H2SO4 + KMnO4 + C4H8 = H2O + CO2 + K2SO4 + MnSO4 + CH3COOH

H_2SO_4 sulfuric acid + KMnO_4 potassium permanganate + (CH_3)_2C=CH_2 isobutylene ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + MnSO_4 manganese(II) sulfate + CH_3CO_2H acetic acid

Balance the chemical equation algebraically: H_2SO_4 + KMnO_4 + (CH_3)_2C=CH_2 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + CH_3CO_2H Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 KMnO_4 + c_3 (CH_3)_2C=CH_2 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 MnSO_4 + c_8 CH_3CO_2H 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 + 8 c_3 = 2 c_4 + 4 c_8 O: | 4 c_1 + 4 c_2 = 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: | 4 c_3 = c_5 + 2 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_5 = 1 and solve the system of equations for the remaining coefficients: c_2 = (2 c_1)/3 c_3 = (5 c_1)/12 - 1/2 c_4 = c_1 + 1 c_5 = 1 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = (5 c_1)/6 - 3/2 Multiply by the least common denominator, 4, to eliminate fractional coefficients: c_2 = (2 c_1)/3 c_3 = (5 c_1)/12 - 2 c_4 = c_1 + 4 c_5 = 4 c_6 = c_1/3 c_7 = (2 c_1)/3 c_8 = (5 c_1)/6 - 6 The resulting system of equations is still underdetermined, so an additional coefficient must be set arbitrarily. Set c_1 = 36 and solve for the remaining coefficients: c_1 = 36 c_2 = 24 c_3 = 13 c_4 = 40 c_5 = 4 c_6 = 12 c_7 = 24 c_8 = 24 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 36 H_2SO_4 + 24 KMnO_4 + 13 (CH_3)_2C=CH_2 ⟶ 40 H_2O + 4 CO_2 + 12 K_2SO_4 + 24 MnSO_4 + 24 CH_3CO_2H

+ + ⟶ + + + +

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

Construct the equilibrium constant, K, expression for: H_2SO_4 + KMnO_4 + (CH_3)_2C=CH_2 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + 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: 36 H_2SO_4 + 24 KMnO_4 + 13 (CH_3)_2C=CH_2 ⟶ 40 H_2O + 4 CO_2 + 12 K_2SO_4 + 24 MnSO_4 + 24 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_2SO_4 | 36 | -36 KMnO_4 | 24 | -24 (CH_3)_2C=CH_2 | 13 | -13 H_2O | 40 | 40 CO_2 | 4 | 4 K_2SO_4 | 12 | 12 MnSO_4 | 24 | 24 CH_3CO_2H | 24 | 24 Assemble the activity expressions accounting for the state of matter and ν_i: chemical species | c_i | ν_i | activity expression H_2SO_4 | 36 | -36 | ([H2SO4])^(-36) KMnO_4 | 24 | -24 | ([KMnO4])^(-24) (CH_3)_2C=CH_2 | 13 | -13 | ([(CH3)2C=CH2])^(-13) H_2O | 40 | 40 | ([H2O])^40 CO_2 | 4 | 4 | ([CO2])^4 K_2SO_4 | 12 | 12 | ([K2SO4])^12 MnSO_4 | 24 | 24 | ([MnSO4])^24 CH_3CO_2H | 24 | 24 | ([CH3CO2H])^24 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])^(-36) ([KMnO4])^(-24) ([(CH3)2C=CH2])^(-13) ([H2O])^40 ([CO2])^4 ([K2SO4])^12 ([MnSO4])^24 ([CH3CO2H])^24 = (([H2O])^40 ([CO2])^4 ([K2SO4])^12 ([MnSO4])^24 ([CH3CO2H])^24)/(([H2SO4])^36 ([KMnO4])^24 ([(CH3)2C=CH2])^13)

Construct the rate of reaction expression for: H_2SO_4 + KMnO_4 + (CH_3)_2C=CH_2 ⟶ H_2O + CO_2 + K_2SO_4 + MnSO_4 + 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: 36 H_2SO_4 + 24 KMnO_4 + 13 (CH_3)_2C=CH_2 ⟶ 40 H_2O + 4 CO_2 + 12 K_2SO_4 + 24 MnSO_4 + 24 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_2SO_4 | 36 | -36 KMnO_4 | 24 | -24 (CH_3)_2C=CH_2 | 13 | -13 H_2O | 40 | 40 CO_2 | 4 | 4 K_2SO_4 | 12 | 12 MnSO_4 | 24 | 24 CH_3CO_2H | 24 | 24 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 | 36 | -36 | -1/36 (Δ[H2SO4])/(Δt) KMnO_4 | 24 | -24 | -1/24 (Δ[KMnO4])/(Δt) (CH_3)_2C=CH_2 | 13 | -13 | -1/13 (Δ[(CH3)2C=CH2])/(Δt) H_2O | 40 | 40 | 1/40 (Δ[H2O])/(Δt) CO_2 | 4 | 4 | 1/4 (Δ[CO2])/(Δt) K_2SO_4 | 12 | 12 | 1/12 (Δ[K2SO4])/(Δt) MnSO_4 | 24 | 24 | 1/24 (Δ[MnSO4])/(Δt) CH_3CO_2H | 24 | 24 | 1/24 (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 = -1/36 (Δ[H2SO4])/(Δt) = -1/24 (Δ[KMnO4])/(Δt) = -1/13 (Δ[(CH3)2C=CH2])/(Δt) = 1/40 (Δ[H2O])/(Δt) = 1/4 (Δ[CO2])/(Δt) = 1/12 (Δ[K2SO4])/(Δt) = 1/24 (Δ[MnSO4])/(Δt) = 1/24 (Δ[CH3CO2H])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium permanganate | isobutylene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | acetic acid formula | H_2SO_4 | KMnO_4 | (CH_3)_2C=CH_2 | H_2O | CO_2 | K_2SO_4 | MnSO_4 | CH_3CO_2H Hill formula | H_2O_4S | KMnO_4 | C_4H_8 | H_2O | CO_2 | K_2O_4S | MnSO_4 | C_2H_4O_2 name | sulfuric acid | potassium permanganate | isobutylene | water | carbon dioxide | potassium sulfate | manganese(II) sulfate | acetic acid IUPAC name | sulfuric acid | potassium permanganate | 2-methylprop-1-ene | water | carbon dioxide | dipotassium sulfate | manganese(+2) cation sulfate | acetic acid