 10 | 1/10 (Δ[K2MnO4])/(Δt) K_2CO_3 | 1 | 1 | (Δ[K2CO3])/(Δt) CH_3COOK | 1 | 1 | (Δ[CH3COOK])/(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 = -1/13 (Δ[KOH])/(Δt) = -1/10 (Δ[KMnO4])/(Δt) = -(Δ[C3H6])/(Δt) = 1/8 (Δ[H2O])/(Δt) = 1/10 (Δ[K2MnO4])/(Δt) = (Δ[K2CO3])/(Δt) = (Δ[CH3COOK])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| potassium hydroxide | potassium permanganate | cyclopropane | water | potassium manganate | pearl ash | potassium acetate formula | KOH | KMnO_4 | C_3H_6 | H_2O | K_2MnO_4 | K_2CO_3 | CH_3COOK Hill formula | HKO | KMnO_4 | C_3H_6 | H_2O | K_2MnO_4 | CK_2O_3 | C_2H_3KO_2 name | potassium hydroxide | potassium permanganate | cyclopropane | water | potassium manganate | pearl ash | potassium acetate IUPAC name | potassium hydroxide | potassium permanganate | cyclopropane | water | dipotassium dioxido-dioxomanganese | dipotassium carbonate | potassium acetate