H2O + CO2 + K2MnO4 = KMnO4 + MnO2 + KHCO3

H_2O water + CO_2 carbon dioxide + K_2MnO_4 potassium manganate ⟶ KMnO_4 potassium permanganate + MnO_2 manganese dioxide + KHCO_3 potassium bicarbonate

Balance the chemical equation algebraically: H_2O + CO_2 + K_2MnO_4 ⟶ KMnO_4 + MnO_2 + KHCO_3 Add stoichiometric coefficients, c_i, to the reactants and products: c_1 H_2O + c_2 CO_2 + c_3 K_2MnO_4 ⟶ c_4 KMnO_4 + c_5 MnO_2 + c_6 KHCO_3 Set the number of atoms in the reactants equal to the number of atoms in the products for H, O, C, K and Mn: H: | 2 c_1 = c_6 O: | c_1 + 2 c_2 + 4 c_3 = 4 c_4 + 2 c_5 + 3 c_6 C: | c_2 = c_6 K: | 2 c_3 = c_4 + c_6 Mn: | c_3 = c_4 + c_5 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_1 = 2 c_2 = 4 c_3 = 3 c_4 = 2 c_5 = 1 c_6 = 4 Substitute the coefficients into the chemical reaction to obtain the balanced equation: Answer: | | 2 H_2O + 4 CO_2 + 3 K_2MnO_4 ⟶ 2 KMnO_4 + MnO_2 + 4 KHCO_3

+ + ⟶ + +

water + carbon dioxide + potassium manganate ⟶ potassium permanganate + manganese dioxide + potassium bicarbonate

Construct the equilibrium constant, K, expression for: H_2O + CO_2 + K_2MnO_4 ⟶ KMnO_4 + MnO_2 + KHCO_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: 2 H_2O + 4 CO_2 + 3 K_2MnO_4 ⟶ 2 KMnO_4 + MnO_2 + 4 KHCO_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 H_2O | 2 | -2 CO_2 | 4 | -4 K_2MnO_4 | 3 | -3 KMnO_4 | 2 | 2 MnO_2 | 1 | 1 KHCO_3 | 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 | 2 | -2 | ([H2O])^(-2) CO_2 | 4 | -4 | ([CO2])^(-4) K_2MnO_4 | 3 | -3 | ([K2MnO4])^(-3) KMnO_4 | 2 | 2 | ([KMnO4])^2 MnO_2 | 1 | 1 | [MnO2] KHCO_3 | 4 | 4 | ([KHCO3])^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])^(-2) ([CO2])^(-4) ([K2MnO4])^(-3) ([KMnO4])^2 [MnO2] ([KHCO3])^4 = (([KMnO4])^2 [MnO2] ([KHCO3])^4)/(([H2O])^2 ([CO2])^4 ([K2MnO4])^3)

Construct the rate of reaction expression for: H_2O + CO_2 + K_2MnO_4 ⟶ KMnO_4 + MnO_2 + KHCO_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: 2 H_2O + 4 CO_2 + 3 K_2MnO_4 ⟶ 2 KMnO_4 + MnO_2 + 4 KHCO_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 H_2O | 2 | -2 CO_2 | 4 | -4 K_2MnO_4 | 3 | -3 KMnO_4 | 2 | 2 MnO_2 | 1 | 1 KHCO_3 | 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 | 2 | -2 | -1/2 (Δ[H2O])/(Δt) CO_2 | 4 | -4 | -1/4 (Δ[CO2])/(Δt) K_2MnO_4 | 3 | -3 | -1/3 (Δ[K2MnO4])/(Δt) KMnO_4 | 2 | 2 | 1/2 (Δ[KMnO4])/(Δt) MnO_2 | 1 | 1 | (Δ[MnO2])/(Δt) KHCO_3 | 4 | 4 | 1/4 (Δ[KHCO3])/(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 = -1/2 (Δ[H2O])/(Δt) = -1/4 (Δ[CO2])/(Δt) = -1/3 (Δ[K2MnO4])/(Δt) = 1/2 (Δ[KMnO4])/(Δt) = (Δ[MnO2])/(Δt) = 1/4 (Δ[KHCO3])/(Δt) (assuming constant volume and no accumulation of intermediates or side products)

| water | carbon dioxide | potassium manganate | potassium permanganate | manganese dioxide | potassium bicarbonate formula | H_2O | CO_2 | K_2MnO_4 | KMnO_4 | MnO_2 | KHCO_3 Hill formula | H_2O | CO_2 | K_2MnO_4 | KMnO_4 | MnO_2 | CHKO_3 name | water | carbon dioxide | potassium manganate | potassium permanganate | manganese dioxide | potassium bicarbonate IUPAC name | water | carbon dioxide | dipotassium dioxido-dioxomanganese | potassium permanganate | dioxomanganese | potassium hydrogen carbonate