H2SO4 + K2Cr2O7 + CH3CHCH2 = H2O + CO2 + K2SO4 + Cr2(SO4)3 + CH3COOH

H_2SO_4 sulfuric acid + K_2Cr_2O_7 potassium dichromate + C_3H_6 cyclopropane ⟶ H_2O water + CO_2 carbon dioxide + K_2SO_4 potassium sulfate + Cr_2(SO_4)_3 chromium sulfate + CH_3CO_2H acetic acid

Balance the chemical equation algebraically: H_2SO_4 + K_2Cr_2O_7 + C_3H_6 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + CH_3CO_2H Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2SO_4 + c_2 K_2Cr_2O_7 + c_3 C_3H_6 ⟶ c_4 H_2O + c_5 CO_2 + c_6 K_2SO_4 + c_7 Cr_2(SO_4)_3 + 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, Cr, K and C: H: | 2 c_1 + 6 c_3 = 2 c_4 + 4 c_8 O: | 4 c_1 + 7 c_2 = c_4 + 2 c_5 + 4 c_6 + 12 c_7 + 2 c_8 S: | c_1 = c_6 + 3 c_7 Cr: | 2 c_2 = 2 c_7 K: | 2 c_2 = 2 c_6 C: | 3 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 = c_1/4 c_3 = c_1/4 - 2/3 c_4 = c_1 + 1 c_5 = 1 c_6 = c_1/4 c_7 = c_1/4 c_8 = (3 c_1)/8 - 3/2 Multiply by the least common denominator, 3, to eliminate fractional coefficients: c_2 = c_1/4 c_3 = c_1/4 - 2 c_4 = c_1 + 3 c_5 = 3 c_6 = c_1/4 c_7 = c_1/4 c_8 = (3 c_1)/8 - 9/2 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 = 9 c_3 = 7 c_4 = 39 c_5 = 3 c_6 = 9 c_7 = 9 c_8 = 9 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 36 H_2SO_4 + 9 K_2Cr_2O_7 + 7 C_3H_6 ⟶ 39 H_2O + 3 CO_2 + 9 K_2SO_4 + 9 Cr_2(SO_4)_3 + 9 CH_3CO_2H

+ + ⟶ + + + +

sulfuric acid + potassium dichromate + cyclopropane ⟶ water + carbon dioxide + potassium sulfate + chromium sulfate + acetic acid

Construct the equilibrium constant, K, expression for: H_2SO_4 + K_2Cr_2O_7 + C_3H_6 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + 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 + 9 K_2Cr_2O_7 + 7 C_3H_6 ⟶ 39 H_2O + 3 CO_2 + 9 K_2SO_4 + 9 Cr_2(SO_4)_3 + 9 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 K_2Cr_2O_7 | 9 | -9 C_3H_6 | 7 | -7 H_2O | 39 | 39 CO_2 | 3 | 3 K_2SO_4 | 9 | 9 Cr_2(SO_4)_3 | 9 | 9 CH_3CO_2H | 9 | 9 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) K_2Cr_2O_7 | 9 | -9 | ([K2Cr2O7])^(-9) C_3H_6 | 7 | -7 | ([C3H6])^(-7) H_2O | 39 | 39 | ([H2O])^39 CO_2 | 3 | 3 | ([CO2])^3 K_2SO_4 | 9 | 9 | ([K2SO4])^9 Cr_2(SO_4)_3 | 9 | 9 | ([Cr2(SO4)3])^9 CH_3CO_2H | 9 | 9 | ([CH3CO2H])^9 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) ([K2Cr2O7])^(-9) ([C3H6])^(-7) ([H2O])^39 ([CO2])^3 ([K2SO4])^9 ([Cr2(SO4)3])^9 ([CH3CO2H])^9 = (([H2O])^39 ([CO2])^3 ([K2SO4])^9 ([Cr2(SO4)3])^9 ([CH3CO2H])^9)/(([H2SO4])^36 ([K2Cr2O7])^9 ([C3H6])^7)

Construct the rate of reaction expression for: H_2SO_4 + K_2Cr_2O_7 + C_3H_6 ⟶ H_2O + CO_2 + K_2SO_4 + Cr_2(SO_4)_3 + 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 + 9 K_2Cr_2O_7 + 7 C_3H_6 ⟶ 39 H_2O + 3 CO_2 + 9 K_2SO_4 + 9 Cr_2(SO_4)_3 + 9 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 K_2Cr_2O_7 | 9 | -9 C_3H_6 | 7 | -7 H_2O | 39 | 39 CO_2 | 3 | 3 K_2SO_4 | 9 | 9 Cr_2(SO_4)_3 | 9 | 9 CH_3CO_2H | 9 | 9 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) K_2Cr_2O_7 | 9 | -9 | -1/9 (Δ[K2Cr2O7])/(Δt) C_3H_6 | 7 | -7 | -1/7 (Δ[C3H6])/(Δt) H_2O | 39 | 39 | 1/39 (Δ[H2O])/(Δt) CO_2 | 3 | 3 | 1/3 (Δ[CO2])/(Δt) K_2SO_4 | 9 | 9 | 1/9 (Δ[K2SO4])/(Δt) Cr_2(SO_4)_3 | 9 | 9 | 1/9 (Δ[Cr2(SO4)3])/(Δt) CH_3CO_2H | 9 | 9 | 1/9 (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) = -1/9 (Δ[K2Cr2O7])/(Δt) = -1/7 (Δ[C3H6])/(Δt) = 1/39 (Δ[H2O])/(Δt) = 1/3 (Δ[CO2])/(Δt) = 1/9 (Δ[K2SO4])/(Δt) = 1/9 (Δ[Cr2(SO4)3])/(Δt) = 1/9 (Δ[CH3CO2H])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| sulfuric acid | potassium dichromate | cyclopropane | water | carbon dioxide | potassium sulfate | chromium sulfate | acetic acid formula | H_2SO_4 | K_2Cr_2O_7 | C_3H_6 | H_2O | CO_2 | K_2SO_4 | Cr_2(SO_4)_3 | CH_3CO_2H Hill formula | H_2O_4S | Cr_2K_2O_7 | C_3H_6 | H_2O | CO_2 | K_2O_4S | Cr_2O_12S_3 | C_2H_4O_2 name | sulfuric acid | potassium dichromate | cyclopropane | water | carbon dioxide | potassium sulfate | chromium sulfate | acetic acid IUPAC name | sulfuric acid | dipotassium oxido-(oxido-dioxochromio)oxy-dioxochromium | cyclopropane | water | carbon dioxide | dipotassium sulfate | chromium(+3) cation trisulfate | acetic acid